Integral methods in science and engineering. Computational and analytic aspects

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Christian Constanda r Paul J. Harris Editors

Integral Methods in Science and Engineering Computational and Analytic Aspects

Editors Christian Constanda Department of Mathematical and Computer Sciences The University of Tulsa 800 S. Tucker Drive Tulsa, OK 74104-9700 USA [email protected]

Paul J. Harris School of Computing, Engineering, and Mathematics University of Brighton Lewes Road Brighton BN2 4GJ UK [email protected]

ISBN 978-0-8176-8237-8 e-ISBN 978-0-8176-8238-5 DOI 10.1007/978-0-8176-8238-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011933571 Mathematics Subject Classification (2010): 34-06, 35-06, 45-06, 65-06, 74-06, 76-06 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper www.birkhauser-science.com

To the memory of Igor Chudinovich and Alain Largillier, former members of the International Steering Committee of the IMSE Consortium

Preface

The international conferences on Integral Methods in Science and Engineering (IMSE) aim to bring together specialists from various research fields who employ integration as an essential tool in their work, and to promote and consolidate the use by the world scientific community of such elegant, powerful, and widely applicable techniques. The first two conferences in the series, IMSE1985 and IMSE1990, were hosted by the University of Texas-Arlington. At the latter, the IMSE consortium was created and charged with organizing these conferences under the guidance of an International Steering Committee. Subsequently, IMSE1993 took place at Tohoku University, Sendai, Japan; IMSE1996 at the University of Oulu, Finland; IMSE1998 at Michigan Technological University, Houghton, MI, USA; IMSE2000 in Banff, AB, Canada; IMSE2002 at the University of Saint-Étienne, France; IMSE2004 at the University of Central Florida, Orlando, FL, USA; IMSE2006 in Niagara Falls, ON, Canada; and IMSE2008 at the University of Cantabria, Santander, Spain. The 2010 meeting, held at the University of Brighton, UK, July 12–14, and attended by participants representing 20 countries from five continents, has confirmed IMSE as a well-established event on the international conference circuit, which gives scientists and engineers from a whole variety of research backgrounds the opportunity to get together and discuss advances in a large class of important mathematical procedures. The organization of IMSE2010 was, as on previous occasions, of a high standard; acknowledging this fact, the participants wish to thank the School of Computing, Engineering, and Mathematics (formerly the School of Computing, Mathematical, and Information Sciences) at the University of Brighton for its financial support and for the administrative assistance given by the staff in the School office. Special thanks are due, in particular, to the members of the Local Organizing Committee: Paul Harris (Division of Mathematics), Chairman, Derek Covill (Division of Engineering), David Chappell (School of Mathematical Sciences, University of Nottingham).

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The next IMSE conference will be hosted by the Federal University of Rio Grande do Sul and the National Institute for Space Research in Porto Alegre, RS, Brazil, in July 2012. Further details will be posted in due course through the link http://sites.google.com/site/imse2012. This volume contains 37 refereed papers from among those presented in Brighton, arranged alphabetically by the first author’s name. The editors would like to thank the staff at Birkhäuser-Boston for their professionalism and efficient handling of the publication process. Tulsa, Oklahoma, USA

Christian Constanda, IMSE Chairman March 2011

The International Steering Committee of IMSE: C. Constanda (The University of Tulsa), Chairman M. Ahues (University of Saint-Étienne) B. Bodmann (Federal University of Rio Grande do Sul) H. de Campos Velho (INPE, Saõ José dos Campos) P. Harris (University of Brighton) A. Kirsch (Karlsruhe Institute of Technology) S. Mikhailov (Brunel University) D. Mitrea (University of Missouri-Columbia) A. Nastase (RWTH Aachen University) E. Pérez (University of Cantabria) S. Potapenko (University of Waterloo) K. Ruotsalainen (University of Oulu)

Contents

A Collocation Method for Cauchy Integral Equations in L2 . . . . . . . . . . . . M. Ahues and A. Mennouni 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 4 5

On a New Definition of the Reynolds Number from the Interplay of Macroscopic and Microscopic Phenomenology . . . . . . . . . . . . . . . . . . . . . 7 B.E.J. Bodmann, M.T. Vilhena, J.R. Zabadal, and D. Beck 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 The Vortex Correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 The Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 A Self-Consistent Monte Carlo Validation Procedure for Hadron Cancer Therapy Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L.N. Burigo, D. Hadjimichef, and B.E.J. Bodmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Particle Transport by GEANT4 Monte Carlo . . . . . . . . . . . . . . . . . . 3 Heavy Ion Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Simulation Results and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A General Analytical Solution of the Advection–Diffusion Equation for Fickian Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Buske, M.T. Vilhena, C.F. Segatto, and R.S. Quadros 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Experimental Data and Turbulent Parameterization . . . . . . . . . . . . .

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4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A Novel Method for Simulating Spectral Nuclear Reactor Criticality by a Spatially Dependent Volume Size Control . . . . . . . . . . . . . . . . . . . . . . . D.Q. de Camargo, B.E.J. Bodmann, M.T. Vilhena, and S.d.Q.B. Leite 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Neutron Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Neutron Transport by Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Particle Filter for Stable Distribution . . . . . . . . . . . . . . . . . . . . . . . H.F. de Campos Velho and H.C. Morais Furtado 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Standard PF Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 New Approach for Particle Filter . . . . . . . . . . . . . . . . . . . . . 2.3 Identifying the Non-Extensive Parameter q . . . . . . . . . . . . 3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Analytical Solution of the Multi-Group Neutron Diffusion Kinetic Equation in One-Dimensional Cartesian Geometry by an Integral Transform Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Ceolin, M.T. Vilhena, and B.E.J. Bodmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimating the Validity of Statistical Energy Analysis Using Dynamical Energy Analysis: A Preliminary Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.J. Chappell and G. Tanner 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Wave Energy Flow in Terms of the Green’s Function . . . . . . . . . . . . 3 Linear Phase Space Operators and DEA . . . . . . . . . . . . . . . . . . . . . . 3.1 Phase Space Operators and Boundary Maps . . . . . . . . . . . 3.2 Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Basis Function Representations and SEA . . . . . . . . . . . . . . 3.4 Spectral Properties of the Transfer Operator . . . . . . . . . . . 4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 37 38 41 42 44 47 47 48 49 50 53 53 55 56

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5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Efficient Iterative Methods for Fast Solution of Integral Operators Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Chen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fast Iterative Methods for the Helmholtz Equation . . . . . . . . . . . . . . 2.1 Iterative Algorithms of Order O(n2 ) . . . . . . . . . . . . . . . . . . 2.2 Iterative Algorithms of Order O(n) . . . . . . . . . . . . . . . . . . . 3 Fast Iterative Methods for an Image Deblurring Model . . . . . . . . . . 3.1 Intermediate Variable Methods . . . . . . . . . . . . . . . . . . . . . . 3.2 Optimization Based Multilevel Methods . . . . . . . . . . . . . . 4 Open Problems and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 79 80 82 83 85 85 88 89

Analysis of Some Localized Boundary–Domain Integral Equations for Transmission Problems with Variable Coefficients . . . . . . . . . . . . . . . . . 91 O. Chkadua, S.E. Mikhailov, and D. Natroshvili 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2 Reduction to Localized Boundary–Domain Integral Equations . . . . 92 2.1 Formulation of the Interface Problems . . . . . . . . . . . . . . . . 92 2.2 Properties of Localized Potentials . . . . . . . . . . . . . . . . . . . 95 2.3 Basic LBDIE Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3 LBDIES for the Dirichlet Transmission Problem . . . . . . . . . . . . . . . 97 4 The Mixed Transmission Problem (TM) . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Analysis of Segregated Boundary–Domain Integral Equations for Mixed Variable-Coefficient BVPs in Exterior Domains . . . . . . . . . . . . . . . . . . . . . . 109 O. Chkadua, S.E. Mikhailov, and D. Natroshvili 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2 Basic Notation and Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3 Mixed Boundary-Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4 Parametrix and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 Segregated BDIEs for the Mixed Problem . . . . . . . . . . . . . . . . . . . . . 118 6 BDIE Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Thermoelastic Plates with Arc-Shaped Cracks . . . . . . . . . . . . . . . . . . . . . . . 129 I. Chudinovich and C. Constanda 1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2 The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3 The Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

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Almost Periodicity in Semilinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 C. Corduneanu 1 A Result in the Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 2 The Semilinear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3 An Integro-Differential System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4 The Semilinear Equation Associated with (13) . . . . . . . . . . . . . . . . 145 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Bubble Behavior Near a Two Fluid Interface . . . . . . . . . . . . . . . . . . . . . . . . 147 G.A. Curtiss, D.M. Leppinen, Q.X. Wang, and J.R. Blake 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Spectral Stiff Problems in Domains with a Strongly Oscillating Boundary 159 D. Gómez, S.A. Nazarov, and E. Pérez 1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 2 The Case ωε = {x : 0 < ν < ε h(τ )} . . . . . . . . . . . . . . . . . . . . . . . . . 163 3 The Case ωε = {x : 0 < ν < ε h(τ /ε )} . . . . . . . . . . . . . . . . . . . . . . . 169 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Spectra and Pseudospectra of a Convection–Diffusion Operator . . . . . . . . 173 H. Guebbaï and A. Largillier 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 3 The Spectrum of Aη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4 Pseudospectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 A Necessary and Sufficient Condition for the Existence of Absolute Minimizers for Energy Functionals with Scale Invariance . . . . . . . . . . . . . . 181 S.M. Haidar 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 1.1 Preliminary Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 2 Optimality Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

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Nonlinear Abel-Type Integral Equation in Modeling Creep Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 L. Hakim and S.E. Mikhailov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 2 Integral Equation Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 3 Numerical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Some Thoughts on Methods for Evaluating a Class of Highly Oscillatory Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 P.J. Harris 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 2 Quadrature Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 2.1 Interpolation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 2.2 Change of Variable Methods . . . . . . . . . . . . . . . . . . . . . . . . 207 3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4 Application to the Solution of Integral Equations . . . . . . . . . . . . . . . 209 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Numerical Experiments for Mammary Adenocarcinoma Cell Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 C.L. Jorcyk, M. Kolev, and B. Zubik-Kowal 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 2 In Vitro and In Vivo Growth of Cell Lines Established from C3(1)/Tag Mice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 4 Numerical Experiments for Mammary Adenocarcinoma Cell Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 5 Concluding Remarks and Future Directions . . . . . . . . . . . . . . . . . . . 222 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Limiting Cases of Subdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 J. Kemppainen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 2 Boundary Integral Solution of TFDE . . . . . . . . . . . . . . . . . . . . . . . . . 226 3 The Upper Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 4 The Lower Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 A New Hybrid Method to Predict the Distribution of Vibro-Acoustic Energy in Complex Built-Up Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 D.N. Maksimov and G. Tanner 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

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2 Direct Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 3 Stochastic Reverberant Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 4 Energy Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 2-D and 3-D Elastodynamic Contact Problems for Interface Cracks Under Harmonic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 O. Menshykov, M. Menshykova, I. Guz, and V. Mikucka 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3 Iterative Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Asymptotic Behavior of Elliptic Quadratic Algebraic Equations with Variable Coefficients, and Aerodynamical Applications . . . . . . . . . . . 253 A. Nastase 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 2 Qualitative Analysis of Elliptic QAE with Variable Free Term . . . . 254 3 Qualitative Analysis of Elliptic QAE with Free Term and Variable Coefficients of the Linear Terms . . . . . . . . . . . . . . . . . . . . . 256 4 Aerodynamical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Artificial Neural Networks for Estimating the Atmospheric Pollutant Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 F.F. Paes, H.F. de Campos Velho, and F.M. Ramos 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 2 Forward Model: LAMBDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 3 Inverse Method: Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 A Theoretical Study of the Stratified Atmospheric Boundary Layer Through Perturbation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 C.C. Pellegrini, M.T. Vilhena, and B.E.J. Bodmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 2 Perturbation Analysis and Governing Equations . . . . . . . . . . . . . . . . 274 3 Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

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Integro-Differential Equations for Stress Analysis in the Bridged Zone of Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 M. Perelmuter 1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 2 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 3 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4 Numerical Analysis of the Interface Bridged Crack . . . . . . . . . . . . . 293 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Design and Performance of Gas–Liquid Cylindrical Cyclone/Slug Damper System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 E. Pereyra, L. Gómez, R. Mohan, O. Shoham, and G. Kouba 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 2 Slug-Damper Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 3 GLCC Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 3.1 Liquid Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 3.2 Rate of Pressure Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 3.3 Liquid Leg Pressure Drop/Flow Rate . . . . . . . . . . . . . . . . . 303 3.4 Gas Leg Pressure Drop/Flow Rate . . . . . . . . . . . . . . . . . . . . 304 4 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 4.1 Liquid Level Control PID . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 4.2 Separator Pressure Control PID . . . . . . . . . . . . . . . . . . . . . . 305 4.3 Pneumatic Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4.4 Control Valve Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4.5 Valve Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 5 Pipeline Slugging Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 6 Overall GLCC-SD System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 7 Slug-Damper Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 8 GLCC-SD Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 On Quasimodes for Compact Operators and Associated Evolution Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 E. Pérez 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 2 Formulation of the Problems and Preliminary Results . . . . . . . . . . . 314 2.1 Approaches to Solutions of Second-Order Evolution Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 3 New Estimates for Discrepancies from the Semigroup . . . . . . . . . . . 319 3.1 The Case of the First-Order Evolution Equation (5) . . . . . 319 3.2 The Case of the Second-Order Evolution Equation . . . . . . 321 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

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Error Estimation by Means of Richardson Extrapolation with the Boundary Element Method in a Dirichlet Problem for the Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 S. Pomeranz 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 3 Comments on Richardson Extrapolation . . . . . . . . . . . . . . . . . . . . . . 328 4 Implementation of an a Posteriori Pointwise Estimator of Richardson Extrapolation Reliability . . . . . . . . . . . . . . . . . . . . . . . 329 5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 5.1 Richardson Extrapolation for Normal Boundary Flux . . . . 331 5.2 Richardson Extrapolation for Interior Potential . . . . . . . . . 333 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Convergence of a Discretization Scheme Based on the Characteristics Method for a Fluid–Rigid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 J. San Martín, J.-F. Scheid, and L. Smaranda 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 2 Semi-discretization in the Time Variable . . . . . . . . . . . . . . . . . . . . . 343 3 Full Discretization in the Time and Space Variables . . . . . . . . . . . . . 345 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 An Efficient Algorithm to Solve the GITT-Transformed 2-D Neutron Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 M. Schramm, C.Z. Petersen, M.T. Vilhena, and B.E.J. Bodmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 3 A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Nonlinear Localized Dissipative Structures for Solving Wave Equations over Long Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 J. Steinhoff, S. Chitta, and P. Sanematsu 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 2 Approach and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 3 Carrier Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 4 Continuous Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 5 Propagation Through Evaporative Atmospheric and Ionospheric Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 Semianalytical Approach to the Computation of the Laplace Transform of Source Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 L.G. Thompson and G. Zhao 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

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Source Functions: Real Time Computational Issues [Ohaeri:91, Thompson91] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 3 Laplace Transform of Products of Source Functions . . . . . . . . . . . . 374 4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Asymptotic Analysis of Singularities for Pseudodifferential Equations in Canonical Non-Smooth Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 V.B. Vasilyev 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 2 Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 3 Multidimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 4 The Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Optimizing Water Quality in a River Section . . . . . . . . . . . . . . . . . . . . . . . . 391 M.A. Vilar, L.J. Alvarez-Vázquez, A. Martínez, and M.E. Vázquez-Méndez 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 2 Mathematical Description of the Problem . . . . . . . . . . . . . . . . . . . . . 392 3 The Discretized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 3.1 Computation of (An+1 , qn+1 ) . . . . . . . . . . . . . . . . . . . . . . . . 397 3.2 Computation of cn+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 3.3 The Discrete Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 4 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 5 Computational Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Boundary Integral Equations for Arbitrary Geometry Shells . . . . . . . . . . . 403 V.V. Zozulya 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 2 The 3-D Equations of Elasticity in Curvilinear Coordinates . . . . . . 404 3 The 3-D Somigliana Identity and Fundamental Solutions . . . . . . . . 406 4 The 2-D Equations of Elasticity in Coordinates Related to the Middle Surface of the Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 5 The 2-D Somigliana Identity and BIE for Arbitrary Geometry Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 6 First Approximation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

List of Contributors

Mario Ahues Université Jean Monnet, Université de Lyon, 23 rue du Dr. Paul Michelon, Saint-Étienne 42023, cedex 2, France, e-mail: [email protected] Lino J. Alvarez-Vázquez Universidad de Vigo, ETSI Telecomunicación, 36310 Vigo, Spain, e-mail: [email protected] Daniel Beck Universidade Federal do Rio Grande do Sul, PPGMap & PROMEC, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] John R. Blake University of Birmingham, Edgbaston, Birmingham B15 2TT, UK, e-mail: [email protected] Bardo E.J. Bodmann Universidade Federal do Rio Grande do Sul, PPGMap & PROMEC, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Lucas N. Burigo Universidade Federal de Rio Grande do Sul, PPGMap & PROMEC, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Daniela Buske Universidade Federal de Pelotas, Instituto de Fisica e Matemática, Campus Capão do Leão, Caixa Postal 354, 96010-900 Pelotas, RS, Brazil, e-mail: [email protected]

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List of Contributors

Dayana Q. de Camargo Universidade Federal do Rio Grande do Sul, PPGMap & PROMEC, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Haroldo F. de Campos Velho Instituto Nacional de Pesquisas Espaciais (INPE), P.O. Box 515, São José dos Campos, SP 12245-970, Brazil, e-mail: [email protected] Celina Ceolin Universidade Federal do Rio Grande do Sul, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] David J. Chappell University of Nottingham, University Park, Nottingham NG7 2RD, UK, e-mail: [email protected] Ke Chen University of Liverpool, Peach Street, Liverpool L69 7ZL, UK, e-mail: [email protected] Subhashini Chitta Flow Analysis, Inc., 256 93rd Street, Brooklyn, NY 11209, USA, e-mail: [email protected] Otar Chkadua A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 2 University Street, Tbilisi 0186, Georgia, e-mail: [email protected] Christian Constanda The University of Tulsa, 800 S. Tucker Drive, Tulsa, OK 74104, USA, e-mail: [email protected] Constantin Corduneanu University of Texas at Arlington, 411 S. Nedderman Drive, Box 19408, Arlington, TX 76019, USA, e-mail: [email protected] Geoff A. Curtiss University of Birmingham, Edgbaston, Birmingham B15 2TT, UK, e-mail: [email protected] Delfina Gómez Universidad de Cantabria, Av. de los Castros s/n, 39005 Santander, Spain, e-mail: [email protected]

List of Contributors

xxi

Luis Gómez The University of Tulsa, 800 S. Tucker Drive, Tulsa, OK 74104, USA, e-mail: [email protected] Hamza Guebbaï Université Jean Monnet, Université de Lyon, 23 rue du Dr. Paul Michelon, Saint-Étienne 42023, cedex 2, France, e-mail: [email protected] Igor A. Guz Centre for Micro- and Nanomechanics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, UK, e-mail: [email protected] Dimiter Hadjimichef Universidade Federal do Rio Grande do Sul, PPGMap & PROMEC, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Salim M. Haidar Grand Valley State University, 1 Campus Drive, Allendale, MI 49401, USA, e-mail: [email protected] Layal Hakim Brunel University West London, Kingston Lane, Uxbridge, Middlesex UB8 3PH, UK, e-mail: [email protected] Paul J. Harris School of Computing, Engineering and Mathematics, University of Brighton, Lewes Road, Brighton BN2 4GJ, UK, e-mail: [email protected] Cheryl L. Jorcyk Boise State University, 1910 University Drive, Boise, ID 83725-1515, USA, e-mail: [email protected] Jukka Kemppainen University of Oulu, P.O.Box 4500, 90014 Oulu, Finland, e-mail: [email protected] Mikhail Kolev University of Warmia and Mazury, Zolnierska 14, 10-561, Olsztyn, Poland, e-mail: [email protected] Gene Kouba Chevron Energy Technology Company, 1400 Smith Street, Houston, TX 77002, USA, e-mail: [email protected]

xxii

List of Contributors

Sergio de Q.B. Leite Comissão Nacional de Energia Nuclear, Rua General Severiano 90, 22294-900 Rio de Janeiro, RJ, Brazil, e-mail: [email protected] David Leppinen University of Birmingham, Edgbaston, Birmingham B15 2TT, UK, e-mail: [email protected] Dmitrii Maksimov University of Nottingham, University Park, Nottingham NG7 2RD, UK, e-mail: [email protected] Aurea Martínez Universidad de Vigo, ETSI Telecomunicación, 36310 Vigo, Spain, e-mail: [email protected] Abdelaziz Mennouni University of Bordj Bou-Arreridj, 34000 Bordj Bou-Arreridj, Algeria, e-mail: [email protected] Oleksandr V. Menshykov Centre for Micro- and Nanomechanics (CEMINACS), University of Aberdeen, King’s College, Aberdeen AB24 3UE, UK, e-mail: [email protected] Maryna V. Menshykova Centre for Micro- and Nanomechanics (CEMINACS), University of Aberdeen, King’s College, Aberdeen AB24 3UE, UK, e-mail: [email protected] Sergey E. Mikhailov Brunel University West London, Kingston Lane, Uxbridge, Middlesex UB8 3PH, UK, e-mail: [email protected] Vita Mikucka Centre for Micro- and Nanomechanics (CEMINACS), University of Aberdeen, King’s College, Aberdeen AB24 3UE, UK, e-mail: [email protected] Ram Mohan The University of Tulsa, 800 S. Tucker Drive, Tulsa, OK 74104, USA, e-mail: [email protected] Helaine C. Morais Furtado Instituto Nacional de Pesquisas Espaciais (INPE), P.O. Box 515, São José dos Campos, SP 12245-970, Brazil, e-mail: [email protected]

List of Contributors

xxiii

Adriana Nastase Aerodynamik des Fluges, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany, e-mail: [email protected] David Natroshvili Georgian Technical University, 77 M. Kostava, 0175 Tbilisi, Georgia, e-mail: [email protected] Sergey A. Nazarov Institute for Problems in Mechanical Engineering, RAN V.O. Bol’shoi pr., 61, 199178 St. Petersburg, Russia, e-mail: [email protected] Fabiana F. Paes Instituto Nacional de Pesquisas Espaciais (INPE), P.O. Box 515, São José dos Campos, SP 12245-970, Brazil, e-mail: [email protected] Cláudio Pellegrini Universidade Federal de São João del-Rei, Praça Frei Orlando 170, 36307-904 São João del-Rei, MG, Brazil, e-mail: [email protected] Mikhail Perelmuter A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Pr. Vernadskogo 101-1, Moscow 119526, Russia, e-mail: [email protected] Eduardo Pereyra The University of Tulsa, 800 S. Tucker Drive, Tulsa, OK 74104, USA, e-mail: [email protected] Eugenia Pérez Universidad de Cantabria, Avda de los Castros s/n, 39005 Santander, Spain, e-mail: [email protected] Claudio Z. Petersen Universidade Federal do Rio Grande do Sul, PPGMap & PROMEC, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Shirley Pomeranz The University of Tulsa, 800 S. Tucker Drive, Tulsa, OK 74104, USA, e-mail: [email protected] Régis S. Quadros Universidade Federal de Pelotas, Instituto de Fisica e Matématica, Campus Capão do Leão, Caixa Postal 354, 96010-900 Pelotas, RS, Brazil, e-mail: [email protected]

xxiv

List of Contributors

Fernando M. Ramos Instituto Nacional de Pesquisas Espaciais (INPE), P.O. Box 515, São José dos Campos, SP 12245-970, Brazil, e-mail: [email protected] Paula Sanematsu University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN 37388, USA, e-mail: [email protected] Jorge San Martín Universidad de Chile, Casilla 170/3-Correo 3, Santiago 8370459, Chile, e-mail: [email protected] Jean-Frano¸is Scheid Université Henri Poincaré, BP239, F-54506 Vandoeuvre-lès-Nancy, Cedex, France, e-mail: [email protected] Marcelo Schramm Universidade Federal do Rio Grande do Sul, PPGMap & PROMEC, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Cynthia F. Segatto Universidade Federal do Rio Grande do Sul, PPGMap, Av. Bento Gonçalves 9500, 91509-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Ovadia Shoham The University of Tulsa, 800 S. Tucker Drive, Tulsa, OK 74104, USA, e-mail: [email protected] Loredana Smaranda Universitatea din Pite¸sti, Str. Targu din Vale Nr. 1, 110040 Pite¸sti, Romania, e-mail: [email protected] John Steinhoff University of Tennessee Space Institute, 411 B.H. Goethert Parkway, Tullahoma, TN 37388, USA, e-mail: [email protected] Gregor Tanner University of Nottingham, University Park, Nottingham NG7 2RD, UK, e-mail: [email protected] Leslie G. Thompson The University of Tulsa, 800 S. Tucker Drive, Tulsa, OK 74104, USA, e-mail: [email protected]

List of Contributors

xxv

Vladimir B. Vasilyev Bryansk State University, Bezhitskaya 14, Bryansk 241036, Russia, e-mail: [email protected] Miguel E.Vázquez-Méndez Universidad de Santiago, Escola Politécnica Superior, 27002 Lugo, Spain, e-mail: [email protected] Miguel A. Vilar Rivas Universidad de Santiago, Escola Politécnica Superior, 27002 Lugo, Spain, e-mail: [email protected] Marco T. Vilhena Universidade Federal do Rio Grande do Sul, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Qian X. Wang University of Birmingham, Edgbaston, Birmingham B15 2TT, UK, e-mail: [email protected] Jorge R. Zabadal Universidade Federal do Rio Grande do Sul, PPGMap & PROMEC, Av. Osvaldo Aranha, 99/4, 90046-900 Porto Alegre, RS, Brazil, e-mail: [email protected] Gang Zhao University of Regina, 3737 Wascana Parkway, Regina, SK, Canada S4S 0A2, e-mail: [email protected] Vladimir V. Zozulya Centro de Investigación Científica de Yucatán, A.C., Calle 43, no. 130, Colonia Chuburna de Hidalgo, C.P. 97200, Mérida, Yucatán, Mexico, e-mail: [email protected] Barbara Zubik-Kowal Boise State University, 1910 University Drive, Boise, ID 83725-1555, USA, e-mail: [email protected]

A Collocation Method for Cauchy Integral Equations in L2 M. Ahues and A. Mennouni

1 Introduction In this paper we present a collocation method based on trigonometric polynomials combined with a regularization procedure, for solving Cauchy integral equations of the second kind, in L2 (0, 2π ). A system of linear equations is involved. We prove the existence of the solution for a double projection scheme, and we perform the error analysis. Some numerical examples illustrate the theoretical results. Cauchy integral equations appear in many applications in scientific fields such as unsteady aerodynamics and aero elastic phenomena, visco elasticity, fluid dynamics and electrodynamics. There is a theoretical study on some Cauchy integral equations in [Mu53]. Many Cauchy integral equations are difficult to solve analytically, and it is required to obtain approximate solutions. In [Se93] the author has studied a reduction of some class of singular integral equations to regular Fredholm integral equations in L p (−1, 1). The purpose of this paper is to approximate the solution of a Cauchy integral equation of the second kind in L2 (0, 2π ), using collocation, trigonometric polynomials and a regularization procedure.

2 Description of the Method For each nonzero real constant μ and each real function f , consider the problem of finding a function ϕ such that M. Ahues Université Jean Monnet, Université de Lyon, F-42023 Saint-Étienne, France, e-mail: [email protected] A. Mennouni University of Bordj Bou-Arreridj, Algeria, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_1, © Springer Science+Business Media, LLC 2011

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M. Ahues and A. Mennouni

μϕ (s) −

 2π ϕ (t)

t −s

0

0 < s < 2π ,

dt = f (s),

(1)

where the integral is understood to be the Cauchy principal value:  2π ϕ (t)

t −s

0



s−ε

dt = lim

ε →0

0

ϕ (t) dt + t −s

 2π ϕ (t) s+ε

t −s

 dt .

Equation (1) is called a Cauchy integral equation of the second kind. Letting T ϕ (s) :=

 2π ϕ (t)

(1) reads as

0

t −s

0 < s < 2π ,

dt,

μϕ − T ϕ = f .

Theorem 1. For each function f ∈ L2 (0, 2π ), (1) has a unique solution ϕ ∈ L2 (0, 2π ), and the Cauchy integral operator T is bounded and skew-Hermitian from L2 (0, 2π ) into itself. Proof. See [PoSt90]. Let X := L2 (0, 2π ), and Xn denote the space spanned by the first 2n + 1 trigonometric polynomials. Define σn to be the orthogonal projection from X onto Xn . Hence, for ψ ∈ L2 (0, 2π ), lim σn ψ − ψ 2 = 0.

n→∞

Let be 0 < sn,1 < sn,2 < · · · < sn,2n+1 < 2π . For each i ∈ [[1, 2n + 1 ]] consider the hat function en,i in C0 (0, 2π ), such that, for each j ∈ [[1, 2n + 1 ]], en,i (sn, j ) = δi, j . Let Yn be the space spanned by these hat functions, which has dimension 2n + 1. Define the interpolation projection operator πn from C0 (0, 2π ) onto Yn :

πn h(s) :=

2n+1

∑

h(sn, j )en, j (s),

h ∈ C0 (0, 2π ).

j=1

We recall that (see [AhLaLi01]): lim πn h − h∞ = 0.

n→∞

Define the regularized operator Tε for ε > 0: Tε ϕ (s) :=

 2π (t − s)ϕ (t) 0

(t − s)2 + ε 2

dt,

0 < s < 2π ,

A Collocation Method for Cauchy Equations

3

which is compact and skew-Hermitian from L2 (0, 2π ) into itself. Let ϕε be the solution of the regularized integral equation (μ I − Tε )ϕε = f , and consider the approximation operator Tε ,n := πn Tε σn . Theorem 2. For n large enough, the operator μ I − Tε ,n is invertible, the constant

βε := sup (μ I − Tε ,n )−1  n

is finite, and the solution ψε ,n of the equation (μ I − Tε ,n )ψε ,n = f , converges to the solution ϕ of (1) if, first, n → ∞ and then ε → 0. Proof. Since Tε is compact, the theory developed in [AhLaLi01] shows that the inverse operator (I − Tε ,n )−1 exists and is uniformly bounded for n large enough. Since

ψε ,n − ϕε = [( μ I − Tε ,n )−1 − (μ I − Tε )−1 ] f = (μ I − Tε ,n )−1 [Tε − Tε ,n ](μ I − Tε )−1 f = (μ I − Tε ,n )−1 [Tε − Tε ,n ]ϕε , we get

ψε ,n − ϕε 2 ≤ βε (Tε − Tε ,n )ϕε 2 → 0 as n → ∞.

Since Tε is skew-Hermitian, (μ I − Tε )−1  ≤

1 , |μ |

independently of ε . Hence, the constant γ := sup (μ I − Tε )−1  ε

is finite and from we get Hence

ϕε − ϕ = (μ I − Tε )−1 [T − Tε ]ϕ ϕε − ϕ 2 ≤ γ [T − Tε ]ϕ 2 → 0 as ε → 0. ψε ,n − ϕ 2 ≤ ϕε − ϕ 2 + ψε ,n − ϕε 2 → 0,

if, first, n → ∞, and then ε → 0. The collocation method leads to the linear system (μ I − Tε σn )ψε ,n (sn,i ) = f (sn,i ),

i ∈ [[1, 2n + 1 ]].

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M. Ahues and A. Mennouni

3 Numerical Example Let μ = −1 and f (s) := −s[sin s+Si (s) cos s−sin(s)Ci (s)−Si (s−2π ) cos s+Ci (2π −s) sin s], where Si is the sine integral function, and Ci is the cosine integral function. Then the exact solution of (1) is ϕ (s) := s sin s. For the regularization process, take ε = 10−4 . For the numerical approximation take n = 9, n = 23 and n = 62. The results are exhibited in Figs. 1, 2 and 3, respectively.

Fig. 1 n = 9

Fig. 2 n = 23

A Collocation Method for Cauchy Equations

5

Fig. 3 n = 62

References [AhLaLi01] Ahues, M., Largillier, A., Limaye, B.V.: Spectral Computations for Bounded Operators, CRC Press, Boca Raton (2001). [Mu53] Mushkelishvili, N.I.: Singular Integral Equations, Noordhoff, Groningen (1953). [PoSt90] Porter, D., Stirling, D.: Integral Equations: A Practical Treatment, from Spectral Theory to Applications, Cambridge University Press (1990). [Se93] Sengupta, A.: A note on a reduction of Cauchy singular integral equation to Fredholm equation in L p , Applied Mathematics and Computation, 56, 97–100 (1993).

On a New Definition of the Reynolds Number from the Interplay of Macroscopic and Microscopic Phenomenology B.E.J. Bodmann, M.T. Vilhena, J.R. Zabadal, and D. Beck

1 Introduction Turbulence is a behavior seen in many fluid flows, which is conjectured to be driven by the inertia to viscosity force ratio, i.e. the Reynolds number. Even though research in turbulence has existed for more than a century there is still no consensus as how to elaborate a self-consistent and genuine theory, which describes the dynamics of a transition from a laminar to a turbulent regime or vice versa, and the geometric flow structure of turbulent phenomena. So far, it is believed that the Navier–Stokes equations model turbulence in an adequate way. However the existence of general solutions in three plus one space–time dimensions is still an open question [Ca07, Ca08, Co01, Co07, Fe06]. With the present discussion we intend to take a step in a new direction and show that a connection between microscopic and macroscopic degrees of freedom may well be the crucial ingredient for progress on the subject. Usually flow phenomena are captured starting from a continuous medium fluid which, in principle, permits us to scale down volume elements to infinitesimal size. As a consequence of using an equation independent of scales implies that the laws that dictate the macroscopic dynamics do not undergo changes while altering reference lengths, or other B.E.J. Bodmann Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] M.T. Vilhena Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] J.R. Zabadal Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] D. Beck Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_2, © Springer Science+Business Media, LLC 2011

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B.E.J. Bodmann et al.

measure quantities. In fact, fluids are made from atoms and molecules which obey microscopic laws and collectively constitute a stochastic system. This system obeys macroscopic laws provided by statistical thermodynamics and hydrodynamics. Both realms can be described phenomenologically by macroscopic observables and material dependent parameters (viscosity, thermal conductivity, specific heat, compressibility, among others), where these parameters hide the microscopic properties. If one traces back the parameters until its microscopic origin, in principle it should be possible to find quantitative macroscopic-microscopic relationships beyond mere phenomenology, where viscosity and length scales among others are macroscopic manifestations with microscopic origin. In this line we reason that turbulence, which is related to the Reynolds number, may be considered an interplay of the dynamics of at least two scales, a macroscopic one and a microscopic one. Thus, the present work is an attempt to show the possibilities that arise from, in our case, a simplified macroscopic-microscopic relationship, which we derive based on a simplified model motivated by Maxwell– Boltzmann transport. This chapter is organized as follows. In the next section we present the microscopic approach and introduce a length scale which relates to vorticity, in Sect. 3 we identify viscosity based on microscopic and thermodynamical quantities and last (Sect. 4) we discuss our findings and give future perspectives.

2 The Vortex Correlator Consider the fluid being composed by a particle ensemble (atoms, molecules or other micro-particles), which may be characterized mechanically by a local particle density n = n(x, y, z,t) and thermally by a local temperature T = T (x, y, z,t). In local equilibrium one  has a well defined relation between the temperature and a ve-

BT locity scale Cth = km (the thermal velocity) where kB is Boltzmann’s constant and particles have an average mass m. Here local equilibrium signifies that there exists a volume sufficiently small that temperature variations or equivalently variations in the velocity distribution are negligible, but that the volume contains still a sufficiently large number of particles as to represent a statistical ensemble. Further, we assume that there exists a particle–particle interaction with associated potential, which may in general be of scalar, vector or tensor type depending on the structure of the particles under consideration and their properties. For the forthcoming discussion we assume for simplicity that the interaction may be sufficiently characterized by a scalar potential Φ . A frequently used phenomenological potential is the Lennard–Jones potential, with its large range attraction and short range repulsion [Ma81]. Once the interaction potential is known or defined, one may calculate the interaction cross section σ , the correlated mean free path λ = (nσ )−1 and mean free propagation time τλ = Cλ . To have a typical path length, below which particles th in the average do not interact, makes evident the discrepancy between a continuous picture where in principle each infinitesimal volume element of the continuum in-

On a New Definition of the Reynolds Number

9

fluences the remainder of the fluid. The microscopic picture suggests a non-dense point set of interaction centers and a complementary dense set of interaction free points, where the microscopic behavior, because of its different topology in comparison to a continuous approach, may give rise to a different collective behavior (ensemble averages) on a macroscopic scale. On the macroscopic scale we understand the velocity fieldv(x, y, z,t) = c as the ensemble average of particle velocities c in a given volume element Δ V centered at r = (x, y, z) at an instant t. The first difficulty arises when trying to capture a typical macroscopic length scale, based on a microscopic property, which shall be related to a strength with which a flow is perturbed in order to present turbulent behavior. To  r,t) = v and consider this end we define the dimensionless velocity vector field G( Cth  →G  δ R , which shall simulate the change in the the field infinitesimally displaced G velocity field by virtue of vorticity. One may establish the relation to the original field by an infinitesimal coordinate transformation, which reads  δ R (r,t) = R G(R  −1r,t) G  = (1 − δ θ G) G((1 − δ θ G)r,t)  + G (r × (∇ × G)))   + δ θ (−G G =G where G are the generators of the transformation R represented as a vector and each  θ is the infinitescomponent contains a 3 × 3 transformation matrix which act on G. imal transformation parameter, i.e. a rotation angle with respect to a given axis θˆ . In component form and using the convention of summing over double indices, this reads Γδ R i = Γi + δ θ j (−εi jmΓm + ε jmn rm ∂n Γi ) (1) where εi jk is the complete antisymmetric Levi-Civita symbol. These findings may be related to vorticity using a concept from differential geometry, i.e., the generating term in (1) shall arise as a closed operator sequence— translation (Γ ), vorticity (Ω ), back translation and vorticity again, around a plaque of infinitesimal size. 

Ωi d Γj ∝ −εi jmΓm + εimn rm ∂nΓj .

(2)

An expression compatible with (2) and for any volume of interest then has the form

Ωi =

1 V

 V

∂Γj (−εi jmΓm + εimn rm ∂nΓj ) d 3 r. ∂t

(3)

One identifies two contributions, an extrinsic one which explicitly depends on the position and a second contribution which is position independent and thus may have only intrinsic origin. The presence of the second term can describe vorticity without the phenomenon of creating eddies (for instance present in shear flows), whereas the first term creates eddies even for a macroscopic velocity field which derives as a gradient from a scalar potential, for which the second term cancels out. A further

10

B.E.J. Bodmann et al.

comment is in order here: the intrinsic term makes sense only if microscopic degrees of freedom exist that constitute the macroscopic field Γ , since it depends only on the velocity field and its temporal variation in different directions. From this quantity (3) one may derive a macroscopic length scale which shall be used in order to define the Reynolds number. One may recognize that Ω contains the Γ fields in a bilinear form, so that the vorticity may be generalized to a correlation like function, henceforth called vorticity correlator    3  ∂Γj  1     (t) − ϒt (t,t ) =  εi jmΓm (t ) + εimn rm ∂n Γj (t ) d r . V V ∂t In the limit t  → t the correlator turns vorticity. Since the vorticity may be related to the angular frequency of an eddy, the correlator may be used to measure how far a particle with velocity Cth propagates across an eddy with non-vanishing correlations. The length scale Λ is then defined by the correlation between time and thermal velocity via the implicit relation 1   Cth τ ϒ0 (t, 0) 2 dt , Λ = τ Cth = √ 0 ϒt (t, 0) 2 where

   1 3   ϒ0 (t, 0) =  Γj (t) (−εi jmΓm (0) + εimn rm ∂nΓj (0)) d r V V 

and the thermal noise limit limτ →0 τ1 0τ ϒϒ0t dt = 2. So far the velocity field v = c, the macroscopic length scale Λ are available from expectation values of a microscopic ensemble. The remaining quantities like the particle density and the viscosity may be determined only from an analysis of the transport equation, i.e., a Navier–Stokes type equation, which may be derived starting from the Maxwell–Boltzmann transport equation.

3 The Transport Equation The Maxwell–Boltzmann equation describes the time evolution of the pseudo-local expectation values in a transport phenomenon. Here pseudo-local signifies local in a macroscopic (continuous) sense but discrete (by particle nature) in the microscopic sense. Its generic form is [Mu79]

  

∂ ∂ ∂O δ O n O + n cμ O − bμ = n . (4) ∂t ∂ rμ ∂ cμ δt Coll A similar equation to the Navier–Stokes one is obtained

by substituting the op erator to represent momentum transport n O = n mCμ Cν = pμν which is also  =c −v. In thermal equilibrium obviously recognized as the pressure tensor; here C

On a New Definition of the Reynolds Number

11

pμν = pδμν holds and since dissipative contributions are no longer at work, the diagonal (equilibrium) contributions to the pressure tensor contribute only to the homogeneous solution of the transport equation. We are interested in the dissipative part of the equation and hence reduce the pressure tensor to the friction pressure tensor   1 2 πμν = nm Cμ Cν − C δμν 3 with zero trace T r{πμν } = 0. The second term, after decomposition and some algebraic manipulations separates a term with constant temperature and a velocity field gradient, from a term that represents heat flux, which is kinetic energy transport

n qμ = m C2Cμ , 2 respectively. For simplicity we ignore possible contributions of an external force field and its resulting acceleration bμ , so that the term still to be determined is the right hand side of (4). A convenient way to simplify the equation is to approximate the collision term by an average friction pressure change

δ πμν πμν ≈ δt τp which renders the original equation a transport relaxation equation [Ch95, Ba08]



  ∂ 1 ∂ vλ 1 ∂ vμ ∂ vν − πμν + 2nkB T + δμν ∂t 2 ∂ rν ∂ r μ 3 ∂ rλ  



πμν 4 1 ∂ q μ ∂ qν 1 ∂ qλ + δμν + = 0. (5) + − 5 2 ∂ rν ∂ rμ 3 ∂ rλ τp The relaxation time constant τ p for phenomenological potentials and for systems not far from equilibrium in (5) may be related to the microscopic cross section in the spirit of Chapman–Cowling [Ma81]. The expression below shows the mechanical relaxation time, for an isotropic two particle interaction central potential:

−1  ∞  π √ 5 1 −u2 7 2 τp = √ e u (1 − cos (θ ))σ (θ , 2Cth u) sin(θ ) d θ du 16 π nCth 0 0 5 1 = √ . 16 π nCth I2 √ Here u is the relative velocity between the collision partners in multiples of 2Cth , θ signifies the scattering angle, and Cth a velocity scale, i.e. the thermal velocity. A local collision operator is responsible for the space–time evolution of the distribution in consideration. The collision term depends in general on microscopic dynamics which in many cases is not exactly known or is too complex to be evaluated analytically. However, for a number of applications there do exist interaction

12

B.E.J. Bodmann et al.

models [Dh07] that are sufficient to capture qualitatively as well as to a certain precision quantitatively properties of the fluid flow. The equation above results in the Navier–Stokes type equation if the following phenomenological identity holds:  



∂ vλ 1 ∂ vμ ∂ vν 1 ∂ vλ πμν = −ηV δμν − 2η + δμν − ∂ rλ 2 ∂ rν ∂ r μ 3 ∂ rλ with η shear and ηV volumetric viscosity, respectively. By comparison one identifies the shear viscosity as

η = kB T τ p =

5nkB T 5n mCth √ √ = . 16 πCth I2 16 π I2

Upon substitution of the found quantities into the traditional Reynolds number definition and replacing the usually employed macroscopic length by the vortex correlator length Λ one arrives at an expression which is characterized by two macroscopicmicroscopic ratios, the correlation length Λ against the mean free path λ and the macroscopic flow velocity v against the thermal velocity Cth besides a factor which is determined from the collision integral and the total collision cross section σT . √ ρΛ v 16 π I2 Λ v Re = = . η 5 σT λ Cth √ For a collision model where the cross section σ (θ , 2Cth u) does not depend on the scattering angle the integral can be solved analytically and is I2 = σπT .

4 Conclusion In the present discussion we established a connection between microscopic and macroscopic lengths and velocities which redefines the traditional Reynolds number. It is evident from its original definition that one needs a reference length in order to render the transport equation non-dimensional. In any case this length scale shall somehow synthesize the influence of boundaries and/or obstacles. Since boundaries select specific solutions from a manifold the velocity field that results from the solution of the transport equation contains this information and may thus be used to define a problem related length scale which we introduced by the vorticity correlator—a macroscopic reference length. We introduced the correlator motivated by the phenomenon that once a flow changes from laminar to turbulent flow perturbations perpendicular to the local flow velocity become important. In order to see what such a contribution looks like we analyzed the changes in the vector field under infinitesimal rotation. Making contact to the vorticity definition and generalizing our expression led to the vorticity correlator which yields only significant contributions if the afore mentioned perturbations are present in the field. These perturbations are

On a New Definition of the Reynolds Number

13

evidently a manifestation of inner and/or outer boundaries present in the problem under consideration. In our approach for vorticity one identifies an extrinsic (position dependent) and intrinsic contribution. The presence of the intrinsic term accounts for vorticity although the macroscopic velocity field derives from a gradient of a scalar potential, for which the curl of the extrinsic term cancels out. An intrinsic term can only be attributed to intrinsic degrees of freedom of a continuous macroscopic field and thus needs further (microscopic) degrees of freedom. In other words the macroscopic field is nothing but a macroscopic mean field from the microscopic point of view. The counterpart to the vorticity correlator—the microscopic length—has its origin in the interpretation of the dissipation parameter (i.e. the viscosity) in terms of particle collisions which through the cross section supplies with the mean free path of the particles that constitute the fluid. At this length the scaling symmetry of macroscopic transport breaks down. It is noteworthy that these lengths may be of macroscopic magnitude (for instance they may be several cm in a gas). The microscopic picture for dissipation circumvents a problem that arises if the fluid in consideration behaves approximately as an ideal fluid. In the classical Reynolds definition this means that the viscosity tends to small values which rises the Reynolds number in contradiction to the fact that without dissipation turbulence will not occur. This is different from the microscopic definition where the mean free path tends to infinity (or at least is huge) which drives the Reynolds number close to zero. A further effect comes from thermal motion in relation to the flow velocity. In a gas the thermal velocity may be orders of magnitude larger than the flow velocity, which means that the thermal noise may destroy coherent structures which are present in turbulence, because particles propagate back and forth in the fluid over lengths larger than the effective displacement length of the fluid. The closer the two velocities are the less is the influence of noise in the flow, and the formation of coherent flow patterns are possible. Such a collective behavior may not be understood from a purely macroscopic and continuous picture. From our findings we reach a new meaning of scale invariance of the macroscopic transport equation—hydrodynamical similarity. Apart from the collision model which enters as a factor, which for a variety of interaction potentials is of the order of magnitude of 100 , there are two relevant ratios responsible for similarity, the vorticity correlation length times the flow velocity as macroscopic expectation values compared to the mean free path times the thermal velocity, i.e. two microscopic reference quantities. Since our considerations are an attempt to approach the turbulence problem from a microscopic-macroscopic interplay (few body interaction—collective mean field dynamics) the present discussion is a first step into a new direction. The theoretical conception presented in this work may be applied to experimental findings as for instance the visualization of the time evolution of flows and can be compared to the simulations based on the Maxwell–Boltzmann transport equation. Such an analysis is necessary to support our new definition which hopefully will prove useful in the future to classify the regimen in flows and may bring benefit for applications as

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for instance in the problem of dispersion of pollution in the atmosphere and water. These challenges define the next steps of future activities.

References [Ba08] Banda, M., Klar, A., Pareschi, L., Seaïd, M.: Lattice-Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier–Stokes equations. Math. Comp., 77, 943–965 (2008). [Ca07] Cao, C., Titi, E.S.: Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Annals of Mathematics, 166, 245–267 (2007). [Ca08] Cao, C., Titi, E.S.: Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J., 57, 2643–2662 (2008). [Ch95] Cheng, M.-C.: An efficient approach to solving the Boltzmann transport equation. Simulation of Semiconductor Devices and Processes, 6, 202–205 (1995). [Co01] Constantin, P.: Some Open Problems and Research Directions in the Mathematical Study of Fluid Dynamics. Mathematics Unlimited and Beyond, Springer Verlag, Berlin, 353– 360 (2001). [Co07] Constantin, P., Levant, B., Titi, E.S.: A note on the regularity of inviscid shell model of turbulence. Physics Review E, 75, 016304-1–016304-10 (2007). [Dh07] Dhama, A.K., McCourt, F.R.W., Dickinson, A.S.: Accuracy of recent potential energy surfaces for the He–N2 interaction I: Virial and bulk transport coefficients. J. Chem. Phys., 127, 054302-1–054302-13 (2007). [Fe06] Fefferman, C.L.: Fluids and singular integrals. Contemporary Math., 411, 39–52 (2006). [Ma81] Maitland, G.C., Rigby, M., Smith, E.B., Wakeham,W.A.: Intermolecular Forces: Their Origin and Determination. Oxford University Press, Oxford (1981). [Mu79] Muncaster, R.G.: On generating exact solutions of the Maxwell–Boltzmann equation. Arch. Rational Mech. Anal., 70, 79–90 (1979).

A Self-Consistent Monte Carlo Validation Procedure for Hadron Cancer Therapy Simulation L.N. Burigo, D. Hadjimichef, and B.E.J. Bodmann

1 Introduction Accelerated heavy ions (3 He, 12 C, among others) nowadays provide an advanced non-invasive procedure for radiotherapy of tumors with risky or impossible access, henceforth called Hadron Cancer Therapy [Kra90, Dur08]. Moreover heavy-ion beams naturally optimize the physical depth-dose profile (known as Bragg curve) with an increased relative biological efficiency in the target volume that as a consequence minimizes damage in the healthy tissue. A further advantage of heavy ions over protons is the positron production from a by-product of nuclear reactions and subsequent decays [Psh05, Psh06]. Thus positron emission tomography (PET) allows for dose verification in real-time. The raster-scan technology was developed at the GSI facility [Gad90, Kra91] and today is, and in the close future will be implemented in several cancer treatment centers all over Europe. While treatment planning in the pioneer stage of the developments was done by experiments with phantoms, in the consolidation phase these may be substituted by computerized treatment planning engines that make use of physical and radio-biological data obtained at GSI and other places. The state of the art so far permits us to treat skull base tumors and tumors close to the spinal chord [Deb00, Dur10, Sch10]. Raster-scan nuclear surgery is performed by dividing the target (tumor) volume into ∼ 102 slices of equal ion beam range, which is controlled by the particle beam energy and each slice is scanned in horizontal and vertical direction by fast magL.N. Burigo Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] D. Hadjimichef Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] B.E.J. Bodmann Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_3, © Springer Science+Business Media, LLC 2011

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netic deflection [Kra94, Kra95]. However, the relative biological effectiveness or cell survival probability cannot be measured for an individual subject, and therefore one has to resort to calculations or simulations. Hence in model approaches the energy deposit (or dose delivery) to the tumor for tissue-like material shall be under control, which includes the energy loss of the primary particle and the contributions of produced projectile fragments [Psh05, Psh06]. Further, physical–biological models for the interaction of ions with living matter shall be available. Treatment planning by simulations means that one determine the necessary energies, positions and intensities for the ion beam foci which shall result in a close to homogeneous biological dose distribution conforming to the target volume. If such a planning were performed by a numerical inversion algorithm the optimization problem would have to handle up to ∼ 105 degrees of freedom for the largest tumors. Whenever one faces a problem with an exorbitant number of degrees of freedom, sampling methods (here Monte Carlo simulation) seem an adequate tool. More specifically, in the present contribution we make use of the GEANT4 platform (versions 8.2 and 9.3 with patch 01) [Ago03] in order to implement the simulations of heavy-ion transport in the medium energy range (100–400 MeV/u) using the Boltzmann equation with electromagnetic and nuclear collision terms of heavy ions and their products in material compositions that are similar to biological tissues. The tallies of the simulation are position dependent energy deposition besides the nuclear fragment abundances, which are important for the real-time position control by PET. Moreover, we focus especially on the question of simulation fidelity and introduce a self-consistency criterion first solving the transport equation by a physical Monte Carlo (which simulates the microscopic interactions of the processes) and validating the solution by a mathematical Monte Carlo (an abstract formal method without direct physical correspondence). We show results of simulations and their validation.

2 Particle Transport by GEANT4 Monte Carlo Computational technology developments no longer impose limitations in computing power, so that close to reality Monte Carlo simulations for treatment planning has turned into a viable option in a clinical setting. Nevertheless approximations and simplifications or idealizations that speed up the calculations are a common and necessary practice [Moh01], but its impact on dosimetry accuracy is not well known yet, so that validation procedures are needed. One of the promising Monte Carlo platforms for this purposes is the GEANT4 program library [Ago03, All06], which provides a general infrastructure for the description of geometry and materials for processes such as particle transport and interaction with matter, and further allows us to implement material response which may be visualized geometrically with tracks and vertices. It remains for the user to implement the details in the specific code that describes the primary event generator, the specific combinatorial geometry set-up with its specific materials and the material response.

A Self-Consistent Monte Carlo Validation Procedure

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The present simulations assume an idealized continuous beam accelerator with mono-energetic 3 He particles, and no beam divergence, which hit a simplified homogeneous target volume with various test materials (we show results for graphite and gelatine). Once the projectile enters the target volume it may loose energy or produce secondary particles so that a cascade of particles have to be transported taking into account the interactions implemented in the library, and in the end the track of each individual ionizing particle through the volume of interest is simulated. A random number generator together with probability distributions for the different types of interaction determines the spatial sampling procedure (distance between interactions and type), i.e. a particle at a given position interacts and leaves with velocity vector v in a certain direction. The particle is then propagated with velocity v over the distance r to the next interaction location, where the type of interaction that will take place is chosen from probability distributions. In addition to the transport with its associated particle property changes, tallies are recorded that give the resulting energy deposit in a certain volume. Therefore, for each simulated interaction the energy balance of the particles before the vertex minus the energy of the outgoing one(s) plus the particle identities are stored. To calculate the accumulated energy deposition in a particular volume, one integrates the contributions from all interactions taking place inside the respective volume. Note that we consider a homogeneous material, so that the dose distribution and energy deposition are proportional, which is not true for heterogeneous materials. As mentioned in the introduction, the present principal interest is besides the proper simulation also the evaluation of simulation uncertainties that may result from idealizations and simplifications besides imprecisions in the physical data bases such as cross sections. However, for the time being the uncertainty associated with tissue characterization is most difficult to quantify. Case studies in this direction have shown [Fra03] that clinical effects are already noticeable for dose errors of 7%. Therefore accurate dose information is required, in other words the simulation reliability has to be evaluated in order to pin down significant errors to data base entries and physical models [Cyg05]. To this end we propose a self-consistency check by a mathematical Monte Carlo that recovers at least some of the input data and thus helps eliminate problems with logical structures in the program, data bases or models that determine details of processes. Here we present the simplest of this kind of test, we reproduce the total nuclear cross section of the processes considered in the simulation.

3 Heavy Ion Transport The heavy-ion flux φX of a certain species X satisfies a transport equation of the Boltzmann type. In our case, species signifies either the primary particle (3 He with initial energy E = 207.92 MeV per nucleon) or the fragments from the target materials (B, Be, He, Li).

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1 ∂ φX (r, Ω , E,t) + Ω · ∇φX (r, Ω , E,t) + ΣtX (E)φX (r, Ω , E,t) v ∂t = qX (r, Ω , E,t). Here, v is the ion velocity, r,t are the space–time coordinates, Ω indicates the directional solid angle, ΣtX is the total macroscopic cross section for processes of particles of type X ∈ {B, Be, He, Li} and qX (r, Ω , E,t) represents the heavy-ion source (accelerator) for primary particles S and particles produced in fragmentations (the integral term). Note that we include a pseudo Bethe–Bloch cross section into ΣtX , which is possible if one uses the relation of linearized energy loss for a given energy to the macroscopic cross section. qX (r, Ω , E,t) =

∑

 ∞

Y =B,Be,He,Li 0

dE 

 4π

d Ω Σ XY (r, E  → E, Ω  → Ω )

× φX (r, Ω  , E  ,t) + δXHe S(r, Ω , E,t). Here δXHe is the Kronecker symbol, that is, 1 for X = He and 0 otherwise; the first term includes contributions from several nuclear reactions, including particles produced from scattering processes;

ΣXY (r, E  → E, Ω  → Ω ) dE d Ω represents the number of ions of species Y emitted in the small energy interval dE at E and small angular range d Ω at Ω from the reaction of ion X at energy E  and direction of motion Ω  , with Σ XY (r, E  → E, Ω  → Ω ) being the corresponding macroscopic cross section. The energy dependent cross sections for each individual processes are provided by the GEANT4 data libraries [Ago03, All06] that contain also a list of relative atomic masses or molar masses of the elements. Note that detailed data for heavyion collisions are still in their consolidation phase so that there are still considerable uncertainties with respect to cross sections. The densities of the materials used in a simulation are specified by the user, however in the case of graphite we show that the physical condition of the material (atomic structure) is probably not properly taken into account, whereas for the water–gelatine mixture the uncertainties are acceptable. Note that the latter has physical properties close to biological tissues. In the further text we show by simulation results that the accuracy of the simulations are compromised because of uncertainties in the input data, of which the cross sections form a crucial part. Differences between simulation for the same problem occurred when comparing the results of version 8.2 with version 9.3, patch 01. With our proposed self-consistency analysis that we will show in the next sections we pin down uncertainties almost to input data only. Once the simulation is free of logical errors or biases (because of idealizations and simplifications) validation of input data may be performed.

A Self-Consistent Monte Carlo Validation Procedure

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4 Simulation Results and Validation Simulations of 106 3 He ions with initial energy E = 207.92 MeV were performed considering two different target materials, graphite (as specified in the GEANT4 data base) and a mixture of water and gelatine (user defined) that approximates the properties of biological tissues. The choice of the two materials was made in order to show one example where considerable differences appear depending on the version of the simulation platform, here GEANT4 version 8.2 and version 9.3 with patch 01, respectively. The geometry, which in more realistic cases could be of combinatorial form was assumed to be a homogeneous cylinder aligned with the direction of the particle beam. Individual particles were transported across the medium from vertex to vertex and all additional particles that appeared in the specific reactions were propagated until their energies were totally lost in the medium. From the tallies that were collected along the transport of individual particles, one may project out the energy deposition with penetration depth, the yield of β + emitters, relevant for online monitoring of the penetration depth of the primary particles and last not least the electromagnetic ionizing processes manifest in the Bragg curve with its characteristic peak at the end. In Fig. 1 we show the Bragg curve which illustrates the advantage over photon treatments due to the enhanced energy deposition at the target position.

Fig. 1 Bragg curves of 3 He ions with initial energy E = 207.92 MeV incident on graphite (left) and water–gelatine (right)

One clearly observes in the case of graphite a considerable difference in the two energy deposition predictions, which by the way refer to electromagnetic processes and not nuclear ones. We attribute this to the fact that the graphite structure depends strongly on the physical conditions of the material production (diamond is an obvious example) which shall enter in the material properties. For the gelatine-water mixture both simulations coincide. As mentioned before, some of the produced nuclear fragments are β + emitters which permit to monitor the position of the Bragg peak that should match the target location (tumor). From the comparison of the Bragg curves (Fig. 1) and the activity of β + emitters (Fig. 2) one observes that the

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Fig. 2 Activity curves of β + emitters in graphite (left) and in water–gelatine (right) produced by 3 He

Fig. 3 Energy deposition of fragments produced by 3 He ions in graphite (left) and in water– gelatine (right)

peak positions are the same. However, again for graphite the two simulations yield considerable differences in the simulated distributions. In Fig. 3 we show the energy deposition by the fragments boron, beryllium, helium and lithium, produced in particle collisions from nuclei of the target material. A comparison to the Bragg curve shows that nuclear contributions to the energy deposition is of the order of 101 % and thus not negligible. Simulation fidelity may now be implemented using a self-consistency criterion. Recalling that heavy-ion transport was implemented by a physical Monte Carlo following the instructions of the Boltzmann equation. In other words an ensemble of particles together with microscopic processes mimics what we believe that happens in the real world. In parallel to particle propagation tallies are recorded that partially are used to show the depth versus energy or versus activity profiles and on the other hand allow for particle counting at a certain position, with velocity in a certain direction and with a specified energy. Note that the present simulation assumes a stationary case so that no explicit time dependence appears in the simulation data. The afore mentioned self-consistency criterion makes use of the same transport equation together with the tallies and further the Bethe–Bloch approach to invert the problem

A Self-Consistent Monte Carlo Validation Procedure

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and solve for the nuclear cross sections by the use of a mathematical Monte Carlo. We are aware of the fact that there exists a collection of solvers for inverse problems which we could have employed here, but for simplicity we used a sample of cross section values for specific energies which were accepted or rejected according to a probability factor (in the spirit of a Boltzmann factor) that is defined in terms of the distance between the left and right hand side of the Boltzmann equation averaged over all energies (this procedure is incremental). Since 106 particles is not sufficient to get good statistics for each fragment of the nuclear reaction we determine the total nuclear cross section σt for all nuclear processes and averaged over the particle energies. For graphite we get the σt = 1.22b with GEANT4 version 9.3 and σt = 1.11b with version 8.2. The values for water plus gelatine are σt = 1.57b and σt = 1.59b with GEANT4 version 9.3 and with version 8.2, respectively.

5 Conclusion In the present contribution we presented an integral method based on the Monte Carlo method to compute the transport together with properties (tallies) of heavy ions through specific matter in the context of the hadron cancer therapy program. We used the same transport problem to implement particle propagation (by a physical Monte Carlo) and to validate simulation fidelity using self-consistency for the inverse problem that reproduced the energy averaged total cross section of the nuclear reactions (by a mathematical Monte Carlo inversion). In other words the same transport equation system was used for simulation and for validation closing thus self-consistency. As our simulation results plus validation show, it is safe to say that simulation fidelity is under control so that it remains to face physical model and data base validation against experimental data, which up to now are unfortunately scarce. The clinical treatment procedure is well established from the technical point of view, so that reliable simulations are necessary for treatment planning and optimization that will also have impact on the economical factor of the therapy. Simulations will substitute the otherwise necessary experimental treatment simulations. Since from the technical point of view the accelerator surgery precision is on the mm level, the simulator has to close up and attain the same or an even better precision. The ultimate end is to obtain a treatment planning system entirely based on a Monte Carlo dose engine, i.e. where the simulator is integrated into the optimization loop (optimization during treatment planning and treatment monitoring and recording for inverse planning). Such a protocol will be necessary to minimize errors in the dose algorithm, which would otherwise lead to wrongly optimized beam set-ups used for treatment [Jer02]. As a future activity, we will consider also time dependence of the beam and thus time dependence of the energy deposition and activities [Kea01, Liu01, Ver01]. This will especially allow to accompany the transients of the treatment during startup and more important at the end and after the treatment, where the β + are still

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present according to their respective life times, manifest in their decay curves. As a new aspect of time dependence it will also be possible to extend the simulation with time dependent geometries that are up to now an unsolved challenge in radiotherapy dose calculations. These extensions are necessary because of motion of the patient or its body, which introduce new uncertainties into the treatment and simulation [Din03], i.e. positioning errors (different positions during treatment compared to the diagnostic image), inter-fraction organ motion (e.g. digestive system), and intrafraction organ movement (e.g. respiration-induced motion). The new aspect to be faced in future works are complex boundary and interface conditions besides the realistic time structure of the beam. Although we believe to have made progress in improving the GEANT4 based simulation engine, however, in order to turn it an instrument for standard clinical use there is still much to be done.

References [Ago03] Agostinelli, S., Allison, J., Amako, K. et al.: GEANT4—a simulation toolkit. Nucl. Instr. Meth. Phys. Res. A, 506, 250–303 (2003). [All06] Allison, J. et al.: Geant developments and applications. IEEE Trans. Nucl. Sci., 53, n. 1, Part 2, 270–278 (2006). [Cyg05] Cygler, J.E., Lochrin, C., Daskalov, G.M., Howard, M., Zohr, R., Esche, B., Eapen, L., Grimard, L., Caudrelier, J.M.: Clinical use of a commercial Monte Carlo treatment planning system for electron beams. Phys. Med. Biol., 50, 1029–1034 (2005). [Deb00] Debus, J., Haberer, T., Schulze-Ertner, D., Jäkel, O., Wenz, F., Enghardt, W., Schlegel, W., Kraft, G., Wannenmacher, M.: Fractionated carbon ion irradiation of skull base tumors at GSI. First clinical results and future perspectives. Strahlenther. Onkol., 176, 211–216 (2000). [Din03] Ding, M., Li, J., Deng, J., Fourkai, E., Ma, C.-M.: Dose correlation for thoracic motion in radiation therapy of breast cancer. Med. Phys., 30, 2520–2529 (2003). [Dur08] Durante, M.: Focus on heavy ions in biophysics and medical physics. New J. Phys., 10, 075002 (2008). [Dur10] Durante, M., Loeffler, J.S.: Charged particles in radiation oncology. Nat. Rev. Clin. Oncol., 7, 37–43 (2010). [Fra03] Fraass, B.A., Smathers, J., Deye, J.: Summary and recommendations of a National Cancer Institute workshop on issues limiting the clinical use of Monte Carlo calculation algorithms for megavoltage external beam radiation therapy. Med. Phys., 30, 3206–3216 (2003). [Gad90] Gademann, G., Hartmann, G.H., Kraft, G., Lorenz, W.-J., Wannenmacher M.: The medical heavy ion therapy project at the Gesellschaft für Schwerionenforschung facility in Darmstadt. Strahlenther. Onkol., 166, 34–39 (1990). [Jer02] Jeraj, R., Keall, P.J., Siebers, J.V.: The effect of dose calculation accuracy on inverse treatment planning. Phys. Med. Biol., 47, 391–407 (2002). [Kea01] Keall, P.J., Siebers, J.V., Arnfield, M., Kim, J.O., Mohan, R.: Monte Carlo dose calculations for dynamic IMRT treatments. Phys. Med. Biol., 46, 929–941 (2001). [Kra90] Kraft, G.: Radiobiological and physical basis for radiotherapy with protons and heavier ions. Strahlenther. Onkol., 166, 10–13 (1990). [Kra91] Kraft, G., Becher, W., Blasche, K., Böhne, D., Fischer, B., Gademann, G., Geissel, H., Haberer, Th., Klabunde, J., Kraft-Weyrather, W., Langenbeck, B., Mänzenberg, G., Ritter, S., Rösch, W., Schardt, D., Stelzer, H., Schwab, Th.: The heavy ion therapy project at GSI. Nucl. Tracks Radiat. Meas., 19, 911 (1991).

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Kraft, G.: Heavy-ion therapy at GSI. Europhys. News, 25, n. 4, 81 (1994). Kraft, G., Arndt, U., Becher, W., Schardt, D., Stelzer, H., Weber, U.: Heavy ion therapy at GSI. Th. Archinal Nucl. Instr. and Meth. in Phys. Res. A, 367, 66 (1995). [Liu01] Liu, H.H., Verhaegen, F., Dong, L.: A method of simulating dynamic multileaf collimators using Monte Carlo techniques for intensity-modulated radiation therapy. Phys. Med. Biol., 46, 2283–2289 (2001). [Moh01] Mohan, R., Antolak, J., Hendee, W.R.: Monte Carlo techniques should replace analytical methods for estimating dose distributions in radiotherapy treatment planning. Med. Phys., 28, 123–126 (2001). [Psh05] Pshenichnov, I.A., Mishustin, I.N., Greiner, W.: Neutrons from fragmentation of light nuclei in tissue-like media: a study with GEANT4 toolkit. Phys. Med. Biol., 50, 5493– 5507 (2005). [Psh06] Pshenichnov, I.A., Mishustin, I.N., Greiner, W.: Distributions of positron-emitting nuclei in proton and carbon-ion therapy studied with GEANT4. Phys. Med. Biol., 51, 6099– 6112 (2006). [Sch10] Schardt, D., Elsässer, T., Schulz-Ertner, D.: Heavy-ion tumor therapy: physical and radiobiological benefits. Rev. Mod. Phys., 82, 383–425 (2010). [Ver01] Verhaegen, F., Liu, H.H.: Incorporating dynamic collimator motion in Monte Carlo simulations: an application in modelling a dynamic wedge. Phys. Med. Biol., 46, 287–296 (2001).

A General Analytical Solution of the Advection–Diffusion Equation for Fickian Closure D. Buske, M.T. Vilhena, C.F. Segatto, and R.S. Quadros

1 Introduction In the last few years there has been increased research interest in searching for analytical solutions for the advection–diffusion equation (ADE). By analytical we mean that no approximation is done along the derivation of the solution. There exists a significant literature regarding this theme. For illustration we mention the works of [Rou55, Smi57, ScFi75, Dem78, Van78, NiHa81, Tag85, Tir89, TiRi94, Sha96, LiHi97, Tir03]. We note that in these works all solutions are valid for very specialized problems having specific wind and eddy diffusivities vertical profiles. Further, also in the literature there is the ADMM (Advection Diffusion Multilayer Method) approach which solves the two-dimensional ADE with variable wind profile and eddy diffusivity coefficient [Mor06]. The main idea relies on the discretization of the Atmospheric Boundary Layer (ABL) in a multilayer domain, assuming in each layer that the eddy diffusivity and wind profile take averaged values. The resulting advection–diffusion equation in each layer is then solved by the Laplace transformation technique. For more details about this methodology see the review work done by [Mor06]. We are also aware of the recent work of [Cos06], dubbed as GIADMT method (Generalized Integral Advection Diffusion Multilayer Technique), D. Buske Universidade Federal de Pelotas, RS, Brazil, e-mail: [email protected] M.T. Vilhena Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] C.F. Segatto Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] R.S. Quadros Universidade Federal de Pelotas, RS, Brazil, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_4, © Springer Science+Business Media, LLC 2011

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which presented a general solution for the time-dependent three-dimensional ADE, again assuming the stepwise approximation for the eddy diffusivity coefficient and wind profile and proceeding further in similar way according the previous work. To avoid this approximation, in this work we report an analytical general solution for this problem, assuming that the eddy diffusivity coefficient and wind profile are arbitrary functions having a continuous dependence on the vertical and longitudinal variables. Without losing generality we specialize the application in micrometeorology, specially for the problem of simulation of contaminant releasing in the ABL. To reach this goal, we first expand the contaminant concentration in a series expansion in terms of a set of orthogonal eigenfunctions. Replacing this expansion in the time-dependent, three-dimensional ADE in Cartesian geometry and by taking moments we obtain a set of two-dimensional ADEs, which are then solved by the Generalized Integral Laplace Transform Technique (GILTT) discussed by [Mor09a, Mor09b, Bus10]. The main idea of this methodology comprises the steps: we again expand the pollutant concentration in series of a set of orthogonal eigenfunctions. After replacing this expansion in the ADE and taking moments, we come out with a matrix ordinary differential equation that is then analytically solved by the Laplace Transform technique [Mor09b].

2 The Analytical Solution In the following we derive the ADE for the simulation of pollutant releasing in the ABL assuming Fickian closure of the turbulence. We must recall that this equation is derived combining the continuity equation ruled by the conservation law with the Fickian closure of turbulence. Indeed, we write the ADE in Cartesian geometry as in [Bla97]:       ∂c ∂c ∂c ∂c ∂ ∂c ∂ ∂c ∂ ∂c +u +v +w = Kx + Ky + Kz (1) ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂y ∂z ∂z subject to the following boundary and initial conditions:

∂c = 0 at z = 0, h, ∂z

(2)

Ky

∂c = 0 at y = 0, Ly , ∂y

(3)

Kx

∂c = 0 at x = 0, Lx , ∂x

(4)

Kz

c(x, y, z, 0) = 0. Here we replace the source term by a source condition quoted as

(5)

A General Analytical Solution of the ADE

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uc(0, y, z,t) = Qδ (y − y0 )δ (z − Hs ).

(6)

We notice that c denotes the mean concentration of a passive contaminant (g/m3 ) and u, v and w are the Cartesian components of the mean wind speed (m/s) in the directions x (0 < x < Lx ), y (0 < y < Ly ) and z (0 < z < h). Q is the emission rate (g/s), h the height of the ABL (m), Hs the height of the source (m), Lx and Ly are the limits in the x and y-axis and far away from the source (m) and δ represents the Dirac delta function. The source position is at x = 0, y = y0 and z = Hs . In order to solve the problem (1), taking advantage of the well-known solution of the two-dimensional problem with advection in the x-direction by the GILTT method [Mor09a], we initially apply the integral transform technique in the y variable. To this aim, we expand the pollutant concentration as M

c(x, y, z,t) =

∑ cm (x, z,t)Ym (y)

(7)

m=0

where Ym (y) are a set of orthogonal eigenfunctions, given by Ym (z) = cos(λm y), and λm = mπ /Ly (m = 0, 1, 2, . . .) are, respectively, the set of eigenvalues. To determine the unknown coefficient cm (x, z,t) for m = 0 : M we began recasting (1) by applying the chain rule for the diffusion terms. Substituting (7) in the resulting L equation and taking moments, meaning applying the operator 0 y ()Yn (y), we obtain M

∑



m=0

−

∂ cm (x, z,t) ∂t

− vcm (x, z,t) + Kx

 Ly 0

∂ 2 cm (x, z,t) ∂ x2

− λm2 cm (x, z,t)

 Ly 0

Ym (y)Yn (y)dy, −u

Ym (y)Yn (y)dy − w  Ly 0

∂ cm (x, z,t) ∂x

∂ cm (x, z,t) ∂z

Ym (y)Yn (y)dy + Kx

∂ 2 cm (x, z,t) + Kz ∂ z2

0

 Ly

0

Ym (y)Yn (y)dy

Ym (y)Yn (y)dy

∂ cm (x, z,t) ∂x

KyYm (y)Yn (y)dy + cm (x, z,t)  Ly

0

 Ly

 Ly 0

 Ly

0

∂ cm (x, z,t) Ym (y)Yn (y)dy + Kz ∂z

 Ly 0

Ym (y)Yn (y)dy

Ky Ym (y)Yn (y)dy 

 Ly 0

Ym (y)Yn (y)dy = 0. (8)

Defining the integrals appearing in the above equation by  Ly 0

 Ly 0

Ym (y)Yn (y)dy = αn,n ;

Ky Ym (y)Yn (y)dy = γm,n ;

using these definitions we recast (8) as

 Ly

 Ly 0

0

Ym (y)Yn (y)dy = βn,n ;

Ky Ym (y)Yn (y)dy = ηm,n ,

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D. Buske et al. M

∑

m=0

 − αn,n

∂ cm (x, z,t) ∂ cm (x, z,t) − u αn,n − v βn,n cm (x, z,t) ∂t ∂x

∂ cm (x, z,t) ∂ 2 cm (x, z,t) ∂ cm (x, z,t) + Kx αn,n + Kx αn,n 2 ∂z ∂x ∂x 2 c (x, z,t) ∂ m − λm2 γm,n cm (x, z,t) + ηm,n cm (x, z,t) + Kz αn,n ∂ z2  ∂ cm (x, z,t) = 0. + Kz αn,n ∂z − w αn,n

(9)

Without losing generality, we consider the application of a pollutant dispersion problem in ABL assuming that the speeds v and w take the null value. We neglect the diffusion component Kx because we assume that the advection is dominant in the x-direction. Further we also consider that Ky has only dependence on the z-direction. After these assumptions, (9) assumes the matrix form ⎡ ⎤ ⎡ ∂ c0 ⎤ ⎡ ⎤ ⎡ ∂ c0 ⎤ 1 0 · · · 0 1 0 ··· 0 ∂t ∂x ⎥ ∂ c1 ⎥ ⎢ 0 1 ··· 0 ⎥⎢ ⎢ 0 1 ··· 0 ⎥⎢ ⎢ ∂∂cx1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ∂ t ⎢ ⎥ u −⎢ . . . . ⎥⎢ − ⎢ ⎥ . . . . ⎥ . ⎥ ⎥+ ⎣ .. .. . . . .. ⎦ ⎢ ⎣ .. .. . . .. ⎦ ⎢ ⎣ .. ⎦ ⎣ .. ⎦ ∂ cM ∂ cM 0 0 ··· 1 0 0 ··· 1 ∂t ∂x ⎡ 2 ⎤ ⎡ ⎡ ⎡ ⎤ ∂ c0 ⎤ ∂ c0 ⎤ 1 0 ··· 0 ⎢ ∂z ⎥ 1 0 ··· 0 ∂z ∂ c1 ⎥ ⎢ 0 1 · · · 0 ⎥ ⎢ ∂ 2 c1 ⎥ ⎢ 0 1 ··· 0 ⎥⎢ ⎢ ⎢ ⎢ ⎥⎢ ⎥ ∂z ⎥ ⎥ ⎥ +Kz ⎢ . . . . ⎥ ⎢ ∂.z ⎥ + Kz ⎢ . . . . ⎥ ⎢ .. ⎥ + . . . . . . . . ⎣ . . . . ⎦ ⎢ .. ⎥ ⎣ . . . . ⎦⎢ ⎣ . ⎦ ⎣ ⎦ ∂ cM 0 0 ··· 1 0 0 ··· 1 ∂ 2 cM ∂z

⎡

1 0 ··· ⎢ 0 1 ··· ⎢ −λm2 Ky ⎢ . . . ⎣ .. .. . .

⎤⎡

∂z

c0 0 ⎢ c1 0⎥ ⎥⎢ .. ⎥ ⎢ .. . ⎦⎣ .

0 0 ··· 1

⎤ ⎥ ⎥ ⎥=0 ⎦

cM

which clearly leads to the ensuing set of M +1 two-dimensional diffusion equations:   ∂ cm (x, z,t) ∂ cm (x, z,t) ∂ ∂ cm (x, z,t) +u = Kz − λm2 Ky (z) cm (x, z,t). (10) ∂t ∂x ∂z ∂z The problem (10) is then solved by the GILTT method. Following the works of [Mor09a, Mor09b] and taking advantage of the well-known solution for the stationary problem with advection in the x direction, we apply the Laplace transformation technique in the t variable (t → r), obtaining the following steady-state problem:   ∂ Cm (x, z, r) ∂ ∂ Cm (x, z, r) = Kz − λm2 Ky Cm (x, z, r). (11) rCm (x, z, r) + u ∂x ∂z ∂z

A General Analytical Solution of the ADE

29

Now we pose the solution of problem (11) in the form L

Cm (x, z, r) =

∑ Cm,l (x, r) ζl (z),

(12)

l=0

where ζl (z) are a set of orthogonal eigenfunctions, given by ζl (z) = cos(λl z), and λl = l π /h, l = 0, 1, 2, . . . is the set of eigenvalues. Substituting (12) into (11) and taking moments, we get the following first order matrix differential equation: dYm (x, r) + G.Ym (x, r) = 0, dx

(13)

for m = 0 : M, where Ym (x, r) is the column vector whose components are {Cm,l (x, r)} for l = 0 : L. The matrix G is defined as G = B−1 1 B2 . The entries of matrices B1 and B2 are given by (b1 )l, j = −

 h 0

u ζl (z) ζ j (z)dz

and  h

(b2 )l, j =

0

Kz ζl (z) ζ j (z)dz − λl2

 h

−(r + λl2 Ky )

0

Kz ζl (z)ζ j (z)dz

 h 0

ζl (z) ζ j (z)dz,

respectively. As in the work of [Mor09b], we solve problem (13) obtaining the following solution:  1 L γ +i∞ cm (x, z,t) = (14) ∑ γ −i∞ Cm,l (x, r) ζl (z)ert dr. 2π i l=0 To overcome the drawback of evaluating the line integral appearing in the above solution, we perform the calculation of this integral by the Gaussian quadrature scheme, namely   K pk pk , (15) cm (x, z,t) = ∑ ak Cm x, z, t t k=0 where ak and pk are, respectively, the weights and nodes of the Gaussian quadrature scheme [StSe66]. Regarding the issue of the adopted Laplace numerical inversion scheme, it is important to mention that this approach is exact if the integrand is a polynomial of degree 2M − 1 in the variable. We are aware of the existence in the literature of methods to invert numerically the Laplace transformed functions [VaAb04, AbVa04], but we restrict our attention in the problem considered to the Gaussian quadrature scheme. The motivation for this choice comes not only from the simplicity of the scheme but also the good results achieved by it.

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3 Experimental Data and Turbulent Parameterization In the following we report the parameterizations adopted for the simulations reported in this chapter. We need to recall that the choice of the turbulent parameterization represents a fundamental aspect for pollutant dispersion modeling [Mor05]. In terms of the convective scaling parameters, the vertical eddy diffusivity can be well formulated by the algebraic formulation proposed by [Deg02]: 0.583w∗ hci ψ 2/3 (z/h)4/3 X ∗ [0.55(z/h)2/3 + 1.03ci ψ 1/3 ( fm∗ )i X ∗ ] 1/2

Kα =

1/3

[0.55(z/h)2/3 ( fm∗ )i

2/3

1/2

+ 2.06ci ψ 1/3 ( fm∗ )i X ∗ ]2

(16)

where n is the non-dimensional frequency, cv,w = 0.36, cu = 0.3, and ( fm∗ )i is the normalized frequency of the spectral peak independent of the stratification, with ( fm∗ )u = 0.67 for the longitudinal component and ( f m∗ )w

= 0.55

 z  h

   −1 8z 4z 1 − exp − − 0.0003 exp h h

for the vertical component. Notice that these eddy diffusivities are functions of not only turbulence but also of distance from the source [Ary95]. On the other hand, we assume that the wind profile is described by a power law, namely [PaDu88]  n z uz = (17) u1 z1 where uz and u1 are the mean wind velocity at the heights z and z1 , while n = 0.1 under unstable conditions. In order to illustrate the aptness of the discussed formulation to simulate contaminant dispersion in the ABL, we evaluate the performance of the proposed solution against experimental ground-level concentration. The experimental data set used to evaluate the performance of the model in unstable conditions were carried out in the northern part of Copenhagen [GrLy84]. It consists of tracer without buoyancy released from a tower at a height of 115 m, and collection of the tracer sampling units at the ground-level positions at the maximum of three crosswind arcs. The sampling units were positioned at two to six kilometers from the point of releasing. The site has a roughness length of 0.6 m.

4 Numerical Results Next we display the numerical results attained by this methodology and proceed a statistical comparison against the GILTT two-dimensional results assuming Gaussian solution in the y-direction (named here as GGILTT), as well the GIADMT results [Cos06] and experimental data set of Copenhagen experiment. We begin, for

A General Analytical Solution of the ADE

31

sake of completeness, defining the statistical index discussed by [Han89]. In fact, the statistical indices are defined by • • • • •

NMSE (normalized mean square error) = (Co −C p )2 /C p Co , COR (correlation coefficient) = (Co −Co )(Cp −C p )/σo σ p , FA2 = fraction of data (%, normalized to 1) for 0.5 ≤ (C p /Co ) ≤ 2, FB (fractional bias) = Co −Cp /0.5(Co +Cp ), FS (fractional standard deviations) = (σo − σ p )/0.5(σo + σ p ),

where the subscripts o and p refer to observed and predicted quantities, respectively, and the overbar indicates an averaged value. The statistical index NMSE represents the quadratic error of the predicted quantities related to the observed ones. FB says if the predicted quantity underestimates or overestimates the observed ones. The best results are expected to have values near to zero for the indices NMSE, FB and FS, and near to 1 in the indices COR and FA2. In Table 1, we show the statistical comparison of the results presented. Upon a closer look at this table, we quickly realize that the best results are the ones obtained by the proposed method (denoted as the 3D-GILTT approach) and the GIADMT results. This is as we expected because both methods are different versions of the same solution [Mor10]. For the problem considered here the Gaussian solution assumption in y-direction, according the results given, is a poor three-dimensional ADE model to simulate pollutant dispersion in the ABL. On the other hand, the scattering diagram appearing in Fig. 1 reinforces our affirmative regarding the good computational performance of the proposed method. Table 1 Statistical comparison between models using the Copenhagen data set GILTTG GIADMT 3D-GILTT

NMSE 0.33 0.15 0.07

COR 0.80 0.87 0.93

FA2 0.87 0.96 0.96

FB FS 0.28 0.09 0.01 −0.09 0.02 0.03

5 Conclusions From the previous discussion, we promptly realize that the general analytical solution encountered for the time-dependent, three-dimensional ADE assuming Fickian closure for the turbulence is an important and promising formulation to simulate contaminant released in the ABL. In fact, besides the good results achieved, it is relevant to reinforce the analytical character of this solution, in the sense that no approximation is done along the solution derivation, except for the line integral approximation of the Laplace inversion of the transformed solution by the Gaussian quadrature scheme. By general solution we mean that the solution is valid for wind

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Fig. 1 Scatter diagram of the observed vs predicted maximum ground-level concentrations normalized by the emission rate. Data between lines correspond to a factor of two

profile and eddy diffusivity coefficient being arbitrary functions of the x and z variables. The generality also extends for the issue of atmospheric stability once the solution is valid for any stability. Keeping in mind that the problem fulfills the hypothesis of the Cauchy–Kowalewski theorem [CoHi89], we are certain to affirm that the problem studied has a solution and its solution is unique. In addition, it is necessary to underline that this sort of equation models problems in several areas of science, and therefore this solution has a broad class of applications, for instance, we mention the work of [CoBa07] in the area of pollutant dispersion in channels. For the purpose of numerical computations one may make use of the Cardinal Theorem of Interpolation Theory that allows the calculation to be made of the solution for any prescribed accuracy by controlling the number of terms in the series summation. Details of this procedure are reported elsewhere [Bod10]. Finally, we notice that because of the reduction of order of ADE reported, we are confident to stress that the form of solution does not depend on the topology of the problem, once the solution for the two- and three-dimensional problems are written in terms of the same set of eigenvalues. To extend the application of the proposed solution to more realistic problems we shall focus our future attention to the task of extending this solution to problems requiring low wind as well time-dependent wind profile and eddy diffusivity coefficient.

A General Analytical Solution of the ADE

33

Acknowledgements The authors thank to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FAPERGS (Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul) for the partial financial support of this work.

References [AbVa04] Abate, J., Valkó, P.P.: Multi-precision Laplace transform inversion. Int. J. for Num. Methods in Engineering, 60, 979–993 (2004). [Ary95] Arya, S. Pal: Modeling and parameterization of near-source diffusion in weak winds. J. Appl. Meteor., 34, 1112–1122 (1995). [Bla97] Blackadar, A.K.: Turbulence and Diffusion in the Atmosphere: Lectures in Environmental Sciences, Springer-Verlag, 185 pp. (1997). [Bod10] Bodmann, B., Vilhena, M.T., Ferreira, L.S., Bardaji, J.B.: An analytical solver for the multi-group two-dimensional neutron-diffusion equation by integral transform techniques. Il Nuovo Cimento C, 33, 199–206 (2010). [Bus10] Buske, D., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: An Analytical Solution for the Transient Two-Dimensional Advection-Diffusion Equation with Non-Fickian Closure in Cartesian Geometry by Integral Transform Technique. Integral Methods in Science and Engineering: Computational Methods, Birkhauser, Boston, 33–40 (2010). [Cos06] Costa, C.P., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: Semi-analytical solution of the steady three-dimensional advection-diffusion equation in the planetary boundary layer. Atmos. Environ., 40, n. 29, 5659–5669 (2006). [CoBa07] Cotta, R.M., Barros, F.P.J.: Integral transforms for three-dimensional steady turbulent dispersion in rivers and channels. Applied Mathematical Modelling, 31, 2719–2732 (2007). [CoHi89] Courant, R., Hilbert, D.: Methods of Mathematical Physics, John Wiley & Sons, New York (1989). [Deg02] Degrazia, G.A., Moreira, D.M., Campos, C.R.J., Carvalho, J.C., Vilhena, M.T.: Comparison between an integral and algebraic formulation for the eddy diffusivity using the Copenhagen experimental dataset. Il Nuovo Cimento, 25C, 207–218 (2002). [Dem78] Demuth, C.: A contribution to the analytical steady solution of the diffusion equation for line sources. Atmos. Environ., 12, 1255–1258 (1978). [GrLy84] Gryning, S.E., Lyck, E.: Atmospheric dispersion from elevated source in an urban area: comparison between tracer experiments and model calculations. J. Appl. Meteor., 23, 651–654 (1984). [Han89] Hanna, S.R.: Confidence limit for air quality models as estimated by bootstrap and jacknife resampling methods. Atm. Env., 23, 1385–1395 (1989). [LiHi97] Lin, J.S., Hildemann, L.M.: A generalised mathematical scheme to analytically solve the atmospheric diffusion equation with dry deposition. Atm. Env., 31, 59–71 (1997). [Mor05] Moreira, D.M., Vilhena, M.T., Tirabassi, T., Buske, D., Cotta, R.M.: Near source atmospheric pollutant dispersion using the new GILTT method. Atmos. Environ., 39, n. 34, 6290–6295 (2005). [Mor06] Moreira, D.M., Vilhena, M.T., Tirabassi, T., Costa, C., Bodmann, B.: Simulation of pollutant dispersion in atmosphere by the Laplace transform: the ADMM approach. Water, Air and Soil Pollution, 177, 411–439 (2006). [Mor09a] Moreira, D.M., Vilhena, M.T., Buske, D., Tirabassi, T.: The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere. Atmospheric Research, 92, 1–17 (2009). [Mor09b] Moreira, D.M., Vilhena, M.T., Buske, D.: On the GILTT Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer. Air Pollution and Turbulence: Modeling and Applications, CRC Press, Boca Raton – Flórida (USA), 179–202 (2009).

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Moreira, D.M., Vilhena, M.T., Tirabassi, T., Buske, D., Costa, C.P.: Comparison between analytical models to simulate pollutant dispersion in the atmosphere. Int. J. Env. and Waste Management, 6, 327–344 (2010). [NiHa81] Nieuwstadt, F.T.M., de Haan, B.J.: An analytical solution of one-dimensional diffusion equation in a nonstationary boundary layer with an application to inversion rise fumigation. Atmos. Environ., 15, 845–851 (1981). [PaDu88] Panofsky, A.H., Dutton, J.A.: Atmospheric Turbulence, John Wiley & Sons, New York (1988). [ScFi75] Scriven, R.A., Fisher, B.A.: The long range transport of airborne material and its removal by deposition and washout-II. The effect of turbulent diffusion. Atmos. Environ., 9, 59–69 (1975). [Rou55] Rounds, W.: Solutions of the two-dimensional diffusion equation. Trans. Am. Geophys. Union, 36, 395–405 (1955). [Sha96] Sharan, M., Singh, M.P., Yadav, A.K.: A mathematical model for the atmospheric dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmos. Environ., 30, n. 7, 1137–1145 (1996). [Smi57] Smith, F.B.: The diffusion of smoke from a continuous elevated poinr source into a turbulent atmosphere. J. Fluid Mech., 2, 49–76 (1957). [StSe66] Stroud, A. H., Secrest, D.: Gaussian Quadrature Formulas, Prentice Hall Inc., Englewood Cliffs, N.J. (1966). [VaAb04] Valkó, P.P., Abate, J.: Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion. Computers and Mathematics with Application, 48, 629–636 (2004). [Tag85] Tagliazucca, M., Nanni, T., Tirabassi, T.: An analytical dispersion model for sources in the surface layer. Nuovo Cimento, 8C, 771–781 (1985). [Tir89] Tirabassi, T.: Analytical air pollution and diffusion models. Water, Air and Soil Pollution, 47, 19–24 (1989). [TiRi94] Tirabassi, T., Rizza U.: Applied dispersion modelling for ground-level concentrations from elevated sources. Atmos. Environ., 28, 611–615 (1994). [Tir03] Tirabassi, T.: Operational advanced air pollution modeling. PAGEOPH, 160, n. 1–2, 5–16 (2003). [Van78] van Ulden, A.P.: Simple estimates for vertical diffusion from sources near the ground. Atmos. Environ., 12, 2125–2129 (1978).

A Novel Method for Simulating Spectral Nuclear Reactor Criticality by a Spatially Dependent Volume Size Control D.Q. de Camargo, B.E.J. Bodmann, M.T. Vilhena, and S.d.Q.B. Leite

1 Introduction Controlled nuclear reactions in a nuclear reactor are one of the energy resources that may contribute to attend the increasing energy demand while minimizing impact on the environment. Because of its efficient energy release per nuclear reaction in comparison to processes that involve chemical reactions for instance (which differ by more than eight orders in magnitude) reactor control and safety is a crucial issue. Evidently, while designing new reactor conceptions or operating existing reactors the microscopic as well as macroscopic response of the nuclear process must be understood in detail and described adequately in terms of mathematical models together with experimental data such as the nuclear reaction cross sections [Sek07]. The physics of the nuclear reactions taking place in a power reactor and its influence on the neutron flux by perturbations from inside or outside the system are known reasonably well. Nevertheless, as the present contribution will show there is still space for progress which is manifest in a variety of recent attempts to create efficient and adequate algorithms that calculate neutron fluxes, as well as other reactor relevant quantities.

D.Q. de Camargo Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] B.E.J. Bodmann Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] M.T. Vilhena Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] S.d.Q.B. Leite Comissão Nacional de Energia Nuclear, Rio de Janeiro, RJ, Brazil, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_5, © Springer Science+Business Media, LLC 2011

35

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Operating a nuclear reactor involves two classes of problems, the steady state where power output is mainly constant, and power changing actions like start-up, shut-down or changes in the power level. While in steady state condition a reactor is operating at or near criticality, i.e. the neutron multiplication factor k is nearly equal to unity, power changes are characterized by the reactivity ρ = (k − 1)/k. Note that the reactivity is a convenient mathematically defined quantity, but cannot be measured directly in practice. Nevertheless, neutron multiplication or reactivity depends on several variables such as the temperature of the nuclear core, its chemical composition and, as the following discussion will show, on energy. Usually the multiplication factor and reactivity are used over the whole energy range. Especially in situations of power change the neutron energy spectrum changes and so do physical quantities such as the average cross sections. Hence in order to model theses situations a spectral description of k and consequently ρ is of need, which is the focus of the present contribution. In principle one may approach the problem using analytical, numerical or stochastic methods [Bod10, Cam09, Lep05]. Since the energy dependence of the cross sections shows oscillations in the regions of the resonances, analytical and numerical implementations of neutron transport equations are less adequate because they usually work with energy groups that average the cross sections over the respective energy interval and thus would need mechanisms to correct spectral changes. From this point of view one seems to be better off using the Monte Carlo method but facing the task of simulating an exorbitant number of neutrons, which is in practice impossible and has to be circumvented otherwise. In the present work we solve this problem by a spatially dependent control volume size. Our Monte Carlo simulation is a physical implementation using an interactive evaluation of a deterministic model based on randomly distributed numbers according to specified probability densities. This technique turns effective when the model is complex and nonlinear, or when it involves an elevated number of conditions (such as geometry with boundaries and interfaces, change of the chemical composition of the materials) and when the integral involves many dimensions. In our particle transport code, the Monte Carlo technique tallies each of the particles along its trajectories until some terminal event such as absorption or escape, amongst others, terminates the history. In this work, the Monte Carlo implementation takes its instructions from an integro-differential neutron transport equation in three dimensions and genuine energy dependence (i.e. no energy groups). The treatment of the cross section is one aspect that differentiates this work from others known in the literature [Bod10, Gon10, Vil08]. Here cross sections are continuous functions of energy, which are obtained by parametrizations and coded as program procedures. The interaction type that a neutron will suffer and the characteristics of their displacement in the environment are randomly estimated by the use of the relevant probability distributions. In order to render the simulation effective, the environment is divided in several smaller volumes, where one of these volumes is chosen and plays the role of a control volume. These small volume elements are to be chosen to reconstruct a smooth neutron distribution for the whole volume. The volume control implementation in

A Novel Method for Simulating Spectral Nuclear Reactor Criticality

37

the environment is a novel aspect in comparison to existing works and makes use of a fixed initial number of neutrons which are inserted in a specified control volume. Its advantage is that the neutron number may be significantly reduced while still obtaining a good simulation. Varying the control volume is further used to simulate processes with different criticality. So far we show how the method works and leave for the future a detailed examination for more realistic cases with respect to geometries and material compositions, necessary for nuclear reactors physics.

2 Neutron Transport Usual calculations from neutron transport express the spatial neutron population in terms of the neutron flux, which is related to the neutron density by the velocity modulus (Φ (r, E,t) = vn(r, E,t)). Instead, here we use the so-called neutron angular density n(r, Ω , E,t), i.e. the neutron number per unit-volume at location r and at instant t, moving in direction of the oriented unit-solid angle Ω with energy E. For criticality analysis usually the neutron number (integration of the angular neutron density over the solid angle, the energy and the volume of interest) at instant t + τNLC and t are compared, where τNLC is the time of the neutron life cycle [CNSC03]. In our case analyses integrate only over the solid angle and volume, but maintain the spectral dependence. The neutron transport equations is derived from the Boltzmann transport equation [Vli08]. 1∂ Φ (r, Ω , E,t) + Ω · ∇Φ (r, Ω , E,t) + Σt (r, E,t)Φ (r, Ω , E,t) v ∂t = q(r, Ω , E,t). Here Σt (r, E,t) is the macroscopic total cross section at time t and location r; and energy E and q(r, Ω , E,t) represents all contributions of neutrons that appear at instant and position (t, r) and moving in the direction Ω with energy E: q(r, Ω , E,t) =

 ∞ 0

dE 

 4π

d Ω Σ (r, E  → E, Ω  → Ω )Φ (r, Ω  , E  ,t) +S(r, Ω , E,t).

(1)

Here S(r, Ω , E,t) represents the effect of an external neutron source. The integral term contains contributions from the nuclear reactions, such as neutron scattering and fission. Hence, the macroscopic cross section Σ (r, E  → E, Ω  → Ω ) dE d Ω represents the number of neutrons emitted in the infinitesimal energy interval dE at E and into the infinitesimal solid angle d Ω at Ω that have come from a reaction of a neutron at energy E  and incoming direction Ω  . If importance of each individual nuclear reaction is taken into account the cross section may be decomposed as

38

D.Q. de Camargo et al.

Σ (r, E  → E, Ω  → Ω ) = ∑ νk (E  )Σk (r, E  )pk (E  → E, Ω  → Ω ),

(2)

k

with νk (E) is the number of secondary neutrons released by reaction of type k (i.e. for elastic and inelastic scattering νs = 1, for knock out (n, 2n) reactions νko = 2; and for fission νk ∈ [2, . . . , 4]), with Σk (r, E) is the macroscopic cross section for reaction k. pk (E  → E, Ω  → Ω ) dE d Ω is the probability that a neutron moving in incoming direction Ω  with energy E  undergoes reaction k, with outgoing neutrons with energy in [E, E + dE] in d Ω with director Ω .

3 Neutron Transport by Monte Carlo In our Monte Carlo simulation we used the fission spectrum χk (E) in order to implement the associated probabilities, pk (E  → E, Ω  → Ω ) =

1 χk (E). 4π

(3)

Furthermore, we simplified (1) and did not consider any external source (S(r, Ω , E,t) ≡ 0). We also admitted no changes in the chemical composition of the nuclear fuel or the moderator. As already mentioned before, the present approach considers the full energy dependence in the cross sections in analytical form. To this end the cross sections obtained from nuclear data-bases were parametrized for energy intervals that permitted an accurate description ( 1%) of the energy dependence. The results of some of these parametrizations are shown in Figs. 1–5, which show the absorption cross sections for 16 O and 235 U, the scattering cross sections for 16 O and 235 U and the fission cross section for 235 U in comparison to the original data [Cha06]. Note the spurious differences between the two. The preference of using parametrizations instead of a data-base is justified by various tests that have shown us that in the present simulation using interpolated data to get the energy corresponding to the cross section is slower in execution time than a single function call. The control over the chain reaction, that is, to control the number of neutrons per energy interval in a generation relative to the number of neutrons in the previous one, is established by the control volume. Since the number of simulated neutrons in a volume of interest is always the same (here 106 ), adjusting the volume size controls whether the multiplication factor is below or above unity. Note that simulating criticality k = 1 is not a trivial task because one has to find the corresponding volume size, which may be obtained only by simulating super-critical and sub-critical setups with subsequent extrapolation. Moreover, it is tedious to show that a reactor can be critical at any (reasonable) power level. The key to this control is the neutron life cycle. Conventionally, this operation is done using six factors that govern the production, leakage and absorption of neutrons, and enable the quantitative description of the components that govern the mul-

A Novel Method for Simulating Spectral Nuclear Reactor Criticality

39

Fig. 1 Absorption cross section for 16 O in the energy range 10−11 MeV ≤ E ≤ 2 × 102 MeV, original data (grey) and parametrization (black)

Fig. 2 Absorption cross section for 235 U in the energy range 10−11 MeV ≤ E ≤ 2 × 102 MeV, original data (grey) and parametrization (black)

tiplication factor in the neutron cycle. Since our simulation contains the full energy dependence, this may be extended to a spectral (i.e. energy-dependent) description for neutron multiplication. More specifically, the relevant factors for neutron multiplication that are fast fission, fast leakage escape, resonance escape probability, thermal non-leakage, thermal utilization and reproduction by thermal neutrons, have in this treatment an explicit energy dependence. The steps of the simulation are as follows: 1. The material compositions are defined, which may be time dependent. In the present case they are assumed as constant and geometries are considered static.

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Fig. 3 Scattering cross section for 16 O in the energy range 10−11 MeV ≤ E ≤ 2 × 102 MeV, original data (grey) and parametrization (black)

Fig. 4 Scattering cross section for 235 U in the energy range 10−11 MeV ≤ E ≤ 2 × 102 MeV, original data (grey) and parametrization (black)

2. Volume control is implemented as to scale the neutron flux in terms of the neutron density. 3. The initial neutron energy spectrum determines the probability of neutrons that appear with a given energy. 4. The position of the generated neutron with initial energy E is determined randomly at position r within the active volume. 5. The energy value determines the cross sections and thus the scheme to score the reaction type. 6. Once the reaction is determined the length of the trajectory, angles and the new position are computed.

A Novel Method for Simulating Spectral Nuclear Reactor Criticality

41

Fig. 5 Fission cross section for 235 U in the energy range 10−11 MeV ≤ E ≤ 2 × 102 MeV, original data (grey) and parametrization (black)

7. All the previous steps have been carried out without considering boundary conditions, manifest in leakage, so that a decision re-directs the program to a. either terminate the history of a neutron and restart at step 4; b. or to determine the new interaction vertex. 8. According to the type of interaction the procedures are as follows: a. For capture, the history of the neutron ends, and the program restarts at step 4; b. For scattering, the new neutron energy is determined, and the program continues at step 6; c. For fission, the number of new neutrons is determined randomly. For each neutron the program starts at step 4.

4 Results The Monte Carlo step so far is a quantity that may be related to a time scale by calibration. A natural time scale is the neutron generation time, i.e. the time required for neutrons from one generation to cause the fissions that produce the next generation of neutrons. This time interval is determined by three time intervals, the time it takes a fast neutron to slow down to thermal energy, the time the thermal neutron exists prior to absorption in the fuel, and the time for a fissionable nucleus to emit a fast neutron after neutron absorption. Fast neutrons may either slow down to thermal energies or leak out of the reactor, which happens in a time interval of 10−4 s–10−6 s depending on the reactor core architecture including the moderator. Fission and fast

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Fig. 6 Identification of the neutron cycle in Monte Carlo steps

neutron production after neutron absorption in the nuclear fuel takes about 10−13 s [CNSC03]. One has to consider further the neutron precursors, which we did not implement yet into the simulation. Also here three time intervals contribute to the time scale, the time it takes a fast neutron to slow down to thermal energy, the time the thermal neutron bounces around before absorption, and the average time from neutron absorption to neutron emission by the precursors, which are typically organized in six groups. The average time for decay of precursors from 235 U is tP = 12.5 s and the average generation time is then typically of the order of magnitude τNLC ∼ 100 ms. Since in the simulation the neutrons have a time stamp, this permits us to analyze such a cycle. In Fig. 6 we show the cyclic behavior, which does not appear in this form in a real reactor, but is an artifice which may be extracted from the tallies of individual neutrons. Figure 7 shows the energy spectrum after a sequence of Monte Carlo steps. One clearly observes more pronounced changes in the higher energy region of the spectrum. As a novel result we show the spectral multiplication factor change with Monte Carlo steps (Fig. 8).

5 Conclusion In the present work we implemented a Monte Carlo simulation for neutron transport in nuclear power reactors. Using a approach which is different from most of those in the literature we consider three spatial dimensions and continuous energy dependence, instead of the usual energy groups. To this end we supply functions with the cross sections in parametrized and analytical form. This allows a detailed analysis of the energy influence on the neutron population and its impact on the neutron life cycle. As novel results we showed the changes in the neutron energy spectrum with the steps of the simulation and further determined the spectral changes in the neutron multiplication factor per Monte Carlo step.

A Novel Method for Simulating Spectral Nuclear Reactor Criticality

43

Fig. 7 Changes in the neutron energy spectrum after a sequence of Monte Carlo steps

The simulation uses a small sample volume in order to reproduce the neutron population in the whole volume, which permits to simulate huge neutron numbers in a macroscopic volume, as in the nuclear reactor core. Variation of the control volume is further used to simulate processes with different criticality. Power changes are usually expressed using reactivity, which in general makes use of negative feedback (as temperature coefficients of reactivity, for instance) to control the nuclear reactor and make it inherently self-controlling and thus safe. Results from our simulation open new pathways the allow to harness the reactors behavior by a new degree of freedom, i.e. its spectral behavior along the neutron life cycle. Since there are significant changes in the neutron energy spectrum a significant influence results on the balance of respective interactions including leakage and especially the process where neutrons escape the resonances. Furthermore, the fact that there is a strong influence in the spectral shape of the spectrum shows that there ∂k is need for a spectral multiplication factor ∂eEf f . Moreover, approaches that make use of energy groups need to associate a specific effective multiplication factor to

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Fig. 8 Spectral multiplication factor per Monte Carlo step. The symbols of the sequence are defined in Fig. 7

each equation representing a specific group. Until now, only one global ke f f was attributed to the set of equations. The question which we did not answer yet but hope to respond to in future is whether the necessary spectral ke f f ,g used per energy group is an average or a total value. Beyond comparison with existing approaches in the literature, we intend to simulate the transients of any given initial power to any specified final power in order to show how for a given power set-up the stationary state is approached. Along this line, our present work may be considered as a first step into a new direction.

References [Bod10]

Bodmann, B.E.J., de Vilhena, M.T., Ferreira, L.S., Bardaji, J.B.: An analytical solver for the multi-group two-dimensional neutron-diffusion equation by integral transform techniques. Il Nuovo Cimento, 33 C, n. 1, 63–70 (2010). [Cam09] Camargo, D.Q., Bodmann, B.E.J., Garcia, R.D.M., Vilhena, M.T.M.B.: A threedimensional collision probability method: Criticality and neutron flux in a hexahedron setup. Annals of Nuclear Energy, 36, 1614–1618 (2009). [Cha06] Chadwick, M.B., et al.: ENDF/B-VII.0: Next generation evaluated nuclear data library for nuclear science and technology. Nuclear Data Sheets, 107, n. 12, 2931–3118 (2006). [CNSC03] Canadian Nuclear Safety Commission: Science and Reactor Fundamentals – Reactor Physics. CNSC, Ottawa, ON, Canada (2003). [Gon10] Gonçalves, G.A., Vilhena, M.T., Bodmann, B.E.J.: Heuristic geometric “eigenvalue universaly” in a one-dimensional neutron transport problem with anisotropic scattering. Kerntechnik, 75, 50–52 (2010). [Lep05] Leppanen, J.: A new assembly-level Monte Carlo neutron transport code for reactor physics calculations, in Mathematics and Computation, Supercomputing, Reactor

A Novel Method for Simulating Spectral Nuclear Reactor Criticality

[Sek07] [Vil08]

[Vli08]

45

Physics and Nuclear and Biological Applications, Palais des Papes, Avignon, France, September 12–15, 2005, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2005). Sekimoto, H.: Nuclear Reactor Theory, COE-INES Tokyo Institute of Technology, ISBN978-4-903054-11-7 C3058 Part II (2007). de Vilhena, M.T., Heinen, I.R., Bodmann, B.E.J.: An analytical solution for the general perturbative diffusion equation by integral transform techniques. Annals of Nuclear Energy, 35, 2410–2413 (2008). Van Vliet, C.M.: Equilibrium and Non-equilibrium Statistical Mechanics, World Scientific, ISBN 10-981-270-478-7 (2008).

Adaptive Particle Filter for Stable Distribution H.F. de Campos Velho and H.C. Morais Furtado

1 Introduction Estimation theory is a central issue for several applications: filtering, signal analysis, image processing, control theory, inverse problem, and data assimilation. There is intensive research into developing a set of techniques for estimating quantities, for example: the least squares approximation, Kalman filter, variational method, Bayesian approach and, more recently, schemes based on artificial intelligence: neural network and fuzzy logic. For all the methods cited, there are some constraints to be considered. For example, in the context of inverse problem, it is important to include some a priori information to apply the least square solution. Some authors use the expression generalized least squares to characterize the latter condition. Kalman filter is another alternative, and it was developed for a linear dynamical system under Gaussian assumptions for describing the probability density function (PDF) for the variables involved. The extended Kalman filter is an adaptation to employ the filter to nonlinear problems. But this strategy can fail under strong nonlinear regimes. Variational methods are standard procedures in estimation problems. The goal is to calculate a minimum of an objective function (a functional), in a similar way to the least square approach, but the gradient of the functional is computed by using the adjoint function associated to the problem. For Kalman filter and the variational method, one quantity remains unknown: the covariance matrix for representing the modeling error. Some schemes were designed to identify this covariance matrix: the use of an estimating operator, as a secondary H.F. de Campos Velho Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, SP, Brazil, e-mail: [email protected] H.C. Morais Furtado Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, SP, Brazil, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_6, © Springer Science+Business Media, LLC 2011

47

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H.F. de Campos Velho and H.C. Morais Furtado

Kalman filter [Ja70], employing an ensemble KF [Ev07], or using a Fokker–Planck equation [BeMeBr00, BeTaBr01]. For strong nonlinear systems and without the Gaussian assumption, the particle filter is an option. In the literature on inverse problems, the particle filter is usually referred to as Bayesian method. The standard reference is the Tarantola’s book [Ta87], but there an extensive literature on this approach (see for example [JaSo05]). The particle filter is a general formulation. However, there are some distributions where some statistical moments are not defined. Under the latter scenario, there will be presented a procedure to apply a particle filter. There are two key issues in our formulation: a new approach for the likelihood function, and a strategy to identify a key parameter in the likelihood operator. A non-extensive form of entropy has been proposed by Tsallis [Ts88], where a free parameter q, called the non-extensivity parameter, has a central role in Tsallis’ thermostatistics. The likelihood function will be described by employing the Tsallis’ statistics [Ts88, Ts99]. Unfortunately, only in a few cases there is an analytical formula to compute the parameter q; one example is fully developed turbulence [RaRoRoBoSaCa01]. This parameter q could be identified by several methods from the estimation theory. Here, the parameter q will be computed from the secondary particle filter. The proposed method is tested with a nonlinear dynamical system already used in other studies [GoSaSm93, ArMaGoCl02].

2 Particle Filter The fundamental idea underlying the Sequential Monte Carlo is to represent the probability density function (PDF) by a set of samples with their associated weights. This set of samples is also referred to particles [GoSaSm93, ArMaGoCl02, Do98]. In the PF, an estimation of a posteriori PDF is obtained by resampling with replacement from a priori ensemble. As the PF does not require us to assume that the PDF is linear or Gaussian, it is applicable to general nonlinear problems. In particular, the PF can be applied to cases in which the relationship between a state and observed data is nonlinear [NaUeHi07]. However, the PF often encounters a problem, sample impoverishment, since this dependence makes convergence results harder to obtain. One way to avoid this problem is to introduce an additional noise to make the particles differ more from each other [GoSaSm93, ChKr04, FuCaMa08]. Two properties are relevant issues for the PF: the Bayes’ theorem, and the Markov property. Theorem 1. For two statistically independent events A and B, with P(B) = 0, the Bayes rule follows: P(B|A)P(A) . P(A|B) = P(B)

Adaptive Particle Filter

49

The Markov process is characterized by the property p(wn |wn−1 , . . . , w2 , w1 ) = p(wn |wn−1 ). Here, the Gordon’s implementation [GoSaSm93] is considered. The PDF p(xn |Ys ) is approximate for a function of empirical PDF (see below). According to the Bayesian view p(wn |Ys ) contains all statistical information available about the state variable wn , based on the information in the measurements Ys : p(wn |Ys ) ≈

M

∑ q˜n

(i)

i=1 M

∑

(i) q˜t

(i)

δ (wn − wn|s ), (1)

=1

i=1 (i)

where δ (.) is the Dirac delta function; q˜t denotes the weight associated with parti(i) cle wn|s .

2.1 Standard PF Algorithm 1. Compute the initial particle ensemble: (i)

{w0|n−1 }M i=0 ∼ pw0 (w0 ) (initial PDF: Gaussian, with zero mean and σ = 5, i.e., N (0, 5)); 2. Compute: (i) rn = p(yn |wn|n−1 ) = pet (yn − h(wn ,tn )) where

pet (z) = exp(−z2 /2)(2 π )−1

and z = yn − h(wn ,tn ). 3. Normalize:

(i)

(i)

r˜n =

rn

( j)

∑M j=1 rn

.

4. Resampling: extract M particles, with substitution, according to (a standard notation is used here—see [ScGuNr05]): (i)

( j)

( j)

Pr{wn|n = wn|n−1 } = r˜n , i = 1, . . . , M. Resampling step [ScGuNr05]:

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H.F. de Campos Velho and H.C. Morais Furtado

– Generate M ordered numbers uk , following: uk = [(k − 1) + u]M ˜ −1 , with: u˜ ∼ U(0, 1) (uniform distribution). – Resampled particles are obtained by producing mi copies of particle x(i) , where:   mi = number of uk ,

uk ∈

i−1

i

∑ r˜n , ∑ r˜n (s)

s=1

(s)

.

s=1

5. Time updating: compute the new particles: (i) (i) (i) (i) (i) wn+1|n = f (wn|n ,tn ) + μn (μn ∈ N (0, 1)), with: wn+1|n ∼ p(wn+1|n |wn|n ), i = 1, . . . , M. 6. Set: tn+1 = tn + Δ t, and go to step 2. The kernel of the algorithm coming from the application of Bayes’ theorem and of the Markov property: p(wn |Yn ) = p(wn |yn ,Yn−1 ) =

p(yn |wn )p(wn |Yn−1 ) ∝ p(yn |wn )p(wn |Yn−1 ) p(yn |Yn−1 )

suggesting the following choice [Sc06]: p(wn |Yn ) ∝ p(yn |wn ) p(wn |Yn−1 ) .          posteriori(wn )

likelihood(wn )

(2)

priori(wn )

2.2 New Approach for Particle Filter Equation (2) is the Bayesian estimate for the state vector wn . The expectation is a recursive procedure (2) convergent to the true value. There is some criticism on the Bayesian estimation (sometimes this criticism is associated to the frequentist school in the statistics community). In general it is related to the a priori choice for the distribution p(wn |Yn−1 ) ≡ πpr (wn ), where there is no scientific basis for the choice. We are going to stay out of this discussion. However, it is important to point out that starting from a distribution πpr (wn ) with finite variance and a likelihood function as described in step 2 (a Gaussian one) it is not possible to reach a distribution p(wn |Yn ) ≡ πpost (wn |Yn ) with an undefined variance. There are many distributions with no defined variance, Cauchy and Lévy distributions are two examples. In the present paper, our development will be focused on stable distributions [No05], the class of nondegenerate distributions. Definition 1. A random variable X is stable if for X1 and X2 independent copies of X, and any positive constants a and b, the random variable aX1 + bX2 has the same distribution as cX + d, with some constants c and d. The distribution is said to be strictly stable if this holds with d = 0.

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51

The central point of our analysis lies in the discussion of the likelihood function p(yn |wn ) ≡ πnoise (y − A(wn )), where y is the observed value. The mathematical model is given by f = A(w) + δ and the realizations are expressed by

η = A(w) + μ where δ and μ are additive white noise with E{δ } = E{η } = 0, and they are mutually independent from f and η . Therefore [Be98]: E{η } = E{ f } = A(wo ). Now, we can determine the likelihood function:

πη (y|w) = πμ (y − A(w)) = πnoise (y − η ). The latter distribution can be computed from Bayes’s theorem:

πpost (w|y) ∝ πμ (y − A(w)) πpr (w). In Sect. 2.1 for standard PF algorithm, in step 2, a Gaussian distribution was chosen to represent the likelihood function. It is possible to justify such a choice due to the central limit theorem [Pa89]. Actually, there is a more general form for this theorem. In the distribution space, there are at least two attractors for stable distributions. If a random variable can be described as a linear combination of a summation of M independent random variables with finite variance, the resulting distribution in the limit M → ∞ is a Gaussian distribution. Another attractor is the Lévy alpha-stable distribution, resulting as a limit of sum of independent and identically distributed random variables with no defined variance. The second form of the central limit theorem is also called Lévy–Gnedenko’s central limit theorem. A non-extensive form of entropy has been proposed by Tsallis [Ts88]:  N k q 1 − ∑ pi Sq (p) = q−1 i=1 where pi is a probability, and q is a free parameter—it is called the non-extensivity parameter. In thermodynamics, the parameter k is known as Boltzmann’s constant. Tsallis’ entropy reduces to the the usual Boltzmann–Gibbs–Shannon formula, N

S(p) = −k ∑ pi log pi , i=1

in the limit q → 1.

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As for the extensive form of entropy, the equiprobability condition produces the maximum for the non-extensive entropy function, and this condition leads to special distributions [Ts99]. q > 1:

 1 − q  x 2 −1/(q−1) + (3) pq (x) = αq 1 − 3−q σ q = 1:

 1 1 1/2 −(x/σ )2 /2 pq (x) = e σ 2π

q < 1:

 1 − q  x 2 1/(q−1) pq (x) = αq− 1 − 3−q σ

(4)

where

σq2

+∞ 2 x [pq (x)]q dx = −∞ ,

+∞ −∞

[pq (x)]q dx

 1/2 Γ 1 q−1 1 q−1  , = 3−q σq π (3 − q) Γ 2(q−1)  5−3q

 1 − q 1/2 Γ 2(1−q) 1  . = σq π (3 − q) Γ 2−q

αq+

αq−

1−q

The distributions above apply if |x| < σq [(3 − q)/(1 − q)]1/2 , and pq (x) = 0 otherwise. For distributions with q < 5/3, the standard central limit theorem applies, implying that if pi is written as a sum of M random independent variables, and in the limit case M → ∞, the probability density function for pi in the distribution space is the normal (Gaussian) distribution. However, for 5/3 < q < 3 the Lévy–Gnedenko central limit theorem applies, so that for M → ∞ the Lévy distribution is the probability density function for the random variable pi . The index in such a Lévy distribution is α = (3 − q)/(q − 1) [Ts99, TsQu07]. Our purpose is to use Tsallis’ thermostatistics (3) or (4) for substituting the Gaussian distribution to represent the likelihood function in step 2 of the PF. The idea is to explore the property of this thermostatistics to access both attractors in the distribution space. Finally, it is necessary to identify the non-extensive parameter q. The next section will describe a scheme for computing it.

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53

2.3 Identifying the Non-Extensive Parameter q The parameter q is estimated by employing a secondary particle filter. Follow the procedure: 1. Initialize the values of q; (i)

{q0|n−1 }M i=0 ∼ pq0 (q0 ) (initial PDF: Gaussian, q = 1); 2. Compute average q; 3. If on average: q = 1, the PDF is Gaussian; if on average: q > 1, the PDF is the Tsallis equation (3); if on average: q < 1, the PDF is the Tsallis equation (4); 4. Resampling the values of q, in accordance with step 4 of the standard PF; 5. q(t + 1)(i) = q(t)(i) , i = 1, . . . , M and go to step 2.

3 Numerical Results The new approach for a particle filter is tested to identify the state vector of the following dynamical filter. Prediction: xt+1 =

xt 25xt + 8 cos(1.2t) + μt . + 2 1 + xt2

(5)

Observation:

xt2 + νt . 20 For the system (5), 1000 particles are employed. For the initial guess: q0 = 1 (Gaussian distribution), where x0 ∼ N(0, σx = 1). yt+1 =

The true distribution for the random variables μt and νt are both a Tsallis distribution with q = 2.5 (see Fig. 1), and the level of noise is 2% of the maximum value of the noiseless data. Table 1 shows the results obtained using different likelihood functions. The smallest error was obtained with the Tsallis’ distribution, with q determined with the adaptive particle filter algorithm described in Sects. 2.2 and 2.3, where we note the convergence to the exact value q = 2.5 (see Fig. 4). Figure 2 displays true and estimated values for the dynamics; the observation is shown by discrete points (squares).

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Fig. 1 Histogram of random numbers for the Tsallis distribution for q = 2.5

Table 1 Errors: Experiments with noise according to Tsallis distribution for q = 2.5 q 1.0 1.5 2.8 2.9 estimated

Distribution Gaussian Tsallis Tsallis Tsallis Tsallis

Average error 6.31348 6.19225 6.97586 7.19318 6.17809

Fig. 2 True state (continous line), estimated state (dot line), and observation points (squares)

Adaptive Particle Filter

55

Figure 3 shows the errors related to the estimated values obtained with the new approach employing Tsallis’ thermostatistics (Fig. 3 right side) with q value identified (see Fig. 4), and Gaussian likelihood (Fig. 3 left side).

Fig. 3 Error: (left side): estimation using Gaussian likelihood function; (right side): new approach for particle filter

Fig. 4 The estimated value for q parameter by adaptive particle filter

4 Conclusion A particle filter is a statistical procedure for dealing with nonlinear problems and there is no assumption on the Gaussian nature of the distribution of the random variables. This approach has received more attention in different fields: control theory, signal processing—including image filtering, inverse problem, and data assimilation (a good and recent review paper was presented by Leeuwen [Le09]), this is a very important class of inverse problem. In the book of Kaipio and Somersalo data assimilation can be understood as belonging to the nonstationary inverse problems [JaSo05]. Sometimes the particle filter is also called a bootstrap filter.

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It is possible to find strategies taking into account distributions with not defined variance, for example, Kaipio and Somersalo [JaSo05] mentioned the Cauchy distribution. However, we cannot find a general procedure to deal with this scenario. We have shown a more general scheme to address the problem. The procedure introduced here can be employed for all applications cited in the previous paragraph. For the worked example, the procedure converges to the exact distribution.

References [Ja70]

Jazwinski, A.H.: Stochastic Processes and Filtering Theory, Academic Press (1970). [Ev07] Evensen, G.: Data Assimilation: The Ensemble Kalman Filter, Springer Verlag (2007). [BeMeBr00] Belyaev, K.P., Meyers, S., O’Brien, J.J.: Fokker–Planck equation application to data assimilation into hydrodynamic models. J. Math. Sci., 99, 1393 (2000). [BeTaBr01] Belyaev, K.P., Tanajura, C.A.S., O’Brien, J.J.: A data assimilation method used with an ocean circulation model and its application to the tropical Atlantic. Appl. Math. Modelling, 25, 655 (2001) [Ta87] Tarantola, A.: Inverse Problem Theory, Elsevier (1987). [JaSo05] Jaipio, K., Somersalo, E.: Statistical and Computational Inverse Problems, Springer Verlag (2005). [Ts88] Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Statistical Physics, 52, 479 (1988). [Ts99] Tsallis, C.: Nonextensive statistics: theoretical, experimental and computational evidences and connections. Braz. J. Phys., 29, 1 (1999). [RaRoRoBoSaCa01] Ramos, F.M., Rosa, R.R., Neto, C. Rodrigues, Bolzan, M.J.A., Sa, L.D.A., Campos Velho, H.F.: Nonextensive statistics and three-dimensional fully developed turbulence. Phys. A, 295, 250 (2001). [GoSaSm93] Gordon, N., Salmond, D., Smith, A.D.: Novel approach to nonlinear/nonGaussian Bayes ian state estimation. IEE Proc., 140, 107 (1993). [ArMaGoCl02] Arulampalam, M.S., Maskell, S., Gordon, N., Clapp, T.: A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE T. Signal Proces., 50, 174 (2002). [Do98] Doucet, A.: On Sequential Simulation-Based Methods for Bayesian Filtering, Tech. Report: Departament of Engineering, University of Cambridge (CB21PZ Cambridge UK), 1998. [NaUeHi07] Nakano, S., Ueno, G., Higuchi, T.: Merging particle filter for sequential data assimilation. Nonlinear Proc. Geoph., 14, 395 (2007). [ChKr04] Chorin, A.J., Krause, P.: Dimensional reduction for a Bayesian filter. Proc. of Nac. Acad. Sci., 101, 15013 (2004). [FuCaMa08] Furtado, H.C.M., Campos, H.F., Macau, E.E.M.: Data assimilation: particle filter and artificial neural networks. Journal of Physics. Conference Series (Online), 135, 012073 (2008). [No05] Nolan, J.P.: Stable Distributions: Models for Heavy Tailed Data, Birkhäuser, Boston, MA (2005). [ScGuNr05] Schön, T.B., Gustafsson, F., Nordlund, P.-J.: Marginalized particle filters for mixed linear/nonlinear state-space models. IEEE T. Signal Proces., 53, 2279 (2006).

Adaptive Particle Filter [Sc06]

[Be98] [Pa89] [TsQu07]

[Le09]

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Schön, T.B.: Estimation of Nonlinear Dynamic Systems: Theory and Applications, Dissertations no. 998 (Linköping Studies in Sciense and Tecnology), 2006. Bertero, M., Boccacci, P.: Inverse Problems in Imaging, Institute of Physics (1998). Papoulis, A.: Probability and Statistics, Pearson Higher Education (1989). Tsallis, C., Queiros, S.M.D.: Nonextensive statistical mechanics and central limit theorems I – Convolution of independent random variables and q-product, in Complexity, Metastability and Nonextensivity (CTNEXT 07) (Editors: S. Abe, H. Herrmann, P. Quarati, A. Rapisarda, and C. Tsallis), American Institute of Physics (2007). Leeuwen, P.J.V.: Particle filtering in geophysical systems. Monthly Weather Review, 137, 4089 (2009).

On the Analytical Solution of the Multi-Group Neutron Diffusion Kinetic Equation in One-Dimensional Cartesian Geometry by an Integral Transform Technique C. Ceolin, M.T. Vilhena, and B.E.J. Bodmann

1 Introduction The Generalized Integral Transform Technique, henceforth named GITT approach, is a well established methodology to solve analytically linear differential equations for a broad class of problems in the area of physics and engineering. By analytical we mean that no approximation is done along the derivation of the solution, except for the truncation of the solution series in numerical computations. The main idea of this approach relies on the construction of a pair of transformations from the Laplacian adjoint terms appearing in the differential equation to be solved. This fact allows us to write the solution as a series expansion in terms of the orthogonal eigenfunctions obtained from the solution of an auxiliary Sturm–Liouville problem constructed from the adjoint terms. The orthogonality of the eigenfunctions completes the pair of transformations. There exists a vast literature about the basic idea of the method and for illustration we mention the references [Cot93, Cot98]. On the other hand, projects on analytical and experimental benchmark analyses of Accelerator Driven Systems [Mai07, Abd06] have given motivation for researchers to focus their attention on the task of determining an exact analytical solution for the neutron diffusion kinetic equation especially for validating existing computational codes. Recent pioneer work in this direction was done by [Oli07, Cor08] and [Dul07]. The present work may be understood as a continuation along this line, C. Ceolin Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] M.T. Vilhena Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] B.E.J. Bodmann Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_7, © Springer Science+Business Media, LLC 2011

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to be more specific we present an analytical solution for the neutron diffusion kinetic equation in Cartesian geometry. For simplicity, and without losing generality, we discuss the one-dimensional, multi-group diffusion kinetic equation with six delayed neutron precursor concentration groups. Bearing in mind that the equation for the delayed neutron precursor concentration is a first-order linear differential equation in the time variable, one unique procedure (i.e. the GITT approach) for the set of equations is made possible introducing a spurious diffusion term with (small) diffusion constant ε . Applying the GITT technique to the modified diffusion kinetic equation, we come out with a matrix differential equation which has a well known solution in the limit ε → 0. Moreover, a comparison of arbitrary but small values of ε to the zero limit solution permits us to evaluate the influence of diffusion of the delayed neutron precursor concentration on the time evolution behavior of the neutron flux. Note that the presence of this diffusion term may seem an arbitrary complication of the equation system, but as the forthcoming discussion will show, the presence of a spatial second-order derivative term opens the possibility to also express the precursors in terms of the same set of eigenfunctions used for the neutron diffusion problem. Thus the neutron flux as well as the precursor concentrations are written as an expansion of orthogonal eigenfunctions of a self-adjoint compact operator related to the diffusion term. Upon application of the GITT technique to the extended diffusion kinetic equation results in a matrix differential equation that may be solved in general even for large matrix order, due to the non-degeneracy of eigenvalues. This is especially an advantage by virtue of the stiff character of this sort of problems, generated by the considerable differences in time scale of the prompt and delayed neutrons. Unlike other methods, the methodology adopted here is robust in that it allows us to work out the one-dimensional diffusion kinetic equation, in a straightforward manner, for problems that consider a significant number of energy groups (up to 200), and further may be extended to multi-dimensional and multi-layered problems.

2 The Analytical Solution Without losing generality, we consider the one-dimensional diffusion kinetic equation in Cartesian geometry and, assuming two neutron energy groups and six delayed neutron precursor concentration groups, 1 ∂ ∂2 φ1 (x,t) = D1 2 φ1 (x,t) − Σa1 φ1 (x,t) v1 ∂ t ∂x   6 +(1 − β ) ν1 Σ f 1 φ1 (x,t) + ν2 Σ f 2 φ2 (x,t) + ∑ λiCi (x,t),

(1)

i=1

1 ∂ ∂2 φ2 (x,t) = D2 2 φ2 (x,t) − Σa2 φ2 (x,t) + Σs12 φ1 (x,t), v2 ∂ t ∂x

(2)

Analytical Solution of the Neutron Diffusion Kinetic Equation

  ∂ Ci (x,t) = −λiCi (x,t) + βi ν1 Σ f 1 φ1 (x,t) + ν2 Σ f 2 φ2 (x,t) , ∂t

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(3)

where i = 1, . . . , 6, for t > 0 and 0 < x < L. Here φ1 (x,t) and φ2 (x,t) denote the fast and the thermal neutron flux, Ci (x,t) is the delayed neutron precursor concentration of the ith group, v is the neutron velocity, D is the diffusion coefficient, Σ a is the absorption cross section, Σ s is the scattering cross section, Σ f is the fission cross section, ν is the average number of neutrons emitted by fission, β is the delayed neutron fraction and λ is the delayed neutron decay constant. The system of equations (1)–(3) is subject to the zero flux boundary conditions

φ (0,t) = φ (L,t) = 0, and the initial conditions

φ1 (x, 0) = φ1,0 (x), φ2 (x, 0) = φ2,0 (x), βi ν1 Σ f 1 β i ν2 Σ f 2 Ci (x, 0) = φ1,0 (x) + φ2,0 (x), λi λi where φ1,0 (x) and φ2,0 (x) are the fast and the thermal neutron flux at time t = 0. In order to solve the equation system (1)–(3) by the spectral method, known as GITT [Cot93, Cot98], we introduce the aforementioned spurious diffusion term in (3) for the precursor concentration, assuming also the homogeneous boundary condition, C(0,t) = C(L,t) = 0.  ∂ ∂2 Ci (x,t) = ε 2 Ci (x,t) − λiCi (x,t) + βi ν1 Σ f 1 φ1 (x,t) ∂t ∂x  +ν2 Σ f 2 φ2 (x,t) , where i = 1, . . . , 6 and ε is a positive small parameter. This assumption allows us to apply the GITT method to solve the problem. To this end we expand the neutron fluxes and delayed neutron precursor concentrations in the following series:  2 ∞ φ1 (x,t) = (4) ∑ ϕ1n (t)χn (x), L n=1  2 ∞ φ2 (x,t) = (5) ∑ ϕ2n (t)χn (x), L n=1  2 ∞ (6) Ci (x,t) = ∑ ξni (t)χn (x), L n=1 where i = 1, . . . , 6, and χn (x) = sin(γn x) are the eigenfunctions that are solutions of an auxiliary Sturm–Liouville problem constructed from the adjoint terms of the original problem. Replacing this ansatz in the system of equations (1)–(3), taking

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moments and applying the orthogonality property of the eigenfunctions, we come out with the matrix equation for the coefficients of the solution expansion ⎞ ⎞ ⎛ ⎞⎛ ⎛ ϕ1n (t) ϕ1n (t) A B C1 · · · C 6 ⎜ D E F · · · F ⎟ ⎜ ϕ2n (t) ⎟ ⎜ ϕ2n (t) ⎟ ⎟ ⎟ ⎜ ⎟⎜ d ⎜ ⎟ ⎜ ⎟⎜ ⎜ ξn1 (t) ⎟ ⎟ = − ⎜ G1 H1 I1 . . . F ⎟ ⎜ ξn1 (t) ⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ dt . . . . . . ⎟ ⎝ .. .. .. . . . .. ⎠ ⎝ .. ⎠ ⎝ .. ⎠

ξn6 (t)

G6 H 6 F . . . I 6

ξn6 (t)

Here, n = 1, . . . , N, where N is the truncation order of the series. Further, A, B, Ci , D, E, Gi , F, Hi and Ii are diagonal matrices. The first-order homogeneous linear matrix equation in compact form is X (t) + A X(t) = 0, with a well known solution X(t) = exp(−At) X(0).

(7)

Recalling that in the present problem the eigenvalues of A are nondegenerate allows us to rewrite the exponential term X(t) = Y exp(−Dg t)Y−1 X(0)

(8)

where Y is the matrix of the eigenvectors of the matrix A and Y−1 its inverse. Dg is the diagonal matrix with the eigenvalues of A. There are a variety of ways at hand from the literature to compute the solution in (7), as shown in a nineteen fold way by [Mol78]. In a more recent publication [Seg08] one finds the application of the combined Laplace transformation technique and matrix decomposition. This method has the advantage of being general, in the sense that it can be applied to solve problems with repeated eigenvalues as shown in [Seg99, Seg08]. Concluding our derivation, we observe that the solution of the problem (1)–(3) is well defined by (4), (5) and (6) where the expansion coefficients of the solution are evaluated by the formula (8) in the limit ε → 0.

3 Numerical Results To show the aptness of the proposed method to handle the diffusion kinetic equation, in the following we solve a problem considering a slab with L = 160 cm and the nuclear parameters given in Tables 1 and 2. We present the influence of the spurious diffusion term on the results encountered for the kinetic equation in the limit where ε goes to zero. In Tables 3 and 4 the results obtained by this methodology for the fast and the thermal neutron flux are shown for a selection of numerical values for 10−3 ≥ ε ≥ 10−7 . From a simple

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Table 1 Nuclear parameters Parameter D [cm] v [cm/s] Σa [cm−1 ] Σg→g+1 [cm−1 ] νΣ f [cm−1 ]

Group 1 1.0 1.0 × 107 0.02 0.01 0.005

Group 2 0.5 3.0 × 105 0.08 0 0.099

Table 2 Delayed neutron parameters i 1 2 3 4 5 6

βi 0.00025 0.00164 0.00147 0.00296 0.00086 0.00032

λi [s−1 ] 0.0124 0.0305 0.111 0.3010 1.1400 3.0100

inspection of the displayed results in Table 3 stability of the solution for the fast neutron flux becomes apparent; especially for ε = 10−6 and ε = 10−7 the solution may be considered exact for applications. Also for the thermal neutrons (see Table 4) one observes a comparable precision already for ε = 10−5 . Therefore, one may conclude that the problem extended by a spurious term solves the diffusion kinetic equation exactly. Table 3 The ε -influence on the fast neutron flux ε 10−3 10−4 10−5 10−6 10−7

φ1 [cm−2 s−1 ] 0.000881554789 0.000881555095 0.000881555125 0.000881555128 0.000881555128

Table 4 The ε -influence on the thermal neutron flux ε φ2 [cm−2 s−1 ] −3 10 0.000109930416 10−4 0.000109930454 0.000109930458 10−5 0.000109930458 10−6 10−7 0.000109930458

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In the following we show the numerical stability of the approach, i.e. the variation of the numerical values when increasing the number of terms in the solution series. It is noteworthy that for a neutron flux precision of the order of 1% only a small number of terms in the expansion is necessary. In Tables 5 and 6, we show the numerical variation of the results obtained for the fast and the thermal neutron flux with increasing number of terms in the solution series up to N = 150. From Table 5, one observes that an accuracy of eight significant digits for the fast neutron flux is attained for a summation of 40 terms. For the thermal neutron flux (see Table 6), we obtain the same precision with only 20 terms. It is evident that the results as shown in Tables 5 and 6 are not a proof of convergence. However, a proof may be set up by following the reasoning of [Bod10], upon which we will elaborate for the present case in a future work. Table 5 Fast neutron flux simulation for increasing N N 2 10 20 30 40 50 100 150

φ1 [cm−2 s−1 ] 0.00088155513 0.00088019437 0.00088019504 0.00088019498 0.00088019499 0.00088019499 0.00088019499 0.00088019499

Table 6 Thermal neutron flux simulation with increasing N N 2 10 20 30 40 50 100 150

φ2 [cm−2 s−1 ] 0.00010993046 0.00010976388 0.00010976394 0.00010976394 0.00010976394 0.00010976394 0.00010976394 0.00010976394

For further illustration we show the plots for the fast and the thermal neutron fluxes in Figs. 1 and 2, respectively. Further, we show in Fig. 3 also the delayed neutron precursor concentration for the group i = 1.

Analytical Solution of the Neutron Diffusion Kinetic Equation

Fig. 1 Fast neutron flux

Fig. 2 Thermal neutron flux

Fig. 3 Neutron precursor concentration for group (i = 1)

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4 Conclusions In the present discussion we derived an analytical solution to the kinetic diffusion equation in terms of a well defined series. Only a few terms are required to reproduce the exact solution of the problem, to a level of precision of a few percent, in the form of a handy formula. Although the considered problem is the simplest one in the context of more complex scenarios, one may affirm that the proposed method is a promising analytical methodology for these kind of problems. The mathematical elegance of this solution together with its explicit dependence of the parameters is manifest in the fact that the stiffness character of the solved problem that has its origin the considerable differences of physical time scales (three orders of magnitude) between the prompt and delayed neutrons has no consequence such as instability, an effect common in numerical approaches of that problem. Moreover, the matrix exponential solution discussed does not pose limitations to handle such problems, mainly the ones requiring matrices of the order up to 1500, where most methods break down. In other words the technique presented here can be applied to more complex and heterogeneous problems, demanding the multi-group model with up to hundred groups of energy. Although we restricted our discussion to a homogeneous problem, an extension two a multi-layered slab and more dimensions may be implemented following the idea of [Bod10]. The precursor equation with its introduction of the spurious term with small diffusion coefficient has two main consequences. First of all, one observes from the attained results that the one dimensional neutron kinetic diffusion equation is correctly modeled as expected, especially in the limit of vanishing diffusion process of the delayed neutron precursors. Nevertheless, taking into account the diffusion term with in general small but unknown numerical value opens a pathway to express the solution in a series expansion for the neutron fluxes and the delayed neutron precursor concentrations in terms of a common basis, a set of orthogonal functions. These functions are the solution of a Sturm–Liouville problem with origin in the self-adjoint property of the diffusion term appearing in the studied problem. Concerning genuine convergence of the solution, one recognizes that the kinetic problem is a special case of the Cauchy–Kowalewski theorem [Cou89], so that the existence of a unique solution is confirmed. As a consequence we realize that, according to [Bod10] taking the framework of the Cardinal Interpolation Theorem as well as taking into account the boundedness of the operator associated to the neutron point kinetic equation, the derived solution in terms of an infinite series is exact, so that for practical (numerical) purposes one can get exact results within a prescribed accuracy, by just controlling the number of terms in the solution series. From the previous discussion, we envisage a natural and obvious extension of this sort of solution for the multi-dimensional neutron kinetic diffusion equation by the GITT approach. The characteristics of the integral transform technique permits an order reduction of this problem to the one dimensional case solved here because of the invariance of the form of the solution regarding the topology of the problem. Application of GITT in the extra dimensions of this multi-dimensional problem and assuming again the same common set of orthogonal functions as basis leads to a

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procedure analogous to the one discussed here. Indeed, the natural interest of our future work shall focus on this topic.

References [Abd06] Ait-Abderrahim, H., Stanculescu, A.: IAEA Coordinate Research Project on Analytical and Experimental Benchmark Analyses of Accelerator Driven Systems. Proceedings of ANS Topical Meeting on Reactor Physics, Paper C074 (2006). [Bod10] Bodmann, B.E.J., de Vilhena, M.T., Ferreira, L.S., Bardaji, J.B.: An analytical solver for the multi-group two-dimensional neutron-diffusion equation by integral transform techniques. Il Nuovo Cimento, 33 C, n. 1, 63–70 (2010). [Cor08] Corno, S.E., Dulla, S., Picca, P., Ravetto, P.: Analytical approach to the neutron kinetics of the non-homogeneous reactor. Progress in Nuclear Energy, 50, 847–865 (2008). [Cot93] Cotta, R.M.: Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Florida (1993). [Cot98] Cotta, R.M. (Org.): The Integral Transform Method in Thermal and Fluids Science and Engineering, Begell House, Inc., New York (1998). [Cou89] Courant, R., Hilbert, D.: Methods of Mathematical Physics – II, John Wiley & Sons (1989). [Dul07] Dulla, S., Ravetto, P., Picca, P., Tomatis, D.: Analytical benchmarks for the kinetics of accelerator-driven systems. Joint International Topical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications, (2007). [Mai07] Maiorino, J.R.: The Participation of IPEN in the AIEA Coordinate Research Projects on Accelerator Driven Systems (ADS). International Nuclear Atlantic Conference – INAC, (2007). [Mol78] Moler, C., Van Loan, C.: Nineteen dubious ways of computing the exponential of a matrix. SIAM Review, 801–836 (1978). [Oli07] Oliveira, F.L., Maiorino, J.R., Santos, R.S.: The analytical benchmark solution of spatial diffusion kinetics in source driven systems for homogeneous media. International Nuclear Atlantic Conference – INAC, (2007). [Seg99] Segatto, C.F., Vilhena, M.T., Gomes, M.G.: The one dimensional LTSn solution in a slab with high degree of quadrature. Annals of Nuclear Energy, 26, 925–934 (1999). [Seg08] Segatto, C.F., Vilhena, M.T., Marona, D.V.: The LTSn solution of the transport equation for one-dimensional Cartesian geometry with c = 1. Kerntechnik, 73, 57–60 (2008).

Estimating the Validity of Statistical Energy Analysis Using Dynamical Energy Analysis: A Preliminary Study D.J. Chappell and G. Tanner

1 Introduction Dynamical energy analysis was recently introduced as a new approach toward determining the distribution of mechanical and acoustic wave energy in complex built up structures [Ta09]. The technique interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. Statistical energy analysis (SEA) is a highly efficient method for analyzing energy distributions in large scale industrial problems, but remains an expert tool since explicit bounds on its validity are generally difficult to establish. In this work we aim to address this problem through a study of the spectral properties of the transfer operator arising in dynamical energy analysis. Through this study one can estimate escape and correlation decay rates for SEA subsystems and use these quantities to indicate the validity of an SEA approach.

2 Wave Energy Flow in Terms of the Green’s Function It is assumed that the system as a whole is characterized by a linear wave equation describing the overall wave dynamics including damping and radiation in a finite domain Ω ⊂ R2 . In this work only stationary problems with continuous, monochromatic energy sources are considered. We split the system into NΩ subsystems and consider the scalar wave equation for acoustic pressure waves in each homogeneous D.J. Chappell University of Nottingham, UK, e-mail: [email protected] G. Tanner University of Nottingham, UK, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_8, © Springer Science+Business Media, LLC 2011

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N

Ω sub-domain Ωi , with local wave velocity ci , i = 1, . . . , NΩ and Ω = i=1 Ωi . Extensions to more complicated systems with different wave operators in different parts of the system can be treated with the same techniques as long as the underlying wave equations are linear, see the discussion in [Ta09]. The general problem of determining the response of a system to external forcing with angular frequency ω at a source point r0 ∈ Ω0 can then be reduced to solving

ˆ r0 ; ω ) = −δ (r − r0 ), (ki2 − H)G(r,

i = 1, . . . , NΩ ,

(1)

with Hˆ = −Δ , G represents the Green function, r ∈ Ωi is the solution point and δ is the Dirac delta distribution. Furthermore, ki = ω /ci + iμi /2 is a complex valued wavenumber, where the imaginary part represents √ a subsystem dependent damping coefficient μi . Throughout this work we take i = −1 unless used as a subscript, in which case it is an index over the number of subsystems. The wave energy density induced by the source is then given as

ε (r, r0 ; ω ) =

|G(r, r0 ; ω )|2 , mi c2i

(2)

where mi is the density of the medium in Ωi . The linear wave operator Hˆ can naturally be associated with the underlying ray dynamics via the eikonal approximation, see for example [Ta09]. Using small wavelength asymptotics, the Green function in (1) may be written as a sum over all classical rays from r0 to r for fixed kinetic energy of the hypothetical ray particle. One obtains [Gu90] G(r, r0 ; ω ) =

π (2π i)(d+1)/2

∑

j:r0 →r

A j exp(iki L j − iν j π /2),

(3)

where L j is the length of the ray trajectory between r0 and r including possible reflections on boundaries. The amplitudes A j may be written as a product of three terms as in [Ta09] due to damping, mode conversion and reflection/transmission coefficients, and geometrical factors. The phase index ν j contains contributions from the reflection/transmission coefficients at interfaces between subsystems and from caustics along the ray path. Analogous representations to (3) have been considered in detail in quantum mechanics [Gu90] and are also valid for general wave equations in elasticity such as the biharmonic equation and the Navier–Cauchy equation, see [Ta07] for an overview. In the latter case G becomes matrix valued. Note that the summation in (3) is typically over infinitely many terms, where the number of contributing rays increases (in general) exponentially with the length of the trajectories included. This gives rise to convergence issues, especially in the case of low or no damping [Ta07].

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The wave energy density (2) can now be expressed as a double sum over classical trajectories and hence ε (r, r0 ; ω ) may be written as

ε (r, r0 ; ω ) = C

∑

j, j  :r0 →r

A j A j exp(iki [L j − L j ] − i[ν j − ν j ]π /2) (4)

= C[ρ (r, r0 ; ω ) + off-diagonal terms], with C = π 2 /(ρi c2i (2π )(d+1) ). The dominant contributions to the double sum arise from terms in which the phases cancel exactly; one thus splits the calculation into a diagonal part ρ (r, r0 ; ω ) = ∑ |A j |2 (5) j:r0 →r

where j = j , and an off-diagonal part. The diagonal contribution gives a smooth background signal and the off-diagonal terms give rise to fluctuations on the scale of the wavelength. The phases related to different trajectories are (largely) uncorrelated and the resulting net contributions to the off-diagonal part are in general small compared to the smooth part, especially when averaging over frequency intervals of a few wavenumbers. It has been shown in [Ta09] that calculating the smooth diagonal part (5) is equivalent to a ray tracing treatment. That is, the smooth part of the energy density can be described in terms of the flow of fictitious non-interacting particles emerging from the source point r0 uniformly in all directions and propagating along ray trajectories. This makes it possible to relate wave energy transport with classical flow equations and thus thermodynamical concepts, which are at the heart of an SEA treatment. In DEA the classical flow is expressed in terms of linear phase space operators as detailed in the next section.

3 Linear Phase Space Operators and DEA 3.1 Phase Space Operators and Boundary Maps A brief outline of the derivation of the DEA flow equations is now given; for details, see [Ta09] and [Ch10]. The time dependence of a density of ray trajectories can be described in terms of a linear phase space operator L τ (X,Y ) = δ (X − ϕ τ (Y )), known as a Perron–Frobenius operator in dynamical systems theory, thus

ρ (X, τ ) =



L τ (X,Y )ρ0 (Y )dY.

(6)

Here X = (r, p) denotes the phase space coordinate with position r and momentum p. The phase space flow ϕ τ (Y ) gives the position of the particle after time τ starting at Y = (r , p ) when τ = 0. Furthermore, ρ0 denotes the initial ray density at time τ = 0 and the domain of integration is over the whole of phase space.

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Consider a source localized at a point r0 emitting waves continuously at a fixed angular frequency ω . Standard ray tracing techniques estimate the wave energy at a receiver point r by determining the density of rays starting at r0 and reaching r after some unspecified time. This may be written in the form

ρ (r, r0 , ω ) =

 ∞  0

w(Y, τ )L τ (X,Y )ρ0 (Y, ω )dY d p d τ ,

(7)

with initial density ρ0 (Y, ω ) = δ (r −r0 )δ (k02 −H(Y )), where H = |p|2 is the Hamilˆ It can be shown that (7) is equivalent to the ton function for the wave operator H. diagonal approximation (5). A weight function w is included to incorporate damping and reflection/transmission coefficients. It is assumed that w is multiplicative, (w(ϕ τ1 (X), τ2 )w(X, τ1 ) = w(X, τ1 + τ2 )), which holds for standard absorbtion mechanism and reflection processes. In order to solve the stationary flow problem (7) a boundary mapping technique is employed. For the time being let us consider a problem with a single (sub-)system Ω = Ω1 with boundary Γ . The boundary mapping procedure involves first mapping the ray density emanating continuously from the source onto the boundary Γ . The (0) resulting boundary layer density ρΓ is equivalent to a source density on the boundary producing the same ray field in the interior as the original source field after one reflection. Secondly, densities on the boundary are mapped back onto the boundary by a boundary transfer operator B(X s ,Y s ; ω ) = w(Y s )δ (X s − φ ω (Y s )), where X s = (s, ps ) represents the coordinates on the boundary (s parameterizes Γ and ps denotes the momentum component tangential to Γ at s), likewise Y s = (s , ps ), and φ ω is the invertible boundary map. Note that convexity is assumed to ensure φ ω is well defined; non-convex regions could be handled by introducing a cut-off function in the shadow zone as in [Le02] or by subdividing the regions further. The stationary density on the boundary induced by the initial boundary distribu(0) tion ρΓ (X s , ω ) can then be obtained using

ρΓ (ω ) =

∞

∑ B n(ω )ρΓ

n=0

(0)

(ω ) = (I − B(ω ))−1 ρΓ (ω ), (0)

(8)

where B n contains trajectories undergoing n reflections at the boundary. The resulting density distribution on the boundary ρΓ (X s , ω ) can then be mapped back into the interior region. One obtains the density (7) after projecting down onto coordinate space.

3.2 Subsystems Recall the splitting into subsystems Ωi , i = 1, . . . , NΩ introduced earlier. The dynamics in each subsystem are considered separately so that both variability in the wave velocity ci and non-convex domains may be handled simply. Coupling between sub-

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elements can then be treated as losses in one subsystem and source terms in another. Typical subsystem interfaces are surfaces of reflection/ transmission due to sudden changes in material parameters or local boundary conditions. We describe the full dynamics in terms of subsystem boundary transfer operators Bi j ; flow between Ω j and Ωi is only possible if Ωi ∩ Ω j = 0/ and one obtains Bi j (Xis , X js ) = wi j (X js )δ (Xis − φiωj (X js )),

(9)

where φiωj is the boundary map in subsystem j mapped onto the boundary of the adjacent subsystem i and Xis are the boundary coordinates of Ωi . The weight wi j contains, among other factors, reflection and transmission coefficients characterizing the coupling at the interface between Ω j and Ωi .

3.3 Basis Function Representations and SEA In order to evaluate (I − B)−1 and solve (8) numerically, it is convenient to express the transfer operator B in a suitable set of basis functions defined on the boundary. In [Ta09] a Fourier basis has been applied, which is a natural choice of a complete basis for problems with periodic boundary conditions. However, a number of difficulties arise with this choice such as slow convergence of quadrature rules for the associated integrals and the treatment of corners on the boundary. In this work a Chebyshev polynomial basis representation with Gauss–Chebyshev quadrature is employed as detailed in [Ch10]. This leads to considerable improvements on the difficulties encountered using a Fourier basis and means the requirement for periodic boundary conditions can be dropped, allowing for much more freedom in the choice of approximation regions. Up to now, the various representations described are all equivalent and correspond to a description of the wave dynamics in terms of the ray tracing ansatz (7). Traditional ray tracing based on sampling ray solutions over the available phase space is rather inefficient. Convergence tends to be fairly slow, especially if the absorption is low and an exponentially increasing number of long paths including multiple reflections need to be taken into account. An SEA treatment emerges when approximating the individual operators Bi j in terms of constant functions only [Ta09]; using, for example, a Fourier basis this would correspond to an approximation in terms of the lowest order basis functions only. In the case of a Chebyshev basis, the associated integrand is not constant due to the presence of the weight function for the inner product in which the Chebyshev basis is orthonormal. An SEA treatment is obtained only after restricting the basis to the lowest order case and omitting the weight for this case, see [Ch10] for details. In the SEA case the matrix obtained from the basis function representation of Bi j is one-dimensional and gives the mean transmission rate from subsystem j to subsystem i. It is thus equivalent to the coupling loss factor used in standard SEA

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equations [Ly95]. The resulting full NΩ -dimensional B matrix yields a set of SEA equations using (8) after mapping the boundary densities back into the interior. An SEA approximation is justified if the ray dynamics within each subsystem are sufficiently chaotic such that a trajectory entering subsystem j forgets everything about its past before exiting Ω j ; SEA can thus be viewed as a Markov approximation of the deterministic dynamics. Thus correlations within the dynamics must decay rapidly on the time scale it takes for a typical ray to leave Ω j . This is discussed further in relation to the spectral properties of the transfer operator in the next section. Conditions typically cited for SEA to be a good approximation include that the subsystem boundaries are sufficiently irregular, the subsystems are dynamically well separated and absorption and dissipation is small. In this case SEA is an extremely efficient method compared to standard ray tracing. However, for subsystems with regular features, such as rectangular cavities or corridor-like elements, incoming rays are directly channeled into outgoing rays, thus rendering the Markov approximation invalid and introducing memory effects. Likewise, strong damping may lead to a significant decay of the signal before reaching the exit channel introducing geometric (system dependent) effects.

3.4 Spectral Properties of the Transfer Operator The spectrum of the transfer operator contains information about the asymptotic behavior of the ray dynamics and in particular the escape and correlation decay rates. As we have discussed above, for SEA to be a good approximation requires correlations within the ray dynamics to decay rapidly on the time scale it takes for a typical ray to leave a particular subsystem. Hence we expect to see high correlation decay rates and low escape rates in cases where SEA works well. We will put this expectation to the test in what follows. We conduct our analysis for each subsystem i = 1, . . . , NΩ and consider the diagonal components of the transfer operator corresponding to self-interactions of subsystems, that is, Bii . Once a basis function representation has been applied Bii will be represented by a square matrix whose size depends on the order at which we truncate the basis expansion. The size also depends on the form of the approximation, for example whether the basis approximation is applied globally as in [Ta09] or if a subsystem boundary is split into a number of approximation regions due to non-smooth boundary conditions at corners or subsystem interfaces as in [Ch10]. As mentioned above, for SEA the matrix representation of Bii corresponds to a single number. The first quantity of interest is the escape rate for a subsystem i for some i = 1, . . . , NΩ . This may be estimated from the maximum eigenvalue λ0 of the DEA matrix representation Bii of Bii , which gives the probability that a ray trajectory will remain in subsystem i (see Chap. 22 of [Cv06]). Therefore if subsystem i is closed and undamped, that is, there is no transfer of energy between it and any other subsystem and the damping coefficient μi = 0, then λ0 = 1. The asymptotic escape

Validity of Statistical Energy Analysis

rate γ is given as

75

γ = − ln λ0 .

(10)

In this work we assume μi = 0 and therefore obtain an estimate of the minimum escape rate for the subsystem i. Incorporating damping would simply correspond to an increase in the escape rate proportional to the level of damping. The second quantity of interest is the correlation decay rate. This may be estimated from the DEA matrix representation Bii of Bii when the subsystem i is hypothetically considered to be closed and we take the damping coefficient μi = 0. In particular, for ergodic ray dynamics the correlation decay rate may be estimated from the gap between the largest two eigenvalues of Bii ([Cv06], Chap. 22). Since we are considering closed subsystems with zero damping, then the maximum eigenvalue λ0 = 1. The correlation decay rate ν may be estimated from the second largest eigenvalue λ1 as ν = − ln λ1 . (11) Note that the quantities computed as described above will be estimates since we are only able to work with a basis approximation of the transfer operator its spectrum is dependent on the underlying function space, see for example [Ba06]. In the next section we study numerically the relative sizes of γ and ν for some examples where we already know how well SEA works.

4 Numerical Examples A variety of two-cavity systems are considered as in Ref. [Ta09] and are shown in Fig. 1. Configuration A features irregular shaped well separated pentagonal subsystems and SEA has been shown to work well [Ta09, Ch10]. In configuration B the size of the interface between the subsystems is increased reducing their dynamical separation and therefore the applicability of SEA. Configuration C includes a rectangular left-hand subsystem channeling rays out of the subsystem and introducing long-range correlations in the dynamics. In addition, the source is further from the intersection of the two subsystems. It has been demonstrated that SEA does not work well for this configuration, particularly for large damping levels [Ta09, Ch10].

Fig. 1 Coupled two-domain systems: configurations A, B and C, respectively

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Figure 2 shows the approximations to the escape and correlation decay rates for the two subsystems in configuration A. The quantities for subsystem 1 (the lefthand region) are referred to as “Escape 1” and “Decay 1”, and analogous names are used for subsystem 2 (the right-hand region). The order of the basis expansion N is plotted on the horizontal axis. Given that SEA works well for this configuration one would expect relatively low escape rates and large correlation decay rates. In fact we see that only for subsystem 2 is ν > γ , for subsystem 1 this is only true for the lowest level of basis approximation employed (N = 4). The reason the results are not more conclusive is possibly due to the ray dynamics in the irregular polygons not being sufficiently chaotic, leading to algebraic, rather than exponential decay of correlations (see for example [Ca99]), that is, ν = 0. From the figure it is evident that the decay rate is decreasing as we compute to a greater degree of accuracy using larger N, and may be converging to zero. It is expected that more conclusive results would be obtained if noise was incorporated in the model. In fact one could view the error in the lower order approximations as a form of noise and in this case it is clear that ν > γ for both subsystems.

Fig. 2 Escape and correlation decay rates for configuration A

Figure 3 shows the approximations to the escape and correlation decay rates for configuration B and is laid out in the same way as Fig. 2. It is clear for both subsystems here that γ > ν and thus the escape rate is large enough so that correlations in the ray dynamics will not have decayed sufficiently before a typical ray exits either subsystem. Therefore the Markov approximation underlying an SEA treatment will

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not be valid, and as we have seen in previous studies SEA will not be a good model in this case [Ta09, Ch10]. Note that the escape rates shown in Figs. 2 and 3 are lower for subsystem 1 than for subsystem 2, which is expected since in both cases the interface between the subsystems forms a smaller part of the total boundary length for subsystem 2 than for subsystem 1.

Fig. 3 Escape and correlation decay rates for configuration B

Figure 4 shows the approximations to the escape and correlation decay rates for configuration C. For subsystem 2 the results are similar to those for configuration A and hence a similar explanation applies. For subsystem 1 the correlation decay rate is zero because 1 is a repeated eigenvalue and hence ν = − ln λ1 = − ln 1 = 0. The reason why 1 is a repeated eigenvalue here is the regular geometry, and in particular the absolute values of the momenta on each pair of opposite sides being time invariant for each admissible trajectory.

5 Conclusions The escape and correlation decay rates of the transfer operator arising in dynamical energy analysis have been studied numerically, with the aim of using them to indicate the validity of an SEA approach. While the results clearly indicated when SEA was not expected to be valid, they were less conclusive in the case where SEA

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Fig. 4 Escape and correlation decay rates for configuration C

works well. These studies suggest that exponential decay of correlations is perhaps too strong an indicator in the models employed here and weaker measures of the decay of correlations in the ray dynamics may prove more suitable.

References [Ba06] Bai, Z-Q.: On the discrete Frobenius–Perron operator of the Bernoulli map. J. Phys. A: Math. Theor., 39, 4945–4953 (2006). [Ca99] Casati, G., Prosen, T.: Mixing property of triangular billiards. Phys. Rev. Lett., 83, n. 23, 4729–4732 (1999). [Ch10] Chappell, D.J., Giani, S., Tanner, G.: Dynamical energy analysis for built-up acoustic systems at high frequencies. J. Acoust. Soc. Amer., submitted. [Cv06] Cvitanovi´c, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G.: Chaos: Classical and Quantum, Niels Bohr Institut, Copenhagen, ChaosBook.org (2009). [Gu90] Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics, Springer, New York (1990). [Le02] Le Bot, A.: Energy transfer for high frequencies in built-up structures. J. Sound. Vib., 250, 247–275 (2002). [Ly95] Lyon, R.H., DeJong, R.G.: Theory and Application of Statistical Energy Analysis, 2nd edn., Butterworth-Heinemann, Boston, MA (1995). [Ta07] Tanner, G., Søndergaard, N.: Wave chaos in acoustics and elasticity. J. Phys. A: Math. Theor., 40, R443–R509 (2007). [Ta09] Tanner, G.: Dynamical energy analysis—Determining wave energy distributions in vibroacoustical structures in the high-frequency regime. J. Sound Vib., 320, 1023–1038 (2009).

Efficient Iterative Methods for Fast Solution of Integral Operators Related Problems K. Chen

1 Introduction The discretization of integral operator related problems inevitably leads to some kind of linear system involving dense matrices. Such large scale systems can be prohibitively expensive to solve. In this paper, we shall first review various works that aimed to solve such systems effectively. We start from the solution of the boundary integral equation for the exterior Helmholtz problem with smooth boundaries for low and medium wavenumbers, solved by conjugate gradients and multigrid methods. We discuss the importance of effective preconditioning in the contexts of fast multipole methods and wavelet methods. Then we present some recent work on restoring images in the framework of inverse deconvolution, where the integral operator induced dense matrix, though structured, can be generated but cannot be computed due to extremely large sizes. No optimal solvers exist for this problem if the nonlinear total-variation semi-norm based regularizer is used. An effective optimization based multilevel method, using the idea of fast multipole like methods, is developed and presented here. Various numerical experiments are also reported. Finally a brief discussion of open challenges is given.

2 Fast Iterative Methods for the Helmholtz Equation The Helmholtz equation, with a Neumann boundary condition

∂φ ∂n

= g,

K. Chen University of Liverpool, UK, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_9, © Springer Science+Business Media, LLC 2011

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∇2 φ + k2 φ = 0,

p∈Ω

(1)

in an infinite domain (i.e. the domain Ω exterior to the surface S = ∂ Ω of some interior domain Ω − ⊂ R3 ) is typically solved by a boundary integral equation reformulation into    ∂ Gk 1 ∂ 2 Gk dSq φ (q) (p, q) + α Lφ (p) = − φ (p) + 2 ∂ nq ∂ n p nq ∂Ω    α ∂ Gk dSq = g(p) + g(q) Gk (p, q) + α (2) 2 ∂ np ∂Ω where nq is the unit normal exterior to the boundary ∂ Ω away from Ω with Gk (p, q) =

eik|p−q| , 4π |p − q|

and k is the wavenumber; see [AmHaWi92]. Discretizing using boundary elements leads to the n × n linear system [AmHaWi92, Ch05] Au = f

(3)

where u = φ ; for collocation method Ai, j = L(pi )ψ j and for Galerkin method Ai, j = (ψi , L(pi )ψ j ).

2.1 Iterative Algorithms of Order O(n2 ) With the traditional method of using piecewise polynomial basis functions {ψ j }, the above matrix A is dense so matrix-vector multiplications Ax costs O(n2 ) operations. In this context, commonly used methods using these multiplications are the following two [Ch05].

Conjugate Gradients Methods (CGM) Assume that A is not symmetric (if it is symmetric, simpler variants can be used). Normal equation approach. The idea is to consider instead of (3) AAT y = f ,

u = AT y,

since AAT is symmetric positive definite. Given an initial guess u(0) with residual r(0) = f − AAT y(0) = f − Au(0) , we obtain α = (r(k) )T r(k) /(p(k) )T AAT p(k) from solving min rAAT =  f − AAT yAAT y

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in the form y = y(k+1) = y(k) + α p(k) with p(k) a new search direction. Here at k = 0, we take p(0) = r(0) and for k ≥ 1, with r(k) = f − AAT y(k) = f − Au(k) , we use the conjugate gradient direction p(k) = r(k) + β p(k−1) with (p(k) )T AAT p(k−1) = 0 i.e.

β =−

(r(k) )T AAT p(k−1) (r(k) )T r(k) = − . (p(k−1) )T AAT p(k−1) (r(k−1) )T r(k−1)

Equivalently let p(k) denote AT p(k) for the purpose of eliminating the intermediate variable y. Then u = u(k) + α p(k) , p(k) = AT r(k) + β p(k−1) with (p(k) )p(k−1) = 0 and α = (r(k) )T r(k) /(p(k) )T AAT p(k) = (r(k) )T r(k) /(p(k) )T p(k) . Algorithm 1 (CGN algorithm) (CGN Algorithm) given x = x0 , r = b − Ax0 and set initially p = AT r, and rnew = rT r (1) (2) (3) (4)

q = Ap αk = rnew /(pT p) Update the solution x = x + αk p r = b − Ax = r − αk q and set rold = rnew (5) Compute rnew = rT r (exit if rnew is small enough) (6) βk = rnew /rold (7) Update the search direction p = AT r + βk p and continue with step (1) for k = k + 1.

(Naive CGN) given y = y0 , r = b − AAT y0 and set initially p = r, and rnew = rT r q = AAT p, (pT q = AT p22 ) αk = rnew /(pT q) Update y = y + αk p r = b − Ax = r − αk q and set rold = rnew (5) Compute rnew = rT r (exit x = AT y if rnew is small) (6) βk = rnew /rold (7) Update the search direction p = r + βk p and continue with step (1) for k = k + 1.

(1) (2) (3) (4)

Generalized minimal residual approach. The generalized minimal residual method by [SaSc96] with m steps of restart aims to solve min r2 =  f − Au2

u∈Vm

where Vm = span(q1 , q2 , . . . , qm ); here q j ’s are columns of an orthogonal matrix Qm from an Arnoldi iteration AQm = Qm Hm + hqm+1 eTm , where em ∈ Rn is the mth unit vector. Further the above minimization is reduced to the simple least squares’ problem Hm+1 y = r(0) 2 eˆ1 , where y ∈ Rm and eˆ1 ∈ Rm is the first unit vector. After finding y, we obtain the next iterate u(m) = u(0) + Qm y.

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Multigrid Method A multigrid method (MGM) utilizes a series of grids ∂ Ω  with L u  = f  ,

 = 1, 2, . . . , L

such that ∂ Ω 1 is the finest grid where we desire to solve and the other coarser grids are here to speed up the computation. See [TrOoSc2001, Ch05]. The idea of a MGM relies on the residual correction principle. If u˜1 is a known approximation of u1 with a nonzero residual r1 = f1 − L1 u˜1 . Then solving the residual equation L2 v2 = R1 r1 or for a nonlinear case L2 w2 = L2 R1 u˜1 + R1 r1 ,

v2 = w2 − R1 u˜1

will result in an improved approximation u¯1 = u˜1 + P1 v2 if the original approximation u˜1 is smooth (not required to be close to u1 ). Here R1 , P1 are, respectively, the restriction and interpolation operators. This is for a two grid between ∂ Ω 1 , ∂ Ω 2 . Repeated use of this idea will lead to a MGM. The commonly used V-cycling MGM can be simply stated as MGM(u1 , f1 , 1) Algorithm 2 (MGM algorithm)

(1) (2) (3) (4) (5) (6) (7) (8) (9)

MGM(uk , fk , k): Pre-smoothing over ∂ Ω k : Lk uk = fk Computation of the residual ∂ Ω k : rk = fk − Lk uk Set k = k + 1 Restriction to coarse grid ∂ Ω k : uk = Rk−1 uk−1 and rk = Rk−1 rk−1 If k = L (coarsest grid), solve Lk vk = rk ‘accurately’; otherwise call the MGM step again: MGM(uk , fk , k) Set k = k − 1 Interpolation of the correction vk = Pk vk+1 Update the fine grid solution uk = uk + vk Return to continue

The most expensive part of the above two methods CGM and MGM is in the matrix-vector products, which may be speeded up.

2.2 Iterative Algorithms of Order O(n) Two excellent ideas for speeding up matrix-vector products shown below lead to fundamentally new and fast methods.

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Fast Multipole Methods (FMM) The FMM makes to the top 10 algorithms [Ci00]. For boundary integral equations, the decay properties of the integral kernel can be utilized analytically to design hierarchical expansions that may be arranged to give the FMM approximation for computing each row of matrix L1 multiplying a vector quickly. Similar algorithms to the FMM can be derived using a function-free and H-matrix approach, see [BrGr97, Fo09, AmPr99, Ba08]. We remark that while the FMM offers a fast solution per iteration for an iterative method, the overall number of iterations required is dependent on the conditioning of an underlying problem. To speed up such iterations effective preconditioning is necessary. However, the computation of a preconditioner is usually restricted by the inaccessibility of matrix elements from far field.

Wavelets Methods Wavelets methods [Da97] offer a revolutionary idea of using bi-orthogonal basis functions, instead of the traditional piecewise polynomials, for the discretization of a boundary integral equation. The near orthogonality has two related consequences on the resulting matrix: first it can be easily preconditioned by its diagonal matrix and second the matrix is almost sparse. In fact a full orthogonality would imply a strictly diagonal matrix in some cases. The drawback of a wavelet method is that the wavelets are not easily constructed, though the complication will be worthwhile. Some recent work by [MaSc07] proposed to work with non-convergent wavelets for solving operator equations. The main idea is that the construction of approximate wavelets is much easier and it remains to test the efficiency of such methods for practical applications. Combining the traditional boundary elements with the wavelet transforms for the purpose of near optimal preconditioning leads an effective method; this can be done explicitly [ChCh02] or implicitly [HaCh05].

3 Fast Iterative Methods for an Image Deblurring Model A rich class of problems involving an integral operator arise from high resolution image processing [Vo02, ChSh05]. Among others, one example is the image deblurring problem (as shown in Fig. 1) of reconstructing image u from z(x, y) = (Ku)(x.y) + η (x, y)

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Fig. 1 An image deblurring problem

given a noisy and blurred image z in Ω = [0, 1]2 ⊂ R2 , where η is unknown but assumed to be a Gaussian white noise with 0 mean and estimable standard deviation σ [Vo02]. Although more recent models exist, the most well-known model for the above problem is due to Rudin–Osher–Fatemi [RuOsFa92]     |∇u|dxdy + |Ku − z|2 dxdy , (4) min E(u) ≡ α u

Ω

Ω

where the first term is the total-variation (TV) semi-norm regularizer with |∇u| =  2 ux + u2y and the integral operator K is assumed to have a spatially-invariant kernel i.e.  k(x − x , y − y )u(x , y )dx dy . (Ku)(x, y) = Ω

The Euler–Lagrange partial differential equation of (4) is the following: − α∇ ·

∇u + K ∗ Ku = K ∗ z |∇u|β

where |∇u|β =



(5)

u2x + u2y + β

(with β > 0 a small regularization parameter) and K ∗ is the adjoint operator of K. Owning to the usually large dimension of a discrete image z (e.g. n × n = 1024 × 1024 ≈ 1 million), the operator K (when discretized as a dense but structured matrix) cannot be directly formed. If u is assumed to have a zero Dirichlet boundary condition, then K will be a block Toeplitz matrix with Toeplitz blocks (BTTB) or if u is assumed to have a periodic boundary condition, then K will be a block circulant matrix with circulant blocks (BCCB). For either case [Vo02], the use of fast Fourier transforms (FFT) can ensure the matrix vector product Ku to be efficient; however, this restricts many possible solvers to be developed. Two kinds of efficient methods for solving (4) are discussed below.

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3.1 Intermediate Variable Methods The first method [HuNgWe08] introduces an intermediate variable v for (4)    γ 1 2 2 min E1 (u, v) ≡ α |∇u| + (u − v) + |Kv − z| dxdy. u,v 2 2 Ω

(6)

The second method [WaYaYihZ08] introduces an intermediate variable w for (4)    1 γ min E2 (u, ω ) ≡ α |ω | + |ω − ∇u|2 + |Ku − z|2 dxdy. (7) u,ω 2 2 Ω For either method, alternating minimization leads to simple solutions: For (6) solving for u is a simple denoising problem and solving for v can be through FFT because the regularizer now adds a constant diagonal to K ∗ K. For (7) solving for ω can be done analytically while the solution of u involves a simpler semi-norm to enable fast solvers. Although elegant, both methods involve many (up to 20) iterations between two sub-problems and hence are non-optimal. Besides, practically, only a nearby problem is solved as γ cannot be taken too large.

3.2 Optimization Based Multilevel Methods Below we consider how to solve (4) directly after discretizing it. Given z ∈ Rn×n , the above model [RuOsFa92] rewritten as     1 2 2 2 E(u) = α ux + uy + (Ku − z) dxdy, min E(u), u 2 Ω can be discretized to give rise to the discrete optimization problem min E(u),

u∈Rn×n

E(u) = α

n−1 n−1 

∑∑

(8)

(ui, j − ui, j+1 )2 + (ui, j − ui+1, j )2

i=1 j=1

+

2 1 n n  n n Ki, j;,m u,m − zi, j , ∑ ∑ ∑ ∑ 2 i=1 j=1 =1 m=1

with α = α /h and h = 1/(n − 1). Here we shall assume that K = (Ki, j;,m ) is a block circulant matrix with circulant blocks (BCCB). This is the case if we adopt the periodic boundary condition [NgBo03, Vo02, ChSh05]. Now solve (8) by the coordinate descent method on the finest level 1:

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⎧ (0) Given u(0) = (ui, j ) = (zi, j ) with l = 0, ⎪ ⎪ ⎪ ⎨ (l) Solve ui, j = argminui, j ∈R E loc (ui, j ) for i, j = 1, 2, . . . , n ⎪ Set u(l+1) = (u(l) ) and repeat the above step with l = l + 1 ⎪ ⎪ i, j ⎩ until a prescribed stopping step on l, where E

loc

 1 (l) (l) 2 (ui, j ) = Ku − z + α (ui, j − ui+1, j )2 + (ui, j − ui, j+1 )2 2  (l) (l) (l) + (ui, j − ui−1, j )2 + (ui−1, j − ui−1, j+1 )2   (l) (l) (l) + (ui, j − ui, j−1 )2 + (ui, j−1 − ui+1, j−1 )2 .

(9)

(10)

This iterative method can be applied over a general level k min J( u + Pk ci, j )

ci, j ∈R

where Pk is an interpolation operator, distributing a single constant over an index block (i, j) on level k and then padding zeros over the rest of the entire grid of level 1 [ChCh10]. Although each subproblem in (9) is only one dimensional, we see that it has an O(n2 ) complexity because the fitting term involves vectors of length n2 and in 2 t , where t = ( j − 1)n + i, wt ∈ Rn is the tth column particular (Ku)i, j = ui, j wt + w t is a vector not involving ui, j (i.e. a weighted sum of all columns of K of K and w except t). u +Pk ci, j ) leads The same complexity problem persists on level k, where minci, j J( to minimization of the local subproblem 1 1 J loc (ci, j ) = α T (ci, j ) + ci, j wt + K u − z2 = α T (ci, j ) + ci, j wt −  z 2 , 2 2 2

(11)

where  z = z − K u is known, the vector wt ∈ Rn denotes the summation of all columns of K corresponding to the entries inside the (i, j) block on level k, and the TV related term T (ci, j ) is defined by 2

T (ci, j ) =

∑



=1

(ci, j − hk1 −1, )2 + v2k1 −1, +

2 −1 

+

∑

=1

(ci, j − hk2 , )2 + v2k2 , +

k2 −1 

∑

m=k1

k2

∑

m=k1

√  2 + 2 (ci, j − vk2 ,2 )2 + hk2 ,2 ,

(ci, j − vm,2 )2 + h2m,2

 (ci, j − vm,1 −1 )2 + v2m,1 −1

Iterative Methods for Integral Problems



vm, = um,+1 − uk, , vk , + hk2 ,2 vk2 ,2 = 2 2 , 2

87

hm, = um+1, − um, , vk , − hk2 ,2 hk2 ,2 = 2 2 . 2

(12)

Clearly since each iteration would take O(n2 ) per block on any level, the overall algorithm will have O(n4 ) at least and is hence not optimal. A new breakthrough on the issue was made in [ChCh10], based on the reorganizing the solution of the above coarse level subproblems. We first observe that the first order condition of (11) takes the form

α T  (ci, j ) + wtT wt ci, j = wtT  z,

(13)

z (for all wt recursively as one deals with partial sums in the fast where wtT wt , wtT  multipole method [BrGr97]) can be worked out, though of complexity O(n2 ), once only. Then we anticipate that after all such quantities are computed and stored first before each multilevel cycle, the local solvers will not be expensive to proceed. Finally our new multilevel method for the combined denoising and deblurring problem for solving (8) is the following: Algorithm 3 Given z and an initial guess u = z, with L + 1 levels, Pre-calculation. 1. Compute all root matrices Tk and wtT wt = Tk 2F for partial sum matrices on level k = 1, 2, . . . , L + 1. Multilevel Iterations. 2. Iteration starts with uold = u ( u contains the initial guess before the first iteration and the updated solution at all later iterations). for ν times on each level k = 1, 2, 3, . . . , L + 1: 3. Compute  z = z − K u and form K T  z via the FFT. for each block on level k, z from K T  z and compute the minimizer c of (13). 4. form each wtT  end block. 5. add all the corrections (from all blocks on level k), u = u + Pk c, where Pk is the interpolation operator distributing ci, j to the corresponding b × b block on level k. end level k. 6. On level k = 1, check the possible patch size for each position (i, j):   patch = (i , j ) : |ui , j − ui, j | < ε for some small ε . First compute the partial sum vector wt related the detected columns. Then implement the piecewise constant update as with Steps 3–5. 7. If  u − uold 2 is small enough, exit with u = u or return to Step 2 and continue with the next multilevel cycle. Two illustrating examples solved by Algorithm 3 are shown in Figs. 2 and 3.

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Fig. 2 Image deblurring example 1 by Algorithm 3: left z and right u

Fig. 3 Image deblurring example 2 by Algorithm 3

4 Open Problems and Challenges In this short paper, we discussed two problems solved by integral methods. Both problems may be studied further. Firstly, for the Helmholtz equation, an optimal way of combining the FMM and wavelets with suitable preconditioning is still to be found out. The more recent interests of many researchers have turned to high frequency modeling (where the wavenumber k is large) and inverse problems where the boundary information (either boundary conditions or the boundary itself) is missing while some measurement of the solution is known. Secondly, for the image deblurring problem, it remains to develop effective iterative methods for cases where a more sophisticated choice of regularizers (e.g. the mean curvature [BrCh10]) is used or a spatially variant blur kernel is used. A more practical problem is the blind deblurring, where the kernel k is not known and must be estimated together with u. See [ChWo98, MoKa08].

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Acknowledgements Most of the results reviewed in this paper were achieved through joint work with collaborators including Paul J. Harris (University of Brighton), Siamiak Amini (University of Salford), Tony F. Chan (University of California at LA) and Raymond H. Chan (Chinese University of Hong Kong).

References [AmPr99] [AmHaWi92]

[Ba08] [BrGr97]

[BrCh10] [ChCh10] [ChCh02] [ChSh05] [ChWo98] [Ch05] [Ci00] [Da97] [Fo09] [HaCh05] [HuNgWe08] [MaSc07] [MoKa08] [NgBo03] [RuOsFa92] [SaSc96]

Amini, S., Profit, A.J.: Analysis of a diagonal form of the fast multipole algorithm for scattering theory. BIT, 39, 585–602 (1999). Amini, S., Harris, P.J., Wilton, D.T.: Coupled Boundary and Finite Element Methods for the Solution of the Dynamic Fluid-Structure Interaction Problem, Lecture Note in Engineering, 77, C.A. Brebbia and S.A. Orszag, eds., Springer, London (1992). Banjai, L., Hackbusch, W.: Hierarchical matrix techniques for low and high frequency Helmholtz equation. IMA J. Numer. Anal., 28, 46–79 (2008). Beatson, R., Greengard, L.: A short course on fast multipole methods, in Wavelets, Multilevel Methods and Elliptic PDEs, Oxford University Press, 1–37 (1997). See also http://math.nyu.edu/faculty/greengar/shortcourse_fmm.pdf. Brito, C., Chen, K.: Multigrid algorithm for high order denoising. SIAM J. Imaging Sci., 3, 363–389 (2010). Chan, R.H., Chen, K.: A multilevel algorithm for simultaneously denoising and deblurring images. SIAM J. Sci. Comput., 32, 1043–1063 (2010). Chan, T.F., Chen, K.: On two variants of an algebraic wavelet preconditioner. SIAM J. Sci. Comput., 24, 260–283 (2002). Chan, T.F., Shen, J.H.: Image Processing and Analysis – Variational, PDE, Wavelet, and Stochastic Methods, SIAM Publications (2005). Chan, T.F., Wong, C.K.: Total variation blind deconvolution. IEEE Trans. Image Process, 7, 370–375 (1998). Chen, K.: Matrix Preconditioning Techniques and Applications, Cambridge University Press, Cambridge (2005). Cipra, B.A.: The Best of the 20th Century: Editors Name Top 10 Algorithms. SIAM News, 33 (2000). Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numerica, 1997, 55–228 (1997). Fong, W., Darve, E.: The black-box fast multipole method. J. Comp. Phys., 228, 8712–8725 (2009). Hawkins, S., Chen, K.: An implicit wavelet approximate inverse preconditioner. SIAM J. Sci. Comput., 27, 667–686 (2005). Huang, Y., Ng, M., Wen, Y.: A fast total variation minimization method for image restoration. SIAM J. Multiscale Modeling and Simulation, 7, 774–795 (2008). Mazya, V., Schmidt, G.: Approximate Approximations, American Mathematical Society, Providence, RI (2007). Money, J.H., Kang, S.H.: Total variation minimizing blind deconvolution with shock filter. Image Vision Comput., 26, 302–314 (2008). Ng, M., Bose, N.K.: Mathematical analysis of super-resolution methodology. IEEE Signal Proc. Magazine, 20, 62–74 (2003). Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D, 60, 259–268 (1992). Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7, 856–869 (1986).

90 [TrOoSc2001] [Vo02]

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Trottenberg, U., Oosterlee, C., Schuller, A.: Multigrid, Academic Press (2001). Vogel, C.R.: Computational Methods for Inverse Problems, SIAM Publications, Philadelphia, PA (2002). [WaYaYihZ08] Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci., 1, 248–272 (2008).

Analysis of Some Localized Boundary–Domain Integral Equations for Transmission Problems with Variable Coefficients O. Chkadua, S.E. Mikhailov, and D. Natroshvili

1 Introduction We consider the basic and mixed transmission problems for scalar second order elliptic partial differential equations with variable coefficients and use the localized parametrices to reduce the problems to direct segregated boundary–domain integral equations. The treatment, by variational methods, of the transmission problems considered in this paper have been investigated in the research literature, and the corresponding uniqueness and existence results are well known (see, e.g., [HW08, LiMa72]). For the special cases, when the fundamental solution is available, the Dirichletand Neumann-type boundary value problems have also been investigated by the classical potential method (see [Mir70, HW08] and the references therein). Our goal here is to show that the problems can be equivalently reduced to some localized boundary–domain integral equations (LBDIEs) and the corresponding localized boundary–domain integral operators (LBDIOs) are invertible which, beside a pure mathematical interest, may also have some applications in numerical analysis for construction of efficient numerical algorithms (see, e.g., [Mik02, MN05, SSA00, ZZA98, ZZA99] and the references therein). In our case, the localized parametrix is represented as the product of a Levi function of the differential operator under consideration and an appropriately chosen O. Chkadua A.Razmadze Mathematical Institute, Tbilisi, Georgia, e-mail: [email protected] S.E. Mikhailov Brunel University West London, UK, e-mail: [email protected] D. Natroshvili Georgian Technical University, Tbilisi, Georgia, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_10, © Springer Science+Business Media, LLC 2011

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localizing function, e.g., a function supported on some neighborhood of the singularity point of the Levi function. Although the kernels of the corresponding localized potentials do not solve the original PDEs, the localized potentials preserve almost all mapping properties of the usual non-localized ones (cf. [CMN09-1, Mik06, CMN11]). However, some unusual properties of the localized potentials appear due to the localization of the kernel functions which have no counterparts in classical potential theory and which need special consideration and analysis. By the direct approach based on Green’s representation formula, we reduce the Dirichlet and mixed transmission problems to the LBDIE system. First we establish the equivalence between the original transmission problems and the corresponding LBDIE systems, which proved to be a quite nontrivial problem and plays a crucial role in our analysis. Afterwards we investigate the Fredholm properties of the LBDIOs and prove their invertibility in appropriate function spaces. In this paper we present analysis for a wider classes of the localizing functions than in [CMN09-L].

2 Reduction to Localized Boundary–Domain Integral Equations 2.1 Formulation of the Interface Problems Let Ω and Ω1 be bounded open domains in R3 , Ω 1 ⊂ Ω and Ω2 := Ω \ Ω 1 . We assume that the interface surface Si = ∂ Ω1 and the exterior boundary Se = ∂ Ω of the composite body Ω = Ω 1 ∪ Ω 2 are sufficiently smooth, say C∞ -regular if not otherwise stated. Clearly, ∂ Ω2 = Si ∪ Se . Throughout the paper n(q) = n(q) (x) denotes the unit normal vector to ∂ Ωq directed outward from the corresponding domain Ωq . Clearly, n(1) (x) = −n(2) (x) for x ∈ Si .

By H r (Ω  ) = H2r (Ω  ) and H r (S) = H2r (S), r ∈ R, we denote the Bessel potential spaces on a domain Ω  and on a closed manifold S without boundary. The subspace r (R3 ). Recall that of H r (R3 ) of functions with compact support is denoted by Hcomp H 0 (Ω  ) = L2 (Ω  ) is a space of square integrable functions in Ω .

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 r (M ) the subspace of H r (S), For a smooth sub-manifold M ⊂ S we denote by H r r  H (M ) := {g : g ∈ H (M ), supp g ⊂ M }, while H r (M ) denotes the spaces of restrictions on M of functions from H r (S), H r (M ) := {rM f : f ∈ H r (S)}, where rM is the restriction operator onto M . Let us consider the differential operators in the domains Ωq Aq (x, ∂x ) u(x) :=

3

∑ ∂xk [ aq(x) ∂xk u(x)],

q = 1, 2,

k=1

where ∂x = (∂1 , ∂2 , ∂3 ), ∂k = ∂xk = ∂ /∂ xk , k = 1, 2, 3, and aq ∈ C∞ (R3 ), 0 < c0 ≤ aq (x) ≤ c1 < ∞, q = 1, 2 . Further, for sufficiently smooth functions (from the space H 2 (Ωq ), say) we introduce the co-normal derivative operator on ∂ Ωq , q = 1, 2, in the usual trace sense: Tq± (x, ∂x ) u(x) :=

3

∑ aq (x) nk

(q)

(x) γq± [∂xk u(x)],

(1)

k=1

where x ∈ ∂ Ωq and the symbols γq+ and γq− denote the trace operators on ∂ Ωq from the domain Ωq+ := Ωq and its complement Ωq− := R3 \ Ω q , respectively. We set H 1, 0 (Ωq± ; Aq ) := {u ∈ H 1 (Ωq± ) : Aq u ∈ H 0 (Ωq± )}, q = 1, 2. The classical co-normal derivative operators given by (1) can be continuously extended to functions from the spaces H 1, 0 (Ωq± ; Aq ) by the (generalized) canonical 1

co-normal derivative operators Tq± : H 1, 0 (Ωq± ; Aq ) → H − 2 (∂ Ωq ) (cf., for example, [Co88, Lemma 3.2], [McL00, Lemma 4.3]) defined as 



Tq± u , w

∂ Ωq

:= ±

 

 ± (± q w) Aq u + Eq (u, q w) dx

(2)

Ωq

1

± for all w ∈ H 2 (∂ Ωq ). Here ± q are continuous linear extension operators, q : 1

H 2 (∂ Ωq ) → H 1 (Ωq± ) which are the right inverse of the trace operators γq± , while Eq (u, v) := aq (x) ∇x u · ∇x v,

∇x := (∂1 , ∂2 , ∂3 ) ,

and the central dot denotes the scalar product in R3 . The symbol g1 , g2 ∂ Ωq in (2) 1

1

− 2 (∂ Ω ) and H 2 (∂ Ω ), coinciddenotes the q q  duality brackets between the spaces H ing with ∂ Ωq g1 (x) g2 (x)dS if g1 , g2 ∈ L2 (∂ Ωq ). Below for these type dualities we will sometimes use the usual integral symbol when this does not lead to confusion. We will also employ the shorter notations γq ≡ γq+ , Tq ≡ Tq+ . We now formulate the Dirichlet and mixed transmission problems:

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Find functions u1 ∈ H 1,0 (Ω1 ; A1 ) and u2 ∈ H 1,0 (Ω2 ; A2 ) satisfying the differential equations (3) Aq (x, ∂x ) uq = fq in Ωq , q = 1, 2, the transmission conditions on the interface

γ1 u1 − γ2 u2 = ϕ0i on Si , T1 u1 + T2 u2 = ψ0i on Si ,

(4) (5)

and one of the following conditions on the exterior boundary: the Dirichlet boundary condition

γ2 u2 = ϕ0e

on Se ,

(6)

or the mixed-type boundary conditions (M)

on SeD ,

(7)

(M) ψ0e

on SeN ,

(8)

γ2 u2 = ϕ0e T2 u2 =

where SeD and SeN are smooth disjoint sub-manifolds of Se : Se = SeD ∪ SeN ,

SeD ∩ SeN = ∅.

We will refer these boundary transmission problems as (TD) and (TM) problems, respectively. For the data in the above formulated problems we assume 1

1

1

1

ϕ0i ∈ H 2 (Si ), ψ0i ∈ H − 2 (Si ), ϕ0e ∈ H 2 (Se ), ψ0e ∈ H − 2 (Se ), 1

1

ϕ0e ∈ H 2 (SeD ), ψ0e ∈ H − 2 (SeN ), (M)

(M)

f q ∈ H 0 (Ωq ), q = 1, 2. Equations (3) are understood in the distributional sense, the Dirichlet-type boundary and transmission conditions are understood in the usual trace sense, while the Neumann-type conditions for the co-normal derivatives are understood in the sense of the canonical co-normal derivatives defined by (2). We recall that the normal vectors n(1) and n(2) in the definitions of the co-normal derivatives T1 u and T2 u on Si have opposite directions. As we mentioned in the introduction, all the transmission problems formulated above have been investigated in the literature using the variational methods and the corresponding uniqueness and existence results are well known (see, e.g., [LiMa72]). Our goal here is to show that the problems can be equivalently reduced to some LBDIEs and to investigate the Fredholm and invertibility properties of the corresponding LBDIOs.

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2.2 Properties of Localized Potentials It is well known that the function Pq1 (x, y) = −

1 4 π aq (y) |x − y|

is a Levi function for the operator Aq (x, ∂x ) (cf. [CMN09-1]). Now we introduce the localized parametrix (localized Levi function) for the operator Aq , Pq (x, y) ≡ Pqχ (x, y) := χ (x − y) Pq1 (x, y), q = 1, 2, where χ is a localizing function (see Appendix A)

χ (x) = χ˘ ( |x| ),

k χ ∈ X1∗ , k ≥ 3.

One can easily check the following relation [CMN09-L], Aq (x, ∂x ) Pq (x, y) = δ (x − y) + Rq (x, y),

q = 1, 2,

where δ (·) is the Dirac distribution and

3 ∂ aq (x) χ (x−y) Rq (x, y) = Rq χ (x, y) = − 4π a1q (y) ∑ − ∂∂y ∂xj |x−y| j j=1  ∂ χ (x−y) ∂ 1 1 +aq (x) ∂ χ∂(x−y) x |x−y| + aq (x) ∂ x ∂ x |x−y| . j

j

j

The function Rq (x, y) possesses a weak singularity of type O(|x − y|−2 ) as x → y if χ is smooth enough, e.g., if χ ∈ X 2 . Let us introduce the localized volume potentials for y ∈ R3 , 

Pq f (y) := Rq f (y) :=

Ωq



Ωq

Pq (x, y) f (x) dx, (9) Rq (x, y) f (x) dx,

and the surface potentials for y ∈ R3 \S, VS(q) g(y) := −



WS(q) g(y) := −

Pq (x, y) g(x) dSx ,

S  S

 Tq (x, ∂x ) Pq (x, y) g(x) dSx ,

based on the localized parametrices Pq . Here S ∈ {Si , Se , ∂ Ω2 }. Note that for the layer potentials we will drop the subindex S when S = ∂ Ωq , i.e.,

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V (q) := V∂ Ωq ,

(q)

W (q) := W∂ Ωq .

If the domain of integration in (9) is replaced with the whole space Ωq = R3 , we employ the notation Pq f = Pq f . Let us also define the corresponding boundary operators generated by the direct values of the localized single and double layer potentials and their co-normal derivatives for y ∈ S VS g(y) := −



(q)

WS(q) g(y) := −

Pq (x, y) g(x) dSx ,

S 

WS  (q) g(y) := −

 Tq (x, ∂x )Pq (x, y) g(x) dSx ,

S 

 Tq (y, ∂y ) Pq (x, y) g(x) dSx ,

S LS±(q) g(y) := rS Tq± (y, ∂y )WS(q) g(y).

(10) (11) (12) (13)

For the pseudodifferential operator (13), we employ also the notation LS(q) := LS+(q) . Note that the kernel functions of the operators (11) and (12) are at most weakly singular if the localizing function χ ∈ X 2 and the surface S is C1,β smooth with β > 0. The mapping properties of the operators (9)–(13) are studied in [CMN09-L]. Further on we assume that the following relation holds on the interface: a2 (x) = κ a1 (x) for x ∈ Si ,

κ = const > 0.

(14)

Finally, we present some auxiliary propositions which play a crucial role in our analysis and which can be proved by extending the arguments similar to those ap3 to plied in the proof of Lemmas 6.3 and 6.4 in [CMN09-L] from the case χ ∈ X1+ 3 . the case χ ∈ X1∗ 3 , and let condition (14) hold. Further, let G ∈ H 0 (Ω ), g ∈ Lemma 1. Let χ ∈ X1∗ q q 1 1 1 − 21 H (Si ), g2 ∈ H 2 (Si ), ge ∈ H − 2 (Se ) and

VS(1) (g1 ) +WS(1) (g2 ) + P1 (G1 ) = 0 in Ω1 , i

i

(ge )+P2 (G2 ) = 0 in Ω2 . VS(2) (g1 )−WS(2) (g2 )+VS(2) e i

i

Then Gq = 0 in Ωq , q = 1, 2, g1 = 0, g2 = 0 on Si , and ge = 0 on Se . 3 , and let condition (14) hold. Further, let G ∈ H 0 (Ω ), g ∈ Lemma 2. Let χ ∈ X1∗ q q 1 1 1  − 12 (SeD ), geN ∈ H  12 (SeN ), and H − 2 (Si ), g2 ∈ H 2 (Si ), geD ∈ H

VS(1) (g1 ) +WS(1) (g2 ) + P1 (G1 ) = 0 in Ω1 , i

i

(geD ) +WS(2) (geN ) + P2 (G2 ) = 0 in Ω2 . VS(2) (g1 ) −WS(2) (g2 ) +VS(2) e e i

i

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97

Then Gq = 0 in Ωq , q = 1, 2, g1 = 0 and g2 = 0 on Si , geD = 0 and geN = 0 on Se .

2.3 Basic LBDIE Relations The Second Green’s identity holds for the operator Aq (x, ∂x ) and u, v ∈ H 1, 0 (Ωq ; Aq ), see, e.g., [Co88, Lemma 3.2], [McL00, Lemma 4.3], 

[v Aq u − u Aq v] dx =

Ωq



[(γq v)Tq u − (γq u)Tq v] dS, q = 1, 2.

∂ Ωq

By the standard limiting procedure near the singular point of the parametrix (see, e.g., [Mir70]), we obtain the following parametrix-based third Green’s identity for arbitrary u = uq ∈ H 1, 0 (Ωq ; Aq ), uq + Rq uq −V (q) Tq uq +W (q) γq uq = Pq Aq uq in Ωq .

(15)

Recall that for layer potentials we drop the subindex S when S = ∂ Ωq . Bearing in mind the properties of the localized potentials given in Appendix B, the trace and co-normal derivative of (15) are given by 1 γq uq +γq Rq uq −V (q) Tq uq +W (q) γq uq = γq Pq Aq uq on ∂ Ωq , 2

(16)

1 Tq uq +Tq Rq uq −W 2

(17)

 (q)

Tq uq +L (q) γq uq = Tq Pq Aq uq on ∂ Ωq .

With the help of these relations one can construct various types of LBDIE systems for the above formulated transmission BVPs.

3 LBDIES for the Dirichlet Transmission Problem Let a pair (u1 , u2 ) ∈ H 1, 0 (Ω1 ; A1 ) × H 1, 0 (Ω2 ; A2 ) be a solution to the transmission Dirichlet problem (3)–(6), i.e., Problem (TD). Assume that the problem right hand sides satisfy the embeddings 1

1

1

ϕ0i ∈ H 2 (Si ), ψ0i ∈ H − 2 (Si ), ϕ0e ∈ H 2 (Se ), fq ∈ H 0 (Ωq ).

(18)

Let us introduce the following combinations of the unknown functions: 1 ψi = rSi (T1 u1 − T2 u2 ), 2 1

1 ϕi = rSi (γ1 u1 + γ2 u2 ), 2 1

1

ψe = rSe T2 u2 .

Then evidently ψi ∈ H − 2 (Si ), ϕi ∈ H 2 (Si ), ψe ∈ H − 2 (Se ).

(19)

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Let us introduce the five-vector function (column matrix function) U (T D) := (u1 , u2 , ψi , ϕi , ψe ) ∈ H(T D) ,

(20)

where 1

1

1

H(T D) := H 1, 0 (Ω1 ; A1 ) × H 1, 0 (Ω2 ; A2 ) × H − 2 (Si ) × H 2 (Si ) × H − 2 (Se ),

(21)

and consider formally the components of U (T D) as unrelated to each other (i.e., segregated). Further, let us employ the third Green identities (15) in Ω1 and Ω2 , the difference of their traces (16) and the sum of their co-normal derivatives (17) on Si , and also the trace (16) on Se . Then after substituting transmission and boundary conditions (4)–(6) and notations (19) we arrive at the following system of direct segregated LBDIEs for the components of the vector function U (T D) = (u1 , u2 , ψi , ϕi , ψe ) , (T D)

u1 + R1 u1 −VS(1) ψi +WS(1) ϕi = F1 i

i

in Ω1 ,

(22)

(T D)

u2 + R2 u2 +VS(2) ψi +WS(2) ϕi −VS(2) ψe = F2 e i

i

in Ω2 ,

(23)

γ1 R1 u1 − γ2 R2 u2 − (VS(1) + VS(2) )ψi + (WS(1) − WS(2) )ϕi + γ2VS(2) ψe e i

i

i

=

i

(T D) (T D) γ1 F1 − γ2 F2 − ϕ0i

on Si ,

(24)

T1 R1 u1 + T2 R2 u2 − (WS  (1) − WS  (2) )ψi + (LS(1) + LS(2) )ϕi − T2VS(2) ψe e i

i

i

(T D)

= T1 F1

i

(T D)

+ T2 F2

− ψ0i on Si ,

(T D) γ2 R2 u2 +γ2VS(2) ψi +γ2WS(2) ϕi −VS(2) ψe = γ2 F2 −ϕ0e e i i

(25)

on Se ,

(26)

where (T D)

F1

(T D)

F2

1 1 = P1 f1 + VS(1) ψ0i − WS(1) ϕ0i , 2 i 2 i 1 1 = P2 f2 + VS(2) ψ0i + WS(2) ϕ0i −WS(2) ϕ0e . e 2 i 2 i

If we introduce the notation K ⎡

I + rΩ R1 1

0

(T D)

(T D)

= [Kk j

−rΩ VS(1) 1

i

]5×5 := rΩ WS(1) 1

i

0

⎤

⎢ ⎥ ⎢ ⎥ rΩ VS(2) rΩ WS(2) −rΩ VS(2) 0 I + r Ω R2 ⎢ ⎥ e 2 2 2 2 i i ⎢ ⎥ ⎢ rS γ1 R1 −rS γ2 R2 −V (1) − V (2) W (1) − W (2) rS γ2V (2) ⎥ , Si Si Si Si Se ⎢ i ⎥ i i ⎢ ⎥ ⎢ r T1 R1 r T2 R2 −W  (1) + W  (2) L (1) + L (2) −r T2V (2) ⎥ Si Si Si Si Si Si Se ⎦ ⎣ Si 0 rSe γ2 R2 rSe γ2VS(2) rSe γ2WS(2) −VS(2) e i

i

Localized BDIEs for Transmission Problems

99

the LBDIEs system (22)–(26) can be rewritten as K

(T D)

U (T D) = F (T D) ,

(27)

where U (T D) ∈ H(T D) is the unknown vector, while F (T D) ∈ F(T D) is the known vector generated by the right hand side functions in (22)–(26) and 1

1

1

F(T D) := H 1, 0 (Ω1 ; A1 ) × H 1, 0 (Ω2 ; A2 ) × H 2 (Si ) × H − 2 (Si ) × H 2 (Se ). The following equivalence theorem holds. 3. Theorem 1. Let conditions (18) hold and χ ∈ X1∗ 1, 0 1, 0 (i) If a pair (u1 , u2 ) ∈ H (Ω1 ; A1 ) × H (Ω2 ; A2 ) solves the Problem (TD), then the five-vector U (T D) ∈ H(T D) given by (20), where ψi , ϕi and ψe are defined by (19), solves LBDIEs system (22)–(26). (ii) Vice versa, if a five-vector U (T D) ∈ H(T D) solves LBDIEs system (22)–(26) and condition (14) holds, then (u1 , u2 ) ∈ H 1, 0 (Ω1 ; A1 ) × H 1, 0 (Ω2 ; A2 ) solves Problem (TD) and relations (19) hold.

Proof. Claim (i) immediately follows from the deduction of (22)–(26). Now, let a five-vector U (T D) ∈ H(T D) solve LBDIEs system (22)–(26). Subtracting from (24) the trace γ1 of (22) and adding the trace γ2 of (23), we prove (4). Similarly, subtracting from (25) the co-normal derivative T1 of (22) and the co-normal derivative T2 of (23), we prove (5). At last, subtracting from (26) the trace γ2 of (23), we prove (6). That is, the transmission conditions on Si and the Dirichlet boundary condition on Se are fulfilled. It remains to show that uq solves the differential equation (3) and that the conditions (19) hold true. Due to the embedding U (T D) ∈ H(T D) , the third Green identities (15) hold. Comparing these identities with the first two equations of the LBDIEs system, (22) and (23), and taking into account transmission conditions (4)–(5) and the Dirichlet boundary condition (6), already satisfied, we arrive at the relations  (1)

VS

i

 VS(2) i

   1 1 (1) [T1 u1 − T2 u2 ] − ψi +WS ϕi − [γ1 u1 + γ2 u2 ] i 2 2 1 [T1 u1 − T2 u2 ] − ψi 2



= P1 ( f1 − A1 u1 ) in Ω1 ,   1 (2) ϕi − [γ1 u1 + γ2 u2 ] −WS i 2 +VS(2) (ψe − T2 u2 ) = P2 (A2 u2 − f2 ) in Ω2 . e

Whence by Lemma 1 we conclude that conditions (19) are satisfied and A1 u1 − f 1 = 0 in Ω1 , This completes the proof.

A2 u2 − f 2 = 0 in Ω2 .

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Due to this equivalence theorem we conclude that the LBDIE system (22)–(26) with the special right hand side which belongs to the space F(T D) is uniquely solvable in the space H(T D) defined by (21). In particular, the corresponding homogeneous LBDIEs system possesses only the trivial solution. By the way, one can easily check that the right hand side expressions in LBDIEs system (22)–(26) vanish if fq = 0 in Ωq , q = 1, 2, ϕ0i = 0 and ψ0i = 0 on Si , and ϕ0e = 0 on Se . Our next aim is to establish the invertibility of the matrix operator generated by the left hand side expressions in the LBDIEs system (22)–(26) both in already introduced and in wider function spaces. Let us introduce the notations 1

1

1

X(T D) := H 1 (Ω1 ) × H 1 (Ω2 ) × H − 2 (Si ) × H 2 (Si ) × H − 2 (Se ) , 1

1

1

Y(T D) := H 1 (Ω1 ) × H 1 (Ω2 ) × H 2 (Si ) × H − 2 (Si ) × H 2 (Se ) . Evidently H(T D) ⊂ X(T D) and F(T D) ⊂ Y(T D) . Due to Theorems 6 and 7 in Appendix B the following operators are bounded if χ ∈ X 3, K

(T D)

: H(T D) → F(T D) : X(T D) → Y(T D) .

(28) (29)

3 Theorem 2. Let χ ∈ X1∗ , and let condition (14) hold. Then the operators (28) and (29) are invertible.

Proof. We can easily see that the upper triangular matrix operator ⎡

I 0

⎢ ⎢0 ⎢ ⎢ (T D) := ⎢ K0 ⎢0 ⎢ ⎢0 ⎣ 0

−rΩ1 VS(1)

rΩ WS(1)

rΩ VS(2)

rΩ WS(2)

i

I

2

i

1

0

i

2

i

0 −VS(1) − VS(2)

WS(1) − WS(2)

0

0

L (1) + L (2)

0

0

0

i

i

i

i

Si

Si

possesses the same mapping properties as the operator K (T D)

K0

: X(T D) → Y(T D) ,

⎤

⎥ ⎥ −rΩ VS(2) ⎥ e 2 ⎥ ⎥ rSi γ2VS(2) ⎥ e ⎥ (2) −rSi T2VSe ⎥ ⎦

(30)

−VS(2) e (T D) ,

(31)

and by Lemma 4 in Appendix B the operator (31) is a compact perturbation of the operator (29). 1 1 For q = 1, 2, the operators VS(q) : H − 2 (S) → H 2 (S) are strongly elliptic pseudodifferential operators of order −1 with strictly positive principal homogeneous sym1 1 bols [ 2 aq (y) |ξ  | ]−1 for ξ  ∈ R2 \ {0} and y ∈ S, while LS(q) : H 2 (S) → H − 2 (S) are strongly elliptic pseudodifferential operators of order 1 with strictly negative principal homogeneous symbols − 12 aq (y) |ξ  | for ξ  ∈ R2 \ {0} and y ∈ S. Therefore

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by standard arguments it can be shown that the operators in the main diagonal in (30) are Fredholm of index zero in the corresponding function spaces. Therefore the operator (29) is also Fredholm with index zero. It remains to show that the null space of the operator (29) is trivial. We proceed as follows. Let U ∈ X(T D) be a solution to the homogeneous system K (T D)U = 0. Then the first two equations of the system imply that U ∈ H(T D) due Theorems 6 and 7, and by the equivalence Theorem 1 we conclude U = 0. Thus the kernel of the operator (29) is trivial and consequently (29) is invertible. To prove invertibility of operator (28), we remark that for any F (T D) ∈ F(T D) a unique solution U ∈ X(T D) of (27) is delivered by the inverse of the operator (29). On the other hand, since F (T D) ∈ F(T D) , the first two lines of the matrix operator K (T D) imply that in fact U ∈ H(T D) and the mapping F(T D) → H(T D) delivered by the inverse of the operator (29) is continuous, i.e., this operator is inverse to operator (28).

4 The Mixed Transmission Problem (TM) Let us consider the mixed-type transmission problem (3), (4), (5), (7), (8), with the right hand sides 1

ϕ0i ∈ H 2 (Si ), 1

1

ψ0i ∈ H − 2 (Si ), 1

ϕ0e ∈ H 2 (SeD ), ψ0e ∈ H − 2 (SeN ), (M)

(M)

1

fq ∈ H 0 (Ωq ), q = 1, 2.

(32)

1

Let us denote by Φ0e ∈ H 2 (Se ) and Ψ0e ∈ H − 2 (Se ) some fixed extensions of the (M) (M) boundary functions ϕ0e and ψ0e from SeD and SeN , respectively, onto the whole (M)

(M)

surface Se , preserving the space. Then rSeD Φ0e = ϕ0e , rSeN Ψ0e = ψ0e . Similar to (19) for the Problem (TD), let us introduce the following combinations of the unknown boundary functions: 1 1 1 1 ψi = (T1 u1 − T2 u2 ) ∈ H − 2 (Si ), ϕi = (γ1 u1 + γ2 u2 ) ∈ H 2 (Si ), 2 2  − 21 (SeD ),  12 (SeN ). ψe = T2 u2 − Ψ0e ∈ H ϕe = γ2 u2 − Φ0e ∈ H

(33)

Further, let us set U (T M) :=(u1 , u2 , ψi , ϕi , ψe , ϕe ) ∈ H(T M) , H

(T M)

:=H

1, 0

(Ω1 ; A1 ) × H

1, 0

(Ω 2 ; A 2 ) × H

(34) − 12

1 2

(Si ) × H (Si )

 − 12 (SeD ) × H  12 (SeN ), ×H and we consider again the components of the vector U (T M) as formally unrelated.

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Let us employ the third Green identities (15) in Ω1 and Ω2 , the difference of their traces (16) and the sum of their co-normal derivatives (17) on Si ; and also the trace (16) on SeD and the co-normal derivative (17) on SeN . Then after substituting the transmission conditions (4)–(5) and the mixed boundary conditions (7)–(8) we arrive at the following system of direct segregated LBDIEs for the components of the vector U (T M) , (T M)

u1 + R1 u1 −VS(1) ψi +WS(1) ϕi = F1 i

i

in Ω1 ,

(35) (T M)

ψe +WS(2) ϕe = F2 u2 +R2 u2 +VS(2) ψi +WS(2) ϕi −VS(2) e e i

i

in Ω2 ,

(36)

γ1 R1 u1 − γ2 R2 u2 − (VS(1) + VS(2) )ψi + (WS(1) − WS(2) )ϕi i

i

i

i

(T M) (T M) +γ2VSe ψe −γ2WSe ϕe = γ1 F1 −γ2 F2 −ϕ0i (2)

(2)

on Si ,

(37)

T1 R1 u1 + T2 R2 u2 − (WS  (1) − WS  (2) )ψi + (LS(1) + LS(2) )ϕi i

i

i

i

(T M) (T M) −T2VSe ψe + T2WSe ϕe = T1 F1 +T2 F2 −ψ0i (2)

(2)

on Si ,

(38)

γ2 R2 u2 +γ2VS(2) ψi +γ2WS(2) ϕi −VS(2) ψe +WS(2) ϕe e e i

i

(T M)

= γ2 F2

− ϕ0e on SeD ,

(39)

T2 R2 u2 +T2VS(2) ψi +T2WS(2) ϕi −WSe (2) ψe +LS(2) ϕe e i

i

(T M)

= T2 F2

− ψ0e on SeN ,

(40)

where 1 1 = P1 f1 + VS(1) ψ0i − WS(1) ϕ0i , 2 i 2 i 1 1 (T M) = P2 f2 + VS(2) ψ0i + WS(2) ϕ0i +VS(2) Ψ0e −WS(2) Φ0e . F2 e e 2 i 2 i (T M)

F1

(41) (42)

As in the case of the problem (TD), we have here the following equivalence theorem. 1

3 and conditions (32) hold. Further, let Φ ∈ H 2 (S ) and Theorem 3. Let χ ∈ X1∗ 0e e 1 (M) (M) − Ψ0e ∈ H 2 (Se ) be some fixed extensions of the boundary functions ϕ0e and ψ0e from SeD and SeN , respectively, onto the whole surface Se . (i) If a pair (u1 , u2 ) ∈ H 1, 0 (Ω1 ; A1 )×H 1, 0 (Ω2 ; A2 ) solves the transmission mixed problem (TM), then the six-vector U (T M) ∈ H(T M) given by (34), where ψi , ϕi , ψe and ϕe are defined by (33), solves the LBDIEs system (35)–(42). (ii) Vice versa, if a six-vector U (T M) ∈ H(T M) solves the LBDIEs system (35)– (42) and condition (14) holds, then the pair (u1 , u2 ) solves the Problem (TM) and the relations (33) hold.

Proof. The claim (i) immediately follows from the deduction of (35)–(42). Now, let a six-vector U (T M) solve the LBDIEs system (35)–(42). Subtracting from (37) the trace γ1 of (35) and adding the trace γ2 of (36), we prove (4). Simi-

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larly, subtracting from (38) the co-normal derivative T1 of (35) and the co-normal derivative T2 of (36), we prove (5). Subtracting from (39) the trace γ2 of (36), we prove (7). Similarly, subtracting from (40) the co-normal derivative T2 of (36), we prove (8). That is, the transmission conditions on Si and the mixed boundary conditions on Se are fulfilled. It remains to show that the (3) and the relations (33) hold true. Due to the embedding U (T M) ∈ H(T M) , the third Green identities (15) hold. Comparing these identities with the first two equations of the LBDIEs system, (35) and (36), and taking into account transmission conditions (4)–(5) and mixed boundary conditions (7)–(8), already satisfied, we arrive at the relations    1 1 [T1 u1 − T2 u2 ] − ψi +WS(1) ϕi − [γ1 u1 + γ2 u2 ] i i 2 2 = P1 ( f1 − A1 u1 ) in Ω1 ,     1 1 [T1 u1 − T2 u2 ] − ψi −WS(2) ϕi − [γ1 u1 + γ2 u2 ] VS(2) i i 2 2 

VS(1)

+VS(2) (−T2 u2 + ψe + Ψ0e ) +WS(2) (γ2 u2 − ϕe − Φ0e ) e e = P2 (A2 u2 − f2 ) in Ω2 . Whence by Lemma 2 we conclude that (3) and (33) are satisfied. (T M)

Denote by K (T M) = [Kk j ]6×6 the localized boundary–domain 6 × 6 matrix integral operator generated by the left hand side of the expression in (35)–(40) and set 1

1

F(T M) := H 1, 0 (Ω1 ; A1 ) × H 1, 0 (Ω2 ; A2 ) × H 2 (Si ) × H − 2 (Si ) 1

1

× H 2 (SeD ) × H − 2 (SeN ) .

(43)

Then the LBDIEs system (35)–(40) is written in matrix form as K

(T M)

U (T M) = F (T M) ,

(44)

where U (T M) is the unknown six-vector function (34), while F (T M) ∈ F(T M) is the known vector function compiled by the right hand side functions in (35)–(40). From Theorem 3 it follows that the LBDIEs system (35)–(40), i.e., (44), is uniquely solvable in the space H(T M) for the special right hand side vector function, which belongs to the space F(T M) defined by (43). One can easily check that the right hand side expressions in LBDIEs system (35)–(40) vanish if f q = 0 in Ωq , q = 1, 2, f1 = 0 and ψ0i = 0 on Si , Φ0e = 0 and Ψ0e = 0 on Se . Now we establish that the operator given by the left hand side of (44) is continuously invertible as an operator both in the function spaces already introduced and in wider function spaces. To this end let us consider the operators K

(T M)

: H(T M) → F(T M) ,

(45)

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: X(T M) → Y(T M) ,

(46)

where 1 1  − 12 (SeD )×H  12 (SeN ) , X(T M) := H 1 (Ω1 )×H 1 (Ω2 )×H − 2 (Si )×H 2 (Si )×H 1

1

1

1

Y(T M) := H 1 (Ω1 )×H 1 (Ω2 )×H 2 (Si )×H − 2 (Si )×H 2 (SeD )×H − 2 (SeN ) . As follows from the mapping properties of the potentials (see Theorems 6 and 7), operators (45) and (46) are bounded. Let us show that operator (46) is Fredholm with zero index and thus (46) and consequently (45) are invertible. Consider the upper triangular operator ⎡

I 0

−rΩ1 VS(1)

rΩ WS(1) 1

⎢ ⎢0 I rΩ WS(2) rΩ VS(2) ⎢ 2 2 i i ⎢ ⎢ 0 0 −V (1)−V (2) W (1)−W (2) Si Si Si Si ⎢ (T M) := ⎢ K0 (1) ⎢0 0 0 LS +LS(2) ⎢ i i ⎢ ⎢0 0 0 0 ⎣ i

00

0

0

i

−rΩ VS(2) e 2

rSi γ2VS(2) e −rSi T2VS(2) e −rSeD VS(2) e

0

0

0

⎤

⎥ ⎥ rΩ WS(2) ⎥ e 2 ⎥ ⎥ −rSi γ2WS(2) e ⎥ ⎥. ⎥ rSi T2WS(2) ⎥ e ⎥ ⎥ rSeD WS(2) ⎦ e (2) rSeN LSe

It is easy to see that, on the one hand, the operator (T M)

K0

: X(T M) → Y(T M) ,

(47)

is bounded, while due to Lemma 4, K

(T M)

(T M)

− K0

: X(T M) → Y(T M)

is a compact operator. On the other hand, as has been mentioned above, in the proof of Theorem 2, the third and forth operators in the main diagonal 1

1

−[VS(1) + VS(2) ] : H − 2 (Si ) → H 2 (Si ) , i

i

1

1

LS(1) + LS(2) : H 2 (Si ) → H − 2 (Si ) i

i

are Fredholm with zero index. Moreover, applying the results of the theory of strongly elliptic pseudodifferential equations on manifolds with boundary (see, for example, [BCN09, Theorem 3.5], [CMN09-1, Lemma 3.4]), we conclude that the last two operators on the main diagonal,  − 12 (SeD ) → H 12 (SeD ) , rSeD VS(2) :H e are also Fredholm operators of index zero.

 12 (SeN ) → H − 12 (SeN ) rSeN LS(2) :H e

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105

Therefore, (47) and consequently (46) is a Fredholm operator with zero index. It remains to show that the null space of operator (46) is trivial. Let U ∈ X(T M) be a solution to the homogeneous equation K (T M)U = 0. Then due to the first two lines of the matrix equation and mapping properties (52), (53) and (54) we see that U ∈ H(T M) and by the equivalence Theorem 3 we conclude U = 0 due to the uniqueness theorem for the problem (TM) in the space H(T M) . Thus the operator (46) is invertible. To prove the invertibility of operator (45), we note that for any F (T M) ∈ F(T M) a unique solution U ∈ X(T M) of (44) is delivered by the inverse to the operator (46). On the other hand, since F (T M) ∈ F(T M) , the first two lines of the matrix operator K (T M) imply that in fact U ∈ H(T M) and the mapping F(T M) → H(T M) delivered by the inverse to the operator (46) is continuous, i.e., this operator gives the inverse to operator (45) as well. Now we can summarize the results obtained above in the following theorem. 3 , and let condition (14) hold. Then the operators (45) and Theorem 4. Let χ ∈ X1∗ (46) are invertible.

Appendix A Classes of Localizing Functions Let us introduce the classes for localizing functions (cf. [CMN09-L]). Definition 1. (i) We say χ ∈ X k for integer k ≥ 0 if χ (x) = χ˘ (|x|), χ˘ ∈ W1k (0, ∞), ρ χ˘ (ρ ) ∈ L1 (0, ∞). (ii) We say χ ∈ X∗k for k ≥ 1 if χ ∈ X k , χ (0) = 1 and

σχ (ω ) :=

1 χs (ω ) > 0 for a.e. ω ∈ R, ω

(48)

where χs (ω ) denotes the sine-transform of χ˘ :

χs (ω ) :=

 ∞ 0

χ˘ (ρ ) sin(ρ ω ) d ρ .

k for k ≥ 1 if χ ∈ X k and ω χ s (ω ) ≤ 1 (iii) We say χ ∈ X1∗ ∗

∀ ω ∈ R.

Note that if χ˘ ∈ W k (0, ∞), k ≥ 1, then χ˘ is continuous due to the Sobolev embedding theorem, and χ (0) = χ˘ (0) is well defined by continuity of χ˘ . Evidently, we have the k1 k2 following embeddings: X k1 ⊂ X k2 , X∗k1 ⊂ X∗k2 , and X1∗ ⊂ X1∗ for k1 > k2 . Since the k are inequality in (48) is to be satisfied only almost everywhere, the classes X∗k , X1∗ k k wider than their corresponding counterparts X+ , X1+ from [CMN09-L]. Some examples of functions χ from these classes are presented in [CMN09-L]. The class X∗k is defined in terms of the sine-transform. The following lemma implied by [CMN09-L, Lemma 3.2] gives an easily verifiable sufficient condition for nonnegative, non-increasing functions to belong to the class X∗k ⊃ X+k .

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Lemma 3. If χ ∈ X k , k ≥ 1, χ˘ (0) = 1, χ˘ (ρ ) ≥ 0 for all ρ ∈ (0, ∞), and χ˘ is a non-increasing function on [0, +∞), then χ ∈ X∗k .

Appendix B Properties of Localized Potentials Here we collect some assertions describing the properties of the localized potentials following from [CMN09-1, CMN09-L]. Theorem 5. The following operators are continuous:  s (Ωq ) → H s+2 (Ωq ), Pq : H

χ ∈ X 1, 1 1 Pq : H s (Ωq ) → H s+2 (Ωq ), − < s < k − , χ ∈ X k , k = 1, 2, 3. 2 2 s ∈ R,

(49) (50)

Continuity of (49) is given by [CMN09-L, Theorem 5.4] while (50) can be proved using [CMN09-L, Lemma 5.9] and [CMN09-1, Theorem 3.8]. Theorem 6. The following operators are continuous:  s (Ωq ) → H s+1 (Ωq ), Rq : H Rq : H s (Ωq ) → H t (Ωq ), 1 t < k− , 2

s ∈ R, χ ∈ X 2 , 1 1 − < s < k− , 2 2 t ≤ s + 1,

(51)

χ ∈ X k , k = 2, 3.

(52)

Continuity of (51) is given by [CMN09-L, Theorem 5.4] while (52) can be proved using the continuity of operator (50) above along with relation (3.28) and Lemma 5.3 from [CMN09-L]. Theorem 6 implies the following statement. Lemma 4. The operators Rq : H 1 (Ωq ) → H t (Ωq ), t < 3/2,

χ ∈ X 2,

1

γq Rq : H 1 (Ωq ) → H t− 2 (∂ Ωq ), t < 3/2, Tq Rq : H 1 (Ωq ) → H

t− 32

(∂ Ωq ), t < 2,

χ ∈ X 2, χ ∈ X3

are compact. Theorem 7. The following localized operators are continuous: 1

VS(q) : H − 2 (S) → H 1, 0 (Ωq± ; Aq ), 1 2

WS(q) : H (S) → H 1, 0 (Ωq± ; Aq ), VS

(q)

:H

− 12

1 2

(S) → H (S),

χ ∈ X2 , χ ∈ X3 ,

χ ∈X , 1

(53) (54) (55)

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1

1

WS (q) : H − 2 (S) → H − 2 (S), 1 2

1 2

WS(q) : H (S) → H (S), ±(q)

LS

1 2

: H (S) → H

− 21

χ ∈ X2 ,

χ ∈ X2 ,

(S) χ ∈ X , 3

(56) (57) (58)

where Ωq+ := Ωq , Ωq− := R3 \Ω¯ q . Theorem 7 follows from [CMN09-L, Theorems 5.10, 5.14]. The following jump properties are given by [CMN09-L, Theorem 5.13]. 1

1

Theorem 8. Let g ∈ H − 2 (S) and h ∈ H 2 (S). Then

γq+VS(q) g = γq−VS(q) g = VS(q) g, χ ∈ X 1 , 1 Tq±VS(q) g = ± g + WS  (q) g, χ ∈ X 2 , 2 1 ± (q) γq WS h = ∓ h + WS(q) h, χ ∈ X 2 , 2 Tq+WS(q) h − Tq−WS(q) h ≡ LS+(q) h − LS−(q) h = h

∂ aq , χ ∈ X 3. ∂ n(q)

Acknowledgements This research was supported by the EPSRC grant EP/H020497/1: “Mathematical analysis of Localized Boundary–Domain Integral Equations for Variable-Coefficient Boundary Value Problems” and partly by the Georgian Technical University grant in the case of the third author.

References [BCN09]

Buchukuri, T., Chkadua, O., Natroshvili, D.: Mixed boundary value problems of thermopiezoelectricity for solids with interior cracks. Integral Equations and Operator Theory, 64, n. 4, 495–537 (2009). [CMN09-1] Chkadua, O., Mikhailov, S., Natroshvili, D.: Analysis of direct boundary–domain integral equations for a mixed BVP with variable coefficient. Part I. Equivalence and invertibility. J. Integral Equations Appl., 21, 499–542 (2009). [CMN09-L] Chkadua, O., Mikhailov, S., Natroshvili, D.: Analysis of some localized boundary– domain integral equations. J. Integral Equations Appl., 21, 407–447 (2009). [CMN11] Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of segregated boundarydomain integral equations for variable-coefficient problems with cracks. Numerical Methods for PDEs, 27, n. 1, 121–140 (2011). [Co88] Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal., 19, 613–626 (1988). [DaLi90] Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4, Integral Equations and Numerical Methods, Springer-Verlag, Berlin (1990). [HW08] Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, Applied Mathematical Sciences, Springer-Verlag, Berlin–Heidelberg (2008). [LiMa72] Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1, Springer-Verlag, New York–Heidelberg (1972).

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O. Chkadua et al. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK (2000). Mikhailov, S.E.: Localized boundary–domain integral formulation for problems with variable coefficients. Int. J. Engineering Analysis with Boundary Elements, 26, 681– 690 (2002). Mikhailov, S.E., Nakhova, I.S.: Mesh-based numerical implementation of the localized boundary–domain integral equation method to a variable-coefficient Neumann problem. J. Engineering Math., 51, 251–259 (2005). Miranda, C.: Partial Differential Equations of Elliptic Type, Second revised edition, Springer-Verlag, New York–Berlin (1970). Sladek, J., Sladek, V., Atluri, S.N.: Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties. Comput. Mech., 24, n. 6, 456–462 (2000). Mikhailov, S.E.: Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficient. Math. Methods in Applied Sciences, 29, 715–739 (2006). Zhu, T., Zhang, J.-D., Atluri, S.N.: A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput. Mech., 21, n. 3, 223–235 (1998). Zhu, T., Zhang, J.-D., Atluri, S.N.: A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems. Eng. Anal. Bound. Elem., 23, n. 5–6, 375–389 (1999).

Analysis of Segregated Boundary–Domain Integral Equations for Mixed Variable-Coefficient BVPs in Exterior Domains O. Chkadua, S.E. Mikhailov, and D. Natroshvili

1 Introduction The direct segregated boundary–domain integral equations for the mixed boundaryvalue problem for a scalar second order elliptic partial differential equation with variable coefficient in an exterior domain in R3 are analyzed in this paper. In the literature the boundary-value problems considered here have been investigated using variational methods in weighted Sobolev spaces, particularly in [Han71, NP73, GN78, Mäu83, Gir87, DL90, Néd01]. For some cases of the PDE with constant coefficients, when the fundamental solution is available, the Dirichlet and Neumann type boundary-value problems in exterior domains were also investigated by the classical potential (indirect boundary integral equation) method, see [NP73, GN78, Gir87, DL90, CC00, Néd01] and the references therein. Our goal here is to show that the mixed problems with variable coefficients can be reduced to some systems of boundary–domain integral equations (BDIEs) and investigate equivalence of the reduction and invertibility of the corresponding boundary–domain integral operators in the weighted Sobolev spaces. To do this, we extend to the exterior domains and weighted spaces the methods developed in [CMN09a] for the interior domains and standard Sobolev (Bessel potential) spaces.

O. Chkadua A. Razmadze Mathematical Institute, Tbilisi, Georgia, e-mail: [email protected] S.E. Mikhailov Brunel University West London, UK, e-mail: [email protected] D. Natroshvili Georgian Technical University, Tbilisi, Georgia, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_11, © Springer Science+Business Media, LLC 2011

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2 Basic Notation and Spaces Let Ω = Ω + be an unbounded (exterior) open three-dimensional region of R3 such that Ω − := R3 \ Ω is a bounded open domain. For simplicity, we assume that the boundary ∂ Ω = ∂ Ω − is a simply connected, closed, infinitely smooth surface. Let ρ (x) := (1 + |x|2 )1/2 be the weight function and a ∈ C ∞ (R3 ) be such that 0 < a0 < a(x) < a1 < ∞,

ρ (x)|∇a(x)| + ρ 2 (x)|Δ a(x)| < C < ∞,

x ∈ R3 . (1)

Let also ∂ j = ∂x j := ∂ /∂ x j ( j = 1, 2, 3), ∇ = ∂x = (∂x1 , ∂x2 , ∂x3 ). We consider below some boundary–domain integral equations associated with the following scalar elliptic differential equation

∂  ∂ u(x)  a(x) = f (x), x ∈ Ω , ∂ xi i=1 ∂ xi 3

Au(x) := A(x, ∂x ) u(x) := ∑

(2)

where u is an unknown function and f is a given function in Ω . In what follows, H s (Ω ) = H2s (Ω ), H s (∂ Ω ) = H2s (∂ Ω ) denote the Bessel potential spaces (coinciding with the Sobolev–Slobodetski spaces if s ≥ 0), H∂sΩ := ∞ (Ω ), {g : g ∈ H s (R3 ), supp g ⊂ ∂ Ω }. For an open set Ω , we denote D(Ω ) = Ccomp ∗ endowed with sequential continuity, D (Ω ) is the Schwartz space of sequentially continuous functionals on D(Ω ), while D(Ω¯ ) is the set of restrictions on Ω¯ of  s (S1 ) = {g : g ∈ H s (S), supp g ⊂ S1 }, functions from D(R3 ). We also denote H s s H (S1 ) = {rS1 g : g ∈ H (S)}, where S1 is a proper submanifold of a closed surface S and rS is the restriction operator on S1 . 1 To make solution of boundary-value problems for (2) in infinite domains unique, we will use weighted Sobolev spaces (see e.g. [Han71, NP73, GN78, Mäu83, Gir87, DL90, Néd01]). Let L2 (ρ −1 ; Ω ) := {g : ρ −1 g ∈ L2 (Ω )} and H 1 (Ω ) be the Beppo– Levi space, H 1 (Ω ) := {g ∈ L2 (ρ −1 ; Ω ) : ∇g ∈ L2 (Ω )}, g2H 1 (Ω ) := ρ −1 g2L2 (Ω ) + ∇g2L2 (Ω ) . Using the corresponding property for the space H 1 (Ω ), it is easy to prove that D(Ω ) is dense in H 1 (Ω ), cf. [Han71, Theorem I.1], [Gir87, Theorem 2.2]. If Ω is unbounded, then the semi-norm |g|H 1 (Ω ) := ∇gL2 (Ω ) is equivalent to the norm gH 1 (Ω ) in H 1 (Ω ), see e.g. [DL90, Ch. XI, Part B, §1]. If Ω is bounded, then H 1 (Ω ) = H 1 (Ω ). If Ω  is a bounded subdomain of an unbounded domain Ω and g ∈ H 1 (Ω ), then g ∈ H 1 (Ω  ). 1 (Ω ) as a completion of D(Ω ) in H 1 (R3 ), H −1 (Ω ) := Let us define H 1 ∗ −1 1 ∗  [H (Ω )] , H (Ω ) := [H (Ω )] , L2 (ρ ; Ω ) := {g : ρ g ∈ L2 (Ω )}. Evidently −1 (Ω ) has a representation g = L2 (ρ ; Ω ) ⊂ H −1 (Ω ). Any distribution g ∈ H

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111

∑3i=1 ∂i gi + g0 , where gi ∈ L2 (Ω ) and g0 ∈ L2 (ρ ; Ω ), which implies that D(Ω ) is −1 (Ω ). dense in H The operator A applied to u ∈ H 1 (Ω ) in the distributional sense is well defined for a ∈ L∞ (Ω ) as Au, ϕ Ω := −a∇u, ∇ϕ Ω = − where

 Ω

a∇u · ∇ϕ dx,

∀ u ∈ H 1 (Ω ), ϕ ∈ D(Ω ),

E(u, ϕ )(x) := a(x)∇u(x) · ∇ϕ (x).

1 (Ω ), we see that A : H 1 (Ω ) → H −1 (Ω ) is conThus by density of D(Ω ) in H tinuous. From the trace theorem (see, e.g., [LM72]) for u ∈ H 1 (Ω ) it follows that if u ∈ 1 H 1 (Ω ± ), then γ ± u ∈ H 2 (∂ Ω ), where γ ± = γ∂±Ω are the trace operators on ∂ Ω ± from Ω . We will use γ for γ ± if γ + = γ − . We will use also notations u± for the traces γ ± u, when this will cause no confusion. For the linear operator A, we introduce the following subspace of H 1 (Ω ), H 1,0 (Ω ; A) := {g ∈ H 1 (Ω ) : Ag ∈ L2 (ρ ; Ω )} endowed with the norm g2H 1,0 (Ω ;A) := g2H 1 (Ω ) + ρ Ag2L2 (Ω ) , cf. [GN78]. For u ∈ H 1 (Ω ) (as well as for u ∈ H 1 (Ω )) the co–normal derivative operators a∂n u on ∂ Ω may not exist in the classical (trace) sense. However if u ∈ H 1,0 (Ω ; A), one can correctly define the (generalized) canonical co–normal derivative T + u ∈ 1 H − 2 (∂ Ω ) similar to, for example, [Cos88, Lemma 3.2], [McL00, Lemma 4.3]) as 



T +u , w

∂Ω

:=



+ + (γ−1 w)Au + E(u, γ−1 w) dx

1

∀ w ∈ H 2 (∂ Ω ),

Ω

1

+ : H 2 (∂ Ω ) → H 1 (Ω ) is a bounded right inverse of the trace operator γ + : where γ−1 1

H 1 (Ω ) → H 2 (∂ Ω ). The symbol g1 , g2 ∂ Ω denotes the duality brackets between  1 1 the spaces H − 2 (∂ Ω ) and H 2 (∂ Ω ), coinciding with ∂ Ω g1 (x)g2 (x)dS if g1 , g2 ∈ 1 L2 (∂ Ω ). The operator T + : H 1,0 (Ω ; A) → H − 2 (∂ Ω ) is continuous and gives the continuous extension on H 1,0 (Ω ; A) of the classical co-normal derivative operator a∂n , where ∂n = n · ∇ and n = n+ is the normal vector on ∂ Ω directed outward with respect to Ω . Similar to the proofs available in [Cos88, Lemma 3.4], [McL00, Lemma 4.3] for H 1,0 (Ω ; A) (see also [Mik08, Mik11] for the more general spaces H 1,t (Ω ; A)), one can prove that for u ∈ H 1,0 (Ω ; A) the first Green identity holds in the form

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T +u , γ +v

 ∂Ω

=



v Au + E(u, v) dx

∀ v ∈ H 1 (Ω ).

(3)

Ω

Then for arbitrary functions u, v ∈ H 1,0 (Ω ; A) we have the second Green identity, 

    v Au − u Av dx = T + u , γ + v ∂ Ω − T + v , γ + u ∂ Ω .

(4)

Ω

3 Mixed Boundary-Value Problem The mixed boundary-value problem in an exterior domain Ω is defined as follows. Find a function u ∈ H 1,0 (Ω ; A) satisfying the conditions A u = f in Ω , r∂ Ω γ + u = ϕ0 on ∂ ΩD ,

(5) (6)

r∂ Ω T + u = ψ0 on ∂ ΩN ,

(7)

D N

where

1

1

ϕ0 ∈ H 2 (∂ ΩD ), ψ0 ∈ H − 2 (∂ ΩN ), f ∈ L2 (ρ ; Ω ).

(8)

Here ∂ Ω = ∂ ΩD ∪ ∂ ΩN , where ∂ ΩD and ∂ ΩN are nonintersecting simply connected submanifolds of ∂ Ω with an infinitely smooth boundary curve  := ∂ ΩD ∩ ∂ ΩN ∈ C∞ . The first Green identity (3) immediately implies the following uniqueness theorem. Theorem 1. The homogeneous version of BVP (5)–(7), i.e. with ϕ0 = 0, ψ0 = 0, f = 0, has only the trivial solution, while the non-homogeneous problem (5)–(7) with ϕ0 , ψ0 and f satisfying (8) has at most one solution in H 1,0 (Ω ; A). Remark 1. We note that the existence of a solution in H 1 (Ω ; A), and thus in H 1,0 (Ω ; A), can be proved using the variational setting and the Lax–Milgram theorem, cf. [GN78, Mäu83, Gir87], where this was done for the Dirichlet and Neumann problems for the Poisson equation.

4 Parametrix and Potentials It is well known, cf. [Mik02, CMN09a], that the function P(x, y) =

−1 , x, y ∈ R3 , 4π a(y) |x − y|

is a parametrix (Levi function) for the operator A(x, ∂x ), i.e.,

(9)

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113

A(x, ∂x ) P(x, y) = δ (x − y) + R(x, y), where

xi − yi ∂ a(x) , x, y ∈ R3 . 3 ∂x 4 π a(y) |x − y| i i=1 3

R(x, y) = ∑

(10)

The parametrix P(x, y) is related to a fundamental solution to the operator A(y, ∂x ) := a(y)Δx with the “frozen” coefficient a(x) = a(y) and A(y, ∂x ) P(x, y) = δ (x − y). If ρ −1 ∇a ∈ L2 (Ω ), i.e., ∇a ∈ L2 (ρ −1 ; Ω ), then for any fixed y ∈ Ω and any ball Bε (y) centered at y with sufficiently small radius ε > 0, we have, P(., y) ∈ H 1,0 (Ω \Bε (y)) and R(., y) ∈ L2 (ρ ; Ω \Bε (y)). Applying the second Green identity (4) in Ω \Bε (y) with v = P(y, ·) and taking usual limits as ε → 0, cf. [Mir70], we get the third Green identity, u + Ru −V (T + u) +W (γ + u) = PAu in Ω

(11)

for any u ∈ H 1,0 (Ω ; A). Here Pg(y) :=





P(x, y) g(x) dx, Rg(y) :=

Ω

R(x, y) g(x) dx

(12)

Ω

are the parametrix-based volume Newton-type and remainder potentials defined for y ∈ R3 , while V g(y) := −



P(x, y) g(x) dSx , W g(y) := −

∂Ω



[Tx P(x, y)]g(x) dSx

(13)

∂Ω

are surface single layer and double layer potentials, defined for y ∈ R3 \∂ Ω . The Newton-type and the remainder potential operator given by (12) for Ω = R3 will be denoted as P and R, respectively. Recall that in the definition of W we assumed Tx = a(x) n(x) · ∇x , where n = n+ is normal vector on ∂ Ω directed outward with respect to Ω . From (9), (10), (12), and (13) one can obtain representations of the parametrixbased potential operators in terms of their counterparts for a = 1, i.e. associated with the Laplace operator Δ , Pg =

1 1 PΔ g , R g = − a a



P g ∂ ∂ a) , j j Δ ∑ 3

(14)

j=1

1 1 V g = VΔ g, W g = WΔ (ag). a a Theorem 2. The following operators are continuous:

(15)

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P : H −1 (R3 ) → H 1 (R3 ), −1 (Ω ) → H 1 (Ω ), P:H : L2 (ρ ; Ω ) → H (Ω ; A), R : H 1 (Ω ) → H 1,0 (Ω ; A), 1,0

: L2 (ρ V :H

− 12

(16) (17) (18)

; Ω ) → H (Ω ),

(19) (20)

(∂ Ω ) → H 1,0 (Ω ; A),

(21)

−1

1

1 2

W : H (∂ Ω ) → H 1,0 (Ω ; A).

(22)

Proof. Let φ ∈ D(R3 ) ⊂ H −1 (R3 ). Then the Newton potential PΔ φ =

−1 4π

 R3

φ (x) dx |x − y|

evidently belongs to H 1 (R3 ) and solves the Poisson equation Δ v = φ in R3 . On the other hand, the Laplace operator from H 1 (R3 ) to H −1 (R3 ) possesses a continuous inverse operator Δ −1 : H −1 (R3 ) → H 1 (R3 ), see e.g. [Han71]. This implies PΔ φ = Δ −1 φ .

(23)

Due to the density of D(R3 ) in H −1 (R3 ), (23) gives a continuous extension of PΔ to the operator H −1 (R3 ) → H 1 (R3 ). The first relation in (14) implies (16) under condition ρ |∇a| < C, and (17) immediately follows. To prove (18), let us denote by g˜ the extension of a function g ∈ L2 (ρ ; Ω ) by zero outside Ω . Evidently g˜ ∈ L2 (ρ ; R3 ) and PΔ g = PΔ g˜ ∈ H 1 (R3 ). Taking into account that   3 ∂ ja APg = g − ∑ ∂ j PΔ g , a j=1 conditions (1) imply (18). Let us prove the continuity of operator (21). For φ ∈ C∞ (∂ Ω ) consider the single layer potential for the Laplace operator VΔ φ =

1 4π

 ∂Ω

1 φ (x)d Γ (x) |x − y|

which evidently belongs to H 1 (Ω ; Δ ) and solves the Dirichlet problem

Δ v = 0 in Ω ,

γ + v = w on ∂ Ω

(24)

for v ∈ H 1 (Ω ; Δ ), where w = γ VΔ φ . By, e.g., [NP73, Lemma 1.1], problem (24) is uniquely solvable and its solution is delivered by a continuous operator 1 Q : H 2 (∂ Ω ) → H 1 (Ω ; Δ ). Thus VΔ φ = Qγ VΔ φ .

Analysis of Segregated BDIEs in Exterior Domains

115 1

1

Taking into account the continuity of the operator γ VΔ : H − 2 (∂ Ω ) → H 2 (∂ Ω ) and 1 1 the density of C ∞ (∂ Ω ) in H − 2 (∂ Ω ) we arrive at the continuity of VΔ : H − 2 (∂ Ω ) → H 1 (Ω ; Δ ). The first relation in (15) implies continuity of (21) under conditions (1). Continuity of (22) is proved by a similar argument. Let us prove continuity of (19). To this end, let us consider the second relation in (14) for a density φ ∈ D(R3 ) and apply the Gauss divergence theorem  1 ∑ ∂y j |x − y| φ (x)∂ j a(x)dx j=1 Ω  3   1 1 =− ∂ x ∑ Ω j |x − y| φ (x)∂ j a(x)dx 4π a(y) j=1

1 R φ (y) = 4π a(y)

3

=−

 

1 4π a(y)

3

∑



j=1 ∂ Ω

1 (γφ (x))∂n a(x)dSx |x − y|

+

1 4π a(y)

3

∑



j=1 Ω

1 ∂ j (φ (x)∂ j a(x))dx, |x − y|

that is, 3

R φ (y) = −V [(γφ )∂n a](y) − ∑ P[∂ j (φ ∂ j a)](y).

(25)

j=1

Due to the density of D(R3 ) in H 1 (Ω ), the continuity of the operators (18) and (21) and conditions (1), relation (25) are valid also for φ ∈ H 1 (R3 ), thus implying (19). For φ ∈ D(R3 ) the representation similar to (25) when Ω = R3 takes the form 3

Rφ (y) = − ∑ P[∂ j (φ ∂ j a)](y).

(26)

j=1

Since D(R3 ) is dense in L2 (Ω ), it is evidently also dense in L2 (ρ −1 ; R3 ). On the other hand, the operator of multiplication with ∂ j a is continuous from L2 (ρ −1 ; R3 ) to L2 (R3 ) due to conditions (1), while the differential operator ∂ j is continuous from L2 (R3 ) to H −1 (R3 ). By (26) and (16) this implies that the operator R : L2 (ρ −1 ; R3 ) → H 1 (R3 ) is continuous. If g ∈ L2 (ρ −1 ; Ω ), then its continuation ˜ with zero to the function g˜ ∈ L2 (ρ −1 ; R3 ) is a continuous operator and Rg = Rg, which implies (20). Let us introduce also the following boundary integral (pseudodifferential) operators of the direct values and of the co-normal derivatives of the single and double layer potentials: V g(y) := −



P(x, y) g(x) dSx , S

(27)

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W g(y) := −



T (x, n(x), ∂x ) P(x, y) g(x) dSx ,

(28)

S

W  g(y) := −



T (y, n(y), ∂y ) P(x, y) g(x) dSx ,

(29)

S

L ± g(y) := T ±W g(y),

(30)

where y ∈ S. They can be also presented in terms of their counterparts for a = 1, i.e. associated with the Laplace operator Δ , see [CMN09a], 1 1 V g = VΔ g, W g = WΔ (ag), a a    ∂ 1   VΔ g, W g = WΔ g + a ∂n a    ∂ 1 WΔ± (ag) L ± g = LΔ (ag) + a ∂n a

(31) (32) (33)

where, as usual, the subscript Δ means that the corresponding surface potentials are constructed by means of the harmonic fundamental solution PΔ (x, y) = −(4 π |x − y|)−1 . It is taken into account that a and its derivatives are continuous in R3 and Lˆ g := LΔ (ag) := LΔ+ (ag) = LΔ− (ag)

(34)

by the Liapunov–Tauber theorem. The mapping properties of the operators (27)–(30) are described in details in [CMN09a]. Particularly, their jump relations are given by the following theorem presented in [CMN09a, Theorem 3.3]. 1

1

Theorem 3. Let g1 ∈ H − 2 (S), and g2 ∈ H 2 (S). Then

γ ±V g1 (y) = V g1 (y), 1 γ ±W g2 (y) = ∓ g2 (y) + W g2 (y), 2 1 T ±V g1 (y) = ± g1 (y) + W  g1 (y), 2 where y ∈ ∂ Ω . Taking trace and co-normal derivative of the third Green identity (11) on ∂ Ω , we obtain, 1 + γ u + γ + Ru − V T + u + W γ + u = γ + PAu 2 1 + T u + T + Ru − W 2



∂Ω

T + u + L∂+Ω γ + u = T + PAu

on ∂ Ω ,

(35)

on ∂ Ω .

(36)

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117

For arbitrary functions u, f , Ψ , Φ , let us consider a more general “indirect” integral relation, associated with (11), u(y) + Ru −V Ψ +W Φ = P f

in Ω ,

(37)

and prove for the weighted spaces the analog of [CMN09a, Lemma 4.1]. 1

1

Lemma 1. Let u ∈ H 1 (Ω ), f ∈ L2 (ρ ; Ω ), Ψ ∈ H − 2 (∂ Ω ), Φ ∈ H 2 (∂ Ω ) satisfy (37). Then u belongs to H 1,0 (Ω ; A) and is a solution of the equation Au = f

in Ω

(38)

and V (Ψ − T + u)(y) −W (Φ − u+ )(y) = 0, y ∈ Ω .

(39)

Proof. First of all, rewriting (37) in the form u = P f − Ru +V Ψ −W Φ , we conclude by Theorem 2 that u ∈ H 1,0 (Ω ; A). Thus we can write the third Green identity (11) for the function u. Subtracting (37) from the identity (11), we obtain −V Ψ ∗ +W Φ ∗ = P[Au − f ] in Ω ,

(40)

where Ψ ∗ := T + u − Ψ , Φ ∗ := γ + u − Φ . Multiplying equality (40) by a(y) we get −VΔ Ψ ∗ +WΔ (aΦ ∗ ) = PΔ [Au − f ] in Ω . Applying the Laplace operator Δ to the last equation and taking into consideration that both the functions in the left-hand side are harmonic surface potentials, while the right-hand side function is the classical Newtonian volume potential, we arrive at (38). Substituting (38) back into (40) leads to (39). The counterpart of [CMN09a, Lemma 4.2] for an unbounded domain Ω takes the following form. 1

Lemma 2. (i) Let Ψ ∗ ∈ H − 2 (∂ Ω ). If V Ψ ∗ = 0 in Ω , then Ψ ∗ = 0. 1 (ii) Let Φ ∗ ∈ H 2 (∂ Ω ). If W Φ ∗ (y) = 0 in Ω , then Φ ∗ (x) = −C/a(x), where C is a constant.

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(iii) Let ∂ Ω = S1 ∪S2 , where S1 and S2 are nonempty nonintersecting simply con − 21 (S1 ), nected submanifolds of ∂ Ω with infinitely smooth boundaries. Let Ψ ∗ ∈ H  12 (S2 ). If Φ∗ ∈ H V Ψ ∗ (y) −W Φ ∗ (y) = 0 in Ω , then Ψ ∗ = 0 and Φ ∗ = 0 on ∂ Ω . Proof. The proofs of items (i) and (iii) coincide with the proofs of their counterparts for an interior domain in [CMN09a, Lemma 4.2]. To prove item (ii), we first remark that the Gauss lemma implies that ΦΔ = −C satisfies the equation WΔ ΦΔ = 0 in the exterior domain Ω for any C = const. Let us check that there is no other solutions of the equation in Ω . By the usual argument, T +WΔ ΦΔ = T −WΔ ΦΔ = 0 on ∂ Ω , which implies WΔ = const in the interior domain Ω − due to the uniqueness up to a constant of the solution of the Neumann problem in H 1 (Ω − ). Then the jump property of WΔ gives ΦΔ = const. Applying the second relation of (15) finalizes the proof of item (ii).

5 Segregated BDIEs for the Mixed Problem 1

Let Φ0 ∈ H 2 (S) be an extension of the function ϕ0 given in the Dirichlet boundary 1 condition (6) from ∂ ΩD to the whole of ∂ Ω and let Ψ0 ∈ H − 2 (S) be an extension of the given function ψ0 in the Neumann boundary condition (7) from ∂ ΩN to the whole of ∂ Ω . We will explore different possibilities of reducing BVP (5)–(7) to a system of Boundary–Domain Integral Equations (BDIEs) and in all of them we represent in (11), (35) and (36) the trace of the function u and in its co-normal derivative as

γ + u = Φ0 + ϕ ,

 12 (∂ ΩN ); ϕ ∈H

T + u = Ψ0 + ψ ,

 − 12 (∂ ΩD ), ψ ∈H

and we will regard the new unknown functions ϕ and ψ as formally segregated of u. Thus we will look for the triplet U = (u, ψ , ϕ ) ∈ H := H

1,0

 − 12 (∂ ΩD ) × H  12 (∂ ΩN ) (Ω ; A) × H

 − 2 (∂ Ω D ) × H  2 (∂ ΩN ). ⊂ X := H 1 (Ω ) × H 1

1

The BDIE System (M11) First, using (11) in Ω , the restriction of (35) on ∂ ΩD , and the restriction of (36) on ∂ ΩN , we arrive at the BDIE system (M11) of three equations for the triplet of unknowns, (u, ψ , ϕ ),

Analysis of Segregated BDIEs in Exterior Domains

r∂ Ω where

N

119

u + Ru −V ψ +W ϕ = F0 in Ω ,  +  r∂ Ω γ Ru − V ψ + W ϕ = r∂ Ω γ + F0 − ϕ0 on ∂ ΩD , D D   T + Ru − W  ψ + L∂+Ω ϕ = r∂ Ω T + F0 − ψ0 on ∂ ΩN ,

(41)

N

F0 := P f +V Ψ0 −W Φ0 in Ω .

We denote the matrix operator of the left-hand side of the systems (M11) as ⎡ ⎤ I +R −V W ⎢ ⎥ ⎢ ⎥ M 11 := ⎢ r∂ ΩD γ + R −r∂ ΩD V r∂ ΩD W ⎥ . ⎣ ⎦ r∂ Ω T + R −r∂ Ω W  r∂ Ω L + N

N

N

The notation (M11) and the corresponding superscripts mean that system includes the integral operators of the first kind on both the Dirichlet and Neumann parts of the boundary. The other BDIE systems below are also denoted, respectively.

The BDIE system (M12) Here we use (11) in Ω and (35) on the whole of ∂ Ω to arrive at the BDIE system (M12) of two equations for the triplet (u, ψ , ϕ ), u + Ru −V ψ +W ϕ = F0 1 ϕ + γ + Ru − V ψ + W ϕ = γ + F0 − Φ0 2

in Ω , on ∂ Ω .

The left-hand side matrix operator of the system is ⎡ ⎤ I + R −V W ⎢ ⎥ M 12 := ⎣ ⎦. 1 + γ R −V I +W 2

The BDIE System (M21) To arrive at the BDIE system (M21) of two equations for the triplet (u, ψ , ϕ ), we use (11) in Ω and (36) on the whole of ∂ Ω , u + Ru −V ψ +W ϕ = F0

in Ω ,

(42)

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1 ψ + T + Ru − W  ψ + L + ϕ = T + F0 − Ψ0 2 The left-hand side matrix operator of the system is ⎡ I +R −V M 21 := ⎣ 1 T +R I −W  2

W L+

on ∂ Ω .

(43)

⎤ ⎦.

The BDIE System (M22) Finally, using (11) in Ω , the restriction of (36) on ∂ ΩD , and the restriction of (35) on ∂ ΩN , we arrive for the triplet (u, ψ , ϕ ) at the BDIE system (M22) of three equations of “almost” the second kind (up to the spaces),

1 ψ + r∂ Ω D 2

u + Ru −V ψ +W ϕ = F0 in Ω ,     T + Ru − W  ψ + L + ϕ = r∂ Ω T + F0 − Ψ0 on ∂ ΩD ,

1 ϕ + r∂ Ω N 2

D

    γ + Ru − V ψ + W ϕ = r∂ Ω γ + F0 − Φ0 on ∂ ΩN . N

The matrix operator of the left-hand side of the system (M22) takes the form ⎤ ⎡ I +R −V W ⎥ ⎢ 1  ⎥ ⎢  ⎥ ⎢ + I −W r∂ Ω L + M 22 := ⎢ r∂ ΩD T R r∂ ΩD ⎥. D 2 ⎥ ⎢ 1 ⎦ ⎣ I +W r∂ Ω γ + R −r∂ Ω V r∂ Ω N N N 2 Remark 2. Note that the second relation (14) means that if a = const outside a bounded subdomain Ω  ⊂ Ω , then the operator R acts only on the restriction rΩ  u. This implies that all the BDIE systems reduce in this case to the BDIEs over Ω  and ∂ Ω , and are supplemented with the integral representations for u in Ω \Ω¯  given by the first equations of the systems. Denoting the right-hand sides of the systems (M11), (M12), (M21) and (M22) as F αβ , the systems can be rewritten as M αβ U = F αβ , where α , β = 1, 2. Due to the mapping properties of the potentials, F αβ ∈ Fαβ , while the operators M αβ : H → Fαβ and M αβ : X → Yαβ are continuous for any α , β = 1, 2. Here we denoted

Analysis of Segregated BDIEs in Exterior Domains

121

1

1

F11 := H

1, 0

(Ω ; A) × H 2 (∂ ΩD ) × H − 2 (∂ ΩN ),

F12 := H

1, 0

(Ω ; A) × H 2 (∂ Ω ),

F21 := H

1, 0

(Ω ; A) × H − 2 (∂ Ω ),

F22 := H

1, 0

(Ω ; A) × H − 2 (∂ ΩD ) × H 2 (∂ ΩN ),

1

1 1

1

1

1

Y11 := H 1 (Ω ) × H 2 (∂ ΩD ) × H − 2 (∂ ΩN ), 1

Y12 := H 1 (Ω ) × H 2 (∂ Ω ), 1

Y21 := H 1 (Ω ) × H − 2 (∂ Ω ), 1

1

Y22 := H 1 (Ω ) × H − 2 (∂ ΩD ) × H 2 (∂ ΩN ).

6 BDIE Analysis Let us first prove the equivalence theorems. 1

1

Theorem 4. Let ϕ0 ∈ H 2 (∂ ΩD ), ψ0 ∈ H − 2 (∂ ΩN ), f ∈ L2 (ρ ; Ω ) and let Φ0 ∈ 1 1 H 2 (∂ Ω ) and Ψ0 ∈ H − 2 (∂ Ω ) be some extensions of ϕ0 and ψ0 , respectively. (i) If a function u ∈ H 1 (Ω ) solves the BVP (5)–(7), then the triplet (u, ψ , ϕ ), where  − 12 (∂ ΩD ), ϕ = γ + u − Φ0 ∈ H  12 (∂ ΩN ), ψ = T + u − Ψ0 ∈ H (44) solves the BDIE systems (M11), (M12), (M21), (M22).  − 12 (∂ ΩD ) × H  12 (∂ ΩN ) solves one of the (ii) If a triplet (u, ψ , ϕ ) ∈ H 1 (Ω ) × H BDIE systems (M11), (M12) or (M22), then this solution is unique and solves all the systems, including (M21), while u solves the BVP (5)–(7) and relations (44) hold. Proof. Item (i) immediately follows from the deduction of the BDIE systems (M11), (M12), (M21), (M22). Using the similarity of Lemma 1 and items (i, iii) of Lemma 2 to their counterparts Lemma 4.1 and Lemma 4.2 (i, iii) in [CMN09a] for the bounded domain Ω , the proof of item (ii) of the theorem follows word-for-word the corresponding proofs of Theorems 5.2, 5.5 and 5.12 in [CMN09a]. The situation with regards to uniqueness and equivalence for system (M21) differs from the one for other systems and from its counterpart BDIE system (T T ) in [CMN09a], particularly because item (ii) of Lemma 2 is different from its analog, [CMN09a, Lemma 4.2(ii)]. Thus system (M21) will be further analyzed elsewhere. To prove the invertibility of the counterparts of the operators M αβ for bounded domains in [CMN09a], we essentially used there the compactness of the operator R : H 1 (Ω ) → H 1 (Ω ) based on the Rellich compactness theorem. However, the latter theorem does not hold for unbounded domains with compact boundaries, and so to cope with this, we will split the operator R into two parts, one of which can be made arbitrarily small while the other one is compact.

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Lemma 3. Let ρ |∇a(x)| → 0 as |x| → ∞. Then for any ε > 0 the operator R can be represented as R = Rc + Rs , where Rc : H 1 (Ω ) → H 1 (Ω ) is compact, while Rs H 1 (Ω )→H 1 (Ω ) < ε . Proof. Let Bη be a ball of radius η centered at 0 such ∂ Ω ⊂ Bη and let μ ∈ D(R3 ) be a cut-off function such that μ = 1 in Bη , μ = 0 in R3 \B2η and 0 ≤ μ (x) ≤ 1 in R3 . Denote Rc g := R[μ g], Rs g := R[(1 − μ )g]. By (25) we have  3     Rs gH 1 (Ω ) =  ∑ P ∂ j [(1 − μ )g∂ j a] 

H 1 (Ω )

j=1

≤ QPH−1 (Ω )→H 1 (Ω ) ,

where   ∑ ∂ j [(1 − μ )g∂ j a]H−1 (Ω ) ≤ 3

Q :=

j=1

3





∑ (1 − μ )g∂ j aL2 (Ω )

j=1

≤ 3gL2 (ρ −1 ;Ω ) ρ ∇aL∞ (R3 \Bη ) ≤ 3ρ ∇aL∞ (R3 \Bη ) gH 1 (Ω ) . Thus Rs H 1 (Ω )→H 1 (Ω ) ≤ 3ρ ∇aL∞ (R3 \Bη ) PH−1 (Ω )→H 1 (Ω ) → 0 as η → ∞ as claimed. Let us prove the claim about the operator Rc . Since the support of μ belongs to B2η , for any fixed η the operator Rc : H 1 (Ω ) → H 1 (Ω ) can be represented  as Rc g = RΩ2η [ μ rΩ g], where Ω2η = Ω B2η and the operator RΩ2η is given by 2η the second relation (12) with Ω replaced by Ω2η . The operator RΩ2η : L2 (Ω2η ) → H 1 (Ω ) is continuous by (20) since L2 (Ω2η ) = L2 (ρ −1 ; Ω2η ) for the bounded domain Ω2η . On the other hand, the restriction operator rΩ : H 1 (Ω ) → H 1 (Ω2η ) = 2η

H 1 (Ω2η ) is continuous while the imbedding of H 1 (Ω2η ) to L2 (Ω2η ) is compact, which implies that the operator Rc : H 1 (Ω ) → H 1 (Ω ) is compact. Lemma 3 implies the following statement. Corollary 1. The operator I + R : H 1 (Ω ) → H 1 (Ω ) is Fredholm with zero index. Theorem 5. If ρ |∇a(x)| → 0 as |x| → ∞, then the following operators are continuous and continuously invertible, M 11 : X → Y11 , M 11 : H → F11 .

(45) (46)

Proof. Let us consider the operator M011 : X → Y11 ,

(47)

Analysis of Segregated BDIEs in Exterior Domains

where

⎡

−V

I

⎢ ⎢ M011 := ⎢ 0 ⎣ 0

−r∂ Ω V D

0

123

W

⎤

⎥ r∂ Ω W ⎥ ⎥, D ⎦ r Lˆ ∂ ΩN

and Lˆ is defined in (34). Evidently operator (47) is continuous. The diagonal operators of the triangular matrix operator M011 are continuously invertible (cf. the proof of [CMN09a, Theorem 5.3]), implying that the operator (47) is continuously invertible as well. Let us now represent R = Rs + Rc by Lemma 3 so that the operator Rs is sufficiently small for the operator ⎤ ⎡ 0 0 Rs ⎥ ⎢ ⎥ ⎢ Ms11 := ⎢ r∂ ΩD γ + Rs 0 0 ⎥ ⎦ ⎣ r∂ Ω T + Rs 0 0 N

to satisfy the inequality Ms11 X→Y11 < (M011 )−1 Y11 →X , where (M011 )−1 is the operator inverse to M011 . Then the operator M011 + Ms11 : X → Y11 is continuously invertible, while the operator Mc11 := M 11 − M011 − Ms11 : X → Y11 is compact by Lemma 3 and by the mapping properties of the operators W  and L + − Lˆ , see [CMN09a, Theorems 3.4, 3.6]. This implies that operator (45) is a Fredholm operator with zero index. Since by Theorem 4 it is also injective, we conclude that it is invertible. To prove that the operator (46) is also invertible we remark that the unique solution U ∈ X of the system M 11U = F 11 ∈ F11 ⊂ Y11 is delivered by the bounded inverse to the operator (45). By (41) of the system and Lemma 1 we conclude that this solution belongs also to H and the mapping F11 → H delivered by the inverse to the operator (45) is continuous, thus producing the operator inverse to operator (46). This completes the proof for the operator M 11 . Theorem 6. If ρ (x)|∇a(x)| → 0 as |x| → ∞, then the following operators are continuous and continuously invertible, M 12 : X → Y12 , M 12 : H → F12 . Proof. To analyze operator M 12 let us consider the auxiliary operator M012 : X → Y12 , where

(48)

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O. Chkadua et al.

⎡ M012

⎢ := ⎣

I

−V

0

−V

⎤

W

⎥ 1 ⎦. I 2

Evidently operator (48) is continuous. Any solution U = (u, ψ , ϕ ) ∈ X of the 1 equation M012 U = F , where F = (F1 , F2 ) ∈ H 1 (Ω ) × H 2 (∂ Ω ) will solve also the following extended system of three equations, u +W ϕ 1 ϕ 2

−

V ψ = F1

in Ω ,

−

V ψ = F2

on ∂ Ω ,

−r∂ Ω V ψ = r∂ Ω F2 D

D

on ∂ ΩD ,

and vice versa. The diagonal operators of the system, I : H 1 (Ω ) → H 1 (Ω ),

−r∂ Ω

D

1 1 1 I : H 2 (∂ Ω ) → H 2 (∂ Ω ), 2  − 12 (∂ ΩD ) → H 12 (∂ ΩD ), V :H

are continuously invertible, implying that the triangular matrix operator of the sys1  12 (∂ ΩN ) to H 1 (Ω )×H 12 (∂ Ω )×H 12 (∂ ΩD ) tem mapping H 1 (Ω )×H − 2 (∂ Ω )× H  − 12 (∂ ΩD ) solves the third equais also invertible. Taking into account that if ψ ∈ H 1  2 (∂ ΩN ), we arrive at the invertibility tion of the system, then ϕ = 2(F2 + V ψ ) ∈ H of the operator (48). The rest of the proof coincides word-for-word with the one for Theorem 5. To prove the counterpart of Theorems 5 and 6 for the operator M 22 , we need the following statement that can be proved similar to [CMN09a, Lemma 5.13 and Corollary 5.14]. Lemma 4. Let ∂ Ω = S1 ∪ S2 , where S1 and S2 are nonempty nonintersecting simply connected submanifolds of ∂ Ω with infinitely smooth boundaries. For an arbitrary triplet 1 1 F = (F1 , F2 , F3 ) ∈ H 1,0 (Ω ; A) × H − 2 (S1 ) × H 2 (S2 ) there exists a unique triplet 1

1

( f∗ , Ψ∗ , Φ∗ ) = CS1 ,S2 F ∈ L2 (ρ ; Ω ) × H − 2 (∂ Ω ) × H 2 (∂ Ω )

(49)

F1 = P f∗ +V Ψ∗ −W Φ∗ in Ω + , F2 = rS1 T + F1 − rS1 Ψ∗ on S1 ,

(50) (51)

F3 = rS2 γ + F1 − rS2 Φ∗ on S2 .

(52)

such that

Analysis of Segregated BDIEs in Exterior Domains

125

Moreover, the operator 1

1

1

1

CS1 ,S2 : H 1,0 (Ω ; A) × H − 2 (S1 ) × H 2 (S2 ) → L2 (ρ ; Ω ) × H − 2 (∂ Ω ) × H 2 (∂ Ω ) is linear and continuous. Theorem 7. The operator M 22 : H → F22

(53)

is continuous and continuously invertible. Proof. By Lemma 4 any right-hand side F = (F1 , F2 , F3 ) ∈ F22 of the equation M 22 U = F

(54)

can be uniquely represented in form (50)–(52), where the triplet ( f∗ , Ψ∗ , Φ∗ ) is given by (49), S1 = ∂ ΩD , S2 = ∂ ΩN , and the operator C∂ ΩD ,∂ ΩN : F22 → L2 (ρ ; Ω ) × 1

1

H − 2 (∂ Ω ) × H 2 (∂ Ω ) is continuous. 1 1 Let us denote by A : H 1 (Ω ; A) → L2 (Ω ; ρ ) × H 2 (∂ ΩD ) × H − 2 (∂ ΩN ) the left-hand side operator of the mixed BVP (5)–(7), which is evidently continuous. By Theorem 1 and Remark 1 (as well as by Theorem 4, e.g. for the system (M11) and Theorem 5), there exists a continuous inverse operator A −1 : 1 1 L2 (ρ ; Ω ) × H 2 (∂ ΩD ) × H − 2 (∂ ΩN ) → H 1,0 (Ω ; A). Then equivalence Theorem 4 for the system (M22) implies that (54) has a solution U = (M 22 )−1 F , where the operator (M 22 )−1 : F22 → H is given by u = A −1 [(C∂ ΩD ,∂ ΩN F )1 , r∂ Ω (C∂ ΩD ,∂ ΩN F )3 , r∂ Ω (C∂ ΩD ,∂ ΩN F )2 ] , D

N

ψ = T + u − (C∂ ΩD ,∂ ΩN F )2 , ϕ = γ + u − (C∂ ΩD ,∂ ΩN F )3 , and is evidently continuous. Thus the operator (M 22 )−1 is the right inverse of the operator (53) but due to the injectivity of the latter implied by the equivalence Theorem 4, the operator (M 22 )−1 is the two-side inverse of it. In the particular case a = 1 in Ω , (5) becomes the classical Laplace equation, the remainder operator R = 0, and the BDIE system (M22) splits into the system of two Boundary Integral Equations (BIEs), r∂ Ω

1

 ψ − W  Δ ψ + LΔ+ ϕ = r∂ Ω T + F0 − r∂ Ω Ψ0 on ∂ ΩD , D D D 2 1  ϕ − VΔ ψ + WΔ ϕ = r∂ Ω F0+ − r∂ Ω Φ0 on ∂ ΩN , r∂ Ω N N N 2

and the representation formula for u in terms of ϕ and ψ , u = F0 +VΔ ψ −WΔ ϕ

in Ω .

(55) (56)

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O. Chkadua et al.

System (55)–(56) can be rewritten in the form MˆΔ22 UˆΔ = FˆΔ ,

(57)

 − 2 (∂ Ω D ) × H  2 (∂ ΩN ), where UˆΔ := (ψ , ϕ ) ∈ H ⎡ ⎤   + 1  r r I − W L Δ Δ ∂ Ω 2 ∂ Ω D  ⎦, MˆΔ22 := ⎣ D −r∂ Ω VΔ r∂ Ω 12 I + WΔ N N  + r T F0 − r∂ Ω Ψ0 1 1 D FˆΔ22 := ∂ ΩD + ∈ H − 2 (∂ ΩD ) × H 2 (∂ ΩN ). r∂ Ω F0 − r∂ Ω Φ0 1

1

N

(58)

N

 − 12 (∂ ΩD )× H  12 (∂ ΩN ) → H − 12 (∂ ΩD )×H 12 (∂ ΩN ) Moreover, the operator MˆΔ22 : H is bounded and injective. Similar to [CMN09a, Theorem 5.18], one can prove the following corollary from Theorem 7. Theorem 8. The operator  − 12 (∂ ΩD ) × H  12 (∂ ΩN ) → H − 12 (∂ ΩD ) × H 12 (∂ ΩN ) MˆΔ22 : H is invertible. Theorem 9. If ρ |∇a(x)| → 0 as |x| → ∞, then the operator M 22 : X → Y22 is continuous and continuously invertible. Proof. Let us consider the auxiliary operator M022 : X → Y22 , where

⎡

I

⎢ ⎢ M022 := ⎢ 0 ⎣ 0

r∂ Ω

D

−V

1 !  2I −W Δ

−r∂ Ω V N

(59) W

⎤

⎥ ⎥ r∂ Ω Lˆ ⎥. D ⎦

1 ! r∂ Ω 2 I + W N

Operator (59) is evidently continuous and can be considered as a matrix blocktriangle operator with the lower diagonal block ⎡ ⎤

! r∂ Ω 12 I − W  Δ r∂ Ω Lˆ D D Mˆ022 := ⎣

1 ! ⎦. −r∂ Ω V r∂ Ω 2 I + W N

N

Taking into account relations (31) and (33), we can represent   1 MˆΔ22 [diag(1, a)g], Mˆ022 g = diag 1, a

Analysis of Segregated BDIEs in Exterior Domains

127

where diag(1, 1/a) and diag(1, a) are diagonal 2 × 2 matrices. The operator MˆΔ22 given by (58) is invertible by Theorem 8. Since 0 < a0 < a(x) < a1 < ∞, this implies the invertibility of the operator  − 12 (∂ ΩD ) × H  12 (∂ ΩN ) → H − 12 (∂ ΩD ) × H 12 (∂ ΩN ) Mˆ022 : H and thus of operator (59). The rest of the proof coincides word-for-word with the one for Theorem 5.  

7 Concluding Remarks Four different segregated direct boundary–domain integral equation systems associated with the mixed (Dirichlet–Neumann) BVP for a scalar “Laplace” PDE with variable coefficient on a three-dimensional unbounded domain have been formulated and analyzed in this paper. Equivalence of three of the BDIE systems to the original BVPs was proved in the case when right-hand side of the PDE is from 1 L2 (ρ ; Ω ), and the Dirichlet and the Neumann data from the spaces H 2 (∂ ΩD ) and 1 H − 2 (∂ ΩN ), respectively. The invertibility of the BDIE operators of these three systems was proved in the corresponding weighted Sobolev spaces. Using the approach of [Mik06], united direct boundary–domain integro-differential systems can be also formulated and analyzed for the BVPs in exterior domains. The approach can be extended also to more general PDEs and to systems of PDEs, while smoothness of the variable coefficients and the boundary can be essentially relaxed, and the PDE right-hand side can be considered in more general spaces, cf. [Mik05]. Employing methods of [CMN09b], one can consider also the localized counterparts of the BDIEs for BVPs in exterior domains. Acknowledgements This work was supported by the grant EP/H020497/1 “Mathematical analysis of localized boundary–domain integral equations for BVPs with variable coefficients” from the EPSRC, UK.

References [CC00]

Chudinovich, I., Constanda, C.: Variational and Potential Methods in the Theory of Bending of Plates with Transverse Shear Deformation, Chapman & Hall/CRC, Boca Raton – London – New York – Washington, DC (2000). [CMN09a] Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary–domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. Journal of Integral Equations and Applications, 21, 499–543 (2009). [CMN09b] Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of some localized boundary– domain integral equations. Journal of Integral Equations and Applications, 21, 405– 445 (2009).

128 [Cos88] [DL90]

[Gir87]

[GN78] [Han71]

[LM72] [Mäu83] [McL00] [Mik02]

[Mik05]

[Mik06]

[Mik08]

[Mik11] [Mir70] [Néd01] [NP73]

O. Chkadua et al. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal., 19, 613–626 (1988). Dautray, R., Lions, J.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4, Integral Equations and Numerical Methods, Springer, Berlin – Heidelberg – New York (1990). Giroire, J.: Étude de Quelques Problèmes aux Limites Extérieurs et Résolution par Équations Intégrales. Thése de Doctorat d’État, Université Pierre-et-Marie-Curie (Paris-VI) (1987). Giroire, J., Nedelec, J.: Numerical solution of an exterior Neumann problem using a double layer potential. Mathematics of Computation, 32, 973–990 (1978). Hanouzet, B.: Espaces de Sobolev avec poids application au probleme de Dirichlet dans un demi espace. Rend. del Sem. Mat. della Univ. di Padova, XLVI, 227–272 (1971). Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1, Springer, Berlin – Heidelberg – New York (1972). Mäulen, J.: Lösungen der Poissongleichung und harmonishe Vektorfelder in unbeshränkten Gebieten. Math. Meth. in the Appl. Sci., 5, 233–255 (1983). McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK (2000). Mikhailov, S.E.: Localized boundary–domain integral formulations for problems with variable coefficients. Engineering Analysis with Boundary Elements, 26, 681–690 (2002). Mikhailov, S.E.: Analysis of extended boundary–domain integral and integrodifferential equations of some variable-coefficient BVP, in: Advances in Boundary Integral Methods — Proc. of the 5th UK Conference on Boundary Integral Methods (Editor: K. Chen), University of Liverpool Publ., Liverpool, UK, 106–125 (2005). Mikhailov, S.E.: Analysis of united boundary–domain integro-differential and integral equations for a mixed BVP with variable coefficient. Math. Methods in Applied Sciences, 29, 715–739 (2006). Mikhailov, S.E.: About traces, extensions and co-normal derivative operators on Lipschitz domains, in: Integral Methods in Science and Engineering: Techniques and Applications (Editors: C. Constanda, S. Potapenko), Birkhäuser, Boston, 149–160 (2008). Mikhailov, S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Analysis and Appl., 378, 324–342 (2011). Miranda, C.: Partial Differential Equations of Elliptic Type, Second edition, Springer, Berlin – Heidelberg – New York (1970). Nédélec, J.-C.: Acoustic and Electromagnetic Equations, Springer-Verlag, New York (2001). Nedelec, J., Planchard, J.: Une méthode variationelle d’éléments finis pour la résolution numérique d’un probléme extérieur dans R3 . RAIRO, 7, n. R3, 105–129 (1973).

Thermoelastic Plates with Arc-Shaped Cracks I. Chudinovich and C. Constanda

1 Prerequisites Bending theories of elastic plates describe the behavior of thin structures under external mechanical and thermal influences. Among the most accurate ones, Mindlintype models provide information not only on the bending and twisting moments generated in the body, but also on the transverse shear forces acting across the thickness. In what follows we use a combination of variational and boundary integral equation techniques to study the mathematical properties and solution of the theory proposed in [ScTa93], when the plate is weakened by an arc-shaped crack. The corresponding results in the absence of the temperature factor can be found in [ChCo06, ChCo00], and [ChCo05]. Consider a homogeneous and isotropic elastic material that occupies a region S¯ × [−h0 /2, h0 /2] in R3 , where S is a domain in R2 and h0 = const is such that 0 < h0  diam S. A generic point in the plate is written in terms of Cartesian coor¯ The displacement vector at x at time t ≥ 0 is dinates as x = (x, x3 ), x = (x1 , x2 ) ∈ S.    denoted by v(x ,t) = (v1 (x ,t), v2 (x ,t), v3 (x ,t))T , where the superscript T indicates matrix transposition, and the temperature is denoted by θ (x ,t). In the plate model investigated in [Co90] it is assumed that v(x ,t) = (x3 u1 (x,t), x3 u2 (x,t), u3 (x,t))T . If thermal effects are also taken into account, then we must consider the “averaged temperature moment”, defined by (see [ScTa93])

u4 (x,t) =

1 h2 h0

h0 /2

x3 θ (x, x3 ,t) dx3 ,

−h0 /2

h2 =

h20 . 12

C. Constanda The University of Tulsa, OK, USA, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_12, © Springer Science+Business Media, LLC 2011

129

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I. Chudinovich and C. Constanda

In this case, the vector-valued function U = (uT , u4 )T , u = (u1 , u2 , u3 )T , satisfies the equation LU(x,t) = B0 ∂t2U(x,t) + B1 ∂t U(x,t) + AU(x,t) = Q(x,t),

(x,t) ∈ S × (0, ∞), (1) where B0 = diag{ρ h2 , ρ h2 , ρ , 0}, ∂t = ∂ /∂ t, ρ > 0 is the constant density of the material, ⎞ ⎛ ⎞ ⎛ 0 0 0 0 h2 ϖ∂1 ⎜ A h2 ϖ∂2 ⎟ ⎜ 0 0 0 0 ⎟ ⎟ ⎜ ⎟, B1 = ⎜ ⎝ 0 0 0 0 ⎠, A = ⎝ 0 ⎠ η∂1 η∂2 0 κ −1 0 0 0 −Δ ⎛

⎞ −h2 μΔ − h2 (λ + μ )∂12 + μ −h2 (λ + μ )∂1 ∂2 μ∂1 A=⎝ −h2 (λ + μ )∂1 ∂2 −h2 μΔ − h2 (λ + μ )∂22 + μ μ∂2 ⎠ , −μ∂1 −μ∂2 −μΔ

∂β = ∂ /∂ xβ , β = 1, 2, Δ is the two-dimensional Laplacian, and κ, ϖ , and η are physical constants, which are expressed in terms of the thermal conductivity k, thermal expansion coefficient α , specific heat ce , density ρ , reference temperature θ0 , and Lamé constants λ and μ by the equalities κ = k(ρ ce )−1 ,

ϖ = (3λ + 2μ )α ,

η = (3λ + 2μ )αθ0 k−1 .

The right-hand side in (1) has the form Q = (qT , q4 )T , where q = (q1 , q2 , q3 )T is a combination of the forces and moments acting on the plate and its faces and q4 is a combination of the averaged heat-source density, temperature, and heat flux on the faces. Without loss of generality [ChCoVe04], we assume that the initial conditions are homogeneous; in other words, U(x, 0) = 0,

∂t u(x, 0) = 0,

x ∈ S.

(2)

Let ∂ S be a simple, closed C2 -curve that divides R2 into interior and exterior domains S+ and S− , let ∂ S0 be an open connected arc of ∂ S which models a crack, and let ∂ S1 = ∂ S \ ∂ S¯0 . Modifying the original meaning of the symbol S, we now set S = R2 \ ∂ S¯0 and assume this infinite domain to be the middle-plane section of the plate. For ν = 0, 1, we write

Σ = S × (0, ∞), Σ ± = S± × (0, ∞), Γ = ∂ S × (0, ∞), Γν = ∂ Sν × (0, ∞). We restrict our attention to the initial-boundary value problem (TCD) that consists in finding the solution of system (1) satisfying initial conditions (2) and the edge-of-the-crack Dirichlet boundary conditions U + (x,t) = F + (x,t),

U − (x,t) = F − (x,t),

(x,t) ∈ Γ0 ,

(3)

Thermoelastic Plates with Arc-Shaped Cracks

131

where the superscripts + and − denote the limiting values of the corresponding functions as (x,t) tends to Γ from inside Σ + or Σ − , respectively. The vector-valued functions F ± = (( f ± )T , f4± )T , f ± = ( f1± , f2± , f3± )T , are prescribed. As a first step in the solution of the variational, or weak, formulation of (TCD), we consider the boundary value problem (TCD p ) to which the former is reduced by the application of the Laplace transformation with respect to t.

2 The Dirichlet Problem We denote the Laplace transform of a function by a superposed hat and the (complex) transformation parameter by p. ¯ l = 0, 1, 2, . . . , be the space of functions U defined in both S¯+ and S¯− Let Cl (S), and satisfying U|S¯+ ∈ Cl (S¯+ ),

U|S¯− ∈ Cl (S¯− ),

(∂ α U)+ (x) = (∂ α U)− (x),

x ∈ ∂ S1 ,

|α | ≤ l,

where the superscripts ± indicate the limiting values of the corresponding functions as x tends to ∂ S from inside S± . We remark that U and its derivatives may have jump discontinuities across arc ∂ S0 . The classical problem (TCD p ) depending on the complex parameter p consists ¯ where C(S) ¯ = C0 (S), ¯ such that in finding Uˆ ∈ C2 (S) ∩C(S), x ∈ S,

ˆ p), ˆ p) + p(B1U)(x, ˆ ˆ p2 B0U(x, p) + (AU)(x, p) = Q(x, ± ± Uˆ (x, p) = Fˆ (x, p), x ∈ ∂ S0 .

(4)

We introduce a number of function spaces that are necessary in the subsequent analysis. Let m ∈ R and p ∈ C. Hm (R2 ) is the standard space of distributions vˆ4 (x), x ∈ R2 , endowed with norm vˆ4 m =



(1 + |ξ | ) |v˜4 (ξ )| d ξ 2 m

1/2

2

,

R2

where v˜4 is the distributional Fourier transform of vˆ4 . Hm,p (R2 ) is the space of three-component vector distributions vˆ = (vˆ1 , vˆ2 , vˆ3 )T which coincides with [Hm (R2 )]3 as a set but is equipped with the norm v ˆ m,p =



(1 + |ξ | + |p| ) |v( ˜ ξ )| d ξ 2

2 m

2

1/2 .

R2

Hm,p (R2 ) = Hm,p (R2 ) × Hm (R2 ), with the norm of the elements Vˆ = (vˆT , vˆ4 )T defined by |Vˆ |m,p = v ˆ m,p + vˆ4 m ;

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I. Chudinovich and C. Constanda

Hm (S± ) and Hm,p (S± ) consist of the restrictions to S± of all vˆ4 ∈ Hm (R2 ) and vˆ ∈ Hm,p (R2 ), respectively, with norms uˆ4 m;S± =

inf

vˆ4 ∈Hm (R2 ):vˆ4 |S± =uˆ4

vˆ4 m ,

u ˆ m,p;S± =

inf

v∈H ˆ m,p (R2 ):v| ˆ S± =uˆ

v ˆ m,p .

Hm,p (S± ) = Hm,p (S± ) × Hm (S± ), with the norm of Uˆ = (uˆT , uˆ4 )T given by ˆ |U|m,p;S± = u ˆ m,p;S± + uˆ4 m;S± . If p = 0, then we write Hm (R2 ) = Hm,0 (R2 ) = [Hm (R2 )]3 ,

Hm (R2 ) = Hm (R2 ) × Hm (R2 ) = [Hm (R2 )]4 ,

Hm (S± ) = Hm,0 (S± ) = [Hm (S± )]3 ,

Hm (S± ) = Hm (S± ) × Hm (S± ) = [Hm (S± )]4 .

The norms on [Hm (R2 )]n and [Hm (S± )]n are denoted by the same symbols  · m and  · m;S± , respectively, for all n = 1, 2, . . . , as this does not cause any ambiguity. We now introduce spaces of functions defined on ∂ S. H1/2 (∂ S) and H1/2,p (∂ S) are the spaces of the traces (see [LiMa72]) on ∂ S of all uˆ4 ∈ H1 (S± ) and uˆ ∈ H1,p (S± ), with norms  fˆ4 1/2;∂ S =

inf

uˆ4 ∈H1 (S+ ):uˆ4 |∂ S = fˆ4

uˆ4 1;S+ ,

 fˆ1/2,p;∂ S =

inf

u∈H ˆ 1,p (S+ ):u| ˆ ∂ S = fˆ

u ˆ 1,p;S± .

H1/2,p (∂ S) = H1/2,p (∂ S) × H1/2 (∂ S), with the norm of Fˆ = ( fˆT , fˆ4 )T given by ˆ 1/2,p;∂ S =  fˆ1/2,p;∂ S +  fˆ4 1/2;∂ S . |F|

For simplicity, we denote by the same symbols γ ± the continuous (uniformly with respect to p ∈ C) trace operators from H1 (S± ) to H1/2 (∂ S), from H1,p (S± ) to H1/2,p (∂ S), and from H1,p (S± ) to H1/2,p (∂ S). If p = 0, then we write H1/2 (∂ S) = H1/2,0 (∂ S) = [H1/2 (∂ S)]3 and H1/2 (∂ S) = H1/2 (∂ S) × H1/2 (∂ S) = [H1/2 (∂ S)]4 . The norms on [H1/2 (∂ S)]n are denoted by  · 1/2;∂ S for all n = 1, 2, . . . . ˆ If U(x,t) = (uˆT (x,t), uˆ4 (x,t))T is defined for x ∈ S, then its restrictions to S± are denoted by Uˆ ± = (uˆT± , uˆ±,4 )T , and we write uˆ4 = {uˆ+,4 , uˆ−,4 }, uˆ = {uˆ+ , uˆ− }, and Uˆ = {Uˆ + , Uˆ − }. Let πν , ν = 0, 1, be the operators of restriction from ∂ S to ∂ Sν , and let γν± = πν γ ± be the trace operators on ∂ Sν . H1 (S), H1,p (S), and H1,p (S) are the spaces of all uˆ4 = {uˆ+,4 , uˆ−,4 }, uˆ = {uˆ+ , uˆ− }, and Uˆ = {Uˆ + , Uˆ − } such that uˆ±,4 ∈ H1 (S± ), uˆ± ∈ H1,p (S± ), Uˆ ± ∈ H1,p (S± ), and γ1+ uˆ+,4 = γ1− uˆ−,4 , γ1+ uˆ+ = γ1− uˆ− , γ1+Uˆ + = γ1−Uˆ − . The norms on H1 (S), H1,p (S), and H1,p (S) are uˆ4 1;S = uˆ+,4 1;S+ +uˆ−,4 1,p;S− , u ˆ 1,p;S = uˆ+ 1,p;S+ +uˆ− 1,p;S− , ˆ 1,p;S = |Uˆ + |1,p;S+ + |Uˆ − |1,p;S− . and |U| ˚ 1,p (S), and H˚1,p (S) denote the subspaces of H1 (R2 ), H1,p (R2 ), and H˚ 1 (S), H 2 H1,p (R ) of all uˆ4 , u, ˆ and Uˆ such that γ0+ uˆ+,4 = γ0− uˆ−,4 = 0, γ0+ uˆ+ = γ0− uˆ− = 0, + ˆ − ˆ and γ0 U+ = γ0 U− = 0. ¯ and H−1 (S), H ˚ −1,p (S) ¯ and H−1,p (S), H˚−1,p (S) ¯ and H−1,p (S) are the H˚ −1 (S) ˚ ˚ duals of H1 (S) and H1 (S), H1,p (S) and H1,p (S), H1,p (S) and H˚1,p (S), respectively, defined by the duality generated by the inner product (· , ·)0 in the corresponding

Thermoelastic Plates with Arc-Shaped Cracks

133

spaces [L2 (S)]n . The norms on these spaces are denoted by  · −1;S¯ and  · −1;S , ·−1,p;S¯ and ·−1,p;S , |·|−1,p;S¯ and |·|−1,p;S . On H−1,p (S) it is also convenient to use the equivalent norm ˆ −1,p;S = |p| q Q

ˆ −1,p;S + qˆ4 −1;S . ˚ 1/2,p (∂ S0 ), and H˚1/2,p (∂ S0 ), respectively, are the subspaces of H˚ 1/2 (∂ S0 ), H H1/2 (∂ S), H1/2,p (∂ S), and H1/2,p (∂ S) of all fˆ4 , fˆ, and Fˆ with compact support in ∂ S0 . H1/2 (∂ S0 ), H1/2,p (∂ S0 ), and H1/2,p (∂ S0 ) are the spaces of the restrictions from ∂ S to ∂ S0 of the elements of H1/2 (∂ S), H1/2,p (∂ S), and H1/2,p (∂ S). The norms on these spaces are defined by  fˆ4 1/2;∂ S0 =  fˆ1/2,p;∂ S0 = ˆ 1/2,p;∂ S = |F| 0

inf

ϕˆ 4 1/2,∂ S ,

inf

ϕˆ 1/2,p;∂ S ,

inf

|Φˆ |1/2,p;∂ S .

ϕˆ 4 ∈H1/2 (∂ S):π0 ϕˆ 4 = fˆ4

ϕˆ ∈H1/2,p (∂ S):π0 ϕˆ = fˆ

Φˆ ∈H1/2,p (∂ S):π0 Φˆ =Fˆ

We consider an operator l0 of extension from ∂ S0 to ∂ S, which maps (continuously and uniformly with respect to p) H1/2 (∂ S0 ) to H1/2 (∂ S), H1/2,p (∂ S0 ) to H1/2,p (∂ S), and H1/2,p (∂ S0 ) to H1/2,p (∂ S). Additionally, we consider operators l ± of extension from ∂ S to S± , which map (continuously and uniformly with respect to p) H1/2 (∂ S) to H1 (S± ), H1/2,p (∂ S) to H1,p (S± ), and H1/2,p (∂ S) to H1,p (S± ). ˆ p) be a classical solution of (TCD p ), and let C0∞ (S) be the subspace Let U(x, ∞ ˆ = 0, x ∈ ∂ S0 . If we take an arbitrary of C0 (R2 ) consisting of all Uˆ such that U(x) ∞ ˆ function W ∈ C0 (S), multiply the first equation in (4) by Wˆ in [L2 (S)]4 , and integrate the result over S or R2 , we obtain the equality ˆ Wˆ )0 , ˆ Wˆ ) = (Q, ϒp (U,

(5)

where 1/2 1/2 ˆ Wˆ ) = a(u, ϒp (U, ˆ w) ˆ + (∇uˆ4 , ∇wˆ 4 )0 + p2 (B0 u, ˆ B0 w) ˆ 0

+ κ −1 p(uˆ4 , wˆ 4 )0 − h2± ϖ (uˆ4 , div w) ˆ 0 + η p(div u, ˆ wˆ 4 )0 , 

a(u, ˆ w) ˆ =2

E(u, ˆ w) ˆ dx, S

B0 = diag{ρ h2 , ρ h2 , ρ }, 2E(u, ˆ w) ˆ = h2 E0 (u, ˆ w) ˆ + h2 μ (∂2 uˆ1 + ∂1 uˆ2 )(∂2 w¯ˆ 1 + ∂1 w¯ˆ 2 ) + μ [(uˆ1 + ∂1 uˆ3 )(w¯ˆ 1 + ∂1 w¯ˆ 3 ) + (uˆ2 + ∂2 uˆ3 )(w¯ˆ 2 + ∂2 w¯ˆ 3 )],

(6)

134

I. Chudinovich and C. Constanda

E0 (u, ˆ w) ˆ = (λ + 2μ )[(∂1 uˆ1 )(∂1 w¯ˆ 1 ) + (∂2 uˆ2 )(∂2 w¯ˆ 2 )] + λ [(∂1 uˆ1 )(∂2 w¯ˆ 2 ) + (∂2 uˆ2 )(∂1 w¯ˆ 1 )]. Equation (5) suggests a natural weak formulation of problem (TCD p ). We say that Uˆ ∈ H1,p (S) is a variational (weak) solution of (TCD p ) if it satisfies γ0+Uˆ = Fˆ + , γ0−Uˆ = Fˆ − , and (5) for any Wˆ ∈ H˚1,p (S). Let Cκ = {p = σ + iζ ∈ C : σ > κ }. Below, c represents all positive constants occurring in estimates, which do not depend on the functions in those estimates or on p ∈ Cκ , but may depend on κ . Theorem 1. For any κ > 0, p ∈ Cκ , Qˆ ∈ H−1,p (S), and Fˆ ± ∈ H1/2,p (∂ S0 ) such that δ Fˆ = Fˆ + − Fˆ − ∈ H˚1/2,p (∂ S0 ), (TCD p ) has a unique solution Uˆ ∈ H1,p (S) and ˆ −1,p;S + |p|(|Fˆ + |1/2,p;∂ S + |δ F| ˆ 1/2,p;∂ S )}. ˆ 1,p;S ≤ c{ Q

|U| 0

L (S), HL (S), and H L For any κ > 0 and k, l ∈ R, the spaces H1,l, 1,k,κ 1,l,k,κ (S) = κ ˆ p) that consist of elements uˆ4 (x, p), u(x, ˆ p), and U(x,

L HL 1,k,κ (S) × H1,l,κ (S)

ˆ , p) : Cκ → [H1 (S)]3 , (i) define holomorphic mappings uˆ4 (· , p) : Cκ → H1 (S), u(· 4 ˆ and U(· , p) : Cκ → [H1 (S)] ; (ii) have norms defined by uˆ4 21,l,κ ;S = sup

∞

σ >κ −∞

u ˆ 21,k,κ ;S = sup

∞

σ >κ −∞

(1 + |p|2 )l uˆ4 (x, p)21,S d τ < ∞,

(1 + |p|2 )k u(x, ˆ p)21,p;S d τ < ∞,

ˆ 1,l,k,κ ;S = u ˆ 1,k,κ ;S + uˆ4 1,l,κ ;S . |U| L L L L L Similarly, H−1,l, κ (S), H−1,k,κ (S), and H−1,l,k,κ (S) = H−1,k,κ (S) × H−1,l,κ (S) ˆ p) that consist of elements qˆ4 (x, p), q(x, ˆ p), and Q(x, ˆ , p):Cκ → [H−1 (S)]3 , (i) define holomorphic mappings qˆ4 (· , p):Cκ → H−1 (S), q(· 4 ˆ , p) : Cκ → [H−1 (S)] ; and Q(· (ii) have norms defined by

qˆ4 2−1,l,κ ;S

q ˆ 2−1,k,κ ;S

∞

= sup

σ >κ −∞

∞

= sup

σ >κ −∞

(1 + |p|2 )l qˆ4 (x, p)2−1,S d τ < ∞,

(1 + |p|2 )k q(x, ˆ p)2−1,p;S d τ < ∞,

ˆ −1,l,k,κ ;S = q |Q| ˆ −1,k,κ ;S + qˆ4 −1,l,κ ;S .

Thermoelastic Plates with Arc-Shaped Cracks

135

L L L L L H1/2,l, κ (∂ S0 ), H1/2,k,κ (∂ S0 ), and H1/2,l,k,κ (∂ S0 ) = H1/2,k,κ (∂ S0 )×H1/2,l,κ (∂ S0 ) ˆ p) that are the spaces of elements fˆ4 (x, p), fˆ(x, p), and F(x, (i) define holomorphic mappings fˆ4 (· , p) : Cκ → H1/2 (∂ S0 ), fˆ(· , p) : Cκ → ˆ , p) : Cκ → [H1/2 (∂ S0 )]4 ; [H1/2 (∂ S0 )]3 , and F(·

(ii) have norms defined by  fˆ4 21/2,l,κ ;∂ S0 = sup

∞

σ >κ −∞

 fˆ21/2,k,κ ;S = sup

∞

σ >κ −∞

(1 + |p|2 )l  fˆ4 (x, p)21/2;∂ S0 d τ < ∞,

(1 + |p|2 )k  fˆ(x, p)21/2,p;∂ S0 d τ < ∞,

ˆ 1/2,l,k,κ ;∂ S =  fˆ1/2,k,κ ;∂ S +  fˆ4 1/2,l,κ ;∂ S . |F| 0 0 0 L ˚L ˚L ˚L ˚L H˚ 1/2,l, κ (∂ S0 ), H1/2,k,κ (∂ S0 ), and H1/2,l,k,κ (∂ S0 ) = H1/2,k,κ (∂ S0 )× H1/2,l,κ (∂ S0 ) ˆ p) that consist of elements δ fˆ4 (x, p), δ fˆ(x, p), and δ F(x,

(i) define holomorphic mappings

δ fˆ4 (· , p) : Cκ → H˚ 1/2 (∂ S0 ),

δ fˆ(· , p) : Cκ → [H˚ 1/2 (∂ S0 )]3 ,

ˆ , p) : Cκ → [H˚ 1/2 (∂ S0 )]4 ; δ F(· (ii) have norms defined by δ fˆ4 21/2,l,κ ;∂ S = sup

∞

σ >κ −∞

δ fˆ21/2,k,κ ;∂ S = sup

∞

σ >κ −∞

(1 + |p|2 )l δ fˆ4 (x, p)21/2,∂ S d τ < ∞,

(1 + |p|2 )k δ fˆ(x, p)21/2,p;∂ S d τ < ∞,

ˆ 1/2,l,k,κ ;∂ S = δ fˆ1/2,k,κ ;∂ S + δ fˆ4 1/2,l,κ ;∂ S . |δ F| L L ˆ± Theorem 2. If κ > 0, l ∈ R, Qˆ ∈ H−1,l+1,l, κ (S) and F ∈ H1/2,l+1,l+1,κ (∂ S0 ) are such that δ Fˆ = Fˆ + − Fˆ − ∈ H˚ L (∂ S0 ), then the weak solution Uˆ of (TCD p ) 1/2,l+1,l+1,κ

L (S) and satisfies belongs to H1,l,l, κ

 ˆ −1,l+1,l,κ ;S + |Fˆ + |1/2,l+1,l+1,κ ;∂ S + |δ F| ˆ 1/2,l+1,l+1,κ ;∂ S . ˆ 1,l,l,κ ;S ≤ c |Q| |U| 0 −1

−1

−1

−1

L L L ˚L Let H1,l,k, κ (Σ ), H−1,l,k,κ (Σ ), H1/2,l,k,κ (Γ0 ), and H1/2,l,k,κ (Γ0 ), κ > 0, l, k ∈ R,

L be the spaces of the inverse Laplace transforms of the elements of H1,l,k, κ (S), L L L ˚ (S), H (∂ S0 ), and H (∂ S0 ), endowed with the norms H −1,l,k,κ

1/2,l,k,κ

1/2,l,k,κ

136

I. Chudinovich and C. Constanda

ˆ 1,l,k,κ ;S , |U|1,l,k,κ ;Σ = |U| ˆ 1/2,l,k,κ ;∂ S , |F|1/2,l,k,κ ;Γ0 = |F| 0

ˆ −1,l,k,κ ;S , |Q|−1,l,k,κ ;Σ = |Q| ˆ 1/2,l,k,κ ;∂ S . |δ F|1/2,l,k,κ ;Γ = |δ F|

We continue to use the symbols γν± , ν = 1, 2, for the trace operators from Σ ± to Γν , and the symbols πν for the operators of restriction from Γ to Γν . We denote by ¯ + ) consisting of all W such that W (x,t) = 0, (x,t) ∈ Γ0 . C0∞ (Σ ) the subspace of C0∞ (R 3 T T L −1 (Σ ) a weak solution of (TCD) if We call U = (u , u4 ) ∈ H1,0,0, κ (i) γ0 u = 0, where γ0 is the trace operator on S × {t = 0}; (ii) γ0+U = F + and γ0−U = F − ;

∞

(iii) for all W = (wT , w4 )T ∈ C0∞ (Σ ), ϒ (U,W ) = (Q,W )0 dt, where 0

ϒ (U,W ) =

∞

1/2

1/2

{a(u, w) + (∇u4 , ∇w4 )0 − (B0 ∂t u, B0 ∂t w)0

0

− κ −1 (u4 , ∂t w4 )0 − h2 ϖ (u4 , div w)0 − η (div u, ∂t w4 )0 } dt. (7)

ˆ p) be the inverse Laplace transform of the weak Theorem 3. Let U(x,t) = L −1U(x, ˆ p) of problem (TCD p ). If κ > 0, l ∈ R, and Q ∈ H L −1 solution U(x, −1,l+1,l,κ (Σ ) and −1 −1 ± L + − L ˚ F ∈H (Γ0 ) are such that δ F = F − F ∈ H (Γ0 ), then 1/2,l+1,l+1,κ

1/2,l+1,l+1,κ

−1

L (Σ ) and U ∈ H1,l,l, κ

|U|1,l,l,κ ;Σ ≤ c{|Q|−1,l+1,l,κ ;Σ + |F + |1/2,l+1,l+1,κ ;Γ0 + |δ F|1/2,l+1,l+1,κ ;Γ }. If, in addition, l ≥ 0, then U is the unique weak solution of problem (TCD).

3 The Neumann Problem We consider the initial-boundary value problem (TCN) that consists in finding a solution of (1) which satisfies the initial conditions (2) and, along the two edges of the crack, the Neumann-type boundary conditions (TU)+ (x,t) = G+ (x,t),

(TU)− (x,t) = G− (x,t),

(x,t) ∈ Γ0 ,

instead of (3), where 

   (TeU)(x,t) (Tu)(x,t) − h2 ϖ n(x)u4 (x,t) (TU)(x,t) = , = (Tθ U)(x,t) ∂n u4 (x,t) T is the boundary moment–stress operator defined by [Co90] ⎞ ⎛ 2 h2 (λ n1 ∂2 + μ n2 ∂1 ) 0 h [(λ + 2μ )n1 ∂1 + μ n2 ∂2 ] T = ⎝ h2 (μ n1 ∂2 + λ n2 ∂1 ) h2 [(λ + 2μ )n2 ∂2 + μ n1 ∂1 ] 0 ⎠ , μ n1 μ n2 μ∂n

(8)

Thermoelastic Plates with Arc-Shaped Cracks

137

n = (n1 , n2 )T is the outward (with respect to S+ ) unit normal to ∂ S, and ∂n is the normal derivative. To keep the notation simple, in what follows we also denote by n the three-component vector (n1 , n2 , 0)T . The superscripts ± denote the limiting values of the corresponding functions as (x,t) tends to Γ from inside Σ ± , respectively. ± ± ± T T ± The functions G± = ((g± )T , g± 4 ) , where g = (g1 , g2 , g3 ) , are prescribed. As in the case of the Dirichlet problem, we apply the Laplace transformation with respect to the time variable to reduce (TCN) to a boundary value problem (TCN p ), establish the unique solvability of the latter, and then, by means of suitable analytic considerations, return to the spaces of originals and investigate the weak solvability of the time-dependent problem. From (1), (2), and (8) we see that the classical Laplace-transformed problem ¯ that satisfies (TCN p ) consists in finding Uˆ ∈ C2 (S) ∩C1 (S) ˆ p), ˆ p) + p (B1U)(x, ˆ ˆ p) + (AU)(x, p) = Q(x, p2 B0U(x, ˆ ± (x, p) = Gˆ ± (x, p), (TU)

x ∈ ∂ S0 .

x ∈ S,

(9)

We introduce a few more useful spaces. H−1/2 (∂ S), H−1/2,p (∂ S), and H−1/2,p (∂ S) are the dual spaces of H1/2 (∂ S), H1/2,p (∂ S), and H1/2,p (∂ S) with respect to the dualities generated by the inner products in L2 (∂ S), [L2 (∂ S)]3 , and [L2 (∂ S)]4 . The norms of gˆ4 ∈ H−1/2 (∂ S), gˆ ∈ H−1/2,p (∂ S), and Gˆ = (gˆT , gˆ4 )T ∈ H−1/2,p (∂ S) are denoted by gˆ4 −1/2;∂ S , ˆ −1/2,p;∂ S = g ˆ −1/2,p;∂ S + gˆ4 −1/2;∂ S . It is also convenient g ˆ −1/2,p;∂ S , and |G| 4 to endow the set [H−1/2 (∂ S)] with the equivalent norm ˆ −1/2,p;∂ S = |p|g ˆ −1/2,p;∂ S + gˆ4 −1/2;∂ S . G

If p = 0, then we simply write H±1/2 (∂ S) = H±1/2,0 (∂ S) = [H±1/2 (∂ S)]3 and H±1/2 (∂ S) = H±1/2 (∂ S)×H±1/2 (∂ S) = [H±1/2 (∂ S)]4 . The norms on [H±1/2 (∂ S)]n are denoted by  · ±1/2;∂ S for all n = 1, 2, . . . . ˚ −1/2,p (∂ S0 ) and H−1/2,p (∂ S0 ), and H˚−1/2,p (∂ S0 ) H˚ −1/2 (∂ S0 ) and H−1/2 (∂ S0 ), H and H−1/2,p (∂ S0 ) are the dual spaces of H1/2 (∂ S0 ) and H˚ 1/2 (∂ S0 ), H1/2,p (∂ S0 ) ˚ 1/2,p (∂ S0 ), and H1/2,p (∂ S0 ) and H˚1/2,p (∂ S0 ) with respect to the duality and H generated by the inner product in the corresponding spaces [L2 (∂ S0 )]n . It is clear ˚ −1/2,p (∂ S0 ), and H˚−1/2,p (∂ S0 ) are subspaces of H−1/2 (∂ S), that H˚ −1/2 (∂ S0 ), H H−1/2,p (∂ S), and H−1/2,p (∂ S). The norms on H−1/2 (∂ S0 ), H−1/2,p (∂ S0 ), and H−1/2,p (∂ S0 ) are denoted by  · −1/2;∂ S0 ,  · −1/2,p;∂ S0 , and | · |−1/2,p;∂ S0 . On H−1/2,p (∂ S0 ) and H˚−1/2,p (∂ S0 ) it is also convenient to use the equivalent norms ˆ −1/2,p;∂ S = |p|g ˆ −1/2,p;∂ S0 + gˆ4 −1/2;∂ S0 , G

0 ˆ −1/2,p;∂ S = |p|g ˆ −1/2,p;∂ S + gˆ4 −1/2;∂ S . G

138

I. Chudinovich and C. Constanda

¯ is the space of pairs of functions Uˆ = {Uˆ + , Uˆ − }, where Uˆ + ∈ C∞ (S¯+ ), C0∞ (S) ∞ ˆ U− ∈ C0 (S¯− ), and their limiting values and those of all their derivatives from inside S+ and S− coincide on ∂ S1 . ¯ mulLet Uˆ be a classical solution of (TCN p ). Choosing an arbitrary Wˆ ∈ C0∞ (S), 2 4 ˆ tiplying the equation in (9) by W in [L (S)] , and integrating the result over S (or, what is the same, over R2 ), we arrive at ˆ Wˆ )0 + (Gˆ + , γ +Wˆ + )0;∂ S − (Gˆ − , γ −Wˆ − )0;∂ S , ˆ Wˆ ) = (Q, ϒp (U, 0 0 0 0

(10)

where ϒp is defined by (6). In view of the above, we say that Uˆ ∈ H1,p (S) is a variational (weak) solution of (TCN p ) if it satisfies (10) for any Wˆ ∈ H1,p (S). Theorem 4. For any κ > 0, p ∈ Cκ , Qˆ ∈ H˚−1,p (S), and Gˆ ± ∈ H−1/2,p (∂ S0 ) such that δ Gˆ = Gˆ + − Gˆ − ∈ H˚−1/2,p (∂ S0 ), problem (TCN p ) has a unique weak solution Uˆ ∈ H1,p (S) and ˆ −1,p;S¯ + δ G

ˆ −1/2,p;∂ S + Gˆ − −1/2,p;∂ S )}. ˆ 1,p;S ≤ c{ Q

|U| 0 We now introduce a few more spaces. L ˚L ˚L ˚L ˚L H˚ −1,l, κ (S), H−1,k,κ (S), and H−1,l,k,κ (S) = H−1,k,κ (S) × H−1,l,κ (S) are the spaces ˆ p) that of all qˆ4 (x, p), q(x, ˆ p), and Q(x, (i) define holomorphic mappings qˆ4 : Cκ → H˚ −1 (S), qˆ : Cκ → [H˚ −1 (S)]3 , and Qˆ : Cκ → [H˚ −1 (S)]4 ; (ii) have norms qˆ4 2−1,l,κ ;S¯ q ˆ 21,k,κ ;S¯

∞

= sup

σ >κ −∞

∞

= sup

σ >κ −∞

(1 + |p|2 )l qˆ4 (x, p)2−1;S¯ d τ < ∞,

(1 + |p|2 )k q(x, ˆ p)2−1,p;S¯ d τ < ∞,

ˆ −1,l,k,κ ;S¯ = q ˆ −1,k,κ ;S¯ + qˆ4 −1,l,κ ;S¯ . |Q| L L H−1/2,l, κ (∂ S0 ), H−1/2,k,κ (∂ S0 ), and L L L H−1/2,l,k, κ (∂ S0 ) = H−1/2,k,κ (∂ S0 ) × H−1/2,l,κ (∂ S0 )

ˆ p) that are the spaces of all gˆ4 (x, p), g(x, ˆ p), and G(x, (i) define holomorphic mappings gˆ4 : Cκ → H−1/2 (∂ S0 ), (ii) have norms

gˆ : Cκ → [H−1/2 (∂ S0 )]3 ,

Gˆ : Cκ → [H−1/2 (∂ S0 )]4 ;

Thermoelastic Plates with Arc-Shaped Cracks

gˆ4 2−1/2,l,κ ;∂ S0

∞

= sup

σ >κ −∞ ∞

g ˆ 2−1/2,k,κ ;S = sup

σ >κ −∞

139

(1 + |p|2 )l gˆ4 (x, p)2−1/2;∂ S0 d τ < ∞,

(1 + |p|2 )k g(x, ˆ p)2−1/2,p;∂ S0 d τ < ∞,

ˆ −1/2,l,k,κ ;∂ S = g ˆ −1/2,k,κ ;∂ S0 + gˆ4 −1/2,l,κ ;∂ S0 . |G| 0 L ˚L H˚ −1/2,l, κ (∂ S0 ), H−1/2,k,κ (∂ S0 ), and L ˚L ˚L H˚−1/2,l,k, κ (∂ S0 ) = H−1/2,k,κ (∂ S0 ) × H−1/2,l,κ (∂ S0 )

ˆ p) that ˆ p), and δ G(x, are the spaces of all δ gˆ4 (x, p), δ g(x, (i) define holomorphic mappings δ gˆ4 : Cκ → H˚ −1/2 (∂ S0 ), δ g(x, ˆ p) : Cκ → 3 4 ˆ ˚ ˚ [H−1/2 (∂ S0 )] , and δ G(x, p) : Cκ → [H−1/2 (∂ S0 )] ; (ii) have norms δ gˆ4 2−1/2,l,κ ;∂ S

∞

= sup

σ >κ −∞ ∞

δ g ˆ 2−1/2,k,κ ;∂ S = sup

σ >κ −∞

(1 + |p|2 )l δ gˆ4 (x, p)2−1/2;∂ S d τ < ∞,

(1 + |p|2 )k δ g(x, ˆ p)2−1/2,p;∂ S d τ < ∞,

ˆ −1/2,l,k,κ ;∂ S = δ g |δ G| ˆ −1/2,k,κ ;∂ S + δ gˆ4 −1/2,l,κ ;∂ S . L L ˆ± Theorem 5. If κ > 0, l ∈ R, and Qˆ ∈ H˚−1,l+1,l, κ (S) and G ∈ H−1/2,l+1,l,κ (∂ S0 ) are such that δ Gˆ = Gˆ + − Gˆ − ∈ H˚ L (∂ S0 ), then the weak solution Uˆ of −1/2,l+1,l,κ

L (S) and problem (TCN p ) belongs to H1,l,l, κ

 ˆ −1,l+1,l,κ ;S¯ + |δ G| ˆ −1/2,l+1,l+1,κ ;∂ S ˆ 1,l,l,κ ;S ≤ c |Q| |U| + |Gˆ − |−1/2,l+1,l+1,κ ;∂ S0 . Let κ > 0 and l, k ∈ R. We introduce a final batch of function spaces. −1 −1 −1 (Σ ), H L (Γ0 ), and H˚ L (Γ0 ) consist of the inverse Laplace H˚ L −1,l,k,κ

−1/2,l,k,κ

−1/2,l,k,κ

L L ˚L transforms of all elements of H˚−1,l,k, κ (S), H−1/2,l,k,κ(∂ S0 ), and H−1/2,l,k,κ (∂ S0 ), respectively; these spaces are equipped with the norms

ˆ −1,l,k,κ ;S¯ , |Q|−1,l,k,κ ;Σ¯ = |Q| |δ G|−1/2,l,k,κ ;Γ

ˆ −1/2,l,k,κ ;∂ S , |G|−1/2,l,k,κ ;Γ0 = |G| 0 ˆ −1/2,l,k,κ ;∂ S . = |δ G|

C0∞ (Σ¯ ) is the space of all U = {U+ ,U− } such that U+ ∈ C0∞ (Σ¯ + ), U− ∈ C0∞ (Σ¯ − ), and γ1+ ∂ α U+ = γ1− ∂ α U− for all multi-indices α . We say that a vector distribution L −1 (Σ ) is a weak solution of (TCN) if U = (uT , u4 )T ∈ H1,0,0, κ

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(i) γ0 u = 0, where γ0 is the trace operator on S × {t = 0}; (ii) for all W = (wT , w4 )T ∈ C0∞ (Σ¯ ) we have

ϒ (U,W ) =

∞

(Q,W )0 dt + L(W ), 0

where ϒ is defined by (7) and ∞

{(G+ , γ0+W+ )0;∂ S0 − (G− , γ0−W− )0;∂ S0 } dt

L(W ) = 0

∞

{(δ G, γ0+W+ )0;∂ S0 + (G− , δ W )0;∂ S0 } dt,

= 0

δ G = G+ − G− , and δ W = γ0+W+ − γ0−W− . Theorem 6. If U = L −1Uˆ is the inverse Laplace transform of the weak solution Uˆ L −1 ± L −1 of (TCN p ), κ > 0, l ∈ R, and Q ∈ H˚−1,l+1,l, κ (Σ ) and G ∈ H−1/2,l+1,l,κ (Γ0 ) are −1 −1 such that δ G = G+ − G− ∈ H˚ L (Γ0 ), then U ∈ H L (Σ ) and −1/2,l+1,l,κ

1,l,l,κ

ˆ 1,l,l,κ ;Σ ≤ c{|Q|−1,l+1,l,κ ;Σ¯ + |δ G|−1/2,l+1,l,κ ;Γ + |G− |−1/2,l+1,l,κ ;Γ }. |U| 0 If, in addition, l ≥ 0, then U is the unique weak solution of problem (TCN).

References [Co90]

Constanda, C.: A Mathematical Analysis of Bending of Plates with Transverse Shear Deformation, Longman, Harlow (1990). [ScTa93] Schiavone, P., Tait, R.J.: Thermal effects in Mindlin-type plates. Quart. J. Mech. Appl. Math., 46, 27–39 (1993). [ChCo06] Chudinovich, I., Constanda, C.: Potential representations of solutions for dynamic bending of elastic plates weakened by cracks. Math. Mech. Solids, 11, 494–512 (2006). [ChCo00] Chudinovich, I., Constanda, C.: Variational and Potential Methods in the Theory of Bending of Plates with Transverse Shear Deformation, Chapman & Hall/CRC, Boca Raton, FL (2000). [ChCo05] Chudinovich, I., Constanda, C.: Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes, Springer, London (2005). [ChCoVe04] Chudinovich, I., Constanda, C., Colín Venegas, J.: The Cauchy problem in the theory of thermoelastic plates with transverse shear deformation. J. Integral Equations Appl., 16, 321–342 (2004). [LiMa72] Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. 1, Springer, Berlin (1972).

Almost Periodicity in Semilinear Systems C. Corduneanu

1 A Result in the Linear Case In our paper [Co11], the linear system, associated to (S), has been investigated: x(t) ˙ = Ax(t) + f (t), t ∈ R,

(1)

where A stands for a constant n by n matrix, with complex entries, while f (t) ∈ APr (R, C n ). The following result has been proven in [Co11]. Theorem 1. Assume the following conditions are satisfied by the system (1): (a) det(A − iω I) = 0, ω ∈ R.

(2)

(b) f ∈ APr (R, C n ).

(3)

Then, for each f ∈ APr (R, C n ), 1 ≤ r ≤ 2, there exists a unique solution of (1), x(t) ∈ APr (R, C n ). Moreover, there exists M > 0, depending only on A and r such that (4) |x(t)|r ≤ M| f |r . The proof is provided in our paper [Co11], except for the inequality (4), which follows from the estimate (4.20) in [Co11]. Namely, it has been proven in [Co11] that |xk |r ≤ m| fk |r , k ≥ 1, (5) where f∼

∞

∑ fk exp(iλkt),

k=1

x∼

∞

∑ xk exp(iλkt).

(6)

k=1

C. Corduneanu The University of Texas at Arlington, TX, USA, and the Romanian Academy, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_13, © Springer Science+Business Media, LLC 2011

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Indeed, from (5) one derives the inequality ∞

∞

k=1

k=1

∑ |xk |r ≤ m ∑ | fk |2,

(7)

which implies, taking into account the norm in APr (R, C n ), the inequality (5), with M = m1/r .

2 The Semilinear Case We will consider now the system (S) in APr (R, C n ), and use Theorem 1 and the contraction mapping principle to obtain the existence and uniqueness of solution. The following hypotheses will be imposed on the “nonlinear” part f x: (c) f : APr (R, C n ) → APr (R, C n ) satisfies the Lipschitz-type condition | f x − f y|r ≤ λ |x − y|r ,

(8)

where λ > 0 is small enough (to be made precise in the proof). We can state now the following Theorem 2. Under assumptions (a) on A, and (c) on f , there exists a unique x ∈ APr (R, C n ), satisfying (1). Remark 1. Since for any function x ∈ AP2 (R, C n ) ⊃ APr (R, C n ), 1 ≤ r < 2, it follows that any solutions of (S) will satisfy the system in Carathéodory’s sense, i.e., almost everywhere. See our book [Co09], where AP2 (R, C n ) is shown to consist of locally square integrable functions and, therefore x˙ ∈ AP2 (R, C ) enjoys the same property. Proof of Theorem 2. One considers the operator T : APr (R, C ) into itself, by letting T x = y, with x and y related by the equation y˙ = Ay + ( f x)(t),

(9)

with y ∈ APr (R, C ) the unique solution of (9) in APr (R, C ), according to Theorem 1. If T x = y and Tu = z, then v = y − z satisfies v˙ = Av + ( f x)(t) − ( f u)(t),

(10)

which is linear in v and of the form (1). Applying (4) to (10), and taking into account (8), one obtains the inequality |v|r = |T x − Tu|r ≤ λ M|x − u|r , for any x, u ∈ APr (R, C n ). Therefore, if we assume further

(11)

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143

λ < M −1 ,

(12)

then it follows that T is a contraction mapping on APr (R, C ). As mentioned above, M = M(A, r). This ends the proof of Theorem 2, on the basis of the Banach principle. Remark 2. Similar results to Theorem 2 are valid for the classical spaces of almost periodic functions like AP1 (R, C n ), AP(R, C n ) or S(R, C n ). Moreover, in case f ∈ S, one obtains a better result, namely, x ∈ AP(R, C n ). It would be interesting to investigate if f ∈ APr (R, C n ) could imply x ∈ APs (R, C n ) with s > r. Such a result would mean an improvement, since APs ⊂ APr for s > r. For the classical spaces see the author’s book [Co73].

3 An Integro-Differential System Let us now consider the system 

x(t) ˙ = R

k(t − s)x(s)ds + f (t),

(13)

where k : R → L (Rn , Rn ), and |k| ∈ L1 (R, R).

(14)

If one assumes that f ∈ APr (R, C n ), then it appears natural to look for a solution x ∈ APr (R, C ) to (13). In [Co11] we have established the inequalities |k ∗ x|r ≤ |k|L1 |x|r , x ∈ APr ,

(15)

with k ∗ x the convolution product as in (13), for r = 1, and  ∞  k(s) exp(−λ j s)ds x j exp(iλ j t), (k ∗ x)(t)  ∑ j=1

(16)

R

in case 1 < r ≤ 2. Let us rewrite (13) in the form (13)

x(t) ˙ = (k ∗ x)(t) + f (t), t ∈ R,

with k ∗ x given by (16). We now substitute into (13) the Fourier series, which leads to the following identity:  ∞  ∞ iλ j x j exp(iλ j t) ∼ ∑ k(s) exp(−iλ j s)ds x j exp(iλ j t) + ∑ f j exp(iλ j t). j=1

R

Further, we obtain the equations

j=1

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iλ j x j = and finally

 R

 k(s) exp(−iλ j s)ds x j + f j , j ≥ 1,

   iλ j − k(s) exp(−iλ j s)ds x j = f j , j ≥ 1.

(17)

R

It is now obvious that by imposing the condition        (d) det iω I − k(s) exp(−iω s)ds  ≥ m > 0, ω ∈ R, R

each x j is uniquely determined, for any f ∈ APr (R, C n ), i.e., for any sequence {λ j } ∈ R. The condition (d) can be rewritten in the form  

 (d) det iω I − k(iω )  ≥ m > 0, ω ∈ R, (18) where k(iω ) is the Fourier transform of the kernel k in (13) or (13) : k(iω ) =

 R

k(s) exp{−iω s}ds, ω ∈ R.

Condition (18) allows us to uniquely determine each x j , j ≥ 1, from the system (17), and also leads to the estimate |x j | ≤ m−1 | f j |, j ≥ 1.

(19)

At this point in the discussion, we are proceed as in Theorem 1 and conclude that f ∈ APr (R, C n ) implies x ∈ APr (R, C n ), where f and x are like in (13) . Moreover, an estimate of the form (4) is valid, with M = m−1/r . Summarizing the above discussion on system (13) , we can state the following result. Theorem 3. Assume the following conditions hold for (13) : k(iω )]| ≥ m > 0, ω ∈ R. (d) | det[iω I − (e) |k| ∈ L1 (R, R), f ∈ APr (R, C n ). for fixed r ∈ [1, 2], and k ∗ x defined by (16). Then there exists a unique solution x ∈ APr (R, C n ), for any f ∈ APr (R, C ), 1 ≤ r ≤ 2. The proof of Theorem 3 being provided above, we shall concentrate on condition (d).

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Remark 3. It is obvious that condition (d) above is the equivalent of condition (a) in Theorem 1, serving the same purpose. It is more complex, and it would be interesting to see whether or not it can be replaced by its weaker counterpart k(iω ))| > 0, ω ∈ R. | det(iω I −

(20)

Actually, (20) must be valid only when ω runs over the set {λk ; k ≥ 1} of Fourier exponents of f . When f is fixed in APr (R, C n ), we deal with much less than required in (20). If we want the result to be valid for any f ∈ APr (R, C n ), then (20) must be verified in full, due to the arbitrariness of the Fourier exponents in case of almost periodic functions. Let us consider this problem. Since | k(iω )| → 0 as |ω | → ∞, on behalf of assumption (e) in Theorem 3, and due to the continuity of the value of a determinant with respect to its entries, we can write | det(iω I − k(iω ))| > m1 , |ω | > M1 .

(21)

Indeed, the dominating term is |ω |n , and (21) follows immediately. But in the closed k(iω )) is a continuous function of ω , and therefore, interval |ω | ≤ M1 , det(iω I − having | det(iω I − k(iω ))| > 0, there is a positive minimum of this quantity, say m2 > 0. These estimates lead to the conclusion k(iω ))| = m > 0, ω ∈ R, inf | det(iω I − with m = min{m1 , m2 }. Hence, condition (d) in Theorem 3 can be replaced with its weaker form (20).

4 The Semilinear Equation Associated with (13) The equation we want to study is x(t) ˙ = (k ∗ x)(t) + ( f x)(t),

k,t ∈ R,

(22)

with k ∗ x defined by (16), and f x as in Theorem 1. Let us formulate now the condition guaranteeing the existence and uniqueness of solution in APr (R, C n ). Theorem 4. Assume the following conditions are satisfied by (22): (g) |k| ∈ L1 (R, R), and inequality (20) takes place; (h) f satisfies hypothesis (c) in Theorem 2, with a fixed r, 1 ≤ r ≤ 2. Then, there exists a unique solution x ∈ APr (R, C n ) to (22). Proof. It is similar to that of Theorem 2, based on contraction principle. We notice that the estimates (19) imply an estimate of the form (4), i.e., |x(t)|r ≤ M| f |r ,

(23)

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for f and x related by (13) . If we denote by T the operator from APr (R, C n ) into itself, defined by y(t) ˙ = (k ∗ y)(t) + ( f x)(t), t ∈ R, then for v = y − z = T x − Tu one obtains v˙ = (k ∗ v) + ( f x)(t) − ( f u)(t), which leads to the inequality |T x − Tu|r ≤ λ M|x − u|r ,

(24)

after using the Lipschitz condition (8). For λ satisfying (12), one obtains the result of Theorem 4, which ends the proof. Remark 4. In dealing with the system (S), we have considered two particular choices for the linear operator A: the matrix case and the convolution type operator. A problem arising in this context is to find other linear operators acting on APr (R, C n ), such that the linear equation x(t) ˙ = (Ax)(t) + f (t) has unique solution in APr (R, C n ), for each f in the same space. Then, the associated semilinear equation will enjoy the same property. Remark 5. It is certain that the equations we have investigated possess other solutions, besides the almost periodic one. It is interesting to build up such solutions and compare them with the one already emphasized by the above results. Remark 6. Another approach to the investigation consists in considering the equation/system of the form x(t) = (k ∗ x)(t) + f (t), (E) instead of (13) . The semilinear case is described by the equation x(t) = (k ∗ x)(t) + ( f x)(t),

(E)

and the condition (20) should be replaced by det(I − k(iω )) = 0, ω ∈ R.

References [Co11] Corduneanu, C.: A scale of almost periodic functions spaces. Differential and Integral Equations, 24 (2011), 1–27. [Co09] Corduneanu, C.: Almost Periodic Oscillations and Waves, Springer, New York (2009). [Co73] Corduneanu, C.: Integral Equations and Stability of Feedback Systems, Academic Press, New York (1973).

Bubble Behavior Near a Two Fluid Interface G.A. Curtiss, D.M. Leppinen, Q.X. Wang, and J.R. Blake

1 Introduction The influence of rigid and free boundaries on the dynamics of bubbles has been researched extensively, both experimentally and theoretically. Experiments by [Be66] showed that the presence of a solid boundary caused the formation of a liquid jet through the bubble, forming a toroidal bubble. This behavior has been observed in many other experiments since, including [Br02, Pi98, To86, La75]. Similar behavior is also observed when a bubble collapses near a free surface. In such conditions bubble jetting may be directed away from the surface, with a counter-jet forming out of the free surface. Experiments using spark-generated bubbles by [Bl81] under free fall conditions and [Ch80] in standard gravity showed this counter-jet to be greatly influenced by the standoff distance. Bubbles formed very close to the surface generate severe vertical surface spikes, and those at greater distances create much smaller and smoother deformations to the surface. Numerical investigations into both of these phenomena have been undertaken using boundary integral methods to great effect. Simply connected simulations were first carried out by [Le76], and [Ta85, Bl86] into cavitation collapse near rigid boundaries. These results showed jet direction in the absence of gravity to be diG.A. Curtiss University of Birmingham, UK, e-mail: [email protected] D.M. Leppinen University of Birmingham, UK, e-mail: [email protected] Q.X. Wang University of Birmingham, UK, e-mail: [email protected] J.R. Blake University of Birmingham, UK, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_14, © Springer Science+Business Media, LLC 2011

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rected toward the wall, and that the expansion phase of the bubbles is of importance for the later collapse phase, in spite of the high level of sphericity observed about the maximum bubble volume. Simulations of behavior near free surfaces were also carried out by [Ta85, Bl87] with some success. More accurate implementations have since modeled pre-toroidal behavior near free surfaces with a high degree of accuracy [Wa96, Pe03, Ro01]. This paper seeks to investigate the behavior of high pressure cavitation bubbles in the vicinity of a two fluid interface, that is, the interface between two immiscible liquids of different densities. Where appropriate this is continued into the toroidal phase using the vortex ring method of [Lu91]. Previously [Kl04a] have used a boundary integral approach to examine the dynamics of a single bubble near a density interface where the authors decoupled the normal velocity of the fluid– fluid interface from the discretized matrix equations using geometrical arguments derived from the equal density case. This does not decrease the time required to calculate the Green’s function block matrices, yet it does significantly decrease the size of the matrix to be inverted. Their results showed various types of behavior in the presence of gravity, including jetting toward and away from the interface, and bubble pinching near the null impulse line, in agreement with the model presented herein. A further advancement has been presented to account for a linearly elastic fluid in the uncavitated layer, by means of a displacement parameter on the fluid–fluid interface [Kl04b]. The results obtained are in agreement with experiments involving PAA samples by [Br01a, Br01b]. In the current work we extend the results of [Kl04a] by implementing a mathematical model to examine the evolution of multiple bubbles on either side of the interface without making any unnecessary assumptions about the fluid flow. In Sect. 2 we introduce the mathematical model, with details of the numerical implementation given in Sect. 3. The model validation is briefly discussed in Sect. 4 and results are presented in Sect. 5. The overall conclusions are made in Sect. 6.

Fig. 1 (a) A sketch of bubbles interacting with a density interface with fluid of density ρ1 in fluid 1 underlying fluid of density ρ2 in fluid 2. (b) The axisymmetric geometry employed in the numerical implementation

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2 Mathematical Formulation The problem we are considering is sketched in Fig. 1(a) with a layer of fluid of density ρ1 underlying a layer of fluid of density ρ2 with ρ1 > ρ2 . In each layer the fluid is assumed to be inviscid, incompressible and irrotational so that the velocity field can be written as the gradient of a potential with

for i = 1, 2 where and

u i = ∇ φi

(1)

∇ · ui = ∇2 φi = 0

(2)

∇ × ui = 0.

(3)

There is at least one bubble in either the upper or lower layer, with an arbitrary number of additional bubbles. Our objective is to examine the interaction of the bubbles with the density interface and with each other. With the above assumptions Bernoulli’s equation is valid everywhere in both exterior fluids which gives p p∞ ∂ φi |ui |2 + + + g(z − zo ) = ∂t 2 ρi ρi

(4)

where t is time, p is pressure, g is the acceleration of gravity, z is the vertical coordinate of a fluid particle with reference to the initial vertical coordinate of the interface zo and p∞ is the far field pressure at zo . The exterior fluid layers are taken as isothermal. Furthermore it is assumed that no mass or heat transfer will occur across the gas/liquid boundaries. Thus the bubble gases obey an adiabatic law. The pressure inside each bubble is given by  γ V0 , (5) p b = pv + p 0 V where pv represents the condensible gas vapor pressure, V, V0 are the bubble volume and initial reference volume, respectively, p0 is the initial reference pressure for the bubble at reference volume and γ is the ratio of specific heats of the incondensible gas (taken as 1.4 in this paper). The pressure of the surrounding liquid at the bubble surface is given by the Young–Laplace condition as p = pb − σb,i ∇ · nb

(6)

where σb,i is the surface tension of a bubble in fluid i and nb is the outward pointing unit normal to the bubble. The pressure jump across the density interface is also calculated using the Young–Laplace condition as p1 = p2 − σI ∇ · n1

(7)

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where σI is the surface tension of the density interface and n1 is the unit normal pointing from fluid 1 into fluid 2. In this paper p∞ is taken as constant, and the early bubble behavior observed is the result of an initially high internal gas pressure, p0  p∞ − pv . This is valid for spark and laser generated bubbles, where the initial pressure may reach many thousands of atmospheres. We proceed by non-dimensionalizing our equations using the following scales: the density of the lower layer, ρ1 , for density; the difference in the far field pressure and the condensible gas vapor pressure, Δ p = p∞ − pv , for pressure; the maximum radius a bubble would reach in an infinite fluid of density ρ1 , Rmax for length; Rmax (ρ1 /Δ p)1/2 for time; (Δ p/ρ1 )1/2 /Rmax for potential; Rmax Δ p for surface tension. Thus along a bubble interface in fluid 1 the Bernoulli equation becomes  γ ∂ φ1 |u1 |2 V0 + = 1−ε − δ (z − z0 ) + σb,1 ∇ · nb (8) ∂t 2 V where ε = p0 /Δ p and δ = ρ1 gRmax /Δ p and all variables are now dimensionless. The corresponding result in fluid 2 is   γ  ∂ φ2 |u2 |2 V0 1 + = 1−ε (9) − δ (z − z0 ) + σb,1 ∇ · nb ∂t 2 ρ V where ρ = ρ1 /ρ2 . Along the density interface we use (4) and (7) to obtain     ∂ φ1 |u1 |2 ∂ φ2 |u2 |2 + −ρ + = δ z (1 + ρ ) − σI ∇ · n1 . ∂t 2 ∂t 2

(10)

3 Numerical Implementation The general problem considered in Fig. 1(a) is fully three-dimensional. For computational convenience we consider the axisymmetric problem sketched in Fig. 1(b) where (b, c) refer to points on a bubble surface in fluid 1, (d, e) refer to points on a bubble surface in fluid 2 and (p, q) refer to points on the interface between fluids 1 and 2. The substantial derivatives on all surfaces are given by D ∂ = + u · ∇, Dt ∂t

(11)

where u = ui on the bubble surface in fluid i and u = (u1 + u2 )/2 on the two fluid interface. Particles on the bubble surfaces and the density interface are treated as material points and are advected according to db = u1 , dt For convenience we define

dd = u2 , dt

dp u1 + u2 = . dt 2

(12)

Bubble Behavior Near a Two Fluid Interface

F(p) = (φ1 (p) + φ2 (p))(1 − ρ ) + (φ1 (p) − φ2 (p))(1 + ρ )

151

(13)

as the density-weighted potential along the density interface which satisfies the evolution equation DF(p) = (1 − ρ ) (u1 · u2 ) − 2(1 + ρ )δ z + 2σI ∇ · n1 . Dt Along the bubble interfaces in fluid 1 we have  γ V0 Dφ1 (b) |u1 |2 = +1−ε − δ (z − z0 ) + σb,1 ∇ · nb Dt 2 V and along the bubble interfaces in fluid 2 we have   γ  V0 Dφ2 (d) |u2 |2 1 − δ (z − z0 ) + σb,2 ∇ · nb . 1−ε = + Dt 2 ρ V

(14)

(15)

(16)

Our numerical procedure is as follows. We place a discrete series of nodal points (b, d, p) along the bubble surfaces in fluid 1 and 2, and along the density interface, respectively. We then time step (using a variable order Runge–Kutta scheme) (12) to update the surface locations and (14)–(16) to update the nodal potentials. The nodal velocities are calculated using ui = ∇φi where we use the boundary integral technique to calculate the normal gradient of the velocity potentials and splining techniques to calculate the tangential derivatives. In particular if ∇2 φ = 0 in some domain D with boundary ∂ D then we can write    ∂ φ (x) ∂ G(x0 , x) − φ (x) dS (17) c(x0 )φ (x0 ) = G(x0 , x) ∂n ∂n ∂D

where G(x0 , x) is the free space Green’s function given by G(x0 , x) = 1/|x − x0 |, ∂ G(x0 , x)/∂ n is the normal derivative of the Green’s function with respect to the outward normal of the fluid domain and x ∈ ∂ D. In our implementation all of the surfaces are represented by quintic splines and are smooth so that c(x0 ∈ D\∂ D) = 4π , c(x0 ∈ ∂ D) = 2π .

(18) (19)

Equation (17) is appropriate in both fluid 1 and fluid 2. In fluid 1 ∂ D is the union of the surfaces of all bubbles located in fluid 1 and of the interface separating fluid 1 and 2. Similarly, in fluid 2 ∂ D is the union of the surfaces of all bubbles located in fluid 2 and of the interface separating fluid 1 and 2. The central portion of the density interface is represented using nodal points. The truncated interface is then extended to infinity using a least squares type approximation [Cu09] with all integrals evaluated analytically along the infinite extension.

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The procedure for discretizing and solving the boundary integral equation given in (17) is well known. Details of the exact numerical implementation, including the non-trivial incorporation of the vortex ring method of [Lu91] for when a bubble becomes toroidal is given in the thesis by Curtiss [Cu09]. Briefly, all quantities of interest along the central portion of the density interface and along the bubble surfaces are represented using quintic splines based upon an arc length formulation. The logarithmic singularities associated with the boundary integral method are removed analytically and all of the resultant integrals are approximated using high-order (typically 20) Gaussian quadrature. In summary, when (17) is discretized along each of the bubble surfaces in fluid 1 and 2, and along the density interface between the fluids, we obtain the following system of equations: ⎤⎡ ⎤ ρ ∂ φ1 (b) Gbc 0 −Gbq 1+ρ DGbq ∂ n1 −1 ⎥⎢ ⎥ ⎢ 0 Gde Gdq ∂ φ2 (d) ⎥⎢ ⎥ ⎢ 1+ρ DGdq ∂ n2 ⎥⎢ ⎥ ⎢ ρ −1 ∂ φ (p) G G 0 DG − 2 π I 2 ⎦ ⎣ ⎦ ⎣ pc pe pq pq ρ +1 ∂ n 2 1−ρ G pc −G pe −2G pq DG pq + 2π 1+ρ Ipq φ1 (p) + φ2 (p) ⎡ ⎤ −1 0 2π Ibc + DGbc ⎤ 2+2ρ DGbq ⎡ −1 ⎢ ⎥ φ1 (b) 0 2 π I + DG DG de de 2+2ρ dq ⎥ ⎢ =⎢ ⎥ ⎣ φ2 (d) ⎦ −1 DG pc DG pe DG ⎣ ⎦ pq 1+ρ F(p) DG pc −DG pe 2π 1+1 ρ Ipq ⎡

(20)

where G is used to indicate the discretized Green’s function and DG the discretized normal derivative of the Green’s function. The block matrix structure is used to emphasize the bubble/bubble, bubble/interface and interface/interface contributions to the overall result with reference to the notation introduced in Fig. 1(b). The RHS of (17) is known at any time step and the resultant set of equations can be inverted to determine the normal velocities ∂ φ1 (b)/∂ n1 , ∂ φ2 (d)/∂ n2 , ∂ φ2 (p)/∂ n2 and the sum φ1 (p) + φ2 (p) and hence to determine the velocity along all interfaces.

4 Validation The interaction between single/multiple bubbles and a density interface can be examined by time-stepping (12) and (14)–(16), with the interfacial velocities determined by inverting (20) and using splining techniques to calculate tangential velocities. The current numerical implementation has been validated by comparison with the case of ρ = 1 with σi = 0 (i.e. the case of a Rayleigh–Plesset bubble in an infinite ambient liquid) and with previous experimental results for the interaction of a bubble with a free surface [Bl81] (i.e. the case of ρ = 0) and a rigid boundary [Br02] (i.e. the case of ρ = ∞). Full validation details are available [Cu09]. Arguably the best validation of the numerical implementation is with reference to

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Fig. 2 Comparisons between experiments near a water/white spirit interface with ρ = 0.76, δ = 0.0147 by [Ch80], used with permission, and the numerical code for h = 0.87 (top) and h = 2.2 (base). The final frames in both cases are where the simulation ends due to either non-axial jet impact (top) or jet disconnection (base), and the resultant time of the final frames for h = 2.2 are displaced. Other parameters are estimated as ε = 100, σb = 0.001

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the experiments of Chahine and Bovis [Ch80] of the interaction of a bubble with a water/white spirit interface with a non-trivial value of ρ = 0.76 as seen in Fig. 2. In this case a spark-generated centimeter-sized bubble is formed in the water layer. Figure 2 compares two of these experiments with BIM simulations acting under gravity with a buoyancy parameter of δ = 0.0147. With the shallow standoff of h = 0.87, the simulation is halted after one oscillation as a non-axial jet impact occurs. In the deeper standoff case of h = 2.2, a buoyancy driven jet forms, leading to an axial jet impact and is hence simulated further using the toroidal bubble model. In both cases excellent agreement is seen between the experiment and the simulation.

Fig. 3 The evolution of the bubble and fluid–fluid interface with h = 0.5 for density ratios ρ = 0, 0.1, 0.2, 0.3, 0.4 with the smaller density ratios showing greater surface deformation and counterjetting in the bubble

5 Results The numerical model above has been used to examine the interaction of a bubble (or bubbles) with a density interface. The direct output from (12) is the time evolution of all of the free surface shapes. In addition, information concerning the pressure field and the velocity field in fluids 1 and 2 can be obtained using (8), (9) and (17) as detailed in [Cu09]. Below we will give some examples of the results that can be obtained using the current numerical method. Figure 3 shows the time evolution of

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a single bubble interacting with a density interface. Figure 4 shows the interaction of two bubbles below a density interface. Figure 5 shows the interaction of two bubbles on opposite sides of a density interface. Figure 6 shows the interaction of two bubbles below and one bubble above a density interface. In all cases the symbol h refers to the standoff distance of the bubble which is distance of the centroid of the initially spherical bubble from the fluid–fluid interface divided by the maximum radius the bubble would obtain in an infinite quiescent ambient liquid given its initial internal pressure. Figure 3 shows the evolution of the density interface and bubble shape for the case of one bubble with h = 0.5 for ρ = 0, 0.1, 0.2, 0.3 and 0.4. The case of ρ = 0 corresponds to the interaction of a bubble with a free surface where an intense upwards jet is formed on the density interface with a corresponding counter-jet on the bubble surface. As the density ratio increases away from the free surface case, the intensity of the free surface jet and the counter-jet decreases. The bubbles become toroidal when the counter-jet pierces the lower face of the bubble. The simulations proceed into the toroidal regime until they become numerically unstable. While it is not shown here, simulations for ρ > 1 approach the known limit for the interaction of a bubble with a rigid boundary in the limit as ρ → ∞.

Fig. 4 Collapse and re-expansion of two bubbles on the same side of a density interface with ρ = 0.5 and h1 = 1.5, h2 = 4.5

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Figure 4 follows the evolution of two bubbles on the same side of a density interface with ρ = 0.5 for the case of h1 = 1.5 and h2 = 4.5. We consider the case of ε = 100 with γ = 1.4 so that initially the bubbles expand spherically to their maximum radius and then they collapse. As they collapse the bubbles strongly interact with each other. While the density interface remains predominantly horizontal throughout jetting occurs in both bubbles during the collapse and then reexpansion phase and ultimately both bubbles become toroidal. The relative strength of the bubble–bubble interaction and the bubble–interface interactions depends on the standoff distances and the density ratio.

Fig. 5 Two bubble interactions about an interface with ρ = 1.3. The initial standoff distances are h1 = 0.75 in the lower layer and h2 = 1.5 in the top layer

An interesting interaction with one bubble on each side of the density interface is shown for ρ = 1.3 in Fig. 5 where the standoff distance of the bubble in the lower layer is h1 = 0.75 and the standoff distance of the bubble in the upper layer is h2 = 1.5. The deformation of both the upper and lower bubble is significantly different at this value of ρ from what would be the case if either of the bubbles was absent. The bubble–bubble interaction and the bubble–interface interaction depends on the value of ρ and increases as the standoff distances decrease. For the case considered the bubble in the upper layer is strongly deformed by the interaction. The upper bubble rapidly becomes toroidal with the bubble jet impacting upon the density

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interface and deforming the lower bubble. This suggests a possible mechanism for the enhancement of mixing across a density interface.

Fig. 6 Three bubble interaction with ε = 100, γ = 1.4, ρ = 1.3 and h1 = 1.5 in the upper layer and h2 = 1.5 and h3 = 4.5 in the lower layer

The case of ρ = 1.3 with one bubble above (h1 = 1.5 in the upper layer) and two bubbles below the density interface (h2 = 1.5, h3 = 4.5 in the lower layer) is shown in Fig. 6. As expected for this value of ρ it is bubble in the upper layer which is most strongly deformed. The downward liquid jet in the upper bubble deflects the density interface downward and deforms the middle bubble. The lower bubble in the dense layer interacts with the bubble above it to form a strong upward axial jet ultimately becoming toroidal.

6 Conclusions We have implemented an axisymmetric multi-bubble numerical model to examine the interaction of bubbles with a density interface. The model is based on the boundary integral method and it has been validated against previous experimental results. A range of interesting physical phenomena have been identified in Figs. 3–6 with further results detailed in the thesis by Curtiss [Cu09]. The model is able to examine bubbles well into the toroidal regime and across the complete range of interfacial density ratios.

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References [Be66] [Bl81] [Bl86] [Bl87] [Br02] [Br01a] [Br01b]

[Ch80]

[Cu09] [Kl04a] [Kl04b] [La75] [Le76] [Lu91] [Pe03] [Pi98] [Ro01] [To86] [Ta85] [Vo80]

[Wa96]

Benjamin, T.B., Ellis, A.T.: The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Phil. Trans. R. Soc. Lond. A, 260, 221–240 (1966). Blake, J.R., Gibson, D.C.: Growth and collapse of a vapour cavity near a free surface. J. Fluid Mech., 111, 123–140 (1981). Blake, J.R., Taib, B.B., Doherty, G.: Transient cavities near boundaries. Part 1. Rigid boundary. J. Fluid Mech., 170, 479–497 (1986). Blake, J.R., Taib, B.B., Doherty, G.: Transient cavities near boundaries. Part 2. Free surface. J. Fluid Mech., 181, 197–212 (1987). Brujan, E.A., Keen, G.S., Vogel, A., Blake, J.R.: The final stage of the collapse of a cavitation bubble close to a rigid boundary. Phys. Fluids, 14, 85–92 (2002). Brujan, E.A., Nahen, K., Schmidt, P., Vogel, A.: Dynamics of laser-induced cavitation bubbles near an elastic boundary. J. of Fluid Mech., 433, 251–281 (2001). Brujan, E.A., Nahen, K., Schmidt, P., Vogel, A.: Dynamics of laser-induced cavitation bubbles near elastic boundaries: Influence of the elastic modulus. J. Fluid Mech., 433, 283–314 (2001). Chahine, G.L., Bovis, A.: Oscillation and collapse of a cavitation bubble in the vicinity of a two-liquid interface. Springer Series in Electrophysics 4 - Cavitation and inhomogeneities in underwater acoustics, 23–29 (1980). Curtiss, G.A.: Non-linear, non-spherical bubble dynamics near a two fluid interface, PhD Thesis, The University of Birmingham (2009). Klaseboer, E., Khoo, B.C.: Boundary integral equations as applied to an oscillating bubble near a fluid–fluid interface. Comp. Mech., 33, 129–138 (2004). Klaseboer, E., Khoo, B.C.: An oscillating bubble near an elastic material. J. Appl. Phys., 96, 5808–5818 (2004). Lauterborn, W., Bolle, H.: Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary. J. Fluid Mech., 72, 391–399 (1975). Lenoir, M.: Calcul numérique de l’implosion d’une bulle de cavitation au voisinage d’une paroi ou d’une surface libre. J. Mécanique, 15, 725–751 (1976). Lundgren, T.S., Mansour, N.N.: Vortex ring bubbles. J. Fluid Mech., 224, 177–196 (1991). Pearson, A.: Hydrodynamics of jet impact in a collapsing bubble. PhD Thesis, The University of Birmingham (2003). Phillip, A., Lauterborn, W.: Cavitation erosion by single laser-produced bubbles. J. Fluid Mech., 361, 75–116 (1998). Robinson, P.B., Blake, J.R., Kodama, T., Shima, A., Tomita, Y.: Interaction of cavitation bubbles with a free surface. J. Appl. Phys., 89, 8225–8237 (2001). Tomita, Y., Shima, A.: Mechanisms of impulsive pressure generation and damage pit formation by bubble collapse. J. Fluid Mech., 169, 535–564 (1986). Taib, B.B.: Boundary integral method applied to cavitation bubble dynamics. Phd Thesis, The University of Wollongong (1985). Vogel, A., Schweigner, P., Frieser, A., Asiyo, M.N., Birngruber, R.: Intraocular Nd:YAG Laser surgery: Light tissue interaction, damage range, and reduction of collateral effects. J. Quant. Elec., 26, 2240–2260 (1990). Wang, Q.X., Yeo, K.S., Khoo, B.C., Lam, K.Y.: Nonlinear interaction between gas bubble and free surface. Computers and Fluids, 25, 607–628 (1996).

Spectral Stiff Problems in Domains with a Strongly Oscillating Boundary D. Gómez, S.A. Nazarov, and E. Pérez

1 Introduction and Preliminaries Let Ω be a bounded domain of R2 with a smooth boundary Γ and let (ν , τ ) be the natural orthogonal curvilinear coordinates in a neighborhood of Γ : τ is the arc length and ν the distance along the outer normal to Γ . Also let  denote the length of Γ and κ(τ ) denote the curvature of the curve Γ at the point τ . We assume that the domain Ω is surrounded by a curvilinear strip ωε of variable width O(ε ) where ε > 0 is a small parameter. Let Ωε be the domain Ωε = Ω ∪ ωε ∪ Γ and Γε the boundary of Ωε . We consider two different types of bands ωε . First, we assume ωε to be defined by ωε = {x : 0 < ν < ε h(τ )} (1) with h a strictly positive function of the variable τ , -periodic and h ∈ C∞ (S ) where S stands for the circumference of length  (see Fig. 1). Second, we assume ωε to be defined by ωε = {x : 0 < ν < ε hε (τ )} (2) where hε (τ ) = h(τ /ε ), with h a positive, P-periodic (P a positive constant), C∞ function in R; namely, a domain with strongly oscillating boundary (see Fig. 2). We consider the spectral Neumann problem in Ωε for a second order differential operator with piecewise constants coefficients:

D. Gómez Universidad de Cantabria, Santander, Spain, e-mail: [email protected] S.A. Nazarov Institute for Problems in Mechanical Engineering, St. Petersburg, Russia, e-mail: [email protected] E. Pérez Universidad de Cantabria, Santander, Spain, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_15, © Springer Science+Business Media, LLC 2011

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⎧ −AΔxU ε = λ ε U ε ⎪ ⎪ ⎪ ⎪ ⎪ −aε −t Δx uε = λ ε ε −1 uε ⎪ ⎪ ⎨ U ε = uε ⎪ ⎪ ⎪ A∂ν U ε = aε −t ∂ν uε ⎪ ⎪ ⎪ ⎪ ⎩ −t aε ∂n uε = 0

in Ω , in ωε , on Γ ,

(3)

on Γ , on Γε .

Fig. 1 A geometrical configuration of Ωε

Fig. 2 A geometrical configuration of Ωε with a strongly oscillating boundary

In (3), A and a are two positive constants while ∂ν and ∂n denote the derivatives along the outward normal vectors ν and n to the curves Γ and Γε , respectively, and t is a parameter such that 0 ≤ t < 1. We study the asymptotic behavior, as ε → 0, of the eigenvalues λ ε of (3), in the range of the low frequencies, and of their corresponding eigenfunctions {U ε , uε }.

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Spectral stiff Neumann problems in domains surrounded by thin bands ωε have been considered in [Go06a] and [Go06b] where the thin band ωε is both stiff and heavy. As a matter of fact, problem (3) where ωε is defined by (1) fits into the case where m = 1 − t in the set of problems with (3)1 , (3)3 , (3)4 , (3)5 , and − aε −t Δ x uε = λ ε ε −t−m uε

in

ωε

(4)

for different values of t and m, provided that t ≥ 0,

t + m ≥ 0,

and either t > 0 or t + m > 0.

The parameters t and t + m reflect the relative stiffness of the band and the dead weight of the band, respectively, in mechanical problems. These stiff problems have been introduced in [Go06a] and are of interest, for instance, in the study of reinforcement problems for solid media and in vibrations for a two-phase system in fluid mechanics. A characterization of the limiting problems for the eigenpairs of (3)1 , (4), (3)3 , (3)4 , (3)5 , for the different values of t and m has been obtained in [Go06a] by means of asymptotic expansions. In [Go06a], we also provide sharp bounds for convergence rates of the eigenpairs {λ ε , {U ε , uε }} in the case where t = 1 and m = 0 by using the so-called inverse-direct reduction method. A different approach for the eigenpairs is provided in [Go06b] for the case where t > 1 and m = 0 where, in addition to the convergence, a complete asymptotic expansion for the eigenpairs has been obtained, and a connection of this problem with Wentzell problems with small parameters has been shown. Asymptotics for the middle and high frequencies have been considered in [GoNa11]. We refer to [Go06a] and [Go06b] for a review of previous works in the literature on the subject which in fact are scarce. In contrast, this is the first time in the literature that the boundary homogenization of a heavy stiff band surrounding a fixed domain is considered. For the homogenization of a soft oscillating band and a stationary problem, see [BuKo87]. For homogenization of spectral problems in domains with oscillating boundary, let us mention [Sa80, Na90, OlSh92] and [LoSa79]. See [Na07] in connection with the elasticity operator and [Na08] for further references. In this paper, we deal with the case where 0 ≤ t < 1 and m = 1 − t, namely with problem (3) where the strip ωε is defined by (1) and (2). In Sect. 2 we consider the shape of ωε in (1) and in Sect. 3 that in (2). We obtain the limiting problems associated for each case and prove the convergence of the eigenvalues with conservation of the multiplicity toward those of the limiting problems using the Rayleigh quotient and contradictory arguments (see, for instance, Chapter III of [At84] and [GoLo99]). Here, since we are dealing with variable domains we need extension operators (10) for the proof. We show that, for 0 < t < 1 and m = 1 − t, the stiffer band does not influence the limit behavior which is the same when t = 0, that is, the band is only heavy and the density takes the value O(ε −1 ) in the band.

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1.1 Statement of the Problem The variational formulation of (3) for (1) and (2) reads: Find λ ε and {U ε , uε } ∈ H 1 (Ωε ), {U ε , uε } = 0, satisfying 

A



Ω

a ∇xU ε · ∇x G dx + t ∇x uε · ∇x g dx ε ωε    1 ε ε ε U G dx + u g dx ∀{G, g} ∈ H 1 (Ωε ). =λ ε ωε Ω

(5)

Here, and in what follows, we identify a function gε in L2 (Ωε ) (H 1 (Ωε ), respectively) with the pair of functions {G, g} where G stands for the restriction of gε to Ω and g for the restriction of gε to ωε . In particular, the eigenelements formed by the eigenvalues λ ε and the corresponding eigenfunctions uε read (λ ε , {U ε , uε }). For each ε > 0, problem (5) is a standard spectral problem in the couple of spaces H 1 (Ωε ) ⊂ L2 (Ωε ), with a discrete spectrum. Let us consider 0 = λ0ε < λ1ε ≤ λ2ε ≤ · · · ≤ λkε ≤ · · · −−−−−−→ ∞, k→∞

that is, the sequence of eigenvalues repeated according to their multiplicities. Let {{Ukε , uεk }}∞ k=0 be the corresponding eigenfunctions which are subject to the orthonormalization condition   1 ε ε Uk Ul dx + uε uε dx = δk,l (6) ε ωε k l Ω where δk,l denotes the Kronecker symbol and {{Ukε , uεk }}∞ k=0 form a basis in both spaces L2 (Ωε ) and H 1 (Ωε ). Let us denote by (·, ·)ε the scalar product defined by the left-hand side of (6) in L2 (Ωε ), that is, ({U, u}, {G, g})ε =

 Ω

U G dx +

1 ε

 ωε

u g dx

∀{U, u}, {G, g} ∈ L2 (Ωε ).

(7)

On the other hand, on account of the continuity of the function h and of the curvature κ, for a certain sufficiently small d > 0, there exist constants c, C1 , C2 and C3 independent of ε such that 0 < c < K(ν , τ ) < C1 |1 − K(ν , τ )| ≤ C2 ε where

and

∀ν ∈ [−d, d], τ ∈ S ,

|1 − K(ν , τ )−1 | ≤ C3 ε K(ν , τ ) = 1 + ν κ(τ )

(8)

∀ν ∈ [0, ε h(τ )], τ ∈ S , (9)

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denotes the Jacobian of the transformation from (x1 , x2 ) to (ν , τ ). Here and in the sequel, c,C,Ci denote different constants independent of ε . In (9), h(τ ) is hε (τ ) in the case of (2). For each F ∈ C∞ (Ω ), let us define the function {F, f˜} ∈ H 1 (Ωε ) where f˜(x) = F(0, τ ) for x ∈ ωε .

(10)

Here, we refer to F(ν , τ ) as the function F(x) written in curvilinear coordinates, and, if no confusion arises, we do not distinguish between a point τ on the boundary Γ and its coordinate along Γ .

2 The Case ωε = {x : 0 < ν < ε h(τ )} In this section we prove the convergence, as ε → 0, of the eigenpairs of (5) toward those of the resulting problem (21) where ωε is defined by (1) (cf. Theorem 1). Let us first introduce notation and results for further use. In this connection, in the following Lh2 (Γ ) denotes the space L2 (Γ ) with the scalar product defined by  Γ

hFG d τ

∀F, G ∈ L2 (Γ ) .

Considering (10), by virtue of (8) and (9), it is clear that

f˜ L2 (ωε ) ≤ Cε 1/2 F L2 (Γ ) , h

(11)

∇x f˜ L2 (ωε ) ≤ Cε 1/2 ∂τ F L2 (Γ ) . h

Let us also introduce the following three inequalities which will be useful throughout this paper:  T Z(0) − 1 Z(t) dt ≤ T 1/2 Z  L2 (0,T ) ∀Z ∈ H 1 (0, T ), T > 0, (12) T 0  ε h(τ ) 0

|u(ν , τ )−u(0, τ )|2 d ν ≤ ε 2 h(τ )2

 ε h(τ ) 0

|∂ν u(ν , τ )|2 d ν

and

U −CU 2L2 (Ω ) + U −CU 2L2 (Γ ) ≤ C ∇xU 2L2 (Ω ) h

∀u ∈ H 1 (ωε ) (13)

∀U ∈ H 1 (Ω )

(14)

where CU denotes the constant defined by    1

U dx + hU d τ . CU =

Ω Γ Ω dx + Γ h d τ The two first inequalities are obtained from the Newton–Leibnitz formula (cf. [Go06a] for instance). In order to prove the Sobolev type inequality (14), we con-

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sider the Steklov-type eigenvalue problem    ˜ ∇xV ∇x G dx = λ V G dx + hV Gd τ Ω

Ω

Γ

∀G ∈ H 1 (Ω ).

(15)

Taking into account that the eigenfunctions corresponding to λ˜ 0 = 0 are the constants and that, for all U ∈ H 1 (Ω ), U − CU is orthogonal to the constants with the scalar product defined by (24), the minimax principle ensures that

2 dx Ω |∇x G|

2 2 Ω |G| dx + Γ h|G| d τ

λ˜ 1 = min

G∈E˜0⊥ G=0



≤

Ω

|U

2 dx Ω |∇xU|

2 −CU | dx + Γ h|U

−CU |2 d τ

,

where E˜0⊥ denotes the space of the functions of H 1 (Ω ) which are orthogonal to the constants. We obtain the following bound for the eigenvalues of (3). Lemma 1 Let λkε be the eigenvalues of (3) with 0 ≤ t < 1 and ωε be defined by (1). For each fixed k ∈ N, and ε sufficiently small, we have 0 < C ≤ λkε ≤ Ck

(16)

where C,Ck are constants independent of ε and Ck → ∞ when k → ∞. Proof. The minimax principle gives the equalities 



a A |∇xV | dx + t |∇x v|2 dx ε ωε Ω   , max 1 {V,v}∈Ek |V |2 dx + |v|2 dx {V,v}=0 ε ωε Ω 2

λkε =

min

Ek ⊂H 1 (Ωε ) dim Ek =k+1

(17)

where the minimum is taken over all the subspaces Ek ⊂ H 1 (Ωε ) with dim Ek = k + 1. ∞ Let {λi }∞ i=0 be the eigenvalues of the Dirichlet problem in Ω and {Vi }i=0 the corresponding eigenfunctions which are assumed to form an orthonormal basis in L2 (Ω ). For each fixed k, let us denote by Ek∗ the particular subspace of H 1 (Ωε ), Ek∗ = [{V0 , 0}, . . . , {Vk , 0}] ⊂ H 1 (Ωε ), where {Vi , 0} denotes the extension of Vi to Ωε by 0 in ωε , for i = 0, 1, . . . , k and [. . . ] stands for a linear space. Then, from (17), we derive  A |∇xV |2 dx λkε ≤ max ∗ Ω = λk . {V,v}∈E 2 k |V | dx {V,v}=0 Ω

This inequality provides the right-hand side of (16). As regards the left-hand side of (16), it is enough to prove it for k = 1, namely λ1ε > C > 0. Let {U1ε , uε1 } be an eigenfunction of (5) corresponding to λ1ε and satisfying (6). Then,

Stiff Problems in Domains with a Strongly Oscillating Boundary

A ∇xU1ε 2L2 (Ω ) +

165

1

∇x uε1 2L2 (ωε ) = λ1ε εt

(18)

and the proof is completed by showing that ∇xU1ε L2 (Ω ) > C > 0 for sufficiently small ε . Introducing in (6) the curvilinear coordinates in the integral in ωε and using inequalities (9) and (13) and the fact that U1ε = uε1 on Γ , we have 1 ≤ C( U1ε 2L2 (Ω ) + U1ε 2L2 (Γ ) + uε1 2L2 (ωε ) + ε ∇x uε1 2L2 (ωε ) ).

(19)

h

In order to estimate the two first terms on the right-hand side of (19), we use (14) and the definition of CU1ε . Then,

U1ε 2L2 (Ω ) + U1ε 2L2 (Γ ) = U1ε −CU1ε 2L2 (Ω ) + U1ε −CU1ε 2L2 (Γ ) + (CU1ε )2 [



Ω

dx +



h



ε 2 2 Γ h d τ ] ≤ C ∇xU1 L2 (Ω ) + (CU1ε ) [

h

Ω

dx +



Γ

h dτ ] .

Besides, taking into account the orthogonality condition of {U1ε , uε1 } to the constants for the scalar product (7) (cf. (6)), (3)3 and inequalities (9) and (12) yields    1 1 ε ε

CU1ε =

hu1 d τ − u dx ≤ Cε 1/2 uε1 H 1 (ωε ) ε ωε 1 Γ Ω dx + Γ h d τ and consequently

U1ε 2L2 (Ω ) + U1ε 2L2 (Γ ) ≤ C( ∇xU1ε 2L2 (Ω ) + ε uε1 2H 1 (ωε ) ).

(20)

h

Finally, from (19), (20), (18), the normalization condition (6) for {U1ε , uε1 } and the boundedness of λ1ε , we deduce 1 ≤ C( ∇xU1ε 2L2 (Ω ) + uε1 2L2 (ωε ) + ε ∇x uε1 2L2 (ωε ) ) ≤ C( ∇xU1ε 2L2 (Ω ) + ε ) and so, for sufficiently small ε , ∇xU1ε 2L2 (Ω ) > 1/(2C). Therefore, the lemma is proved. Estimate (16) indicates the order of magnitude of the so-called low frequencies; that is, for fixed k, λkε = O(1), and its asymptotic behavior as ε → 0 and that of the corresponding eigenfunctions {Ukε , uεk } has been predicted by means of matched asymptotic expansions in [Go06a] where the limiting spectral problem

−AΔxV = μV in Ω , (21) A∂ν V = μ hV on Γ , has been obtained formally. The weak formulation of (21) reads: Find μ and V ∈ H 1 (Ω ), V = 0, satisfying the integral identity

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A

Ω

∇xV · ∇xW dx = μ

 Ω



VW dx +

Γ

hVW d τ

∀W ∈ H 1 (Ω ).

(22)

Problem (22) has a nonnegative discrete spectrum (see [Go06a] for details). Let k→∞

0 = μ0 < μ1 ≤ μ2 ≤ · · · ≤ μk ≤ · · · −−−−−−→ ∞ be the eigenvalues of (22) with the usual convention of repeated eigenvalues. Let {Vk }∞ k=0 be the corresponding eigenfunctions which are subject to the orthonormalization condition   Vk Vl dx + hVk Vl d τ = δk,l (23) Ω

Γ

and form a basis in H 1 (Ω ). The eigenfunctions associated with μ0 = 0 are the constants. Let us denote by (·, ·)0 the scalar product defined by the left-hand side of (23) in H 1 (Ω ), that is, 

(V,W )0 =

Ω



V W dx +

Γ

hV W d τ

∀V,W ∈ H 1 (Ω ).

(24)

The main convergence result for the low frequencies of (3) with 0 ≤ t < 1 is stated in the following theorem. Theorem 1. Let λkε be the eigenvalues of (3) with 0 ≤ t < 1 and ωε defined by (1). For each fixed k ∈ N ∪ {0}, the sequence λkε converges toward the eigenvalue μk of (22) as ε → 0. In addition, for each sequence {Ukε , uεk } of eigenfunctions of (5), {Ukε , uεk } satisfying the normalization condition (6), we can extract a subsequence (still denoted by ε ) such that Ukε → Vk∗ weakly in H 1 (Ω ), as ε → 0, where Vk∗ is an eigenfunction of (22) corresponding to μk and the set {Vk∗ }∞ k=0 forms an orthonormal basis in H 1 (Ω ) for the scalar product defined by (24). Proof. Taking into account (16), the orthonormalization condition (6) and the fact that (λkε , {Ukε , uεk }) is an eigenelement of (5), we can extract a subsequence (still denoted by ε ) such that for each k = 1, 2, 3 . . . ε →0

λkε −−−−−−→ λk∗

ε →0

and Ukε −−−−−−→Vk∗ weakly in H 1 (Ω ),

where λk∗ > 0 and Vk∗ is certain function in H 1 (Ω ). Moreover, the following estimates hold: (25)

uεk 2L2 (ωε ) ≤ ε and ∇x uεk 2L2 (ωε ) ≤ Ck ε t . We prove that, for each fixed k, (λk∗ ,Vk∗ ) is an eigenelement of (22). 1 First, we note that {Vk∗ }∞ k=0 are orthonormal in H (Ω ) with the scalar product ∗ (24) and therefore V = 0. To prove it, we write

Stiff Problems in Domains with a Strongly Oscillating Boundary

1 ε

 ωε

uεk uεl dx =

1 ε

  ε h(τ ) Γ

1 + ε +

1 ε

0

167

uεk (ν , τ ) uεl (ν , τ )(K(ν , τ ) − 1) d ν d τ

  ε h(τ ) Γ

(uεk (ν , τ ) − uεk (0, τ )) uεl (ν , τ ) d ν d τ

0

  ε h(τ ) Γ

0

uεk (0, τ ) (uεl (ν , τ ) − uεl (0, τ )) d ν d τ +

 Γ

huεk uεl d τ

and, using (25), (9), (13) and the fact that uεk converges to Vk∗ strongly in L2 (Γ ), we deduce that the three first terms on the right-hand side converge toward zero as ε → 0. Then, we take limits in (6), as ε → 0, and we get  Ω

Vk∗Vl∗ dx +

 Γ

hVk∗Vl∗ d τ = δk,l .

In order to identify (λk∗ ,Vk∗ ), for any fixed W ∈ C∞ (Ω ), we consider the variational formulation (5) for λ ε = λkε , U ε = Ukε , uε = uεk and the test function {G, g} = {W, w} ˜ where w˜ is defined by (10), and introduce the curvilinear coordinates in ωε : 

A

 

ε h(τ ) a ∇xUkε · ∇xW dx + t ∂τ uεk ∂τ wK ˜ −1 d ν d τ ε Γ 0 Ω     ε h(τ ) 1 ε ε ε = λk Uk W dx + uk wK ˜ dν dτ . ε Γ 0 Ω

(26)

Taking into account estimates (9), (11), (13) and (25) and that 0 ≤ t < 1, we pass to the limit in (26) and obtain     (27) A ∇xVk∗ · ∇xW dx = λk∗ Vk∗W dx + hVk∗W d τ . Ω

Ω

Γ

Therefore, since C∞ (Ω ) is dense in H 1 (Ω ) and Vk∗ = 0, it follows that Vk∗ is an eigenfunction of (22) corresponding to the eigenvalue λk∗ . Therefore, we have {λk∗ : k ∈ N ∪ {0}} ⊂ {μk : k ∈ N ∪ {0}}, and since the multiplicity of each eigenvalue is finite, we deduce that λk∗ → ∞ as k → ∞. Now, we prove by induction that for each k, λk∗ = μk . It is clear for k = 0. Let us prove it for k = 1. Since the eigenfunctions V0∗ and V1∗ corresponding to λ0∗ = 0 and λ1∗ are orthonormal in H 1 (Ω ) with the scalar product (24), we have that μ1 ≤ λ1∗ . In order to prove λ1∗ ≤ μ1 , for each ε > 0, we consider {Φ ε , φ ε } ∈ H 1 (Ωε ) the solution of 

A







a 1 ∇x Φ ε · ∇x G dx + t ∇x φ ε · ∇x g dx + Φ ε G dx + φ ε g dx ε ωε ε ωε Ω Ω    1 V1 G dx + v˜1 g dx ∀{G, g} ∈ H 1 (Ωε ) (28) = (μ1 + 1) ε ωε Ω

168

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where V1 is an eigenfunction, of norm 1, corresponding to μ1 and v˜1 is defined by (10). Taking {G, g} = {Φ ε , φ ε } in (28) and using the normalization of V1 and (11), we obtain that for sufficiently small ε

∇x Φ ε 2L2 (Ω ) +

1 1

∇x φ ε 2L2 (ωε ) + Φ ε 2L2 (Ω ) + φ ε 2L2 (ωε ) ≤ C, εt ε

(29)

and we can extract a subsequence (still denoted by ε ) such that Φ ε converges weakly in H 1 (Ω ) toward some function Φ ∈ H 1 (Ω ). To identify Φ , for any fixed W ∈ C∞ (Ω ), we consider the variational formulation (28) with the test function {G, g} = {W, w} ˜ where w˜ is defined by (10) and pass to the limit using (9), (11), (13), (29) and the fact that 0 ≤ t < 1. Then, by density, we get 

A



Ω



∇x Φ · ∇xW dx + ΦW dx + hΦ W d τ Γ   Ω  ∀W ∈ H 1 (Ω ) = (μ1 + 1) V1W dx + hV1W d τ Ω

Γ

(30)

and taking into account the uniqueness of problem (30) we identify Φ as the eigenfunction V1 corresponding to μ1 . Also, taking {G, g} = {Φ ε , φ ε } in (28), we prove the convergence, as ε → 0, 

A

Ω

|∇x Φ ε |2 dx +



a εt

ωε

|∇x φ ε |2 dx +

 Ω

|Φ ε |2 dx +



1 ε

ωε

|φ ε |2 dx → μ1 + 1. (31)

Now, let us define {Ψ ε , ψ ε } = {Φ ε , φ ε } − Cε where Cε is the constant defined by



1 1

Ω dx + ε ωε dx

Cε =

Ω

Φ ε dx +

1 ε

 ωε

 φ ε dx .

By construction, {Ψ ε , ψ ε } is orthogonal to the constants with the scalar product (7). Moreover, the constants Cε converge toward zero when ε → 0. Indeed, by virtue of (9), (13), (29) and the weak convergence of Φ ε toward V1 in H 1 (Ω ), which is orthogonal to the constants with the scalar product (24), we get  Ω

Φ ε dx +

1 ε



ε →0

ωε

φ ε dx −−−−−−→





Ω

V1 dx +

Γ

hV1 d τ = 0.

Hence, denoting by Rε {G, g} the Rayleigh quotient ε

R {G, g} =

A



Ω

|∇x G|2 dx+ εat



ωε

Ω

|∇x g|2 dx+ 1

|G|2 dx+ ε

ωε



Ω

|G|2 dx+ ε1



ωε

|g|2 dx

|g|2 dx

for all {G, g} ∈ H 1 (Ωε ), and using (28), (29), (9), (11), (13) and the convergence of Φ ε toward V1 weakly in H 1 (Ω ) (cf. (31)), we have lim Rε {Ψ ε , ψ ε } = lim Rε {Φ ε , φ ε } = μ1 + 1.

ε →0

ε →0

Stiff Problems in Domains with a Strongly Oscillating Boundary

169

But the minimax principle allows us to write λ1ε +1 ≤ Rε {Ψ ε , ψ ε } and, taking limits as ε → 0, λ1∗ + 1 ≤ μ1 + 1. Thus, the result λk∗ = μk holds for k = 1. Let us assume that λi∗ = μi holds for 0 ≤ i ≤ k. Since the eigenfunctions {Vi∗ }ki=1 corresponding to {λi∗ }ki=1 are orthonormal with the scalar product (24), it is clear ∗ . In order to prove λ ∗ ≤ μ ε ε 1 that μk+1 ≤ λk+1 k+1 , we consider {Φ , φ } ∈ H (Ω ε ) k+1 the solution of (28) with μ1 ,V1 , v˜1 replaced by μk+1 ,Vk+1 , v˜k+1 , respectively, where Vk+1 is an eigenfunction of (22), with norm 1, corresponding to μk+1 and such that it is orthogonal to {Vi∗ }ki=0 with the scalar product (24), and v˜k+1 is defined by (10). We use the same argument as in (28) to prove that Φ ε converges toward Vk+1 weakly in H 1 (Ω ) as ε → 0, and Rε {Φ ε , φ ε } converges to μk+1 + 1. Now, we define k

{Ψ ε , ψ ε } = {Φ ε , φ ε } − ∑ ({Φ ε , φ ε }, {Uiε , uεi })ε {Uiε , uεi }, i=0

which is orthogonal to {{Uiε , uεi }}ki=0 with the scalar product (7) and satisfies ε →0

({Ψ ε , ψ ε } − {Φ ε , φ ε }, {Ψ ε , ψ ε } − {Φ ε , φ ε })ε −−−−−−→ 0 and

∇x (Ψ ε − Φ ε ) 2L2 (Ω ) + Then

1 ε →0

∇x (ψ ε − φ ε ) 2L2 (ωε ) −−−−−−→ 0. εt

lim Rε {Ψ ε , ψ ε } = lim Rε {Φ ε , φ ε } = μk+1 + 1.

ε →0

ε →0

ε + 1 ≤ Rε {Ψ ε , ψ ε }, taking limits as ε Since the minimax principle gives λk+1 ∗ ∗ =μ ∗ we obtain λk+1 + 1 ≤ μk+1 + 1 and so λk+1 k+1 . Therefore, the result λk

→ 0, = μk

holds for k = 0, 1, 2 . . . We have proved the result for the eigenvalues for a certain subsequence ε , but taking into account that for any sequence it is possible to extract a subsequence satisfying the same result, the statement holds for the sequence ε . In addition, the 1 fact that the set {Vk∗ }∞ k=0 forms a basis in H (Ω ) is obtained by contradiction. In∗ ∞ deed, let us assume that {Vk }k=0 is not a basis in H 1 (Ω ). Let V ∗ an eigenfunction of (22) corresponding to an eigenvalue μk of multiplicity mk , μk < μk+1 . Then, {V1∗ ,V2∗ , . . . ,Vk∗ ,V ∗ } are eigenfunctions corresponding to {μ1 , μ2 , . . . , μk } which is a contradiction with the assumption of the repeated eigenvalues. Therefore, the theorem is proved.

3 The Case ωε = {x : 0 < ν < ε h(τ /ε )} In this section we study the asymptotic behavior of the eigenvalues of (3) and their corresponding eigenfunctions when 0 ≤ t < 1 and the band ωε is a thin domain with strongly oscillating boundary. The aim is to show the convergence, as ε → 0, of the eigenpairs of (5) when ωε is defined by (2) toward those of the resulting problem (34) (cf. Theorem 2).

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Throughout this section, the constant h˜ denotes the average of h, namely 1 h˜ = P

 P 0

h(η ) d η .

As is well known, the following convergence holds (cf. Chapter V of [Sa80] for instance): hε → h˜ weakly in L2 (Γ ) as ε → 0. (32) We previously proved that the bound for the eigenvalues of (3) given by Lemma 1 also holds in this case. Lemma 2 Let λkε be the eigenvalues of (3) with 0 ≤ t < 1 and ωε defined by (2). For each fixed k ∈ N, and ε sufficiently small, we have 0 < C ≤ λkε ≤ Ck ,

(33)

where C,Ck are constants independent of ε and Ck → ∞ when k → ∞. Proof. We observe that the proof of the right-hand side of (16) in Lemma 1 is independent of the domain ωε . To prove the left-hand side of (33) we use the technique in Lemma 1 with minor modifications. Considering problem (15) with h the constant ˜ from (14) we have h, ˜ − CU 2 2 ≤ C ∇xU 2 2

U − CU 2L2 (Ω ) + h U L (Γ ) L (Ω )

∀U ∈ H 1 (Ω )

where CU denotes the constant defined by    1 ˜ dτ , CU =

U dx + hU

˜ Ω Γ Ω dx + Γ h d τ ˜ Thus, if {U ε , uε } is an eigenfunction of (5) that is to say, CU ≡ CU when h(τ ) = h. 1 1 ε corresponding to λ1 satisfying (6), we have ˜ ε − CU ε 2 2 + rε + rε 1 = U1ε − CU1ε 2L2 (Ω ) + h U 1 1 2 1 L (Γ ) where 

r1ε

=



ε ˜ Ω U1 dx + h Γ

U1ε d τ

˜ Γ| |Ω | + h|

2 and

r2ε =

1 ε

 ωε

|uε1 |2 dx − h˜

 Γ

|U1ε |2 d τ .

On account of the normalization of {U1ε , uε1 }, the variational formulation (5) and the boundedness of λ1ε , we can extract a subsequence (still denoted by ε ) such that U1ε converges toward some function V1∗ weakly in H 1 (Ω ) as ε → 0. Then, from the orthogonality condition of {U1ε , uε1 } to the constants given by (6), (3)3 , (9), (13), (32) and the strong convergence of U1ε to V1∗ in L2 (Γ ) yields

Stiff Problems in Domains with a Strongly Oscillating Boundary

   ε h (τ ) ε 1 1 ε r1 = uε1 (ν , τ )(K(ν , τ ) − 1) d ν d τ ˜ |Ω | + h|Γ | ε Γ 0 1 + ε

  ε hε ( τ ) Γ 0

(uε1 (ν , τ ) − uε1 (0, τ )) d ν d τ +



Γ

˜ 1ε d τ (hε − h)U

171

2

ε →0

−−−−−−→ 0.

Similar considerations allow us to assert that r2ε also converges to zero as ε → 0 and, in consequence, ˜ ε − CU ε 2 2 + oε (1) ≤ C ∇xU ε 2 2 + oε (1), 1 = U1ε − CU1ε 2L2 (Ω ) + h U 1 1 L (Ω ) 1 L (Γ ) where oε (1) → 0 as ε → 0. Therefore, proceeding as in Lemma 1 and using (18), for ε sufficiently small, λ1ε ≥ A ∇xU1ε 2L2 (Ω ) > c > 0 and the left-hand side of (33) holds, which concludes the proof. Estimate (33) allows us to assert that the low frequencies λkε for fixed k are of order O(1). In fact, on account of (33) and the normalization of the eigenfunctions (6), for each fixed k, we can extract a subsequence (still denoted by ε ) such that ε →0

λkε −−−−−−→ λk∗

ε →0

and Ukε −−−−−−→Vk∗ weakly in H 1 (Ω ),

where λk∗ is some constant and Vk∗ is some function of H 1 (Ω ). In order to identify (λk∗ ,Vk∗ ), we take limits in the variational formulation (5) with the test functions {G, g} = {W, w} ˜ where W ∈ C∞ (Ω ) and w˜ is defined by (10). Arguments similar to those in the proof of Theorem 1 (cf. (25)–(27)) as well as the convergence (32) lead us to the limit problem: Find μ and V ∈ H 1 (Ω ), V = 0, satisfying the integral identity    ∀W ∈ H 1 (Ω ). VW dx + h˜ VW d τ (34) A ∇V · ∇W dx = μ Ω

Ω

Γ

This problem is the variational formulation of

−AΔ xV = μ V in Ω , ˜ on Γ A∂ν V = μ hV (cf. (21) to compare). We note that the limiting problem depends only on the average of h or equivalently on the area of the unit cell. Problem (34) has a nonnegative discrete spectrum that we denote by {μk }∞ k=0 . be the corresponding eigenfunctions which are subject to the orthonorLet {Vk }∞ k=0 malization condition  Ω

Vk Vl dx + h˜

 Γ

Vk Vl d τ = δk,l

(35)

and form a basis in H 1 (Ω ). The following result establishes the convergence for the low frequencies of (3) in domains with strongly oscillating boundary.

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Theorem 2. Let λkε be the eigenvalues of (3) with 0 ≤ t < 1 and ωε defined by (2). For each fixed k ∈ N ∪ {0}, the sequence λkε converges toward the eigenvalue μk of (34) as ε → 0. In addition, for each sequence {Ukε , uεk } of eigenfunctions of (5), {Ukε , uεk } satisfying the normalization condition (6), we can extract a subsequence (still denoted by ε ) such that Ukε → Vk∗ weakly in H 1 (Ω ), as ε → 0, where Vk∗ is an eigenfunction of (34) corresponding to μk and the set {Vk∗ }∞ k=0 forms an orthonormal basis in H 1 (Ω ) for the scalar product defined by the left-hand side of (35). Proof. We rewrite the proof of Theorem 1 with minor modifications (let us to refer those outlined in the proof of Lemma 2). Acknowledgements This work has been partially supported by the Spanish grant MTM200912628. The work of S.A. Nazarov has also been partially supported by the RFBR: 09-01-00759.

References [At84] Attouch, H.: Variational Convergence for Functions and Operators, Pitmann (1984). [BuKo87] Buttazzo, G., Kohn, R.V.: Reinforcement by a thin layer oscillating thickness. Appl. Math. Optim., 16, 247–261 (1987). [Go06a] Gómez, D., Lobo, M., Nazarov, S.A., Pérez, E.: Spectral stiff ptoblems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues. J. Math. Pures Appl., 85, 598–632 (2006). [Go06b] Gómez, D., Lobo, M., Nazarov, S.A., Pérez, E.: Asymptotics for the spectrum of the Wentzell problem with a small parameter and other related stiff problems. J. Math. Pures Appl., 86, 369–402 (2006). [GoLo99] Gómez, D., Lobo, M., Pérez, E.: On the eigenfunctions associated with the high frequencies in systems with a concentrated mass. J. Math. Pures Appl., 78, 841–865 (1999). [GoNa11] Gómez, D., Nazarov, S.A., Pérez, E.: Spectral stiff problems in domains surrounded by thin stiff and heavy bands: local effects for eigenfunctions. Netw. Heterog. Media, 6, 1–35 (2011). [LoSa79] Lobo, M., Sanchez-Palencia, E.: Sur certaines propriétés spectrales des perturbations du domaine dans les problèmes aux limites. Comm. Partial Differential Equations, 4, n. 10, 1085–1098 (1979). [Na90] Nazarov, S.A.: Binomial asymptotic behavior of solutions of spectral problems with singular perturbations. Mat. Sb., 181, n. 3, 291–320 (1990) (English transl: Math. USSR-Sb., 69, n. 2, 307–340 (1991)). [Na07] Nazarov, S.A.: Eigenoscillations of an elastic body with a rough surface. Prikl. Mekh. Tekhn. Fiz., 48, n. 6, 103–114 (2007) (English transl: J. Appl. Mech. Tech. Phys., 48, n. 6, 861–870 (2007)). [Na08] Nazarov, S.A.: Asymptotic behavior of solutions and the modeling of problems in the theory of elasticity in a domain with a rapidly oscillating boundary. Izv. Ross. Akad. Nauk Ser. Mat., 72, n. 3, 103–158 (2008) (English transl: Izv. Math., 72, n. 3, 509–564 (2008)). [OlSh92] Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization, North-Holland (1992). [Sa80] Sanchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory, Springer-Verlag (1980).

Spectra and Pseudospectra of a Convection–Diffusion Operator H. Guebbaï and A. Largillier

1 Introduction In this paper we study the spectral stability for a nonselfadjoint convection–diffusion operator on an unbounded 2-dimensional domain starting from a result about the pseudospectrum. Our goal is to study the spectrum of the following convection– diffusion operator A Au := −Δ u − (y, x) · ∇u + (x2 + y2 )u defined on L2 (Ω ), where Ω is an unbounded open set of R2 and under a Dirichlet boundary condition. Our study is based upon pseudospectral theory because its tools are easier to handle and give better results when compared to those of spectral theory (see [Bo99, ReTr94, Tr92], and [Tr97]). In fact, it has been established that the spectrum of a sequence of differential operators may be unstable when going to the limit, unlike the pseudospectrum which is known to be stable (see [Da00] and [Da95]). For ε > 0 the pseudospectrum spε (A) of A is the union of the spectrum of A and the set of all z ∈ C such that (zI − A)−1  ≥ ε −1 . Equivalently [RoSi96], sp ε (A) =



sp (A + D).

D≤ε

If A is a normal operator, its pseudospectrum is equal to the ε -neighborhood of its spectrum. We also exploit the fact that the spectrum of an operator is separated into two sets: The point spectrum, sp p (A) which is composed of all the eigenvalues of A and the essential spectrum, sp ess (A) which is composed of all λ ∈ C such that the operator λ I − A is injective but not surjective. We conclude by a result on the stability of the spectrum obtained through pseudospectral theory. H. Guebbaï Université Jean Monnet, Université de Lyon, Saint-Étienne, France, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_16, © Springer Science+Business Media, LLC 2011

173

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H. Guebbaï and A. Largillier

2 Formulation of the Problem Let Ω be the open set of R2 defined by   Ω := (x, y) ∈ R2 : x > 0 and − x < y < x . Let A be the differential operator defined by Au := −Δ u − (y, x) · ∇u + (x2 + y2 )u. The universe of our discourse is the Hilbert space L2 (Ω ) of complex-valued (classes of) function defined a.e. on Ω . We consider the Hermitian form ϕ defined by

ϕ ( f , g) :=

 Ω

∇ f · ∇gdxdy −

 Ω

(y, x) · ∇ f gdxdy +

 Ω

(x2 + y2 ) f gdxdy.

The quadratic form associated with ϕ is Q(u) := ∇u2L2 (Ω ) −

 Ω

(y, x) · ∇uudxdy +

and verifies

ϕ ( f , g) =

 Ω

(x2 + y2 )|u|2 dxdy,

1 4 k ∑ i Q( f + ik g). 4 k=1

By the Cauchy–Schwarz inequality,   1     2 ∇uL2 (Ω ) + (x2 + y2 )|u|2 .  (−yu, −xu) · ∇u ≤ 2 Ω Ω

Hence ϕ is a sectorial form defined on the linear space V := H01 (Ω ) ∩ {u ∈ L2 (Ω ) : (x2 + y2 )u ∈ L2 (Ω )}. A is the operator associated with ϕ [Ka80] and its domain is D(A) := H 2 (Ω ) ∩ H01 (Ω ) ∩ {u ∈ L2 (Ω ) : (x2 + y2 )u ∈ L2 (Ω )}. Our goal is to determine the spectrum of A. We remark that D(A) includes a Dirichlet boundary condition. Consider first the eigenvalue problem: Find λ ∈ C and u ∈ D(A) \ {0} such that −Δ u − y∂x u − x∂y u + (x2 + y2 )u = λ u on Ω , u = 0 on ∂ Ω . Define the family (Ωη )0 η0 , the essential spectrum sp ess (Aη ) is included in (5η 2 /4,CPF −2 2 the point spectrum sp p (Aη ) is included in [CPF + η , +∞), • if η ≤ η0 , Aη has no essential spectrum and the point spectrum is included in −2 [CPF + η 2 , +∞).

Proof. For all η ∈ (0, 1) and all u ∈ D(Aη ), 1 1 Re Aη u, u = [ Aη u, u + Aη u, u] = [ Aη u, u + u, Aη u]. 2 2 But  Ωη

−Δ u(x, y)u(x, y)dxdy = −



+ 

0=

Ωη

Ωη



∇u(x, y).∇u(x, y)dxdy =

y∂x (uu)(x, y) dxdy =

 Ωη

∂ Ωη

∂u (x, y) u(x, y) d σ

  ∂ν =0

 Ωη

over ∂ Ω η

|∇u(x, y)|2 dxdy,

 y ∂x u(x, y)u(x, y) + u(x, y)∂x u(x, y) dxdy

and 

0= Thus

Ωη

x∂y (uu)(x, y) dxdy =

 Ωη

 x ∂y u(x, y)u(x, y) + u(x, y)∂y u(x, y) dxdy.

Spectra and Pseudospectra of a Convection–Diffusion Operator

Aη u, u =



177

|∇u|2 + yu∂x u + xu∂y u + (x2 + y2 )|u|2 .

Ωη

Similarly,

u, Aη u =



|∇u|2 − yu∂x u − xu∂y u + (x2 + y2 )|u|2 ,

Ωη

where Re Aη u, u = ∇u2L2 (Ωη ) +

 Ωη

−2 (x2 + y2 )|u(x, y)|2 dxdy ≥ (CPF + η 2 )u2L2 (Ωη ) .

Thus, for all λ ∈ R, −2 (Aη − λ I)uL2 (Ωη ) ≥ [CPF + η 2 − λ ]uL2 (Ωη ) . −2 −2 + η 2 and thus sp p (Aη ) is included in [CPF + Hence Aη is injective for all λ < CPF 2 1 2 2 η , +∞). Let H = H0 (Ωη ), g ∈ L (Ωη ) and λ ∈ (−∞, 5η /4). The sesquilinear form defined on H by    ∇u.∇v + 5(x2 + y2 )/4 − λ uv dxdy ϕλ (u, v) :=

Ωη

verifies |ϕλ (u, v)| ≤ ∇uL2 (Ωη ) ∇vL2 (Ωη ) +CuL2 (Ωη ) vL2 (Ωη ) , where and

  C := sup 5(x2 + y2 )/4 : (x, y) ∈ Ωη + |λ |,   |ϕ (u, u)| ≥ min 1, (5η 2 /4 − λ ) u2H .

Since, for all g ∈ L2 (Ωη ), the semilinear form L : H → C; v →

 Ωη

gv dxdy

is continuous, it follows from the Lax–Milgram theorem that the equation ϕλ (u, v) = L(v) for all v ∈ H, has a unique solution u in H. Let g ∈ L2 (Ωη ). Consider the problem (P) Find u ∈ L2 (Ωη ) : Aη u − λ u = g on Ωη , u = 0 on ∂ Ωη . xy

xy

xy

Multiply the equation by e 2 , set g (x, y) := g(x, y)e 2 , u (x, y) := u(x, y)e 2 and let g ∈ L2 (Ωη ). Then (P) is equivalent to  Find u ∈ L2 (Ωη ) : − Δ u + 5 (x2 + y2 )

u − λ u = g on Ωη , u = 0 on ∂ Ωη . (P) 4

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H. Guebbaï and A. Largillier

The sesquilinear form

ϕλ (u, v) =

 Ωη

∇u.∇v +

5 4

(x2 + y2 ) − λ uvdxdy

 has a unique solution u , is an inner product on L2 (Ωη ) for λ < 5η 2 /4. Hence (P) xy and (P) has a unique solution u defined by u(x, y) := u (x, y)e− 2 .

4 Pseudospectra In this section we establish relations between different spectra and pseudospectra  in order to localize the spectrum of A. The equalities sp p (A0 ) = sp p (Aη ) and sp ε (A0 ) =

0 0,



  √ √ √ √ 2r   − λt ωt ∗ ∗  e u − u(t) ≤ C max (1 + λ ) + r , r t (19) e   H r∗ and   √ −√λ t du    λe (t) u +  dt H



√ √ √ √ √ √ 2r 2r √ ∗ eω t , ∗ ) r∗ t + r∗ . ≤ C max (1 + λ ) + r λ , ( λ + r r∗ r∗ (20) √ (ii) For (ϕ , ψ ) = (u, λ u) and any t > 0,  √   λt  e u − u(t)

H

≤ C max



√ √ 2r (1 + λ ) + r∗ ∗ r



eω t ,

2r √λ t √ ∗ (√λ +√r∗ )t e , r te r∗

(21)

and  

√ √  √ √λ t 2r ∗ eω t ,  λ e u − du (t) ≤ C max (1 + λ ) + r  dt H r∗

√ √ √ √ √ √ 2r √ √λ t √ ∗ ) r ∗ te( λ + r∗ )t + r ∗ e λ t . (22) λ e , ( λ + r r∗ In addition, estimates (19) and (20) also hold for the norms in H of I(r∗ )

∑ αk e−

k=1

√

λi(r∗ )+k t

ui(r∗ )+k − e−

√ λt

u

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and its time derivative, while estimates (21) and (22) hold for the norms in H of I(r ∗ )

∑ αk e

√

λi(r∗ )+k t

ui(r∗ )+k − e

√ λt

u

k=1

and its time derivative. In the bounds above (19)–(22), C and ω are constants independent of λ , r, r∗ and t, with ω appearing in (11).

3 New Estimates for Discrepancies from the Semigroup As is well known, the solution of (10) (cf. also (5), (12) and (18)) and, more precisely, of the nonhomogeneous equation ⎧ ⎨ d u¯ + A u¯ = f¯(t) (23) ⎩ dt ¯ u(0) = ϕ¯ for given ϕ¯ ∈ H, T > 0, and f¯ ∈ C1 ([0, T ], H), is provided by ¯ = e−A t ϕ¯ + u(t)

 t 0

e−A (t−s) f¯(s) ds,

∀t ∈ [0, T ].

(24)

In this section we show that (24) allows us to obtain certain bounds for the dis¯ crepancies between the solutions u(t) of (10) when the initial data are linear combinations of eigenfunctions (or quasimodes) and the standing waves constructed from the quasimode (u, λ ) in Theorems 1–3. The bounds that we obtain in this section are different from those in Sect. 2; they are simpler but they involve, among other parameters, the inverse of the distance from λ to the spectrum of A, that is, d(λ , σ (A))−1 : see Remark 2 in this connection. Problem (5), which is the simplest one, is considered in Sect. 3.1. As a matter of fact, bounds (7) and (8) in Theorem 1 are replaced by (25) in Theorem 4. We proceed in a similar way for the second-order evolution problems (12) and (18) in Sect. 3.2. Let us refer to Sects. III.6.2–III.6.3 and V.3.8 in [Ka66] for the general theory and formulas that we use throughout this section. See also Sect. III.8 of [SaSa89] in this connection.

3.1 The Case of the First-Order Evolution Equation (5) Throughout this section, under the hypotheses of Theorem 1, we consider (5), namely, (23) and (24) for A ≡ ±A, and for suitable scalar ϕ¯ ≡ ϕ and f¯(t) ≡ f (t).

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Theorem 4. Let us consider the hypotheses in Theorem 1 on the operator A, on the quasimode (u, λ ), the eigenelements {(λi(r∗ )+k , ui(r∗ )+k )}k=1,2,...,I(r∗ ) and the numbers r, r∗ and λ . Then, for ϕ = u, and for any t > 0, the solution u± (t) of (5) satisfies (25) e∓λ t u − u± (t)H ≤ d(λ , σ (A))−1 e∓λ It − e∓At L (H) r . For the positive sign in (5), the right hand side in (25) is d(λ , σ (A))−1 r. Also, for any t > 0 we have   I(r ∗ )     ∓λ t ∓λi(r∗ )+k t ui(r∗ )+k  e u − ∑ αk e   k=1 H 2r + d(λ , σ (A))−1 e∓λ It − e∓At L (H) r . (26) ≤ eAL (H) t r∗ Proof. As is well known, u± (t) = e∓At u. Let us consider the function e∓λ t u which satisfies d ∓λ t e u ± Ae∓λ t u = ±(Au − λ u)e∓λ t dt and takes the value u at t = 0. Let us denote by w the “rest”, that is, w = Au − λ u with a norm in H bounded by r. Then, the function e∓λ t u − u± (t) is the solution of problem (23) for f = e∓λ t (±w) and ϕ = 0. Consequently, formula (24) reads e∓λ t u − u± (t) =

 t

 t

e−(±A)(t−s) e∓λ s (±w)ds = e∓At e±(A−λ I)s ds (±w) 0 0 t = e∓At (A − λ I)−1 [e±(A−λ I)s 0 w = (A − λ I)−1 (e∓λ It − e∓At )w , (27)

and, taking norms in (27), we obtain (25). In order to prove (26), we consider the function ∓λ t

e

I(r∗ )

u−

∑ αk e∓λi(r∗ )+k t ui(r∗ )+k .

k=1

It suffices to realize that it satisfies (23) for the same f (t) above, namely, f (t) = I(r∗ ) ±(Au − λ u)e∓λ t , and for ϕ = u − ∑k=1 αk ui(r∗ )+k with a norm less or equal than I(r∗ )

2r(r∗ )−1 . Let us denote w∗ ≡ u − ∑k=1 αk ui(r∗ )+k . Therefore, (24) takes the form ∓λ t

e

u−

I(r ∗ )

 t

k=1

0

∑ αk e∓λi(r∗ )+k t ui(r∗ )+k = e∓At w∗ +

e−(t−s)(±A) e∓λ s (±w)ds

whose norm is bounded by the right hand side of (26). Considering (6) with ω + = 0, we obtain d(λ , σ (A))−1 r on the right hand side of (25). Hence, the theorem is proved.

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3.2 The Case of the Second-Order Evolution Equation Throughout this section we consider the case of the second-order equation, namely, (23) for the operator A in (9), depending on the sign accompanying I. For the sake of simplicity, throughout the section we perform the proof for problem (18) and we state the results for problem (12) without proof (cf. comments after the proof of Theorem 5). Theorem 5. Let us consider the hypotheses in Theorem 3 on the operator A, the quasimode (u, λ ), the eigenelements {(λi(r∗ )+k , ui(r∗ )+k )}k=1,2,...,I(r∗ ) , and the num√ bers r, r∗ and λ . Then, for ϕ = u, ψ = ± λ u, and for any t > 0, the solution u(t) of (18) and its derivative satisfy   √ √ √     u(t) − e± λ t u +  − du (t) ± λ e± λ t u  dt  H   H √ √   ≤ (1 + 2 λ + λ ) d(λ , σ (A))−1 + 1 e± λ I t − e−A t 

L (H)

r . (28)

In addition, we also have  ∗  √

I(r )  √ √ √  ± λi(r∗ )+k t A L (H) t 2r(1 + λ ) ± λt  ∗ ui(r∗ )+k − e u ≤ e + r  ∑ αk e  k=1  r∗ H

  √ √   + (1 + 2 λ + λ )d(λ , σ (A))−1 + 1 e± λ I t − e−A t  

L (H)

r,

(29)

and the same bound holds for the time derivative of the discrepancy above. Here, A is the operator defined by (9) for the positive sign accompanying I, and I is the identity operator on H. Proof. We follow the notations used in the proof of Theorem 4 for w and w∗ . That is, w = Au − λ u, while w∗ denotes the discrepancy between u and the linear combination of eigenfunctions u∗ arising in the definition of quasimode (1)–(2), namely w∗ = u − u∗ . They satisfy wH ≤√r and w∗ H ≤ 2r(r∗ )−1 . Let us consider the function e± λ t u and u(t) the solution of (18) in the statement of the theorem. We compute

√ 2 √ √ d ± λt ± λt ± λt − A e u = (−Au + λ u)e = −e w. dt 2 ¯ with components Consequently, the vector function u(t) u1 (t) = u(t) − e±

√ λt

u,

u2 (t) = −

√ √ du (t) ± λ e± λ t u dt

satisfies the problem (23) for the data

ϕ¯ = (0, 0)τ ,

√  τ f¯(t) = 0, −e± λ t w ,

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and formula (24) reads  t

¯ = u(t)

−A (t−s)

e

f¯(s) ds = e−A t

 t

0

eA s e±

√

λI s

0

= e−A t

 t

(0, −w)τ ds e(A ±

√ λ I )s

0

(0, −w)τ ds . (30)

Here, and in what follows, the superscript τ denotes the transpose vector. Performing the integration in (30), we have √  √  ¯ = (A ± λ I )−1 e± λ I t − e−A t (0, −w)τ u(t) and, therefore, √ √ −1 ± λ It ¯ u(t) − e−A t L (H) r . H ≤ (A ± λ I ) L (H) e

(31)

√ Let us obtain a bound for (A ± λ I )−1 L (H) in terms of (A − λ I)−1 L (H) and consequently, in terms of d(λ , σ (A))−1 . As is known, √ ¯ H (A ± λ I )−1 w . (A ± λ I ) L (H) = sup ¯ H w ¯ w∈H √

−1

(32)

¯ =0 w

√ ¯√= (w1 , w2 )τ , let us analyze (A ± λ I )−1 w For w ¯ H ,√ by considering ¯ = v¯ ≡ (v1 , v2 )τ . That means that w ¯ = (A ± λ I )v¯ and con(A ± λ I )−1 w sequently, (w1 , w2 ) satisfy √ w1 = ± λ v1 + v2

and

√ w2 = Av1 ± λ v2 .

√ √ √ Thus, w2 = Av1 ± λ (∓ λ v1 + w1 ), or equivalently, Av1 − λ v1 = w2 ∓ λ w1 , which determines v1 and v2 in terms of the resolvent of A at λ . Namely, √ v1 = (A − λ I)−1 [w2 ± λ w1 ]

and

√ √ v2 = w1 ∓ λ ( (A − λ I)−1 [w2 ± λ w1 ] ).

√ Therefore, from (32), (A ± λ I )−1 L (H) is bounded by sup ¯ w∈H ¯ =0 w

√ √ (1 + λ )(A − λ I)−1 w2 H + ( λ + λ )(A − λ I)−1 w1 H + w1 H ¯ H w

√ √ (A − λ I)−1 w2 H (A − λ I)−1 w1 H ≤ (1 + λ ) sup + ( λ + λ ) sup +1 w2 H w1 H w2 ∈H w1 ∈H w2 =0

w1 =0

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323

and we have the estimate for the resolvent of A in terms of that of A: √ √ (A ± λ I )−1 L (H) ≤ (1 + 2 λ + λ )(A − λ I)−1 L (H) + 1 √ = (1 + 2 λ + λ ) d(λ , σ (A))−1 + 1 . (33) Now, from (31) and (33) we obtain estimate (28). √ Similar bounds are derived, respectively, for the discrepancies between e± λ t u √ I(r∗ ) and ∑k=1 αk e± λi(r∗ )+k t ui(r∗ )+k , and their time derivatives. √ τ  Indeed, considering (23) for f¯(t) = 0, −e± λ t w and 

ϕ¯ =

I(r ∗ )

I(r∗ )

k=1

k=1

∑ αk ui(r∗ )+k − u , ± ∑ αk



√

λi(r∗ )+k ui(r∗ )+k ∓ λ u

τ ,

we apply (24) and we have   ∗

 I(r ) √ √ √ √  ± λi(r∗ )+k t A L (H) t 2r ± λt  ∗ ui(r∗ )+k − e u ≤ e (1 + λ ) + r  ∑ αk e   k=1 r∗ H √ √ + (A ± λ I )−1 L (H) e(± λ I t − e−A t L (H) r . The same bound is obtained for the time derivative of the discrepancy. Hence, from (33) and the bound above, (29) holds, and the theorem is proved. Let us state the main results for the case of the second-order evolution equation arising in (12). We avoid here the proofs which imply extending real Hilbert spaces and operators acting on these spaces to complex Hilbert spaces and associated operators (cf. Sect. VI.I in [Mi69] for the kind of construction). Below, we consider the assumptions in Theorem 2. Let us consider the solution u(t) of (12) for the initial data ϕ = α u, ψ = β u for the value of the constants α and β either 0 or 1. Theorem 6. Let us consider the hypotheses in Theorem 2 on the operator A, the I(r∗ ) quasimode (u, λ ), the eigenelements {(λi(r∗ )+k , ui(r∗ )+k )}k , and the numbers r, r∗ and λ . Then, for ϕ = α u, ψ = β u, with the value of the constants α and β either 0 or 1, and for any t > 0, the solution u(t) of (12) satisfies    √   √ sin ( λ t)   u u(t) − α cos ( λ t) + β √   λ H   √ √ √  du   +  (t) + α λ sin ( λ t)u − β cos ( λ t)u  dt H

   √ √ β   ≤ α+√ r. 1 + 2 λ + λ ) d(λ , σ (A))−1 + 1 e−i λ I t − e−A t  L (H∗ ) λ depending on α and β .

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In addition, in the case where α = 1 and β = 0, we have   ∗  I(r )  √    ∑ αk cos( λi(r∗ )+k t)ui(r∗ )+k − cos( λ t)u    k=1 H

≤ eA L (H) t



   √ √ 2r  −i λ I t −1 −A t  λ + λ )d( λ , σ (A)) + 1 e − e r, + 1 + 2   r∗ L (H∗ )

and the same estimate holds for its time derivative. Also, in the case where α = 0 and β = 1, the inequality above holds for  √ I(r∗ ) sin( λi(r∗ )+k t) sin( λ t) ui(r∗ )+k − √ u ∑ αk  λ k=1 λi(r∗ )+k and its√time derivative, that we replace the term 2r(r∗ )−1 multiplying eA L (H) t √ √ once −1 by 2r( λ r∗ ( λ − r∗ )) . Here, H∗ is the complex Hilbert space extension of H, i is the imaginary unit, A is the operator in (9) for the negative sign accompanying I, and I is the identity operator on H∗ . Remark 1. Let us observe that depending on the problem under consideration, the bounds in Theorem 6 and Theorem 5 containing the norms of A and eA t can be expressed in terms of the operator A. Remark 2. Comparing the bounds in Theorem 1 (Theorem 2 and Theorem 3 respect.) with those in Theorem 4 (Theorem 6 and Theorem 5 respect.), it should be emphasized that in the case where the distance between two consecutive eigenvalues is unknown, the bounds in Theorems 1, 2 and 3 are more precise. As a matter of fact, once the quasimode has been constructed, the bounds on the right hand sides of (7)–(8), (14)–(17) and (19)–(22) are well determined in terms of well known parameters. In addition, they provide precise bounds for the time t in which we can have real approaches between solutions and standing waves in terms of these parameter (see [Pe11]). Consequently, the results in this paper highlight the interest of the technique developed in [Pe11] related to singularly perturbed problems. Acknowledgements This work has been partially supported by the Spanish grant MICINN: MTM2009-12628.

References [Ka66] [La99]

Kato, T.: Perturbation Theory for Linear Operators, Springer-Verlag, New York (1966). Lazutkin, V.F.: Semiclassical asymptotics of eigenfunctions, in: Partial Differential Equations V (Editor: M.V. Fedoryuk), Springer-Verlag, Heidelberg, 133–171 (1999).

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[LoPe10] Lobo, M., Pérez, E.: Long time approximations for solutions of wave equations associated with Steklov spectral homogenization problems. Math. Meth. Appl. Sci., 33, 1356–1371 (2010). [Mi69] Mikhlin, S.G.: Mathematical Physics, an Advanced Course, North-Holland, Leningrad (1969). [Pe08] Pérez, E.: Long time approximations for solutions of wave equations via standing waves from quasimodes. J. Math. Pure Appl., 90, 387–411 (2008). [Pe11] Pérez, E.: Long time approximations for solutions of evolution equations from quasimodes: perturbation problems. Math. Balkanica (N.S.), 25 (Fasc. 1–2), 95–130 (2011). [SaSa89] Sanchez-Hubert, J., Sanchez-Palencia, E.: Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer, Heidelberg (1989).

Error Estimation by Means of Richardson Extrapolation with the Boundary Element Method in a Dirichlet Problem for the Laplace Equation S. Pomeranz

1 Introduction Richardson extrapolation can be used to improve the accuracy of numerical solutions for the normal boundary flux and the interior potential resulting from the boundary element method applied to a Dirichlet problem for the Laplace equation. Using numerical results related to the Richardson extrapolation, a technique will be developed that predicts the reliability of the Richardson extrapolation results. This paper builds on many other papers that give results on topics such as enhancing the boundary element method with extrapolation, collocation convergence, and related error estimation (see, e.g., [KaNi87, NiKa89, RüZh98, XuZh96, YaSl88, Ya90]). The main goal of this paper is to use Richardson extrapolation, even when the precise form of the error terms is not available, in order to obtain more accurate numerical approximations and, in some sense, to be able to verify that the approximations are indeed more accurate.

2 Problem Statement The problem to be solved is the interior Dirichlet problem for Laplace’s equation on  D, a bounded domain in the plane, i.e., to find u ∈ C(D) C2 (D) such that u(P) = 0, u(P) = u0 (P),

P ∈ D, P ∈Γ,

(1)

S. Pomeranz The University of Tulsa, OK, USA, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_30, © Springer Science+Business Media, LLC 2011

327

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S. Pomeranz

where Γ is the boundary of domain D, and u0 ∈ C(Γ ) is a given function. The domain may have a smooth boundary or may be a polygonal domain with corners. The direct boundary integral method [AtHa01], [Pa06], which involves the fundamental solution for Laplace’s equation in the plane [Co00] yields    1 ∂ ln |P − Q| ∂ u(Q) − ln |P − Q| dSQ , P ∈ Γ . (2) u0 (P) = u0 (Q) π Γ ∂ nQ ∂ nQ Once (2) is solved for q ≡ ∂ u/∂ n, the unknown normal boundary flux, then u, the unknown potential in the interior of domain D, can be determined using    1 ∂ ln |P − Q| − q(Q) ln |P − Q| dSQ , P ∈ D. u(P) = u0 (Q) 2π Γ ∂ nQ

3 Comments on Richardson Extrapolation The object of Richardson extrapolation is to find a computationally inexpensive way to combine previously computed lower-order (less accurate) numerical results in a way that produces formulas with higher-order (more accurate) numerical results. It is stated that the method is extremely useful when there is a reliable estimate of the form of the discretization error as a function of the grid length [Sm87]. However, even if such information is not available, under quite general conditions Richardson extrapolation can improve the accuracy of numerical results. The following material is a brief description of Richardson extrapolation as used in this paper. Let q denote the unknown exact quantity that is desired. Let q1 and q2 denote two numerical approximations to q that are computed using the same formula (and at the same grid point) but with different, sufficiently small positive grid spacings, h1 and h2 , respectively. If the dominant term in the discretization error is proportional to h p , for some known positive number p, then we obtain q − q1 = Ah1p + higher-order terms, q − q2 =

p Ah2 +

higher-order terms,

(3) (4)

where A denotes a constant of proportionality and the higher-order error terms are assumed negligible. Taking a linear combination of (3) and (4) and solving for q, yields p p h q1 − h 1 q2 . q ≈ q˜ ≡ 2 p h2 − h1p The Richardson extrapolation result is given by q. ˜ If the value of p, the order of the dominant error term, is unknown, then three approximations, using grid spacings h1 , h2 , and h3 , respectively, can be used to obtain

Error Estimation by Means of Richardson Extrapolation

q ≈ q˜ ≡

329

q1 q3 − q22 , q1 − 2q2 + q3

(5)

where it is assumed that h1 /h2 = h2 /h3 ≡ c, for some constant c > 1. Further, the order of the dominant error term can be approximated by   1 ln qq23 −q −q2 . (6) p≈ ln c In the numerical example investigated in Sect. 5, the value of p is not known. Richardson extrapolation is applied to the model problem. Using [BuFa05, #16, p. 87], the Richardson extrapolation convergence can be described as if it were convergence from Aitken’s 2 method. That convergence result is stated now in Theorem 1: Theorem 1. Suppose that the sequence of approximations {qn }∞ n=1 converges to the limit q such that qn+1 − q ≡ λ < 1, 0 ≤ lim n→∞ qn − q for some constant λ . Then the associated sequence of iterates {q˜n }∞ n=1 , where q˜n ≡

qn qn+2 − q2n+1 , qn − 2qn+1 + qn+2

n = 1, 2, . . . ,

converges to q faster than {qn }∞ n=1 in the sense that lim

q˜n − q

n→∞ qn − q

= 0.

Now consider approximating the unknown quantity, for example, the normal boundary flux, q, (or the interior potential) using three successive grid spacings, h1 > h2 > h3 , chosen sufficiently small. The approximations q1 , q2 , and q3 are computed using h1 , h2 , and h3 , respectively, and are such that (q2 − q)/(q1 − q) ≈ (q3 − q)/(q2 − q). Further assume that the errors qi − q, for i = 1, 2, 3, all have the same sign. Theorem 1 states that if q1 , q2 , and q3 belong to the sequence converging to q as described, then Richardson extrapolation based on q1 , q2 , and q3 will give a more accurate approximation than that given by q1 , q2 , or q3 .

4 Implementation of an a Posteriori Pointwise Estimator of Richardson Extrapolation Reliability Recall that the original problem to be solved is the interior Dirichlet problem for the  Laplace equation (1), to find u ∈ C(D) C2 (D) such that

330

S. Pomeranz

u(P) = 0, u(P) = u0 (P),

P ∈ D, P ∈Γ,

where Γ is the boundary of domain D and u0 ∈ C(Γ ) is a given function. For the numerical experiments, the domain was selected to be the unit square, D ≡ (0, 1) × (0, 1). Therefore, the boundary of D is not smooth, and, consequently, the theory for domains with smooth boundaries does not apply. Nevertheless, it is still of interest to apply Richardson extrapolation (5) to boundary element results. Let p be the order of the dominant error term, as defined in (3) and (4). One issue that should be resolved is how to determine if the Richardson extrapolation numerical results are valid. Numerical results for the p value estimates (6) can be used to predict those grid points at which the Richardson extrapolation results are not accurate. The p values should be positive numbers. The p value estimates that are complex numbers (with nonzero imaginary parts), negative, or too small (relative to the neighboring values) can be interpreted as warning flags, i.e., indicators that predict the grid points at which the Richardson extrapolation results are not accurate (relative to the neighboring results). This provides a rejection criterion: The corresponding Richardson extrapolation results should be rejected. These values are not accurate either due to the boundary element method approximations themselves being inaccurate (for example, near corners of the domain) or due to the Richardson extrapolation being inaccurate (for example, if the grid spacings are too large). We cannot say with certainty that some Richardson extrapolation results are good, but we can say with certainty that specific Richardson extrapolation results are bad. For the purposes of describing the numerical experiments reported here, the following terminology is used. A “bad Richardson extrapolation value” at a grid point (either on the boundary, when the normal boundary flux is of interest; or in the interior, when the potential (primary unknown) is of interest) is one for which the fine grid result is more accurate than the Richard extrapolation result. This means that the error magnitude for the fine grid result is smaller than that for the Richardson extrapolation result, which should not be the case. Similarly, a “bad p value” at a grid point (either on the boundary or in the interior) is one that is complex (with nonzero imaginary part), negative, or too small. Since the p value estimate is the numerically approximated order of the dominant error term, it should be a positive number, and not too small in regions for which the solution is reasonably smooth. The bad Richardson extrapolation values are known in Sect. 5, since a test problem is used with known exact solutions for both the normal boundary flux and the interior potential. However, in applications the bad Richardson extrapolation values would not be known, and the bad p values, which are known, would be used as warning flags, i.e., predictors of points with poor Richardson extrapolation results.

Error Estimation by Means of Richardson Extrapolation

331

5 Numerical Experiments The numerical example chosen for the model problem (1) has the exact solution u(x, y) = π ey cos(x − π /7) + e(1−π x) cos(π y − π /2) +

1 π e5x cos(5y − π /2). 100

The square domain is D = {(x, y)| 0 < x < 1, 0 < y < 1}. The Dirichlet boundary data used were those given on the boundary by the exact solution. A Mathematica© notebook implementing the boundary element method was written by the author of this paper. The notebook implemented collocation in the classical direct boundary element method [GaKoWa03]. Piecewise constant (discontinuous) elements (basis functions) for the primary unknown and its normal boundary flux were used. The collocation points were selected as the midpoints of each element (subinterval) in a uniform element grid on the boundary of the square. Note that the geometry nodes that delineate the end points of each boundary element are different from the collocation nodes, which are at the element midpoints. The boundary nodes were numbered sequentially, counterclockwise from the lower left vertex. Initially, a relatively coarse uniform geometry grid was used on the boundary. The number of geometry x-nodes on a horizontal edge of the square domain was set equal to the number of geometry y-nodes on a vertical edge, denoted by nxnodes and nynodes, respectively. The boundary grid was then uniformly refined in such a way so as to have the collocation nodes in the coarse boundary grid remain as collocation nodes in the two other grids, the refined grids, that were used to construct the Richardson extrapolation results. The numbers of boundary geometry nodes were taken successively as nxnodes = nynodes = 10, 28, and 82, respectively, for the coarse, intermediate, and fine boundary grids. The interior grid was similarly refined for Richardson extrapolation of the interior potential values, where there was a 9 × 9 interior coarse grid (81 interior nodes). The 81 interior nodes were numbered sequentially from the lower left, from left to right and then from bottom to top. The focus of the numerical results that will now be described is (i) to use Richardson extrapolation to improve the numerical results for the normal outward boundary flux and, separately, (ii) to use Richardson extrapolation to improve the numerical results for the interior potential. A main interest in both (i) and (ii) is to be able to get information on the validity of these Richardson extrapolation results, especially since precise values of the parameters involved in the error terms are not known.

5.1 Richardson Extrapolation for Normal Boundary Flux The numerical results for normal boundary flux errors computed with a coarse grid (nxnodes = nynodes = 10), intermediate grid (nxnodes = nynodes = 28), fine grid

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(nxnodes = nynodes = 82), and associated Richardson extrapolation and the resulting numerical estimates for the p values are given in Table 1. These numerical errors for the normal boundary flux obtained from executing the Mathematica boundary element notebook three times, using the coarse, intermediate, and refined boundary grids, are compared with the Richardson extrapolation error that is computed using these three data sets. Specifically, the Richardson extrapolation errors should be compared node-by-node with the fine grid errors to see which are smaller in magnitude. In particular, see the data in bold font in Table 1, which will now be discussed. The numerical justification for application of Richardson extrapolation, with the expectation that Richardson extrapolation will improve the numerical results, is provided most clearly by Table 1 and Fig. 1. The Richardson extrapolation results should be better than the fine grid results. However, the Richardson extrapolation results at the nine boundary nodes numbered 10, 17, 18, 21, 22, 23, 27, 28, and 36 are not as accurate as the corresponding fine grid results. This is known here since a test problem with a known solution is used. However, in an actual application, this information would be desired, but not available. The utility of the numerical p value estimates can now be demonstrated. These a posteriori estimates are available and predict the locations of bad Richardson extrapolation results. In Fig. 1, it can be observed that there are bad p values at the ten boundary nodes numbered 1, 9, 10, 17, 18, 19, 21, 27, 28, and 36. The p values predict the locations at which the Richardson extrapolation results are bad. There are discrepancies, though, at the following boundary nodes. Boundary nodes numbered 1, 9, and 19 are detected as bad by the p value estimates but are not bad Richardson nodes. However, this is desirable because, although the magnitudes of the Richardson extrapolation errors are smaller than the magnitudes of the fine grid errors at these boundary nodes, both the fine grid results and the Richardson extrapolation results are very inaccurate. Note from Fig. 1 that these boundary nodes that have been detected are at or near corners of the domain, and these are points at which the boundary element results themselves are inaccurate. Therefore, it is good that the p values detected these boundary nodes, and local grid refinement could be performed near these nodes in order to improve the accuracy of the boundary element results. On the other hand, in Fig. 1, it can be observed that at boundary nodes numbered 22 and 23 the p value estimates do not predict anything problematic, however these are bad Richardson extrapolation nodes. This is not troublesome because both the fine grid normal boundary flux results and the Richardson extrapolation normal boundary flux results are relatively accurate and the differences in relative accuracy are not significant.

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5.2 Richardson Extrapolation for Interior Potential The numerical results for interior potential errors computed with a coarse grid (nxnodes = nynodes = 10), intermediate grid (nxnodes = nynodes = 28), fine grid (nxnodes = nynodes = 82), and associated Richardson extrapolation and the resulting numerical estimates for the p values are given in Table 2.

Fig. 1 Boundary flux nodes at which Richardson extrapolation normal boundary flux is bad (R), compared with boundary nodes at which the p values predict bad Richardson extrapolation results (p)

Mathematica graphics of the solution for the interior potential obtained from the Richardson extrapolation interior potential results and the exact solution are displayed in Figs. 2 and 3. These numerical errors for the interior potential obtained from executing the Mathematica boundary element notebook three times, using the coarse, intermediate, and refined boundary grids, are compared with the Richardson extrapolation error that is computed using these three data sets at the end of the calculations (i.e., the Richardson extrapolation for the normal boundary flux is not used here). Specifically, the Richardson extrapolation errors should be compared node-by-node with

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Table 1 Table of normal boundary flux errors and p estimates

node nxnodes = 10 # 1 −1.01 × 101 2 1.76 × 10−1 3 3.95 × 10−2 4 3.39 × 10−2 5 3.15 × 10−2 6 3.80 × 10−2 7 5.03 × 10−2 8 1.95 × 10−1 9 −8.43 10 1.06 × 10−1 11 1.29 × 10−1 12 9.81 × 10−2 13 8.34 × 10−2 14 5.36 × 10−2 15 1.37 × 10−2 16 −3.04 × 10−2 17 3.48 × 10−2 18 −1.25 19 1.93 × 101 20 −7.80 × 10−2 21 3.86 × 10−3 22 5.40 × 10−3 23 1.11 × 10−2 24 1.72 × 10−2 25 2.44 × 10−2 26 5.87 × 10−2 27 3.76 28 −4.97 × 10−1 29 7.14 × 10−2 30 3.41 × 10−2 31 4.20 × 10−2 32 4.60 × 10−2 33 4.89 × 10−2 34 5.52 × 10−2 35 9.22 × 10−2 36 2.30 × 10−1

Normal Boundary Flux Errors nxnodes = 28 nxnodes = 82 −8.79 1.23 × 10−2 6.29 × 10−3 4.36 × 10−3 3.95 × 10−3 4.75 × 10−3 7.35 × 10−3 1.46 × 10−2 −7.18 7.87 × 10−2 1.73 × 10−2 1.31 × 10−2 1.07 × 10−2 6.66 × 10−3 1.41 × 10−3 −3.66 × 10−3 −6.67 × 10−3 8.84 × 10−2 1.81 × 101 −5.18 × 10−4 −5.88 × 10−4 5.28 × 10−5 7.30 × 10−4 1.55 × 10−3 2.79 × 10−3 5.48 × 10−3 3.98 5.02 × 10−2 4.89 × 10−3 4.51 × 10−3 5.04 × 10−3 5.55 × 10−3 6.08 × 10−3 7.41 × 10−3 1.35 × 10−2 7.89 × 10−2

−8.93 1.94 × 10−3 9.06 × 10−4 5.89 × 10−4 5.18 × 10−4 6.27 × 10−4 1.01 × 10−3 2.17 × 10−3 −7.32 1.24 × 10−2 2.40 × 10−3 1.63 × 10−3 1.27 × 10−3 7.68 × 10−4 1.45 × 10−4 −4.32 × 10−4 −6.97 × 10−4 −3.31 × 10−4 1.81 × 101 −1.44 × 10−4 −1.48 × 10−4 −7.10 × 10−5 1.33 × 10−5 1.21 × 10−4 2.96 × 10−4 7.22 × 10−4 3.95 3.59 × 10−3 7.03 × 10−4 5.44 × 10−4 5.81 × 10−4 6.46 × 10−4 7.41 × 10−4 9.82 × 10−4 1.98 × 10−3 1.32 × 10−2

p Estimate Richardson Extrapolation −8.91 2.04 + 2.86i 1.24 × 10−3 2.51 −1.35 × 10−4 1.66 3.78 × 10−5 1.87 2.98 × 10−5 1.90 4.34 × 10−5 1.90 −8.60 × 10−5 1.74 1.25 × 10−3 2.44 −7.31 2.03 + 2.86i 1.25 × 10−1 −0.81 1.11 × 10−4 1.84 −1.49 × 10−4 1.82 −1.25 × 10−4 1.86 −7.78 × 10−5 1.89 2.67 × 10−7 2.07 9.20 × 10−6 1.93 −1.45 × 10−3 1.76 + 2.86i 5.20 × 10−3 2.47 + 2.86i 1.81 × 101 2.66 + 2.86i −1.42 × 10−4 4.85 −1.88 × 10−4 2.11 + 2.86i −7.39 × 10−5 3.43 −3.98 × 10−5 2.43 −2.21 × 10−5 2.18 −2.79 × 10−5 1.97 2.55 × 10−4 2.20 3.951 1.86 + 2.86i 7.25 × 10−3 2.24 + 2.86i 4.23 × 10−4 2.52 −7.09 × 10−5 1.83 −2.99 × 10−5 1.93 −3.02 × 10−5 1.92 −1.87 × 10−5 1.90 −1.63 × 10−5 1.83 −3.43 × 10−6 1.75 −3.72 × 10−2 0.76

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Table 2 Table of interior potential errors and p estimates

node nxnodes = 10 # 1 8.62 × 10−4 2 −2.40 × 10−3 3 −1.44 × 10−3 4 −1.35 × 10−3 5 −1.36 × 10−3 6 −1.50 × 10−3 7 −1.83 × 10−3 8 −3.21 × 10−3 9 −1.61 × 10−4 10 −2.31 × 10−3 11 −2.92 × 10−3 12 −2.57 × 10−3 13 −2.37 × 10−3 14 −2.39 × 10−3 15 −2.67 × 10−3 16 −3.35 × 10−3 17 −4.61 × 10−3 18 −4.80 × 10−3 19 −3.44 × 10−3 20 −3.72 × 10−3 21 −3.24 × 10−3 22 −2.97 × 10−3 23 −2.95 × 10−3 24 −3.25 × 10−3 25 −4.00 × 10−3 26 −5.47 × 10−3 27 −6.13 × 10−3 28 −4.18 × 10−3 29 −4.35 × 10−3 30 −3.70 × 10−3 31 −3.33 × 10−3 32 −3.22 × 10−3 33 −3.39 × 10−3 34 −3.95 × 10−3 35 −5.18 × 10−3 36 −5.75 × 10−3 37 −4.40 × 10−3 38 −4.56 × 10−3 39 −3.87 × 10−3 40 −3.45 × 10−3

Interior Potential Errors nxnodes = 28 nxnodes = 82 −1.94 × 10−4 −1.60 × 10−4 −1.49 × 10−4 −1.43 × 10−4 −1.44 × 10−4 −1.55 × 10−4 −1.79 × 10−4 −2.27 × 10−4 −3.29 × 10−4 −2.94 × 10−4 −2.66 × 10−4 −2.38 × 10−4 −2.22 × 10−4 −2.24 × 10−4 −2.47 × 10−4 −3.05 × 10−4 −4.17 × 10−4 −6.12 × 10−4 −4.49 × 10−4 −3.65 × 10−4 −3.12 × 10−4 −2.85 × 10−4 −2.82 × 10−4 −3.08 × 10−4 −3.78 × 10−4 −5.24 × 10−4 −8.05 × 10−4 −5.51 × 10−4 −4.33 × 10−4 −3.63 × 10−4 −3.25 × 10−4 −3.12 × 10−4 −3.26 × 10−4 −3.80 × 10−4 −5.03 × 10−4 −7.57 × 10−4 −5.79 × 10−4 −4.57 × 10−4 −3.83 × 10−4 −3.40 × 10−4

−1.49 × 10−5 −1.59 × 10−5 −1.53 × 10−5 −1.49 × 10−5 −1.50 × 10−5 −1.59 × 10−5 −1.81 × 10−5 −2.22 × 10−5 −2.83 × 10−5 −3.11 × 10−5 −2.76 × 10−5 −2.48 × 10−5 −2.33 × 10−5 −2.34 × 10−5 −2.58 × 10−5 −3.16 × 10−5 −4.32 × 10−5 −6.43 × 10−5 −4.82 × 10−5 −3.87 × 10−5 −3.31 × 10−5 −3.01 × 10−5 −2.97 × 10−5 −3.23 × 10−5 −3.97 × 10−5 −5.51 × 10−5 −8.55 × 10−5 −5.93 × 10−5 −4.63 × 10−5 −3.87 × 10−5 −3.45 × 10−5 −3.31 × 10−5 −3.45 × 10−5 −4.01 × 10−5 −5.32 × 10−5 −8.04 × 10−5 −6.23 × 10−5 −4.89 × 10−5 −4.09 × 10−5 −3.63 × 10−5

p Estimate Richardson Extrapolation −4.08 × 10−5 1.62 + 2.86i −5.93 × 10−6 2.50 1.78 × 10−7 2.06 3.92 × 10−7 2.04 4.54 × 10−7 2.04 −5.19 × 10−9 2.07 −6.83 × 10−7 2.12 −7.11 × 10−6 2.44 −2.21 × 10−4 −0.53 + 2.86i 8.54 × 10−6 1.85 −4.24 × 10−6 2.20 −3.49 × 10−6 2.18 −2.95 × 10−6 2.16 −2.95 × 10−6 2.17 −3.45 × 10−6 2.18 −4.67 × 10−6 2.19 −6.54 × 10−6 2.20 1.80 × 10−5 1.85 1.39 × 10−5 1.83 −3.73 × 10−6 2.12 −3.64 × 10−6 2.14 −3.40 × 10−6 2.14 −3.43 × 10−6 2.15 −3.87 × 10−6 2.16 −4.81 × 10−6 2.16 −6.07 × 10−6 2.14 2.71 × 10−5 1.82 1.78 × 10−5 1.82 −3.87 × 10−6 2.11 −3.69 × 10−6 2.12 −3.50 × 10−6 2.13 −3.48 × 10−6 2.13 −3.74 × 10−6 2.14 −4.35 × 10−6 2.14 −5.36 × 10−6 2.13 2.55 × 10−5 1.82 1.86 × 10−5 1.82 −3.95 × 10−6 2.10 −3.67 × 10−6 2.11 −3.40 × 10−6 2.12 (continued on the next page)

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Table 2 (Continued)

node nxnodes = 10 # 41 −3.23 × 10−3 42 −3.16 × 10−3 43 −3.26 × 10−3 44 −3.69 × 10−3 45 −3.67 × 10−3 46 −4.06 × 10−3 47 −4.34 × 10−3 48 −3.78 × 10−3 49 −3.39 × 10−3 50 −3.09 × 10−3 51 −2.76 × 10−3 52 −2.29 × 10−3 53 −1.55 × 10−3 54 −5.30 × 10−4 55 −3.24 × 10−3 56 −3.79 × 10−3 57 −3.51 × 10−3 58 −3.25 × 10−3 59 −2.96 × 10−3 60 −2.52 × 10−3 61 −1.62 × 10−3 62 2.51 × 10−4 63 2.55 × 10−3 64 −2.51 × 10−3 65 −3.20 × 10−3 66 −3.20 × 10−3 67 −3.11 × 10−3 68 −2.97 × 10−3 69 −2.68 × 10−3 70 −1.87 × 10−3 71 3.48 × 10−4 72 3.27 × 10−3 73 −2.20 × 10−4 74 −2.42 × 10−3 75 −2.38 × 10−3 76 −2.40 × 10−3 77 −2.46 × 10−3 78 −2.52 × 10−3 79 −2.49 × 10−3 80 −1.09 × 10−3 81 1.34 × 10−3

Interior Potential Errors nxnodes = 28 nxnodes = 82 −3.17 × 10−4 −3.09 × 10−4 −3.18 × 10−4 −3.59 × 10−4 −4.73 × 10−4 −5.29 × 10−4 −4.32 × 10−4 −3.74 × 10−4 −3.37 × 10−4 −3.08 × 10−4 −2.76 × 10−4 −2.28 × 10−4 −1.49 × 10−4 −4.21 × 10−5 −4.09 × 10−4 −3.72 × 10−4 −3.46 × 10−4 −3.25 × 10−4 −3.00 × 10−4 −2.59 × 10−4 −1.69 × 10−4 2.43 × 10−5 3.89 × 10−4 −2.51 × 10−4 −3.02 × 10−4 −3.15 × 10−4 −3.14 × 10−4 −3.08 × 10−4 −2.85 × 10−4 −2.11 × 10−4 1.66 × 10−5 6.20 × 10−4 −1.81 × 10−4 −2.69 × 10−4 −2.92 × 10−4 −3.07 × 10−4 −3.27 × 10−4 −3.54 × 10−4 −3.81 × 10−4 −3.51 × 10−4 1.55 × 10−4

−3.37 × 10−5 −3.28 × 10−5 −3.37 × 10−5 −3.80 × 10−5 −5.01 × 10−5 −5.68 × 10−5 −4.62 × 10−5 −4.00 × 10−5 −3.60 × 10−5 −3.29 × 10−5 −2.95 × 10−5 −2.43 × 10−5 −1.58 × 10−5 −4.15 × 10−6 −4.37 × 10−5 −3.96 × 10−5 −3.70 × 10−5 −3.48 × 10−5 −3.23 × 10−5 −2.80 × 10−5 −1.85 × 10−5 2.40 × 10−6 4.18 × 10−5 −2.62 × 10−5 −3.19 × 10−5 −3.36 × 10−5 −3.38 × 10−5 −3.34 × 10−5 −3.12 × 10−5 −2.34 × 10−5 9.35 × 10−7 6.65 × 10−5 −1.59 × 10−5 −2.82 × 10−5 −3.12 × 10−5 −3.32 × 10−5 −3.57 × 10−5 −3.90 × 10−5 −4.24 × 10−5 −3.99 × 10−5 1.55 × 10−5

p Estimate Richardson Extrapolation −3.24 × 10−6 2.12 −3.23 × 10−6 2.12 −3.40 × 10−6 2.13 −3.80 × 10−6 2.13 1.44 × 10−5 1.84 1.61 × 10−5 1.83 −3.90 × 10−6 2.11 −3.56 × 10−6 2.11 −3.17 × 10−6 2.11 −2.83 × 10−6 2.11 −2.51 × 10−6 2.11 −2.18 × 10−6 2.11 −1.72 × 10−6 2.14 −9.48 × 10−7 2.32 1.05 × 10−5 1.86 −3.86 × 10−6 2.12 −3.43 × 10−6 2.12 −2.90 × 10−6 2.10 −2.37 × 10−6 2.09 −1.77 × 10−6 2.08 −9.52 × 10−7 2.06 4.83 × 10−8 2.12 −2.44 × 10−5 1.67 −1.44 × 10−6 2.10 −4.10 × 10−6 2.16 −3.22 × 10−6 2.12 −2.54 × 10−6 2.09 −1.90 × 10−6 2.07 −1.09 × 10−6 2.04 3.59 × 10−7 1.99 1.61 × 10−7 2.78 −7.99 × 10−5 1.42 −2.32 × 10−4 −1.31 2.20 × 10−6 1.99 6.06 × 10−6 1.89 7.91 × 10−6 1.85 1.04 × 10−5 1.81 1.45 × 10−5 1.76 2.23 × 10−5 1.67 1.83 × 10−4 0.79 −3.32 × 10−6 1.94

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Fig. 2 Mathematica plot of exact interior potential

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Fig. 3 Mathematica plot of BEM interior potential

Fig. 4 Interior nodes at which Richardson extrapolation potential is bad (R), compared with interior nodes at which the p values predict bad Richardson extrapolation results (p)

the fine grid errors to see which are smaller in magnitude. In particular, see the data in bold font in Table 2, which will now be discussed. The numerical justification for application of Richardson extrapolation, with the expectation that Richardson extrapolation will improve the numerical results, is provided most clearly by Table 2 and Fig. 4. The Richardson extrapolation results should be better than the fine grid results. However, the Richardson extrapolation results at the five interior nodes numbered 1, 9, 72, 73, and 80 are not as accurate

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as the corresponding fine grid results. This is known here since a test problem with a known solution is used. However, in an actual application, this information would be desired, but not available. Again, this demonstrates the utility of the numerical p value estimates. These a posteriori estimates are available and predict the locations of bad Richardson extrapolation results. In Fig. 4, it can be observed that there are bad p values at the five interior nodes numbered 1, 9, 72, 73, and 80. The p values predict the locations at which the Richardson extrapolation results are bad. For this numerical example, the interior nodes at which the Richardson extrapolation values are bad and the interior nodes at which the p value estimates predict bad Richardson extrapolation results turn out to coincide. Further, these interior nodes are all near corners of the domain, where the boundary element method performs poorly. Therefore, the p values again can serve as a posteriori warning flags, i.e., indicators that the Richardson extrapolations results are suspect. In such cases the grid can be locally refined or some other corrective actions can be taken. Acknowledgements The author acknowledges support from a University of Tulsa Faculty Summer Fellowship.

References [AtHa01]

Atkinson, K., Han, W.: Theoretical Numerical Analysis–A Functional Analysis Framework, Sections 11.1–11.2, 342–362; Chapter 12, 405–435, Springer-Verlag, New York (2001). [BuFa05] Burden, R.L., Faires, J.D.: Numerical Analysis, Eighth edition, Section 2.5, 83–87; Section 4.2, 179–186, Thomson–Brooks/Cole (2005). [Co00] Constanda, C.: Direct and Indirect Boundary Integral Equation Methods, Chapter 1, 1–53, Chapman & Hall/CRC, Boca Raton (2000). [GaKoWa03] Gaul, L., Kogl, M., Wagner, M.: Boundary Element Methods for Engineers and Scientists, Springer, Berlin (2003). [KaNi87] Kaitai, L., Ningning, Y.: The Extrapolation Method for Boundary Finite Elements, IMA Preprint Series, http://ima.umn.edu/preprints/Jan87-Dec87/287.pdf. [NiKa89] Ning, Y., Kai-tai, L.: An extrapolation method for bem. J. Comput. Math., 7, n. 2, 217–225 (April 1989). [Pa06] Panahi, M.: The boundary element method for numerical solution of the Laplace equation. Int. J. Appl. Math., 19, n. 4, 403–410 (2006). [RüZh98] Rüde, U., Zhou, A.: Multi-parameter extrapolation methods for boundary integral equations. Adv. Comput. Math., 9, 173–190 (1998). [Sm87] Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods, Third edition, Oxford University Press, Oxford, 249 (1987). [XuZh96] Xu, Y., Zhao, Y.: An extrapolation method for a class of boundary integral equations. Math. Comp., 65, n. 214, 587–610 (April 1996). [YaSl88] Yan, Y., Sloan, I.H.: On integral equations of the first kind with logarithmic kernels. J. Integral Equations Appl., 1, 549–579 (1988). [Ya90] Yan, Y.: The collocation method for first-kind boundary integral equations on polygonal regions. Math. Comp., 54, n. 189, 139–154 (Jan. 1990).

Convergence of a Discretization Scheme Based on the Characteristics Method for a Fluid–Rigid System J. San Martín, J.-F. Scheid, and L. Smaranda

1 Preliminaries In this chapter, we present our latest results concerning the convergence of a numerical method to discretize the equations modeling the motion of a rigid solid immersed into a viscous incompressible fluid using the characteristics technique. Before stating these results, let us introduce the continuous model of our problem. We assume that the fluid–rigid system occupies a bounded and regular domain O ⊂ R2 and that the solid is a ball of radius 1 whose center, at time t, is denoted by ζ (t). The fluid fills the part Ω (t) = O \ B(ζ (t)) at time t. The velocity field u(x,t) and the pressure p(x,t) of the fluid, the center of mass ζ (t) and the angular velocity ω (t) of the ball satisfy the following Navier–Stokes system coupled with Newton’s laws:   ∂u ρf (1) + (u · ∇)u − μΔ u + ∇p = ρ f f, x ∈ Ω (t),t ∈ [0, T ], ∂t div u = 0, x ∈ Ω (t),t ∈ [0, T ], (2) (3) u = 0, x ∈ ∂ O,t ∈ [0, T ], (4) u = ζ  (t) + ω (t)(x − ζ (t))⊥ , x ∈ ∂ B(ζ (t)),t ∈ [0, T ], mζ  (t) = −

 ∂ B(ζ (t))

σ n dΓ + ρs

 B(ζ (t))

f(x,t)dx,t ∈ [0, T ],

(5)

J. San Martín Universidad de Chile, Santiago, Chile, e-mail: [email protected] J.-F. Scheid Université Henri Poincaré, Nancy, France, e-mail: [email protected] L. Smaranda Universitatea din Pite¸sti, Romania, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_31, © Springer Science+Business Media, LLC 2011

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J ω  (t) = −



(x − ζ (t))⊥ · σ n d Γ + ρs

∂ B(ζ (t))



(x − ζ (t))⊥ · f(x,t)dx,t ∈ [0, T ].

B(ζ (t))

(6)

In the above system, σ = −pId + 2 μ D(u) denotes the Cauchy stress tensor with D(u) = (∇u + ∇uT )/2 and ∇uT means the transpose of ∇u. The positive constant μ is the dynamic viscosity of the fluid and the constants m and J are the mass and the moment of inertia Throughout this chapter, we will use the  of the rigid  body.  2 for all x = x1 ∈ R2 . The system (1)–(6) is completed with notation x⊥ = −x x1 x2 initial conditions: (7) u(x, 0) = u0 (x), x ∈ Ω (0),

ζ (0) = ζ 0 ∈ R2 ,

ζ  (0) = ζ 1 ∈ R2 ,

ω (0) = ω0 ∈ R.

(8)

One important hypothesis of our problem is that the density ρ f of the fluid and the density ρs of the solid are constant, but different, that is,

ρ f = ρs . The fluid-structure interaction problem (1)–(8) is characterized by the strong coupling between the nonlinear equations of the fluid and those of the structure, as well as the fact that the equations of the fluid are written in a variable domain in time, which depends on the displacement of the structure. Various authors have proposed a number of different techniques to solve the governing equations on moving domains, such as the level set method [OsSe88], the fictitious domain method [GPHJP00, GPHJP01], the immersed boundary method [Pe02] and the Arbitrary Lagrangian Eulerian (ALE) method [MoGl97, Ma99, FoNo99, Ga01, LeTa08, SMST09]. The numerical convergence of Navier–Stokes equations, when the domain is independent of time, has been considered in [Pi82, Su88, AcGu00]. The convergence of numerical methods based on finite elements with a fixed mesh for a two dimensional fluid–rigid body problem has been considered in [SMSTT04, SMSTT05] where the densities of the fluid and the solid are equal (i.e. ρ f = ρs ). The main result presented in this chapter is the convergence of two numerical schemes for the generalized case where the densities of the fluid and the solid are not equal (i.e. ρ f = ρs ). The convergence results are given in Theorems 1 and 2 below, and they are concerned with the semi-discretization of the time variable and the full discretization in time and space variables, respectively. The complete proofs of these results could be found in our recent papers [SMSS10a, SMSS10b]. We now introduce the notation and functional spaces that we shall be using. Throughout this chapter, we shall use the classical Sobolev spaces H s (O), H0s (O), H −s (O), s  0 and the space of Lipschitz continuous functions C0,1 (O) on the closure of O. We also define    2 2 f dx = 0 . L0 (O) = f ∈ L (O)  O

The usual inner product in

L2 (O)2

will be denoted by

Discretization Scheme for a Fluid–Rigid System



(u, v) =

O

u · v dx

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∀u, v ∈ L2 (O)2 .

(9)

If A is a matrix, we denote by AT its transpose. For any 2 × 2 matrices A, B ∈ M2×2 , we denote by A : B their inner product A : B = Trace(AT B), and by |A| the corresponding norm. For convenience, we use the same notation as in (9) for the inner product in L2 (O, M2×2 ), that is, 

(A, B) =

O

A : B dx

∀ A, B ∈ L2 (O, M2×2 ).

For ζ ∈ O, we introduce the space of rigid functions in B(ζ ) = {x ∈ R2 : |x− ζ | ≤ 1},

 (10) K (ζ ) = u ∈ H01 (O)2 | D(u) = 0 in B(ζ ) , the space of rigid functions in B(ζ ) which are divergence free in the whole domain O,

 (11) K (ζ ) = u ∈ K (ζ ) | div u = 0 in O and the space of the pressure

 M(ζ ) = p ∈ L02 (O) | p = 0 in B(ζ ) .

(12)

In the remainder of this chapter we assume that any velocity field in K (ζ ) will be extended by zero outside of O. According to Lemma 1.1 of [Te83, pp. 18], for any u ∈ K (ζ ), there exist lu ∈ R2 and ωu ∈ R such that u(y) = lu + ωu (y − ζ )⊥

∀y ∈ B(ζ ).

Let us define the density ρ by the piecewise constant function

ρs if x ∈ B(ζ ), ρ (x) = ρ f if x ∈ O \ B(ζ ). We note that by using the above definitions, for any u, v ∈ K (ζ ), we have (ρ u, v) =

 O\B(ζ )

ρ f u · v dx + Mlu · lv + J ωu ωv .

The spaces (10), (11) are specific to our problem. In fact, if the solution u of (1)–(8) is extended by u(x,t) = ζ  (t) + ω (t)(x − ζ (t))⊥

∀x ∈ B(ζ (t)),

then we easily see that u(t) ∈ K (ζ (t)). In the remainder of this chapter, the solution u of (1)–(8) will be extended as above.

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An important feature of our numerical method is that we use the characteristic function whose level lines are the integral curves of the velocity field. More pre : [0, T ]2 × O → cisely (see, for instance, [Pi82, Su88]) the characteristic function ψ O is defined as the solution of the initial value problem ⎧ ⎨ dψ  (t; s, x) = u(ψ  (t; s, x),t) ∀t ∈ [0, T ], dt ⎩  (s; s, x) = x. ψ It is well known that the material derivative Dt u = ∂ u/∂ t + (u · ∇)u of u at the instant t0 satisfies  d   (t;t0 , x),t) | . Dt u(x,t0 ) = u(ψ t=t0 dt Remark 1. If

ζ ∈ H 2 (0, T )2 ,

ω ∈ H 1 (0, T ),

u ∈ C([0, T ]; K (ζ (t))),

then by using a classical result of Liouville (see, for instance, [Ar92, p. 251]), we see that det Jψ = 1, where we have denoted by Jψ = ( ∂∂ ψy ji )i, j the jacobian matrix of the transformation  (y). y → ψ 

Let us now state the weak formulation of the system (1)–(8), which we use to discretize the problem in time. Proposition 1. Assume that       u ∈ L2 0, T ; H 2 (Ω (t))2 ∩ H 1 0, T ; L2 (Ω (t))2 ∩C [0, T ]; H 1 (Ω (t))2 , p ∈ L2 0, T ; H 1 (Ω (t)) , ζ ∈ H 2 (0, T )2 , ω ∈ H 1 (0, T ) and that u is extended by u(x,t) = ζ  (t) + ω (t)(x − ζ (t))⊥

∀x ∈ B(ζ (t)).

Then (u, p, ζ , ω ) is the solution of (1)–(8) if and only if for all t ∈ [0, T ], u(·,t) ∈ K (ζ (t)), p(·,t) ∈ M(ζ (t)) and (u, p) satisfies    d   (t), ϕ + a(u, ϕ ) + b(ϕ , p) = (ρ f(t), ϕ ) ∀ϕ ∈ K (ζ (t)), (13) u ◦ψ ρ dt b(u, q) = 0

∀q ∈ M(ζ (t)),

where the bilinear forms a(·, ·) and b(·, ·) are defined as follows: a(u, v) = 2μ

 O

D(u) : D(v) dx

∀ u, v ∈ H 1 (O)2

(14)

Discretization Scheme for a Fluid–Rigid System

and b(u, p) = −

 O

div(u)p dx

343

∀u ∈ H 1 (O)2 , ∀ p ∈ L02 (O).

For the proof of Proposition 1 we refer the reader to [QuVa94, Ch.12]. In the remainder of this chapter, we suppose that f and u0 satisfy u0 ∈ H 2 (Ω )2 , div(u0 ) = 0 in Ω , u0 (y) = ζ 1 + ω0 (y − ζ 0 )⊥ on ∂ B(ζ 0 ),

f ∈ C([0, T ]; H 1 (O)2 ), u0 = 0 on ∂ O,

(15)

where ζ 0 , ζ 1 ∈ R2 , ω0 ∈ R and Ω = O \ B(ζ 0 ). Let us also assume that the corresponding solution (u, p, ζ , ω ) of problem (1)–(8) satisfies ⎧ 2 2 1 2 2 ⎪ ⎨ u ∈ C([0, T ]; H (Ω (t)) ) ∩ H (0, T ; L (Ω (t)) ), (16) Dt2 u ∈ L2 (0, T ; L2 (Ω (t))2 ), u ∈ C([0, T ];C0,1 (O)2 ), ⎪ ⎩ 1 3 2 2 p ∈ C([0, T ]; H (Ω (t))), ζ ∈ H (0, T ) , ω ∈ H (0, T ), and

dist (B(ζ (t)), ∂ O) > 0 ∀t ∈ [0, T ].

(17)

Remark 2. The hypotheses (16) and (17) imply the existence of η > 0 such that dist (B(ζ (t)), ∂ O) > 3η

∀t ∈ [0, T ].

2 Semi-discretization in the Time Variable By using the weak formulation (13), (14), let us derive a semi-discrete version of our system. For N ∈ N∗ we denote Δ t = T /N and tk = kΔ t for k = 0, . . . , N. Denote by (uk , ζ k ) ∈ K (ζ k ) ∩C0 (O)2 × O the approximation of the solution of (1)–(8) at the time t = tk . From now on we shall use the notation   (tk ;tk+1 , x) ∀x ∈ O. X(x) =ψ We approximate the position of the rigid ball at instant tk+1 by ζ k+1 which is defined by ζ k+1 = ζ k + uk (ζ k )Δ t. We then define the characteristic function ψ associated with the semi-discretized velocity field as the solution of ⎧ ⎨ d ψ (t;t , x) = uk (ψ (t;t , x)) − uk (ζ k ) ∀t ∈ [tk ,tk+1 ], k+1 k+1 dt (18) ⎩ ψ (tk+1 ;tk+1 , x) = x − uk (ζ k )Δ t

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and we denote

k

X (x) = ψ (tk ;tk+1 , x)

∀x ∈ O.

(19)

In (18), the velocity field is extended by zero outside of the domain O. We next define uk+1 ∈ K (ζ k+1 ) as the solution of the following Stokes type system: uk



ρ

k+1 u

k+1 − uk ◦ Xk

Δt

 , ϕ + a(uk+1 , ϕ ) = (ρ k+1 fk+1 , ϕ )

∀ ϕ ∈ K (ζ k+1 ), (20)

where fk+1 = f(tk+1 ) and ρ k+1 is defined by

ρs if x ∈ B(ζ k+1 ), k+1 ρ (x) = ρ f if x ∈ O \ B(ζ k+1 ). Equation (20) can be rewritten by using a mixed formulation. It is clear that (20) is equivalent to the following system:   k uk+1 − uk ◦ X , ϕ + a(uk+1 , ϕ ) + b(ϕ , pk+1 ) ρ k+1 Δt

b(uk+1 , q) = 0

= (ρ k+1 fk+1 , ϕ )

∀ ϕ ∈ K (ζ k+1 ), (21)

∀ q ∈ M(ζ k+1 ),

(22)

(ζ k+1 ) × M(ζ k+1 ).

∈K of unknowns It is well known (see, for example, [GiRa79, Corollary I.4.1., pp. 61]) that the mixed formulation (21), (22) is a well-posed problem, provided that the spaces K (ζ ), M(ζ ) and the bilinear form b satisfy an inf–sup condition. The fact that this inf–sup condition is satisfied in our case follows from the result below (for the proof see, for instance [GiRa79, pp. 81]): (uk+1 , pk+1 )

Lemma 1. Suppose that ζ ∈ O is such that d(ζ , ∂ O) = 1 + η , with η > 0. Then there exists a constant β > 0, depending only on η and on O, such that for all q ∈ M(ζ ) there exists u ∈ K (ζ ) with  O

div(u) q dx ≥ β uH 1 (O)2 qL2 (O) .

In addition, we have uk+1 ∈ C0 (O)2 (for more details, see [SMSS10b]). Let us now state the first main result concerning the convergence of the semidiscrete scheme (21), (22) (for the proof of the next theorem, we refer the reader to [SMSS10b]): Theorem 1. Suppose that O is an open smooth bounded domain in R2 , f and u0 satisfy (15) and (u, p, ζ , ω ) is a solution of (1)–(8) satisfying (16) and (17).

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Then there exist two positive constants C and τ ∗ not depending on Δ t such that for all 0 < Δ t  τ ∗ the solution (uk , pk , ζ k ) of the semi-discretization problem (21), (22) satisfies   sup |ζ (tk ) − ζ k | + u(tk ) − uk L2 (O)2  CΔ t. 1kN

The key features used in the proof of the above theorem are some of the properties of the characteristic functions associated with the semi-discretized velocity field which are given in the following lemma (more details and the complete proof of this result could be found in [SMSS10b]): Lemma 2. For any k ∈ {0, . . . , N}, the characteristic function ψ defined in (18), (19) satisfies the following properties:  k i) X B(ζ k+1 ) = B(ζ k ); ii) If we extend by ρ f the density field ρ k outside of O, we have k

ρ k+1 = ρ k ◦ X ; iii) For any f ∈ L2 (R2 ) such that f = 0 in R2 \ O, we have      f ◦ ψ (t;tk+1 , ·) 2 ≤  f L2 (O) ∀t ∈ [tk ,tk+1 ]. L (O)

3 Full Discretization in the Time and Space Variables In order to discretize the problem (21), (22) with respect to the space variable, let us introduce two families of finite element spaces which approximate the spaces K (ζ ) and M(ζ ) defined in (10) and (12). To this end, we consider the discretization parameter 0 < h < 1. Let Th be a quasi-uniform triangulation of the domain O. We denote by Wh the P1 -bubble finite elements space associated with Th for the velocity field in the Stokes problem and by Eh the P1 -finite elements space for the pressure. Then, we define the following finite elements spaces for a conforming approximation of the fluid–rigid system: Kh (ζ ) = Wh ∩ K (ζ ) Mh (ζ ) = Eh ∩ M(ζ )

∀ζ ∈ O, ∀ζ ∈ O.

In order to define the approximate characteristics, let us denote by Fh the P2 -finite element space associated with the triangulation Th and we introduce the space: Rh (ζ ) = {∇⊥ ϕh : ϕh ∈ Fh , ϕh = 0 on ∂ O} ∩ K (ζ )

∀ζ ∈ O,

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 − ∂ ϕ /∂ y where ∇⊥ ϕh = ∂ ϕ h/∂ x . 

h

We denote P(ζ ) the orthogonal projection from L2 (O)2 onto Rh (ζ ), i.e. if u ∈ 2 L (O)2 then P(ζ )u ∈ Rh (ζ ) such that (u − P(ζ )u, rh ) = 0 for all rh ∈ Rh (ζ ). Let N be a positive integer. We denote Δ t = T /N and tk = kΔ t for all k ∈ {0, . . . , N}. Assume that the approximate solution (ukh , pkh , ζ kh ) of (1)–(8) at t = tk is known. We describe below the numerical scheme allowing to determinate the k+1 k+1 k+1 approximate solution (uk+1 ∈ R2 h , ph , ζ h ) at t = tk+1 . First, we compute ζ h by ζ k+1 = ζ kh + ukh (ζ kh )Δ t. h We now consider the approximated characteristic function ψ kh defined as the solution of ⎧ ⎨ d ψ k (t;tk+1 , x) = P(ζ k )uk (ψ k (t;tk+1 , x)) − P(ζ k )uk (ζ k ) ∀t ∈ [tk ,tk+1 ], h h h h h h dt h ⎩ k k k ψ h (tk+1 ;tk+1 , x) = x − uh (ζ h )Δ t

and we define

k

Xh (x) = ψ kh (tk ;tk+1 , x)

∀x ∈ O.

(23)

(24)

We observe that since div (P(ζ kh )ukh (ψ kh (t;tk+1 , ·))−P(ζ kh )ukh (ζ kh )) = 0 and ∇(x− we get det Jψ k = 1. h We now split the mesh into the union of four different types of triangle’s subsets. We first introduce Ah as the union of all triangles intersecting the ball B(ζ kh ), i.e. ukh (ζ kh )Δ t) = Id,

Ah =



T.

T ∈Th ◦ ◦ T ∩B(ζ kh )=0/

We denote by Qh the union of all triangles such that all their vertices are contained in Ah . The triangles of Th are then split into the following four categories: • • • •

F1 is the subset of Th formed by all triangles T ∈ Th such that T ⊂ B(ζ kh ). F2 is the subset formed by all triangles T ∈ Th \ F1 such that T ⊂ Qh . F3 is the subset formed by all triangles T ∈ Th such that T ∩ Qh = 0/ and T ⊂ Qh . F4 = Th \ (F1 ∪ F2 ∪ F3 ).

We introduce two approximated density functions ρhk and ρ kh as follows:

ρs if x ∈ B(ζ kh ), ρs if x ∈ Qh , k k ρh (x) = ρ h (x) = k ρ f if x ∈ F4 . ρ f if x ∈ O \ B(ζ h ), With these notations, we consider the following mixed variational fully discrete k+1 k+1 k+1 formulation: Find (uk+1 h , ph ) ∈ Kh (ζ h ) × Mh (ζ h ) such that

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347

  k k+1 k k+1 uh − uh ◦ Xh k+1 , ϕ + a(uk+1 ρh h , ϕ ) + b(ϕ , ph ) Δt k+1 k+1 = (ρ k+1 h fh , ϕ ) ∀ϕ ∈ Kh (ζ h ), (25)

b(uk+1 h , q) = 0

∀q ∈ Mh (ζ k+1 h ),

(26)

where fk+1 is the L2 (O)2 -projection of fk+1 = f(tk+1 ) on (Eh )2 . h We can now state the second result of this chapter concerning the convergence of the fully discrete scheme (25), (26) (for the proof of this result, we refer the reader to [SMSS10b]): Theorem 2. Let O be a convex domain with a polygonal boundary. Suppose that f and u0 satisfy the conditions (15) and that (u, p, ζ , ω ) is a solution of (1)–(8) satisfying the regularity properties (16) and such that (17) holds. Let C0 > 0 be a fixed constant. Then there exist two positive constants C and τ ∗ independent of h and Δ t such that for all 0 < Δ t ≤ τ ∗ and for all h ≤ C0 Δ t 2 we have   sup |ζ (tk ) − ζ kh | + u(tk ) − ukh L2 (O)2 ≤ CΔ t. 1≤k≤N

The key to the proof of the previous convergence result are the properties of the characteristic functions associated with the fully discretized velocity field given in the following lemma (the proof of this result is analogous to the proof of Lemma 2 and could be found in [SMSS10b]): Lemma 3. For any k ∈ {0, . . . , N} and h ∈ (0, 1), the characteristic function ψ kh defined in (23), (24) satisfies the following properties:  k k (i) Xh B(ζ k+1 h ) = B(ζ h ); k

(ii) If we extend by ρ f the density field ρhk outside of O, then ρhk+1 = ρhk ◦ Xh ; (iii) For any f ∈ L2 (R2 ) such that f = 0 in R2 \ O, we have    f ◦ ψ k (t;tk+1 , ·) 2 2 ≤  f  2 2 ∀t ∈ [tk ,tk+1 ]. h L (O) L (O) Acknowledgements J. San Martín was partially supported by Grant Fondecyt 1090239 and BASAL-CMM Project. J.-F. Scheid gratefully acknowledges the Program ECOS-CONICYT (Scientific cooperation project between France and Chile) through grant C07-E05. L. Smaranda was partially supported by Grant RP-2, no. 6/01.07.2009 of CNCSIS-UEFISCSU.

References [AcGu00]

Achdou, Y., Guermond, J.-L.: Convergence analysis of a finite element projection/Lagrange–Galerkin method for the incompressible Navier–Stokes equations. SIAM J. Numer. Anal., 37, 799–826 (2000).

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[SMSS10b]

[SMSTT04]

[SMSTT05]

[SMST09]

[Su88] [Te83]

J. San Martín et al. Arnold, V.: Ordinary Differential Equations, Springer-Verlag, Berlin (1992). Formaggia, L., Nobile, F.: A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math., 7, 105–131 (1999). Gastaldi, L.: A priori error estimates for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math., 9, 123–156 (2001). Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier–Stokes Equations, Springer-Verlag, Berlin (1979). Glowinski, R., Pan, T.-W., Hesla, T., Joseph, D., Périaux, J.: A distributed Lagrange multiplier/fictitious domain method for the simulation of flow around moving rigid bodies: application to particulate flow. Comput. Methods Appl. Mech. Engrg., 184, 241–267 (2000). Glowinski, R., Pan, T.-W., Hesla, T.I., Joseph, D.D., Périaux, J.: A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys., 169, 363–426 (2001). Legendre, G., Takahashi, T.: Convergence of a Lagrange–Galerkin method for a fluid–rigid body system in ALE formulation. M2AN Math. Model. Numer. Anal., 42, 609–644 (2008). Maury, B.: Direct simulations of 2D fluid-particle flows in biperiodic domains. J. Comput. Phys., 156, 325–351 (1999). Maury, B., Glowinski, R.: Fluid-particle flow: a symmetric formulation. C. R. Acad. Sci. Paris Sér. I Math., 324, 1079–1084 (1997). Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys., 79, 12–49 (1988). Peskin, C.S.: The immersed boundary method. Acta Numer., 11, 479–517 (2002). Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier–Stokes equations. Numer. Math., 38, 309–332 (1982). Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin (1994). San Martín, J., Scheid, J.-F., Smaranda, L.: A time discretization scheme of a characteristics method for a fluid–rigid system with discontinuous density. C. R. Math. Acad. Sci. Paris, 348, 935–939 (2010). San Martín, J., Scheid, J.-F., Smaranda, L.: A modified Lagrange–Galerkin method for a fluid–rigid system with discontinuous density. Submitted to Numer. Math., (2010). San Martín, J., Scheid, J.-F., Takahashi, T., Tucsnak, M.: Convergence of the Lagrange–Galerkin method for a fluid–rigid system. C. R. Math. Acad. Sci. Paris, 339, 59–64 (2004). San Martín, J., Scheid, J.-F., Takahashi, T., Tucsnak, M.: Convergence of the Lagrange–Galerkin method for the equations modelling the motion of a fluid–rigid system. SIAM J. Numer. Anal., 43, 1536–1571 (2005). San Martín, J., Smaranda, L., Takahashi, T.: Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time. J. Comput. Appl. Math., 230, 521–545 (2009). Süli, E.: Convergence and nonlinear stability of the Lagrange–Galerkin method for the Navier–Stokes equations. Numer. Math., 53, 459–483 (1988). Temam, R.: Problèmes mathématiques en plasticité, Gauthier-Villars, Montrouge (1983).

An Efficient Algorithm to Solve the GITT-Transformed 2-D Neutron Diffusion Equation M. Schramm, C.Z. Petersen, M.T. Vilhena, and B.E.J. Bodmann

1 Introduction In the last few years special attention has been devoted to searching analytic solutions for the diffusion equation. We are aware of literature for this sort of solution for specialized topics dealing with the simulation of pollutant dispersion in the atmosphere. For illustration we cite the works of [BuViMoTi07, BuViMoTi07a, MoViBuTi09]. On the other hand, the literature is scarce regarding analytical solutions for the neutron diffusion equation, except for very specialized problems [MaSaCaRoAnOlLe07]. Work on analytical solutions to the one-dimensional and two-dimensional two-group neutron diffusion equation for either homogeneous or heterogeneous sheets by the well-known GITT technique [MoViBuTi09] has recently emerged in the literature. The key feature of this methodology is that is it uses an expansion of the fast and thermal fluxes in a series written in terms of a set of orthogonal eigenfunctions. Replacing these expansions in the original equation and taking moments, yields a second-order matrix differential equation, known in the framework of this methodology as the GITT transformed problem. At this point, we shall recall that the solution of this sort of problem is given by a linear combination of the sine and cosine functions of the square root of a M. Schramm Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] C.Z. Petersen Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] M.T. Vilhena Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] B.E.J. Bodmann Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_32, © Springer Science+Business Media, LLC 2011

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matrix. This means a function composed of two matrix functions. Efficient algorithms to evaluate this type of solution are not available in the literature except for very specific problems, mainly the ones requiring low-order matrices. Therefore the standard procedure which is usually adopted to solve this second-order matrix differential equation has been to reduce the order of this equation to a system of first-order matrix differential equations which has a well-known solution. For more details see elsewhere [MoVa78]. The drawback of this procedure relies on the fact that the resulting matrix has its order doubles. This imposes a limitation on the ability of this technique to handle problems requiring high-order matrices. In order to circumvent this difficulty, we report an efficient approach from the computational point of view. After diagonalizing the matrix, we define a new variable in such a way that the matrix appearing in the new transformed equation is diagonal. As a consequence this new system of equations is a set of uncoupled equations which can be solved using standard results for second-order linear differential equations with constant coefficients. Once the solution of the transformed problem is known, the solution of the neutron diffusion equation for a homogeneous rectangle can be determined. This solution technique can be extended for the heterogeneous sheet, following the idea in the work of [BoViFeBa10]. Finally, we complete our analysis by reporting on a criterion of convergence for the proposed solution using the Cardinal Theorem of Interpolation Theory, which permits us to obtain results with any prescribed accuracy. Moreover the Cauchy– Kowalewski theorem guarantees the existence and uniqueness of the obtained solution, so that a finite number of terms in the series solution gives a suitable and robust algorithm to generate exact results to within a chosen accuracy. We shall remark that this sort of solution is appropriate for validating the physical model. The question of benchmarks for numerical approaches becomes obsolete once an analytical solution is available.

2 Mathematical Formulation We consider the two-group neutron diffusion equation in a homogeneous rectangle (0 < x < M and 0 < y < L), −Dg

∂ 2 φg (x, y) ∂ 2 φg (x, y) − Dg + Σ Rg φg (x, y) 2 ∂x ∂ y2 =

2 2 1 χg ∑ νΣ f g φg (x, y) + ∑ Σgg φg (x, y) , (1) keff g =1 g =1 g =g

subject to the boundary conditions

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φg (0, y) = φg (M, y) = 0, φg (x, 0) = 0, ∂ φg φg (x, L) (x, L) = − , ∂y d where d is the extrapolated distance. Here g is the index that denotes the energy group (1 for fast, 2 for thermal); φg (x, y) is the neutron scalar flux of group g of energy; Dg is the diffusion coefficient of group g of energy; ΣRg is the macroscopic removal cross section of group g of energy; Σ ag is the macroscopic absorption cross section of group g of energy; Σ gg is the macroscopic scattering cross section of group g of energy to group g of energy; Σ f g is the macroscopic fission cross section of group g of energy; keff is the effective multiplication factor; ν is the average number of neutrons liberated by fission; χg is the integrated fission spectrum of group g of energy and Σ Rg = Σag + Σ gg . In order to solve (1) by the GITT approach, we initially expand the fast and the thermal flux in a series expansion in terms of a set of orthonormal eigenfunctions ψi (x) = sin(λi x) with the respective eigenvalues λi = iπ /M for i = 1, . . . , Nmax :

φg (x, y) =

Nmax

ψi (x)ϕgi (y)

i=1

Ni2

∑

1

(2)

.

Here Ni denotes the norm. Replacing (2) in (1) and taking moments, that is, multi plying the resulting equation by the operator 0M ψ j (x)(·)dx, we come out with the following linear second-order matrix differential equation Y  (y) +UY (y) = 0, which is known as the GITT transformed equation. Here Y (y) is the column vector Y (y) = (ϕ11 (y) . . . ϕ1N (y)ϕ21 (y) . . . ϕ2N (y))T and the matrix U has the form ⎛

A − λ12 0 ⎜ 0 A − λ22 ⎜ ⎜ ⎜ ⎜ ⎜ U =⎜ ⎜ C 0 ⎜ ⎜ 0 C ⎜ ⎜ ⎝

B 0 ..

..

. A − λQ2

..

νΣ

. B

0 D − λ12 0 D − λ22 ..

. C

νΣ

⎞

0 B

.

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

D − λQ2

where A = D1 kfeff1 − ΣDR11 , B = D1 kfeff2 , C = ΣD122 and D = − ΣDR22 . Recalling that the eigenvalues of matrix U are distinct since the operator associated to the neutron diffusion equation is self-adjoint, the next step is to diagonalize the matrix U to obtain

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Y  (y) + PDP−1Y (y) = 0,

(3)

where P in the matrix of the eigenvectors of the U matrix, P−1 is its inverse and D is the diagonal matrix of the eigenvalues of the U matrix. Defining a new available R(y) = P−1Y (y), (3) becomes R (y) + DR(y) = 0.

(4)

Recalling that the matrix D is diagonal, the matrix equation (4) reduces to a set of uncoupled linear second-order differential equations with constant coefficients,  rgi (y) + γi2 rgi (y) = 0

for g taking values 1 and 2 and for i = 1, . . . , Nmax . The well-known solution to this equation is rgi (y) = αgi cos(γi y) + βgi sin(γi y), where αgi and βgi are the integration constants. Applying the same series expansion for the boundary conditions, we come out with identical boundary conditions for the ϕgi (y). Therefore, the solutions of the neutron diffusion equation for the fast and thermal neutron are completely determined by the application of the boundary conditions and evaluating Y (y) as Y (y) = PR(y). Although this solution is determined for homogeneous rectangle, its extension to heterogeneous problem can be easily done following the procedure of [BoViFeBa10]. To show the efficiency of the proposed algorithm and to evaluate the solution, in what follows we discuss a criterion for convergence with a prescribed accuracy. In fact, the cardinal theorem of interpolation theory states: “A square-integrable function ω = ψi (x)dx ∈ L2 r

with eigenvalues λi and i = 1, . . . , Nmax which is limited in by (mΣ T g )−1 has an exact solution for an finite expansion”. More specifically, knowing that the behavior of the fast and thermal neutron fluxes are determined by the value of the total cross sections for each group (ΣT g ), we can infer that between two successive neutron interactions, by which we mean that for each neutron mean free path (λg ) is given by λg = ΣT−1 g , the neutron flow is unchanged. The basic idea is to define that the convergence criterion consists of determining an integer multiple of the neutron mean path such that beyond this value, the neutron flux value is almost constant. So far, the choice of m is related to the number of terms (Nmax ) in the series for the region of interest, which depends on the convergence of the solution. On the other hand, we use the Parseval theorem [Go05] to estimate the solution error. In the framework of this theorem, we can select m in such a way that the solution coincides with the exact solution to within a prescribed error. In the next section, we illustrate this procedure, attaining results by this methodology with an error of < 1% where only two terms in the series are sufficient.

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3 A Case Study To complete our analysis, we solve the two-group neutron diffusion equation in a homogeneous rectangle with L = M = 200 cm, with the nuclear parameters values depicted in Table 1 [AbHa08]. Table 1 Nuclear data for the case study Material properties Dg (cm) Σag (cm−1 ) ν Σ f g (cm−1 ) Σgg (cm−1 ) keff

Energy group (g) 1 1.35 0.001382 2.41 0.000242 0.0023 = 1.000008

2 1.08 0.00569 2.41 0.00428 0

In Table 2, we report the fast and thermal neutron fluxes values given by the proposed solution with an relative error of < 1% and by summing just two terms in the solution series at symmetric points in the interior of rectangle including its center. In Fig. 1 we display the fast neutron flux which shows, as expected, that the value of the neutron flux takes its maximum at the center and decreases to near to zero at the boundary. Table 2 Values of the fast and thermal neutron fluxes at some points of the domain Point of the domain (x, y) (50, 50) (50, 100) (50, 150) (100, 50) (100, 100) (100, 150) (150, 50) (150, 100) (150, 150)

Fast neutron flux [N/cm2 s] 0.197212 0.268498 0.168486 0.278900 0.379713 0.238275 0.197212 0.268498 0.168486

Thermal neutron flux [N/cm2 s] 0.167536 0.228091 0.142740 0.236932 0.322570 0.201865 0.167536 0.228091 0.142740

4 Concluding Remarks This paper presented an efficient method which generates analytical solutions for neutron diffusion in two dimensions. In the future these solutions will be incorpo-

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Fig. 1 Fast neutron flux for physical parameters given in Table 1

rated into a program library and serve as generic solutions for multi-group neutron flux in a multi-region geometry. The principal steps employed are the generalized integral transform technique and diagonalizing the equation system. We note that existence and uniqueness are guaranteed by the Cauchy–Kowalewsky theorem. Further, the convergence of the solution is under control by a new interpretation of the Cardinal Theorem of Interpolation theory, where the macroscopic cross section plays the role of a sampling density and the reconstruction of the solution follows the common procedure in signal processing as originally introduced by Shannon [Go05]. Thus it is fair to say that we have found an analytical solution to the problem. For any numerical implementation one only needs the desired precision which may be immediately transformed into a truncation index of the solution series. The only task to be which needs to be carried out for applications is to determine numerically the GITT eigenvalues and to substitute the physical parameters and boundary conditions into the stored solutions which may then be calculated directly. In order to get comparable precision, numerical or stochastic procedures will be more time consuming because they have to execute a numerical algorithm for each energy group and region, respectively. This is a clear advantage, if the influences of modifications in geometry and material composition on the solution are to be examined, which in the present procedure may be carried out in an analytical fashion. Acknowledgements This work was sponsored by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq – Brazil).

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References [AbHa08]

Aboander, A.E., Hamada, Y.M.: Generalized Runge–Kutta method for two- and three-dimensional space-time diffusion equations with a variable time step. Annals of Nuclear Energy, 35, 1024–1040 (2008). [BoViFeBa10] Bodmann, B., Vilhena, M.T., Ferreira, L.S., Bardaji, J.B.: An analytical solver for the multi-group two dimensional neutron-diffusion equation by integral transform techniques. Il Nuovo Cimento della Società Italiana di Fisica, C. Geophysics and Space Physics, 1, 1–10 (2010). [BuViMoTi07] Buske, D., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: Simulation of pollutant dispersion for low wind in stable and convective Planetary Boundary Layer. Atmosphere Environment, 41, 5496–5501 (2007). [BuViMoTi07a] Buske, D., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: An analytical solution of the advection–diffusion equation considering non-local turbulence closure. Environmental Fluid Mechanics, 41, 5496–5501 (2007). [Ce10] Ceolin, C.: Solução Analítica da Equação Cinética de Difusão Multigrupo de Nêutrons em Geometria Cartesiana Unidimensional pela Técnica da Transformada Integral, M.Sc. Dissertation, PROMEC/UFRGS (in Portuguese), Brazil (2010). [Co93] Cotta, R.M.: Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Florida, USA (1993). [CoMi97] Cotta, R., Mikhaylov, M.: Heat Conduction Lumped Analysis, Integral Transforms, Symbolic Computation, John Wiley & Sons, Lane, Chinchester, England (1997). [Co98] Cotta, R.M.: The Integral Transform Method in Thermal and Fluids Science and Engineering, Begell House Inc., New York, USA (1998). [DuHa76] Duderstadt, J.J., Hamilton, L.J.: Nuclear Reactor Analysis, John Wiley & Sons, New York, USA (1976). [Go05] Goodman, J.W.: Introduction to Fourier Optics, Inc., Hardcover, USA (2005). [He75] Henry, A.: Nuclear-Reactor Analysis, Chap. 7, The MIT Press, Cambridge Massachusetts, USA (1975). [La65] Lamarsh, J.R.: Introduction to Nuclear Reactor Theory, Addison-Wesley Publishing Company, Inc., Massachusetts, USA (1965). [MaSaCaRoAnOlLe07] Maiorino, J.R., Santos, A., Carluccio, T., Rossi, P.C.R., Antunes, A., Oliveira, F., Lee, S.M., The participation of IPEN in the AIEA coordinate research projects on accelerator driven systems (ADS), in: International Nuclear Atlantic Conference – INAC (2007). [MoViBuTi09] Moreira, D.M., Vilhena, M.T., Buske, D., Tirabassi, T.: The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere. Atmospheric Research, 92, 1–17 (2009). [MoVa78] Moler, C., Van Loan, C., Nineteen dubious ways of computing the exponential of a matrix. SIAM Review, 45, 801–836 (1978). [ViCoMoTi08] Vilhena, M.T., Costa, C.P., Moreira, D., Tirabassi, T.: A semi-analytical solution for the three-dimensional advection diffusion equation considering non-local turbulence closure. Atmospheric Research, 63 (2008).

Nonlinear Localized Dissipative Structures for Solving Wave Equations over Long Distances J. Steinhoff, S. Chitta, and P. Sanematsu

1 Introduction There have been many conventional approaches to discretize and solve time-dependent wave equations on Eulerian grids. For short pulses or waves, these are, of course, limited by the requirement that a sufficient number of grid cells must span the pulse to accurately solve the equations, and that the propagation distance cannot be very long compared to the pulse width. (Our method applies equally well to pulses and waves and we will use the term “pulse width” to also denote the wave length.) These approaches to the wave equation problem involve, as usual, formulating the governing partial differential equations (pde’s), discretizing them and solving them as accurately as possible on feasible computational grids, assuming smooth enough solutions. For smooth, long pulses that span over many grid cells, these methods are well known to converge to the correct solution as the number of points across the pulse becomes large [1]. Even then, solutions degrade over moderate propagation distances (a number of pulse widths). As a result, they will not be feasible for most problems involving thin pulses convecting over moderate distances. Further, adaptive, unstructured grids cannot significantly improve the solution for realistic problems with many thin, time-dependent pulses (or intersecting waves). In such cases, adding grid points in many locations will only result in long computation time and complexity. J. Steinhoff University of Tennessee Space Institute, Tullahoma, TN, USA, e-mail: [email protected] S. Chitta Flow Analysis, Inc., Brooklyn, NY, USA, e-mail: [email protected] P. Sanematsu University of Tennessee Space Institute, Tullahoma, TN, USA, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_33, © Springer Science+Business Media, LLC 2011

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Another approach that has been widely used for this purpose is Lagrangian ray tracing, which requires complex interpolation routines to accommodate the addition of markers (when wave surfaces propagate and expand, reducing resolution). Unlike ray tracing, our new Eulerian finite difference method, “Wave Confinement” (WC), uses a continuous phase field, which can automatically compute multiple waves. WC allows propagation of short pulses as coherent phase surfaces, for indefinitely long distances, without numerical dissipation. Such accurate propagation will greatly extend the potential of WC to simulate many important wave propagation cases (with minimum computational cost and effort), including over the horizon (OTH) radar and cell phone communications, with effects such as reflections from complex terrain and buildings, as well as ionospheric and atmospheric refraction. Also, it will allow for the inclusion of turbulence models, interference and dispersion for synthetic aperture radars (SAR) propagation. WC is based on a previously successful technique, “Vorticity Confinement”, which allows the propagation of thin, concentrated vortices over arbitrarily long distances, and keeps many of the Eulerian finite difference properties of the original fluid dynamic solution method [2, 3]. Both methods employ nonlinear solitary waves based on a regular lattice and use the same new idea to generate discrete solutions for these thin features. The basic scheme involves an “inner” and an “outer” solution as shown in Fig. 1. The inner solution represents the very thin, physical pulse. (It could be only several centimeters wide.) This obviously could not be represented, in general, on a computational grid spanning several thousand kilometers, nor does it have to. We take an approach similar to shock capturing in compressible flow. There, the actual shock can be only millimeters wide, but the computational shock is several grid cells wide. In our case, the computational pulse may be several grid cells wide, and hence span several kilometers which is not a problem as long as it is very thin compared to the overall dimension of the problem, (∼1000 km), and as long as it carries the correct total energy (and other important integrated quantities) and the arrival time (or speed) is correct.

Fig. 1 Inner and outer solution

As in shock capturing, there are a few cases (such as combustion in fluid dynamics and nonlinear effects in waves) where details of the internal structure must be modeled or solved for, but for most cases the computed “captured” shock is adequate. Similar scenarios occur in pulse propagation where nonlinear effects may

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be important. For many problems where thin, concentrated pulses or wave fronts must be numerically propagated over long distances, the main interest is in the far field, where the amplitude integrated normally through the pulse at each point along the pulse surface, together with the motion of the centroid surface (which we use to represent the wave fronts) are important, rather than the details of the internal structure.

2 Approach and Methodology The basic idea of WC is to treat thin features, such as short waves, as a type of “weak solutions” of the governing pde. Basically, the features are treated as multidimensional nonlinear solitary waves that “live” on the computational lattice, while preserving the essential physics of the pulse. This approach has been taken in condensed matter physics for some years [4] where the computational objects have these necessary properties, and are known as “nonlinear diffusive solitary waves” [5]. The basic point is that if we can formulate a pde such that the solutions are qualitatively what we desire (thin surfaces), propagate with correct speed, conserve the desired quantities and are stable to the inevitable perturbations from the discrete computation, then they will serve as useful “basis functions”. As a simple example, we start with the 1-D scalar wave equation with constant wave speed, c, ∂t2 φ = c2 ∂x2 φ . (1) Evidently, when the above equation is numerically solved using conventional discretization schemes based on Taylor expansions, there will be numerical errors, which will grow for linear discretization. When we confine the pulse solution to 2–3 grid cells, which is our goal, the derivatives of φ and hence these “errors” will be large. Also, corresponding to the small number of grid points within the pulse, there will be only a small number of quantities that we can conserve. However, this is sufficient for our goal. Adding a term

∂t ∂x2 F(φ ) to (1) that vanishes at the boundaries, along with sufficient derivatives, will not affect the conservation of these quantities, which include the total amplitude and centroid speed. We then have ∂t2 φ = c2 ∂x2 φ + ∂t ∂x2 F. (2) To preserve the essential physics of (1) for a short convecting pulse, F should be homogeneous of degree one, like (1), so that it does not depend on the magnitude of φ . This is an important distinction of the WC equation. Many nonlinear equations use non-homogeneous terms for a nonlinear term [6]. In fact, Cahn and Hilliard also used a nonlinear term under a second derivative as in our equation, but one that

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was not homogeneous, and represented a model for the free energy for multi-phase phenomena [7]. Equation (2) produces a stable solution, which can be easily derived in the convecting frame of the pulse. The pde then becomes the heat equation

and F=

∂t φ = ∂x2 F

(3)

 α  2 ∂ ψ −λψ ψ2 x

(4)

where ψ = φ −1 . The above form of F is proven to produce stable solutions and is described in [8, 9]. When (3) converges, the pulse relaxes to the form

φ → φ0 sech [γ (x − ct)] √ where γ = λ and φ0 is an arbitrary constant. One possible discretized formulation of the pde given in (2) can be written as

δn2 φ = ν 2 δ j2 φ + a δn δ j2 F

(5)

where δn f n = f n − f n−1 , δn2 f n = f n − 2 f n−1 + f n−2 , a = Δht2 , ν = cΔh t , Δ t is the time step, and h is the grid cell size. Many conventional schemes can be put in this form, where F adds a (typically linear) stabilizing dissipation. However, the role of F is very different here. The “Confinement” term, F, is defined as Fjn = μφ jn − εΦ nj , where Φ is a nonlinear function of φ (given below) and μ is a diffusion coefficient that can model numerical discretization effects in a conventional wave equation solution (we assume physical diffusion is much smaller). For the last term, ε is a numerical coefficient that, together with μ , controls the size and relaxation time scales of the confined features. For this reason, we refer to the two terms as confinement terms. Upon Taylor expansion, we must be able to recover the pde given by (4) in the fine grid limit. In the semi-discrete limit, or if a number of these “Confinement” steps are taken for each convection step, then the above form should result. However, even in the semi-discrete case, the “peak” or maximum value of the amplitude will change periodically by a small amount as the centroid moves across each grid cell. This is necessary because the sum of φ j is conserved. Results very close to these are also found with convection steps that are not small. (Similar small periodic variations are found in other solitary wave ansätze to nonlinear wave equations [10]. These are often termed “wobble”.) There are many possibilities for Φ . A simple class is 2



Φn =

∑l (φ˜ln )−ρ N

−1/ρ

φ˜ n = φ n + δ sign(φ n ).

(6)

The above sum is over a set of N neighboring grid nodes near and including the node where Φ is computed. A small positive constant (δ ∼ 10−8 ), is added with the

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sign of φ n to prevent problems due to finite precision. The above term can be used in multi-dimensional formulations with the sum in (6) taken over nearest neighbors. If ρ = 1, Φ is the harmonic mean of φ on the local stencil. Other forms could also be used, however, we use only ρ = 1 or ρ = 2. The two parameters, ε and μ (> 0), are determined by the two small scales of the computation, h and Δ t, since we want the small features to relax to their solitary wave shape in a small number of time steps and to have a support of a small number of grid cells. Thus, even though h may be small, the Laplacian will be large and so will the total effect. At convergence, the solution to the above equation that vanishes in the far field is then (modulo a small variation), φ → φ0 sech [γ ( j − j0 ± ν n)], where j0 is an arbitrary constant (since (5) is translation invariant). With propagation in a smooth external field, this relation is still approximately satisfied, as verified by computations and heuristic arguments [11]. The main constraint on the confinement term, F, as in advection, is that it forces an initially isolated, short-range pulse that propagates with a single maximum to remain short range and also not develop any additional maxima. The behavior of a solution computed using (5) is shown in Fig. 2. When μ = 0 and ε = 0, the solution becomes unstable. A small amount of diffusion (μ = 0.2) will prevent the unstable behavior but the pulse quickly spreads, rendering the method useless except for short distance computation, especially with multiple waves. This problem can be resolved by adding the confinement term. The dissipative/dispersive effects are automatically balanced with those of the nonlinearity to produce stable localized structures [12]. The pulse then remains thin for a range of values of ε .

Fig. 2 Behavior of wave equation solution with confinement parameters

Nonlinear equations would ordinarily introduce phase shifts into the solution when the waves pass through each other. However, a very important feature of WC is that the waves do not suffer a “phase shift” when they pass through each other in spite of its nonlinearity. This is an obvious requirement for the equation we want to simulate—the linear wave equation. Even though the phase shift is a 1-D effect,

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such a phase shift would generally show up as a kink in two waves in 2 or 3 dimensions that are passing through each other. Results for the centroid trajectories for two pulses passing through each other (in 1-D) are presented in Fig. 3. There, the computed centroids are plotted for forward and backward propagating waves with periodic boundary conditions. It can be seen that there is no phase shift, to plottable accuracy.

Fig. 3 Centroid propagation for forward and backward propagating pulses

Fig. 4 Spherical wave front propagation

This lack of nonlinear interaction is due to the fact that both the Laplacian and time derivative operator operate on the nonlinear term which cause the change of integrals of interest (amplitude and centroid) to vanish in the interaction region. For 3-D, we simply substitute a multi-dimensional Laplacian into the original wave equation and use a multi-dimensional harmonic mean, where we sum inverse values of φ over each point and the N immediately neighboring grid points on the multi-dimensional grid. An expanding spherical wavefront in a cube with periodic

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boundary conditions is shown in Fig. 4 as a numerical example. It can be seen that the wave retains its shape on the grid even after multiple collisions.

3 Carrier Function Developments that allow wave fronts to capture and propagate other variables, in addition to total amplitude and centroid speed, are described in this section. WC can be used to generate constant arrival time (eikonal phase) surfaces accurately by storing the centroid arrival time at each grid point. These variables can be required to compute interference. Also, multiple arrival times can be easily accommodated, which is difficult using eikonal schemes. Some recent developments in eikonal methods [13] can treat multiple arrival times, but these methods require extra independent variables and complex data management schemes are used to control memory requirements. For WC, it is required that the wave front pulse has almost completely passed the grid point to accurately compute these properties. For very short pulses, or wavelengths, it becomes difficult to accommodate multiple arrival times when the waves are close to each other. To overcome this problem, a scalar field, φ , is used as a carrier function that can act as wave packet, which carries the required details of the propagating quantity. It is then not necessary for a pulse to entirely pass a grid point to capture the properties. The “computed” scalar pulse, which propagates according to the actual wave equation without numerical errors, can be used to “carry” (not propagated by themselves, but as a product of actual scalar and the variable of interest) an array of variables which will have the same propagation path. These variables can be used to describe additional properties of the pulse or wave, such as the propagation direction vector, actual (sub-grid scale) thickness, etc. Propagation of directions has proven to be very beneficial in capturing diffraction. For example, to propagate the directions in 2D, the wave equation is solved for three quantities, φ , φ1 and φ2 , as     δn2 φln = ν 2 ∇2 φln + μδn− ∇2 φln − εδn− ∇2 Φln where

φ1 = φ ,

φ2 = sx φ ,

φ3 = sy φ ,

and sx and sy are the normalized components of the propagating direction. The updated propagation vector is then

φ2n+1 =  sn+1 x 2  2 , φ2n+1 + φ3n+1

sn+1 =  y

φ3n+1 2  2 . φ2n+1 + φ3n+1

In the test computation described, the initial conditions for φ are given as

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φi,n=0 j = A0 sech [γ ri, j ] ,

φi,n=1 j =

A0 sech [γ (ri, j − ν )] A1

with initial directions, sx =

i − i0 , ri, j

sy =

j − j0 ri, j

 where the radius of the wave is ri, j = (i − i0 )2 + ( j − j0 )2 with origin at the center of the domain (i0 , j0 ). The parameters ν , μ , and ε are in the same range for all three quantities. Here, ν = 0.23, μ = 0.2, and ε = 0.3. φ1 and φ2 behave as the scalar function itself during wave collision. In Fig. 5, directions for the thin wave are shown at four different time steps. During interaction, directions temporarily are the mean values of the waves that are together. Then, they subsequently take their original forms after a short relaxation time.

Fig. 5 Propagation of directions using carrier approach

4 Continuous Waves WC has previously proven to be efficient for pulse-like wave propagation and has been recently extended to accurately simulate propagation of continuous waves (CW’s). A constant is first added to the amplitude so that it is non-negative. This addition simplifies the algorithm to study periodic waves with constant and varying amplitude. As a first example, we take an initial condition

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φ j0 = φ j1 =

365

2 + sin 2,

2π j λs



, 5λs ≤ j ≤ 15λs otherwise

where λs is the wave length. The wave was constrained to 5λs ≤ j ≤ 15λs because of the use of periodic boundary conditions. When the wave spans the entire domain, we cannot clearly see the interactions between the two traveling waves because they result in a single standing wave. It can be seen in Fig. 6 that WC propagates these periodic waves for many time steps with no plottable numerical dissipation and no phase shift.

Fig. 6 Propagation and interaction of sinusoidal wave with λs = 50, ν = 0.23, μ = 0.2, and ε = 0.205

To avoid effects due to the symmetry of waves with constant amplitude, a case was run with periodic waves of varying amplitude. Results are shown in Fig. 7 for the initial condition

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 2π j 2π j 3 1 , 5λs ≤ j ≤ 15λs + cos 2 + sin 4 4 λs λc φ j0 = φ j1 = 2, otherwise where λs and λc correspond to the wave length of the sine and cosine waves, respectively. An important characteristic of WC for continuous waves is that the wave maintains its initial shape, which can further help in capturing interference and dispersion effects. As well as for applications for single pulse propagation, WC uses a very simple scheme to propagate continuous waves over long distances with minimal dissipation. This can be a major advantage over higher order schemes when simulating propagation of electromagnetic waves in satellite, cell phone and OTH communications.

Fig. 7 Propagation and interaction of periodic wave with varying amplitude and λs = 25, λc = 125, ν = 0.23, μ = 0.2, and ε = 0.205

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5 Propagation Through Evaporative Atmospheric and Ionospheric Ducts In this section, the applicability of WC for wave propagation in an inhomogeneous medium is described. This type of propagation is important for OTH radar, which involves propagation of electromagnetic waves through the ionosphere which undergoes many variations, for example, due to diurnal solar activity. Ionospheric layers act as a reflecting medium that causes radio waves to be returned to earth at considerable distances from the transmitter. Each point on the wavefront follows a curved path and the degree of curvature depends on the angle of incidence, plasma frequency of the layer and frequency of the incident signal. The discretized equation given in (5) is used, but ν is not constant and must be inside the Laplacian to maintain conservation.

Fig. 8 Plane wave propagation through varying index of refraction.

As an example, WC is illustrated using an example height profile for the refractive index: ν ( j) = 0.23 − 0.0023 ∗ | j − j0 | where j0 = 30. A curved configuration for the earth surface is employed, which is immersed in the uniform Cartesian grid. For a plane wave (again 2-D), propagating through regions where the index of refraction varied by a factor of 2, there was a large refraction effect, yet numerical errors in the speed were insignificant, to plottable accuracy. It is observed that the pulse trajectory is correct, with no diffusion or dispersion, when compared to accurate ray tracing computations. Also, unlike ray tracing schemes, which suffer from scarcity of grid nodes in far field, WC can still capture waves as smooth surfaces without complex logic involving continual

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addition of new rays and interpolation. A comparison is shown in Fig. 8 in which the smooth contours are calculated by the confinement method and compared with ray tracing (depicted as circular “blobs”). It can be seen that in the far field, ray tracing techniques cannot continue to describe the wave as a smooth surface without adding markers and interpolation. Parabolic equation solvers are widely used to propagate flow over both ionospheric layers and atmospheric ducts. These solvers are usually paraxial and can only be used to capture waves that are very close to the propagation direction. Many developments have been made to allow the paraxial equation to include much wider angles, but they involve complicated logic [14]. In such cases, WC can easily accommodate propagation through any angle in one computation and does not need any complex logic to include multiple waves.

References 1. Anderson, D., Tannehill, J., Pletcher, R.: Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York (1984). 2. Steinhoff, J., Lynn, N., Wang, L.: Large eddy simulation using vorticity confinement, in: Implicit Large Eddy Simulation: Computing Turbulent Flow Dynamics (Editors: F.F. Grinstein, L.G. Margolin, W.J. Rider), Cambridge University Press (2006). 3. Steinhoff, J., Lynn, N., Wang, L., Wenren, Y., Fan, M.: Turbulent flow simulations using vorticity confinement, in: Implicit Large Eddy Simulation: Computing Turbulent Flow Dynamics (Editors: F.F. Grinstein, L.G. Margolin, W.J. Rider), Cambridge University Press (2006). 4. Bishop, A., Krumhansl, J., Trullinger, S.: Solitons in condensed matter: A paradigm. Physica D: Nonlinear Phenomena, 1, n. 1, 1–44 (1980). 5. Kuramoto, Y., Mori, H.: Dissipative Structures and Chaos, Springer-Verlag (1998). 6. Rosenau, P., Hyman, J., Staley, M.: Multidimensional compactons. Physical Review Letters, 98, n. 2, 24101 (2007). 7. Cahn, W., Hilliard, J.: Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics, 28, n. 2, 258 (1958). 8. Steinhoff, J., Chitta, S.: Long-time solution of the wave equation using nonlinear solitary waves, in: Integral Methods in Science and Engineering: Computational Aspects (Editors: C. Constanda, M. Perez), Springer Verlag (2009). 9. Steinhoff, J., Chitta, S.: Long distance wave computation using nonlinear solitary waves. Journal of Computational and Applied Mathematics, 234, n. 6, 1826–1833 (2010). 10. Peyrard, M., Kruskal, K.: Kink dynamics in the highly discrete sine-Gordon system. Physica D: Nonlinear Phenomena, 14, n. 1, 88–102 (1984). 11. Steinhoff, J., Haas, S., Xiao, M., Lynn, N., Fan, M.: Simulating small scale features in fluid dynamics and acoustics as nonlinear solitary waves, in: Proceedings of the 41st AIAA Aerospace Sciences Meeting and Exhibit, (January 2003). 12. Lax, P.: Hyperbolic systems of conservation laws II. Communications on Pure and Applied Mathematics, 10, n. 2, 537–566 (1957). 13. Vinje, V., Iversen, E., Gjoystdal, H.: Traveltime and amplitude estimation using wavefront construction. Geophysics, 58, n. 8, 1157–1166 (1993). 14. Bamberger, A., Engquist, B., Halpern, L., Joly, P.: Higher order paraxial wave equation approximations in heterogeneous media. SIAM Journal on Applied Mathematics, 48, n. 1, 129– 154 (1988).

Semianalytical Approach to the Computation of the Laplace Transform of Source Functions L.G. Thompson and G. Zhao

1 Introduction The method of “Sources and Sinks” has has found wide application to the solution of the diffusion equation for linear conduction of heat in infinite, semi-infinite and finite solids; see, for example, [Carslaw88]. Since reservoir flow of a slightly compressible, constant viscosity fluid is governed by the diffusion equation, fundamental solutions of the one-dimensional diffusion equation are routinely used to build solutions to three-dimensional flow problems using the Newmann product method [Newmann:36]. These solutions can be rapidly generated for constant-rate production (or injection) to a uniform-flux line or plane in an anisotropic homogeneous reservoir; (see for example, [Grin:73]). However, for variable-rate, variable-flux or complex-geometry (heterogeneous reservoir) problems, solutions are most easily generated by manipulating Laplace transforms of the product solutions and inverting to the time domain using a numerical inversion procedure such as the Stehfest algorithm [STEHFEST:70]. Raghavan and Ozkan [RagOzkan94] presented complete analytical expressions for the Laplace transforms of the 3-D source functions. For flow in rectangular parallelepiped reservoirs, their expressions involve triple infinite series; for example, the Laplace transform of the dimensionless fundamental solution (γ¯) for flow in a sealed rectangular parallelepiped reservoir was given as (see Section 2.4, Eq. 1 of [RagOzkan94])

γ¯ =

∞

∞

∞

∑ ∑ ∑

k=−∞m=−∞n=−∞

{S1,1,1 + S2,1,1 + S1,2,1 +S2,2,1 + S1,1,2 + S2,1,2 + S1,2,2 + S2,2,2 }

(1)

L.G. Thompson The University of Tulsa, OK, USA, e-mail: [email protected] G. Zhao University of Regina, SK, Canada, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_34, © Springer Science+Business Media, LLC 2011

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where

Si, j,l

  √  exp − u (x˜Di − 2kxeD )2 + (y˜D j − 2myeD )2 + (˜zDl − 2nzeD )2  = (x˜Di − 2kxeD )2 + (y˜D j − 2myeD )2 + (˜zDl − 2nzeD )2

(2)

for i, j, l = 1 or 2. Here u is the dimensionless Laplace variable, and the terms with subscript D represent various dimensionless distances in the x-, y- and z-directions. Note that if we were to use the Stehfest algorithm to numerically invert (1) to the (dimensionless) time domain, we would have to evaluate (2) at real values of the Laplace variable that could vary widely in magnitude. In particular, for very small values of u, many terms would have to be evaluated for the triple series of (1) to converge. In order to enable greater efficiency in their numerical computation, Raghavan and Ozkan simplified their triple infinite sums to equivalent expressions involving the sums of single and double infinite series. Even with these simplifications, it is not clear that the resulting series always converge rapidly. In this work, we explore an alternative method for efficiently computing the Laplace space functions that involves only the computation of three (single) infinite series. The number of actual terms involved in the evaluation of each series is small (typically less than 20 terms), so the method is attractive for application to large systems of equations in Laplace Space.

2 Source Functions: Real Time Computational Issues [Ohaeri:91, Thompson91] The one-dimensional fundamental solutions for a point source at the instant t = τ , located at x = x in a 1-D “slab” reservoir of length Lx (see [Grin:73]) are summarized in Table 1. Note that for systems with finite length, two alternative, mathematically equivalent expressions are available; in the first form, time is in the denominator of the exponential exponent, while in the second form, time is in the numerator of the exponential exponent. This has a significant impact on the rate of convergence of the the alternative infinite series at a given value of elapsed time, t − τ . According to [OBER:73], the elliptic theta functions θ2 (z,t) and θ3 (z,t) are defined as   ∞ 1 (z + n)2 n (3a) θ2 (z,t) = √ ∑ (−1) exp − t π t n=−∞     ∞ 1 2 2 t (3b) = ∑ cos ((2n + 1) π z) exp −π n + 2 n=0 and

= =

Semi-infinite with sealing fault at x = 0

Constant pressure at x = 0 and x = Lx

=

=

=

Sealing fault at x = 0, constant pressure at x = Lx =

=

Constant pressure at x = 0, sealing fault at x = Lx =

Sealing faults at x = 0 and x = Lx

=

Semi-infinite with constant pressure at x = 0

=

=

Infinite reservoir

Reservoir Boundary Conditions

1 2Lx

1 2Lx

n=1

√ 1 2 πη (t−τ )

x

Source Function   2 (x−x ) exp − 4η (t−τ )      2 2 (x−x ) (x+x ) √ 1 exp − 4η (t−τ ) − exp − 4η (t−τ ) 2 πη (t−τ )      2 2 (x−x ) (x+x ) √ 1 exp − 4η (t−τ ) + exp − 4η (t−τ ) 2 πη (t−τ )      2 2 n=∞ (x−x +2nLx ) (x+x +2nLx ) √ 1 exp − − exp − ∑ 4η (t−τ ) 4η (t−τ ) 2 πη (t−τ ) n=−∞           n=∞ nπ (x−x ) nπ (x+x ) π 2 n2 η (t−τ ) 1 cos − cos exp − ∑ Lx Lx Lx Lx2 n=1      2 2   n=∞ (x−x +2nLx ) (x+x +2nLx ) √ 1 exp − 4η (t−τ ) + exp − 4η (t−τ ) ∑ 2 πη (t−τ ) n=−∞        2 2  n=∞ nπ (x−x ) nπ (x+x ) π n η (t−τ ) 1 1 + cos + cos exp − ∑ 2 Lx Lx Lx Lx n=1       +2nL 2  +2nL 2 n=∞ x−x x+x ( ( x) x) √ 1 − exp − 4η (t−τ ) (−1)n exp − 4η (t−τ ) ∑ 2 πη (t−τ ) n=−∞        2 n=∞ (2n+1)π (x−x ) (2n+1)π (x+x ) π 2 (n+ 12 ) η (t−τ ) − cos exp − ∑ cos 2 2Lx 2Lx Lx n=1       +2nL 2  +2nL 2 n=∞ x−x x+x ) ( ( x x) n √ 1 exp − + exp − (−1) ∑ 4η (t−τ ) 4η (t−τ ) 2 πη (t−τ ) n=−∞        2   n=∞ (2n+1)π (x−x ) (2n+1)π (x+x ) π 2 (n+ 12 ) η (t−τ ) + cos exp − ∑ cos 2Lx 2Lx L2

Table 1 Instantaneous 1-D point source functions for various boundary conditions

Semianalytical Laplace Transforms 371

372

L.G. Thompson and G. Zhao



∞ 1 (z + n)2 θ3 (z,t) = √ exp − ∑ t π t n=−∞ ∞

=

∑ εn cos (2nπ z) exp





−π 2 n2 t ,

(4a) (4b)

n=0

respectively, where



εn =

1 2

for n = 0 for n = 1, 2, 3 . . .

We will denote the leading term of (3a) and (4a) (i.e., when n = 0) as θL (z,t) ,  2 z 1 . (5) θL (z,t) = √ exp − t πt In terms of these functions, the point source solutions of Table 1 can be written as shown in Table 2. Equations (3a), (3b) and (4a), (4b) are each expressed in alternative equivalent forms. A question now arises as to which form should be used at any particular value of z and t. For small values of the time group t, the series given by (3b) (or (4b)) converges very slowly; for small values of n, the exponential term in of (3b) (or (4b)) is close to 1, and trigonometric terms oscillate in value between ±1. Under this condition, literally millions of terms may be required to obtain accurate values of this form of the series. On the other hand, for small values of t, successive terms in the series given by (3a) (or (4a)) approach zero rapidly as n increases. The

at large values of t is reversed; for large t, successive values of situation exp −π 2 n2t → 0 rapidly as n increases, whereas n must become very large before 2

exp(− (z+n) ) → 0. t This behavior suggests the following algorithm for deciding on which series representation to use for the source function at a given value of the time group, t. We would like to use the series representation whose terms disappear at the smallest value of n. In each of the series forms, the argument of the exponential terms governs the rate of disappearance of successive terms in the series. For the series given by (3a) or (4a), the series will converge when   (z + n) ˆ 2 ≤ ε, exp − t where ε is a small number. (On computers, “machine epsilon,” εM is defined as the largest number such that the machine cannot distinguish between the numerical values of (1 + εM ) and 1; for PC’s this value is about 10−18 for double precision computations. In our work, we set ε = 10εM ) Solving for the value of n at which this occurs, and denoting this value n, ˆ we have √ nˆ ≥ −t ln ε − z

=

= = = = =

Semi-infinite with sealing fault at x = 0; (Lx is arbitrary, Lx > 0 )

Constant pressure at x = 0 and x = Lx

Sealing faults at x = 0 and x = Lx

Constant pressure at x = 0, sealing fault at x = Lx

Sealing fault at x = 0, constant pressure at x = Lx

Semi-infinite with constant pressure at x = 0; (Lx is arbitrary, Lx > 0 ) =

Infinite reservoir (Lx is arbitrary, Lx > 0 )

Reservoir Boundary Conditions

Table 2 Instantaneous 1-D point source functions for various boundary conditions

     (x−x ) η (t−τ ) (x+x ) η (t−τ ) − θ3 , θ , 3 2Lx 2Lx L2 L2

x

     (x−x ) η (t−τ ) (x+x ) η (t−τ ) + θ3 θ3 2Lx , L2 2Lx , L2

x

     (x−x ) η (t−τ ) (x+x ) η (t−τ ) θ2 , θ , − 2 2Lx 2Lx L2 L2

x

     (x−x ) η (t−τ ) (x+x ) η (t−τ ) θ2 + θ2 2Lx , L2 2Lx , L2

x

x

1 2Lx 1 2Lx 1 2Lx 1 2Lx 1 2Lx

x

x

x

x

x

x

x

     (x−x ) η (t−τ ) (x+x ) η (t−τ ) + θL θL 2Lx , L2 2Lx , L2

1 2Lx

x

     (x−x ) η (t−τ ) (x+x ) η (t−τ ) θL , θ , − L 2Lx 2Lx L2 L2

Source Function, Δ psx   (x−x ) η (t−τ ) 1 2Lx θL 2Lx , L2

Semianalytical Laplace Transforms 373

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L.G. Thompson and G. Zhao

Similarly, the series given by (3b) (or (4b)) will converge for values of n such that

exp −π 2 n˜ 2t ≤ ε . Let n˜ denote the value of n for which this occurs; then 1 − ln ε . n˜ ≥ π t If nˆ < n, ˜ we use (3a) (or (4a)); otherwise, we use (3b) (or (4b)). Using this approach, accurate values for each series can be obtained by summing only a few terms.

3 Laplace Transform of Products of Source Functions In well testing applications, we are most interested in finding the Laplace Transform with respect to time of the time integral of products of the previously defined source functions. That is, a typical unit constant-rate pressure drop solution assumes the form

t 5.615 Δ pcu (x, y, z,t) = Δ psx Δ psy Δ psz d τ , φ ct 0

with Laplace transform

Δ pcu =

5.615 φ ct

∞

Δ psx Δ psy Δ psz

e−uτ dτ . u

(6)

0

Each of the source functions in (6) approach a unique functional behavior at late times. Table 3 summarizes the late-time behaviors of each of the instantaneous source functions and the time range for which this late-time behavior is valid. Equation (6) is written as 5.615 Δ pcu = φ ct

t

Δ psx Δ psy Δ psz

e−uτ dτ +Ψ , u

(7)

0

where

Ψ=

∗

5.615 φ ct

∞

Δ psx Δ psy Δ psz

t∗

e−uτ dτ . u

The value of t ∗ is determined as follows. If the system contains at least one constant pressure boundary in any coordinate direction, then t ∗ is the minimum time at which any of the source functions approaches zero. If there are no constant pressure boundaries, t ∗ is the minimum time at which all of the source functions attain their late-time functional forms. In (7), the first integral is determined numerically

Semianalytical Laplace Transforms

375

Table 3 Late time behavior of 1-D point source functions for various boundary conditions Reservoir Boundary Conditions

Late Time f (t)

For t >

Infinite system

√1 2 πηx t

Semi-infinite with constant pressure at x = 0

0

Semi-infinite with sealing fault at x = 0

√1 πηx t

Constant pressure at x = 0 and x = Lx

0

−4 Lπx2lnη ε

Sealing faults at x = 0 and x = Lx

1 Lx

L2 ln ε − πx2 η x

Constant pressure at x = 0, sealing fault at x = Lx

0

−4 Lπx2lnη ε

0

L2 ln ε −4 πx2 η x

2

Sealing fault at x = 0, constant pressure at x = Lx

(x−x ) − 4η ln(1−ε )   2 2 (x−x ) (x+x ) max − 4ηx ln(1−ε ) , − 4ηx ln(1−ε )   2 2 (x−x ) (x+x ) max − 4ηx ln(1−ε ) , − 4ηx ln(1−ε ) 2

x

2

x

Table 4 Late time behavior of 1-D point source functions for various boundary conditions x-Boundaries

y-Boundaries

z-Boundaries

Ψ

Both infinite

Both infinite

Both finite sealed

E1 (ut ∗ ) 5.615 √ φ ct Lx Ly Lz 4π u Λx Λy

Semi-infinite, sealed

Both infinite

Both finite sealed

E1 (ut ∗ ) 5.615 √ φ ct Lx Ly Lz 2π u Λx Λy

Both infinite

Semi-infinite, sealed

Both finite sealed

E1 (ut ∗ ) 5.615 √ φ ct Lx Ly Lz 2π u Λx Λy

Semi-infinite, sealed

Semi-infinite, sealed

Both finite sealed

Semi-infinite, sealed

Both finite sealed

Both finite sealed

Both finite sealed

Both finite sealed

Both finite sealed

E1 (ut ∗ ) 5.615 √ φ ct Lx Ly Lz π u Λx Λy √ ∗ 5.615 erf c( ut ) 3 φ ct Lx Ly Lz √ Λx u 2 5.615 exp(−ut ∗ ) φ ct Lx Ly Lz u2

using piecewise Chebyshev Integration [Press92] as described below. In Table 4, we present appropriate expressions for the function Ψ for combinations of sealed and infinite boundary conditions; in this table we use the simplifying notation

Λχ =

ηχ L2χ

where χ = x, y, or z. Also, E1 (z) is the exponential integral, defined as E1 (z) =

∞ −t e z

t

dt

(|arg z| < π )

and erfc(z) is the complementary error function defined as

376

L.G. Thompson and G. Zhao

∞

2 erf c (z) = √ π

e−t dt. 2

z

References [Stegun:72] and [FUNLIB:76] provide numerical approximations for the rapid estimation of each of these functions.

t

Considering the first term in (7), i.e.,

∗

−uτ

Δ psx Δ psy Δ psz e u d τ , the value of the

0

integrand decays rapidly from very large values at τ = 0+; this presents a challenge when trying to determine the value of the integral accurately. We use the following method. We approximate the integral via

t

∗

e−uτ dτ ≈ Δ psx Δ psy Δ psz u

t

∗

Δ psx Δ psy Δ psz

tmin

0

e−uτ dτ u

(8)

where tmin is the value of time before which all of the source functions in the product are practically zero. Considering the leading term in the early-time form of the elliptic theta functions (5), this can only be nonzero on a computer if  2 z exp − > εmin t where εmin is the minimum floating point number that can be represented on a computer (approximately 10−308 on a PC). Thus, each source function will only be computationally nonzero if −z2 . (9) t> ln (εmin ) We set tmin to the maximum value of the right side of (9) for each of the source functions that comprise the integrand. Since the integrand in (8) exhibits large variations in value over the interval, we split the interval into log cycles and evaluate the integral as

t

∗

tmin

e−uτ Δ psx Δ psy Δ psz dτ = u

i ncycles 10 tmin

∑

i=1

Δ psx Δ psy Δ psz

10i−1 tmin

t

∗

Δ psx Δ psy Δ psz

+ n 10 cycles tmin

where

 ncycles = int log10



t∗ tmin

here int (x) is the integer part of x without rounding.

e−uτ dτ u

 ;

e−uτ dτ , u

Semianalytical Laplace Transforms

377

On each integration interval, we approximate the integrand and the integral using Chebyshev polynomials (see [Press92]); i.e., if Tn (x) represents the Chebyshev polynomial of degree n, Tn (x) = cos (n arccos (x)) , then, Chebyshev approximation to a function f (x) in the normalized interval [−1, 1] is given by N−1 1 f (x) ≈ ∑ ck Tk (x) − c0 2 k=0 where N is the maximum degree of Chebyshev polynomial used in the approximation, and   



 π k − 12 π j k − 12 2 N cos c j = ∑ f cos N k=1 N N for j = 0, 1, . . . , N − 1. The approximate integral is given by

f (x) dx ≈

N−1

1

∑ Ck Tk (x) − 2 C0

k=0

where Cj =

c j−1 − c j+1 for j > 0, 2j

and C0 is an integration constant. Details are given in [Press92]. We obtained excellent results using Chebyshev polynomials of maximum degree 32, and using the Stehfest algorithm with Stehfest parameter equal to 16 for the numerical inversion. Example applications of the method can be found in [Zhao:2002].

4 Summary In this paper we presented a computationally efficient method of evaluating the numerical Laplace transforms of products of source (or Elliptic Theta) functions. The method is fast because it relies on the evaluation of only single infinite series which can be expressed in alternative rapidly convergent forms depending on the values of the space and time parameters. The Laplace transform integral is split into early-time and late-time parts. The late-time part is evaluated analytically, while the early-time part is evaluated numerically using Chebyshev approximation. The resulting scheme has been successfully applied to the solution of complex reservoir flow problems. Acknowledgements This work was sponsored by the Tulsa University Petroleum Reservoir Exploitation Projects (TUPREP). The authors wish to thank the supporting member companies.

378

L.G. Thompson and G. Zhao

References [Stegun:72]

Abramowitz, M., Stegun I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, United States Department of Commerce, National Bureau of Standards Applied Mathematics Series 55 (1972). [Carslaw88] Carslaw, H., Jaeger, J.: Conduction of Heat in Solids, Claredon Press, Oxford (1988). [FUNLIB:76] Wayne Fullerton, L.W.: Portable special function routines, in: Portability of Numerical Software (Editor: W. Cowell), Springer-Verlag, New York, 452–483 (1976). [Grin:73] Gringarten, A.C., Ramey, H.J.: Use of source and Greens functions in solving unsteady flow problems in reservoirs. Society of Petroleum Engineers Journal, 13, 285–296 (1973). [Newmann:36] Newmann, A.B.: Heating and cooling rectangular and cylindrical solids, Ind. & Eng. Chem., 28, 545–548 (1936). [OBER:73] Oberhettinger, F., Badii, L.: Table of Laplace Transforms, Springer-Verlag, New York (1973). [Ohaeri:91] Ohaeri, C.U., Vo, D.T.: Practical solutions for interactive horizontal well test analysis, Paper SPE 22729 presented at the 66th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Dallas, TX, October 6–9, (1991). [Press92] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipies in c – The Art of Scientific Computing, Cambridge University Press (1992). [RagOzkan94] Raghavan, R., Ozkan, E.: A Method for Computing Unsteady Flows in Porous Media, Pitman Research Nodes in Mathematics Series, Longman Scientific & Technical (1994). [STEHFEST:70] Stehfest, H.: Numerical inversion of Laplace transforms. Communications of ACM, 13, n. 1, 47–49 (1970). [Thompson91] Thompson, L., Manrique, J., Jelmert, T.: Efficient algorithms for computing the bounded reservoir horizontal well pressure response, Paper SPE 21827, Presented at the Rocky Mountain Regional Meeting and Low-Permeability Reservoirs Symposium Held in Denver Colorado, (April 15–17, 1991). [Zhao:2002] Zhao, G., Thompson, L.G.: Semi-analytical modeling of complex geometry reservoirs, SPEREE, ASME (2002).

Asymptotic Analysis of Singularities for Pseudodifferential Equations in Canonical Non-Smooth Domains V.B. Vasilyev

”Although this may seem a paradox, all exact science is dominated by the idea of approximation.” Bertrand Russell

1 Introduction Pseudodifferential operator theory was apparently developed no later than half a century ago [ViEs65, Es81, Mi62, Mi86, Ma62, CaZy57, Ho83, Tr83, Ta83, ReSc83, St66], but it is not so young because its basic achievements were invented in 1960s or 1970s. The main point of this theory is a symbolic calculus for pseudodifferential operators and boundary value problems for pseudodifferential equations on manifolds with a smooth boundary. This is not so good for manifolds with a nonsmooth boundary although there are many studies on this problem (see, for example, the papers of B.-W. Schulze and his colleagues). The first paper in this direction was Kondratiev’s paper of 1967, in which the general boundary value problem for a partial differential operator in a cone was studied. But in this paper, and almost all of he following studies, the conical singularity is geometrically treated as direct product of a manifold with a smooth boundary and a half-line. The author suggests another point of view (for elliptic equations in this time) related to the concept of wave factorization of elliptic symbol [Va00, Va98]. From this point of view the conical singularity is something indivisible, and it cannot be reduced to other cases which were studied earlier. “Curiously enough,” V.A. Kondratiev said to me (in answer to my question) “there is nothing interesting any more in the theory of boundary value problems in domains with non-smooth boundary” (December 2004). This urged the author to collect his own outlines into one but still unfinished fragment. The auV.B. Vasilyev Bryansk State University, Russia, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_35, © Springer Science+Business Media, LLC 2011

379

380

V.B. Vasilyev

thor hopes that in the future it will be possible to obtain a “non-smooth” version of Sternin’s results [St66] by the wave factorization method.

2 Two-Dimensional Case We will begin with the two-dimensional case. The problem is to find out what does an operator, which we multiply with a characteristic function which is positive on the y-axis, represent in Fourier images. Analytically, we have  1, x = 0, y > 0, m (x, y) = 0, otherwise. It is clear that such a multiplier in Fourier images is convolution with a distribution, which is homogeneous of degree −2, as long as the function m (x, y) is homogeneous of degree 0 ([Mi77, Mi62, Mi86, Ma62, CaZy57]), and, consequently, we can think of it as a function, defined on a unit circle, which is touched by the y-axis at a single point, the “north pole”. We can use the standard method: the multiplier m (x, y) “blurs” the angle with size α , then we find an appropriate distribution (this function has been found already [BoMa48, Vl64]) and pass to a limit with α → 0. Remark 1. In the author’s work [Va98] the Gakhov operator G was introduced, which was defined in the entire complex plane. Of course, it is desirable to define this operator in real variables, and for this we must cut out the singularity using a special method. In a Hilbert transform the singularity is cut off by “cross”. Using a two-dimensional Gakhov operator we probably should use some “hyperbolic” cut off. This work will not focus on this (Gakhov’s operator works well); maybe N. Kasumov will deal with this problem as he has some experience [Ka92]. The angle of α is the set {(x, y) : y > a |x| } , a = cot α , and thus the asymptotic needed is (α → 0), i.e. a → ∞. The distribution, appropriate to this angle multiplier is the function 1 δ (ξ ) + Ka (ξ1 , ξ2 ) , 2 (1) 1 a , Ka (ξ1 , ξ2 ) = 2 2π ξ12 − a2 (ξ2 + i 0)2 where ξ = (ξ1 , ξ2 ), and δ (ξ ) is the Dirac’s mass. Now we just have to find lim 2 πa 2 ξ 2 −a1 2 ξ 2 in distribution sense. Let ϕ (ξ ) ∈ x→∞ 1 2  2 S R (Schwartz class of infinitely differentiable, rapidly decreasing at infinity, functions) a 2π 2

 R2

ϕ (ξ1 , ξ2 ) d ξ1 d ξ2 a = 2 2 2 2 2π ξ1 − a ξ2

 R2

ϕ (ξ1 ,t/a) d ξ1 dt. ξ12 − t 2

Pseudodifferential Equations in Canonical Non-Smooth Domains

381

Taking the limit, we have 1 a→∞ 2π 2 lim

 R2

1 ϕ (ξ1 , t/a) d ξ1 dt = 2 2π ξ12 − t 2



ϕ (ξ1 , 0) d ξ1 dt ξ12 − t 2   +∞  +∞ dt 1 d ξ1 . (2) = 2 ϕ (ξ1 , 0) 2 2π −∞ −∞ ξ1 − t 2 R2

Now calculate the inner integral. The function under the integral sign is even, so  +∞ −∞

dt =2 ξ12 − t 2

 +∞ 0

1 dt = ξ12 − t 2 ξ1



+∞ 0

=

dt + ξ1 − t

 +∞ 0

dt ξ1 + t



 1 1 ξ1 + t ∞ πi ln = ln (−1) = . ξ1 ξ1 − t 0 ξ1 ξ1

Using this in (2), we have a a→∞ 2π 2 lim

 R2

ϕ (ξ1 , ξ2 ) d ξ1 d ξ2 i = 2 2 2 2π ξ1 − a ξ2

 R2

ϕ (ξ1 , 0) d ξ1 . ξ1

(3)

Formula (3) for distributions means that lim

a→∞

a 1 i 1 = P ⊗ δ (ξ2 ) , 2 2 2 2 2π ξ1 − a ξ2 2π ξ1

(4)

where the designation of the functional P is taken from Vladimirov’s books [Vl81, Vl79], and ⊗ means direct multiplication of distributions. Distribution (4) corresponds to a semi-infinite crack (of course, with the addition of 12 δ (ξ )). In order to control this, find another asymptotic of the function (1) when a → 0. What we obtain corresponds to the half-plane y > 0 (instead of the “point” this will be the “upper unit half-circle”). We now have what is needed to compare to this result. If we take b = 1/a, then we can write 1 b→0 2π 2 lim

 R2

ϕ (t/b, ξ2 ) d t d ξ2 t 2 − ξ22   +∞  +∞ 1 dt d ξ2 = 2 ϕ (0, ξ2 ) 2 2π −∞ −∞ t 2 − ξ2

or, for distributions, a 1 1 1 = δ (ξ1 ) ⊗ P , 2 2 2 2 a→∞ 2π ξ − a ξ 2π i ξ2 1 2 lim

which is the same result, as the one we had in half-plane (see [ViEs65, Es81]).

(5)

382

V.B. Vasilyev

3 Multidimensional Case Now about a crack in a multidimensional space. We are interested in the asymptotics of the function (see [Va98]) aΓ (m/2) 2π

m+2 2

1

m/2 , 2 2  2 |ξ | − a (ξm + i0)

(6)

where ξ  = (ξ1 , . . . , ξm−1 ) and Γ is Euler’s function. A crack occurs when a → +∞ and when a → 0. In the former case, lim

a→∞

a Γ (m/2) 2π

m+2 2

Rm

ϕ (ξ1 , . . . , ξm−1 , ξm ) d ξ

m/2 |ξ  |2 − a2 ξm2

Γ (m/2)

= lim

2π

a→∞

=



m+2 2

a Γ (m/2) 2π

m+2 2





ϕ (ξ  ,t/a) d ξ  d t

m/2 Rm |ξ  |2 − t 2 ⎛     +∞ ϕ ξ ,0 ⎝

Rm−1

−∞

⎞ dt |ξ  |2 − t 2



m/2

⎠ dξ .

We now calculate the integral  +∞ −∞

dt

=2 2  | ξ | − t 2 m/2

 +∞ 0

dt

. 2  |ξ | − t 2 m/2

Here we should distinguish between the cases where m is odd and even. First consider the odd sequence m = 3, 5, 7, . . .. With the help of a formula from [GrRy71] we can write  +∞ 0



dt |ξ  |2 − t 2



= m/2

1 |ξ  |m−1

(m−3)/2

∑

k=0

+∞      2k+1

 2 2  2  (2k + 1) |ξ | − t k C(m−3)/2 t 2k+1

0

=

1 i |ξ  |m−1

(m−3)/2

∑

k=0

1 . Ck 2k + 1 (m−3)/2

For simplicity, let (m−3)/2

∑

k=0

1 >1. k C(m−3)/2 ≡ bm 2k + 1

(m = 3, 5, 7, . . .)

(b3 = 1, b5 = 4/3, b7 = 6/5 etc.). Then for odd numbers m,

Pseudodifferential Equations in Canonical Non-Smooth Domains

lim

a Γ (m/2) 2π

a→∞



m+2 2

Rm



ϕ (ξ  , ξm ) d ξ Γ (m/2) bm

m/2 = m+2 2π 2 |ξ  |2 − a2 ξm2

383



ϕ (ξ  , 0) Rm

| ξ  |−1

d ξ .

(7)

For distributions, relation (7) will be lim

a→∞

a Γ (m/2) 2π

The meaning of P

1 m/2

m+2 2

(|ξ  |2 −a2 ξm2 ) Γ (m/2) bm 1 = P ⊗ δ (ξm ) , m = 3, 5, 7, . . . . m+2  |ξ |m−1 iπ 2

1 |ξ  | m−1

(8)

will be explained separately.

Now consider the even sequence m = 2, 4, 6, . . .. Using the formula [GrRy71] and taking m/2 = k, we have  +∞ 0

+∞   t 



k =  2 2 |  | − t 2 k−1  2  2 | 2 (k − 1) ξ ξ | |ξ | −t 0 dt

+

2 (k − 1) − 1

 +∞

2 (k − 1) |ξ  |2

0

dt

|ξ  |2 − t 2

, k = 2, 3, . . . , k−1

or, after the calculations, we will obtain the recurrent formula  +∞ 0

dt

| ξ  |2 − t 2

k =

2k − 3

 +∞

2 (k − 1) |ξ  |2

0



dt | ξ  |2 − t 2



, k = 2, 3, . . . . k−1

In the plane case (m = 2) the above calculations lead to  +∞ 0

dt πi . = | ξ  | 2 − t2 2|ξ |

Consequently,  +∞ 0



 +∞ 0



 +∞ 0



dt | ξ  |2 − t 2 dt | ξ  |2 − t 2 dt | ξ  |2 − t 2

2 =

2·2−3 2 (2 − 1) |ξ  |3

·

πi π , = 2 4 | ξ  |3

3 =

(2 · 3 − 3) (2 · 2 − 3) 1 π i · , · 2 (3 − 1) 2 (2 − 1) |ξ  |5 2

4 =

(2 · 4 − 3) (2 · 3 − 3) (2 · 2 − 3) 1 π i · , · 2 (4 − 1) 2 (3 − 1) 2 (2 − 1) |ξ  |7 2

which allows us to write the general formula

384

V.B. Vasilyev

 +∞ 0

k dt πi 2l − 3 ,

k = k  2k−1 ∏ 2 | ξ | 2 l=2 l − 1 | ξ  | − t2

(9)

or remembering that k = m/2,  +∞ 0



dt | ξ  |2 − t 2



= m/2

πi

m/2

2m/2 |ξ  |m−1

2l − 3 . l=2 l − 1

∏

(10)

For brevity designate m/2

2l − 3 = Am l=2 l − 1

∏

(m = 4, 6, 8, . . .)

(c4 = 1, c6 = 3/2, c8 = 5/2, etc.), and for even numbers     Γ m2 cm i ϕ (ξ  , ξm ) d ξ ϕ (ξ  , 0) d ξ  a Γ ( m2 ) lim m+2 = ,

m−1 m/2 a→∞ 2π 2 Rm (2π )m/2 Rm−1 |ξ  | | ξ  |2 − a2 ξm2

(11)

or, in distributions, lim

a Γ ( m2 )

1

m+2

m/2

(| ξ  |2 −a2 ξm2 )   Γ m2 cm i 1 = P ⊗ δ (ξm ) , m/2  |ξ |m−1 (2π )

a→∞ 2π 2

m = 4, 6, 8, . . . .

(12)

At last, it is possible to calculate the distribution for the half-space xm > 0 (a → 0): lim

aΓ ( m 2)

a→0 2π



m+2 2

Rm

= lim

Γ ( m2 )bm−1

b→∞

=



m+2 2π 2

b→∞

= lim

ϕ (ξ  , ξm ) d ξ  d ξm

m/2 | ξ  |2 − a2 ξm2

Γ ( m2 )



m+2 2π 2

Γ

( m2 )

2π

m+2 2

 +∞ −∞

ϕ (ξ ) d ξ

m/2 | bξ  |2 − ξm2   t ϕ tb1 , . . . , m−1 b , ξm d t1 . . . d tm−1 d ξm

m/2 |t  |2 − ξm2 ⎛ ⎞ Rm

Rm

⎜ ϕ (0, . . . , 0, ξm ) ⎝



Rm−1



d t |t  |2 − ξm2

⎟

m/2 ⎠ d ξm ,

where t  = (t1 , . . . ,tm−1 ). We now calculate the inner integral using spherical coordinates:

Pseudodifferential Equations in Canonical Non-Smooth Domains





Rm−1

d t |t  |2 − ξm2

m/2

= ωm−1

 +∞ 0

385

rm−2 d r (r2 − ξm2 )

m/2

,

where ωm−1 is the surface area of the unit sphere in (m − 1)-dimensional space;

ωm−1 =

m−2

2π 2 Γ ( m−2 2 )

(see, for example, [Fi68]).

Using the relevant formula from [GrRy71] we obtain (m ≥ 4)  +∞ 0

rm−2 d r (r2 − ξm2 ) 

 +∞

rm−2 d r

(ξm2 − r2 ) +∞    m − 3 +∞ rm−4 d r rm−3  − m−2  m − 2 0 (ξ 2 − r2 ) m−2 2 (m − 2) (ξm2 − r2 ) 3 0 m

= i−m = i−m

1 im

=

m/2

m−3 m−2

Hence, if we set

m/2

0

 +∞

rm−4 d r (ξm2 − r2 )

0

 +∞

m−2 2

.

rm−2 d r (ξm2 − r2 )

0

then Im (ξm ) = −

>1.

m−2 2

≡ Im (ξm ) ,

m−3 Im−2 (ξm ) . m−2

(13)

For m = 3,  +∞ 0

rdr (r2 − c2 )

+∞  −1/2   =− = −c2 3   (r2 − c2 ) 2 0 i 1 1 = = − ; I3 (c) = ic c c 1

3 2

(for c is the same as c + i0). For m = 4 use the recurrent formula (13),  +∞

I4 (c) = (we recall that I2 (c) = we see that

0

r2 d r 2 (r2 − c2 )

1 πi = − · I2 (c) = − 2 2c

 +∞ r2 d r = 2π ci ). Further, using the recurrent formula (13), 0 c2 −r2

2 21 I5 (c) = − I3 (c) = − , 3 3c      3 1 3 3 πi π − =− − i , I6 (c) = − I4 (c) = − 4 4 4c 4 4 c   2 1 4 4 I7 (c) = − I5 (c) = − − , 5 5 3 c

386

V.B. Vasilyev

   πi 3 5 5 − − . I8 (c) = − I6 (c) = − 6 6 4 4c Consequently, we can write (m ≥ 5)   m−5   2  1   Im (c) = − m−3 m odd m−2  − m−4  · · · − 3  c , m−3 m−5 3 πi Im (c) = − m−2 − m−4 · · · − 4 − 4c , m even In view of the above we have lim

a Γ (m/2) 2π

a→0

m+2 2

 Rm

ϕ (ξ ) d ξ

m/2 |ξ  |2 − a2 ξm2

Γ (m/2) ωm−1

 +∞

ϕ (0, . . . , 0, ξm ) Im (ξm ) d ξm m+2 −∞ 2π 2  +∞ Γ (m/2) ϕ (0, . . . , 0, ξm ) d ξm = 3/2 m  m−1  dm , ξm π i Γ 2 −∞ =

where

im

(14)

⎧ m = 3, ⎪ ⎪ −1, ⎨ πi − , m = 4, dm =  2m−3   m−5   2  − − · · · − , m ≥ 5 odd, ⎪ ⎪   m−4   33   π i  ⎩  m−2 m−5 − · · · − − , m > 5 even. − m−3 m−2 m−4 4 4

The formula (14) for distributions is lim

a→0

a Γ (m/2) 2π

m+2 2

1 m/2 | ξ  |2 −a2 ξm2

(

)

=

  Γ (m/2) dm 1  m−1  δ ξ  ⊗ P . ξm 2

π 3/2 im Γ

(15)

Now we can see, that (15), with the accuracy of a constant, gives us the known formula for half-space. Consequently 1 Γ (m/2) dm  m−1  = . 2π i 2

π 3/2 im Γ

(16)

The correlation (16) is easily verified for m = 3, 4, 5, 6.

4 The Pyramid In this section we talk about asymptotics to a polyhedral angle, with its simplest variant being a pyramid. The equation of such a pyramid x3 > a |x1 | + b | x2 | , which contains two parameters a, b, that correspond to the length of the relevant axis. If we set these parameters to 0 or ∞, then we can obtain other types of singularities. The distribution, which corresponds to a such pyramid, is [Va98]

Pseudodifferential Equations in Canonical Non-Smooth Domains

Ka,b (ξ1 , ξ2 , ξ3 ) =

387

ξ  2  3 2 . 2 2 (2π ) ξ1 − a ξ3 ξ2 − b2 ξ32 4iab

3

a. Consider the case a = const, b → ∞ (it is intuitively understandable that we should obtain a singularity of an infinite crack type as opposed to half-infinite ones from Sect. 2): lim

b→∞



ξ3 ϕ (ξ1 , ξ2 , ξ3 ) d ξ1 d ξ2 d ξ3    ξ1 − a2 ξ32 ξ22 − b2 ξ32 (2π )  4i a1 b ξ3 ϕ (ξ1 , ξ2 , ξ3 ) d ξ1 d ξ2 d ξ3    = lim 3 3 b→∞ (2π ) a ξ1 − ξ32 ξ22 − b2 ξ32 R

t  ξ3 ϕ d t d ξ2 d ξ3 , ξ , ξ 2 3 a b 4i 2 1  = lim  3 t b→∞ (2π ) R3 − ξ32 ξ22 − b2 ξ32 b2

t  ξ3 ϕ a1 b , ξ2 , ξ3 d t d ξ2 d ξ3 4i b2    = lim b→∞ (2π )3 R3 t 2 − b2 ξ32 ξ22 − b2 ξ32

t τ  τϕ a1 b , ξ2 , b d t d ξ2 d τ 4i   = lim b→∞ (2π )3 R3 (t 2 − τ 2 ) ξ22 − τ 2    +∞ τ dtdτ  d ξ2 ,  2 ϕ (0, ξ2 , 0) = −∞ R2 (t 2 − τ 2 ) ξ2 − τ 2 4i a b

3

R3

where a1 = 1/a and abξ1 = t. Calculating the integral gives    +∞  +∞ τ dtdτ dt τdτ =  2 . 2 2 2 2 2 2 ξ22 − τ 2 −∞ −∞ t − τ R (t − τ ) ξ2 − τ The formula in brackets was calculated in Sect. 2:  +∞ −∞

dt =2 t2 − τ2

 +∞ 0

dt πi =− . t2 − τ 2 τ

Then  R2

τ dtdτ  = −π i  (t 2 − τ 2 ) ξ22 − τ 2

 +∞

dτ

2 −∞ ξ2 − τ 2

= −π i ·

π i π2 = , ξ2 ξ2

and hence lim

b→∞

4i a b



(2π )3

For distributions,

R3

i ξ3 ϕ (ξ1 , ξ2 , ξ3 ) d ξ1 d ξ2 d ξ3 = x 2π

 +∞ ϕ (0, ξ2 , 0) −∞

ξ2

d ξ2 .

388

V.B. Vasilyev

4 i a b ξ3  = 3 2 (2 π ) ξ1 − a2 ξ32 ξ22 − b2 ξ32

lim

b→∞

i 1 δ (ξ1 ) ⊗ P ⊗ δ (ξ3 ) . 2π ξ2

b. Now consider a → ∞, b = const. In an analogous manner, we can obtain the formula (here it is only given for distributions): i 1 4 i a b ξ3 P ⊗ δ (ξ2 ) ⊗ δ (ξ3 ) .  2 = 2 2 2 π ξ1 ξ2 − b ξ3

  (2 π )3 ξ12 − a2 ξ32

lim

a→∞

c. Now consider the case a = const, b = 0 (wedge or twofold angle). lim

b→0



4i a b (2π )3

R3

ξ3 ϕ (ξ1 , ξ2 , ξ3 ) d ξ1 d ξ2 d ξ3    ξ12 − a2 ξ32 ξ22 − b2 ξ32 

ξ3 ϕ (ξ1 , ξ2 , ξ3 ) d ξ1 d ξ2 d ξ3  2   c→∞ (2π ) ξ1 − a2 ξ32 c2 ξ22 − ξ32 R3  4i a ξ3 ϕ (ξ1 ,t1 /c, ξ3 ) d ξ1 d t d ξ3  2   = lim c→∞ (2π )3 R3 ξ1 − a2 ξ32 t 2 − ξ32  +∞   ξ3 ϕ (ξ1 , 0, ξ3 ) dt 4i a d ξ1 d ξ3 = 2 −∞ t 2 − ξ3 (2 π )3 R2 ξ12 − a2 ξ32    4i a ξ3 ϕ (ξ1 , 0, ξ3 ) −π i = d ξ1 d ξ3 ξ3 (2 π )3 R2 ξ12 − a2 ξ32 4i a c

lim

= that is,

3

a 2π2



R2

ξ3 ϕ (ξ1 , 0, ξ3 ) d ξ1 d ξ3 , ξ12 − a2 ξ32

ξ  2  3 2  = δ (ξ2 ) ⊗ Ka (ξ1 , ξ3 ) 2 2 (2 π ) ξ1 − a ξ3 ξ2 − b2 ξ32 4iab

lim

3

b→0

(see formula (1)). Analogously (a → 0, b = const) gives lim

a→0

4iab (2 π )

3



ξ12 − a2 ξ32

ξ  3 2  = δ (ξ1 ) ⊗ Kb (ξ2 , ξ3 ) . ξ2 − b2 ξ32

In addition, using previous calculations and results from Sect. 2, we have lim a→0 b→0

4iab (2 π )

3



ξ12 − a2 ξ32

ξ 1 1    3 2 = δ ξ ⊗P , 2 2 2π i ξ3 ξ2 − b ξ3

This last result corresponds to the half-space x3 > 0.

ξ  = (ξ1 , ξ2 ) .

Pseudodifferential Equations in Canonical Non-Smooth Domains

389

References [Vl81]

Vladimirov, V.S.: Equations of Mathematical Physics, Nauka, Moscow (1981) (Russian). [Vl79] Vladimirov, V.S.: Distributions in Mathematical Physics, Nauka, Moscow (1979) (Russian). [ViEs65] Vishik, M.I., Eskin, G.I.: Convolution equations in bounded domains. Russian Math. Surveys, 20, 89–152 (1965) (Russian). [Es81] Eskin, G.: Boundary Value Problems for Elliptic Pseudodifferential Equations, AMS, Providence, RI (1981). [Va00] Vasil’ev, V.B.: Wave Factorization of Elliptic Symbols: Theory and Applications, Kluwer Academic Publishers, Dordrecht–Boston–London (2000). [Mi77] Mikhlin, S.G.: Linear Partial Differential Equations, Vysshaya Shkola, Moscow (1977) (Russian). [Mi62] Mikhlin, S.G.: Multidimensional Singular Integrals and Integral Equations, Fizmatgiz, Moscow (1962) (Russian). [Mi86] Mikhlin, S.G., Prössdorf, S.: Singular Integral Operators, Akademie-Verlag, Berlin (1986). [Ma62] Malgrange, B.: Singular integrtal operators and uniqueness of the Cauchy problem. Matematika, 6, 87–129 (1962) (Russian). [CaZy57] Calderon, A.P., Zygmund, A.: Singular integral operators and differential equations. Amer. J. Math., 79, 901–921 (1957). [Va98] Vasilyev V.B.: Regularization of multidimensional singular integral equations in nonsmooth domains. Trans. Moscow Math. Soc., 59, 73–105 (1998) (Russian). [BoMa48] Bochner, S., Martin, W.T.: Several Complex Variables, Princeton University Press, Princeton, NJ (1948). [Vl64] Vladimirov, B.S.: Methods of Function Theory of Several Complex Variables, Nauka, Moscow (1964) (Russian). [Ka92] Kasumov, N.: Calderon–Zygmund theory for kernels with non-point singularities. Mat. Sb., 183, 89–104 (1992) (Russian). [GrRy71] Gradstein, I.S., Ryzhik, I.M.: Handbook of Integrals, Sums, Series and Products, Nauka, Moscow (1971) (Russian). [Fi68] Fikhtengol’ts, G.M.: Calculus, vol. 3, Nauka, Moscow (1968) (Russian). [Ho83] Hörmander, L.: Analysis of Partial Differential Operators, vols. 1–4, Springer-Verlag, Berlin (1986–1988). [Tr83] Treves, F.: Introduction to Pseudodiferential and Fourier Integral Operators, vols. 1, 2, Plenum Press, New York–London (1982). [Ta83] Taylor, M.: Pseudodifferential Operators, Princeton Univ. Press, Princeton, NJ (1981). [ReSc83] Rempel, S., Schulze, B.-W.: Index Theory of Elliptic Boundary Value Problems, Akademie-Verlag, Berlin (1982). [Bo71] Boutet de Monvel, L.: Boundary problems for pseudodifferential operators. Acta Math., 126, 11–51 (1971). [St66] Sternin, B.Yu.: Elliptic and parabolic problems on manifolds with boundary consisting of components of different dimension. Trans. Moscow Math. Soc., 15, 346–382 (1966) (Russian).

Optimizing Water Quality in a River Section M.A. Vilar, L.J. Alvarez-Vázquez, A. Martínez, and M.E. Vázquez-Méndez

1 Introduction Since early times, rivers have been not only sources of life but also water discharge receivers (both from industrial and urban origin) from the human settlements on their banks. This fact brings with it that pollutant matter concentration surpasses healthy levels in some sections of the rivers. In our paper, we use mathematical modeling and optimal control theory to simulate one of most common strategies in the pollution reduction of a river section: clear water injection into the channel from a nearby reservoir. In this process of increasing the river flow by controlled releases of water from reservoirs, the main problem consists (once the injection point has been chosen by geophysical reasons) of finding the minimum quantity of water which needs to be injected into the river section in order to purify it to a desired level. We formulate this environmental problem as a hyperbolic optimal control problem with control constraints. The state system is given in terms of section-averaged velocity of water and wet section by the 1D inviscid shallow water system coupled with the pollutant concentration equation. In this work, we deal with a unique generic pollutant: pathogenic microorganisms or chlorides (because in an urban sceM.A. Vilar Universidad de Santiago, Lugo, Spain, e-mail: [email protected] L.J. Alvarez-Vázquez Universidad de Vigo, Spain, e-mail: [email protected] A. Martínez Universidad de Vigo, Spain, e-mail: [email protected] M.E. Vázquez-Méndez Universidad de Santiago, Lugo, Spain, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_36, © Springer Science+Business Media, LLC 2011

391

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M.A. Vilar et al.

nario they are the most commonly used pollutants for water quality analysis), and can be easily scaled to several pollutants. In our case, we take the flux of injected water as the control parameter of the optimal control problem, and the constraints correspond to technological bounds, while the objective function is related to the total quantity of released water and to the fulfillment of water quality constraints. For the numerical solution we combine finite difference and finite element techniques for the time-space discretization, and a gradient-free method (the Nelder– Mead algorithm) to solve the discrete optimization problem. Finally, computational results for a realistic problem are provided.

2 Mathematical Description of the Problem Consider a river L meters in length, with O tributaries (located at points e1 , . . . , eO ) flowing into the river, V waste-water discharges (located at points v1 , . . . , vV ) coming from purifying plants, and one point p where clear water is discharged from a nearby reservoir (a diagram of a generic example can be seen in Fig. 1). Bearing in mind that we are interested in controlling pollution in the river section corresponding to [p, L] for a time interval of T seconds, we are going to consider only one-dimensional changes along the direction of flow in the river. Thus, for each (x,t) ∈ [0, L] × [0, T ] we will denote by A(x,t) the area of the river cross-section occupied by water (usually known as wet section) which is assumed to remain nonnegative for any point x ∈ [0, L] and for any time t ∈ [0, T ]; denote by u(x,t) the averaged velocity in the wet section x meters from the source and t seconds from the moment that control is initiated; denote by q(x,t) the flow rate across the section (that is, q(x,t) = A(x,t)u(x,t)); and denote by c(x,t) the quantity of a generic pollutant in the wet section (that is, c(x,t) = A(x,t)ρ (x,t), with ρ (x,t) the averaged pollutant concentration). The evolution of the wet area A(x,t), the flow rate q(x,t) and the quantity of pollutant c(x,t) is given—as can be seen, for instance, in [Al06]—by the following hyperbolic system: ⎧ O V ∂A ∂q ⎪ ⎪ ⎪ + = Qδ (x − p) + ∑ q j δ (x − e j ) + ∑ pk δ (x − vk ), ⎪ ⎪ ∂t ∂x ⎪ j=1 k=1 ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ =g1 (x,t) ⎪

 ⎪ ⎪ ∂ q ∂ q2 ∂η ⎪ ⎪ ⎪ + + gA = QW cos(γ )δ (x − p) ⎪ ⎪ ∂x ⎪ ⎨ ∂t ∂ x A O

V

+ ∑ q jU j cos(α j )δ (x − e j ) + ∑ pkVk cos(βk )δ (x − vk ) + S f , ⎪ ⎪ ⎪ j=1 k=1 ⎪ ⎪   ⎪ ⎪ ⎪ =g2 (x,t) ⎪ ⎪ ⎪ O V ⎪ ⎪ ∂ c ∂ qc ⎪ ⎪ + + kc = ∑ n j δ (x − e j ) + ∑ mk δ (x − vk ), ⎪ ⎪ ∂t ∂x A ⎪ j=1 k=1 ⎪ ⎪ ⎪   ⎩ =g3 (x,t)

(1)

Optimizing Water Quality in a River

393

in (0, L) × (0, T ), with the boundary conditions A(L,t) = AL (t),

q(0,t) = q0 (t),

c(0,t) = c0 (t)

(2)

q(x, 0) = q0 (x),

c(x, 0) = c0 (x)

(3)

in [0, T ], and the initial conditions A(x, 0) = A0 (x),

in [0, L], and where δ (x − b) denotes the Dirac measure at point b ∈ [0, L]; for j = 1, . . . , O, e j ∈ (0, L) is the point where the mouth of the jth tributary is located, q j (t) is the corresponding flow rate, U j (t) is its velocity, α j is the angle between the jth tributary and the main river, and n j (t) is its mass pollutant flow rate; for k = 1, . . . ,V , vk ∈ (0, L) is the point where the kth waste-water discharge is located, pk (t) is the corresponding flow rate, Vk (t) is its velocity, βk is the angle between the kth discharge and the river, and mk (t) is its mass pollutant flow rate; p ∈ (0, L) is the point where clear water is discharged, Q(t) is the corresponding flow rate (which will be our control), W (t) is its velocity, and γ is the angle between the discharge and the river (it is worthwhile remarking here that, since we are injecting clear water, this term does not appear in the second member of the pollutant equation); g stands for the acceleration due to gravity; S f denotes the bottom friction stress, which can be given, for instance, by the Chézy law; η (x,t) = H(x,t) + b(x) is the height of water with respect to a fixed reference level, where H(x,t) represents the height of the water column and b(x) describes the geometry of the river bottom; and k(x,t) is the pollutant loss rate.

Fig. 1 Diagram of a generic river

At first sight we can see four unknowns in state system (1)–(3): A(x,t), q(x,t), η (x,t), and c(x,t). However, it is obvious that, if the river geometry is known, A(x,t) can be derived from η (x,t). In effect, for each x ∈ [0, L], the geometry of the river gives us a smooth, strictly increasing and positive function S(., x) satisfying S(0, x) = 0 and S(H(x,t), x) = A(x,t) in [0, L] × [0, T ]. Specific characterizations of S for particular geometries can be found, for instance, in [Be06]. So, since we are dealing with a system of balance laws whose conservative variables are A, q and c, if we write η in terms of A, the non-conservative unknown η in (1) can be suppressed.

394

M.A. Vilar et al.

Explicitly, we have

∂η ∂ (x,t) = [B(A(x,t), x)] + b (x), ∂x ∂x

(4)

where, for each x ∈ [0, L], B(., x) denotes the inverse of the function S(., x) (i.e. B(A(x,t), x) = H(x,t)), which is also a smooth, strictly increasing and positive function. Moreover, we can write A(x,t)

∂η ∂ (x,t) = [G(A(x,t), x)] − F(A(x,t), x) + A(x,t)b (x), ∂x ∂x

(5)

where  B(A,x)

S(r, x) dr,

G(A, x) = 0

F(A, x) =

 B(A,x) ∂S 0

∂x

(r, x) dr = −

 A ∂B 0

∂x

(s, x) ds.

Then, the second equation of (1) can be rewritten as 



∂ q ∂ q2 ∂ + +g G(A, x) − F(A, x) + Ab (x) ∂t ∂x A ∂x = QW cos(γ )δ (x − p) + g2 .

(6)

Since B(·, x) is a strictly increasing function and B(0, x) = 0, it is easy to prove that G(·, x) is strictly increasing on [0, ∞) and G(0, x) = 0. On the other hand, realistic  defined by solutions correspond to A ≥ 0. Hence, we can replace G by G ⎧ ⎨ G(A, x) if A > 0,  x) = (−∞, 0] if A = 0, G(A, ⎩ Ø if A < 0.  x) is a maximal monotone graph in R × R for each x ∈ (0,L). Then Notice that G(·,  x) there exists a lower semicontinuous convex proper function φ (·, x) such that G(·, is the subdifferential of φ (·, x) (see, for instance, [Br73]). Hence we can rewrite (6) in the more useful way:



 ∂ q ∂ q2 ∂ζ + +g − F(A, x) + Ab (x) ∂t ∂x A ∂x (7) = QW cos(γ )δ (x − p) + g2 , ˜ x) for a.e. (x,t) ∈ (0, L) × (0, T ). with ζ (x,t) ∈ G(A(x,t), Now, recalling the mathematical formulation of the environmental problem, for technical reasons we can only consider the fluxes in the set of admissible controls:

Optimizing Water Quality in a River

395

Uad = {Q ∈ L2 (0, T ) : 0 ≤ Q ≤ Qmax },

(8)

since we are just injecting (not extracting) clear water, and the quantity of injected water must be bounded. In order to formulate the control problem we consider as the cost functional the total amount of clear water injected through the point p together with a measure (in the region of the river starting from point p) of the contaminant concentration which remains higher than the fixed threshold cmax . Thus, we define the cost function: J(Q) =

ε 2

 T

Q(t)2 dt +

0

μ 2

 T L 0

p

(c(x,t) − cmax )2+ dx dt

(9)

where ε and μ are two weight parameters, and (c − cmax )+ denotes the positive part of c − cmax , that is, (c − cmax )+ = max{c − cmax , 0}. So the problem, denoted by (P), of the optimal water injection for the purification of a polluted section in a river consists of finding the control flux Q ∈ Uad of injected clear water in such a way that the state system (1)–(3) is satisfied and which minimizes the cost function J given by (9). Thus, the problem can be written in short as (P)

min J(Q).

Q∈Uad

The authors have derived in [Al09] a formal first-order optimality condition by means of adjoint state techniques. In this paper we will center our attention on the numerical resolution of the optimal control problem (P).

3 The Discretized Problem For technical reasons (flow control mechanisms cannot act upon water flow in a continuous way, but discontinuously at short time periods) we look for admissible controls Q in the piecewise-constant L2 (0, T ) functions. So, for the time interval [0, T ] we choose a number K ∈ N, consider the time step Δ τ = T /K, and define the discrete times τm = mΔ τ , for m = 0, . . . , K. Thus, a function Q ∈ Uad which is constant at each subinterval determined by the grid {τ0 , τ1 , . . . , τK } is completely fixed by the set of values QΔ τ = (Q0 , Q1 , . . . , QK−1 ) ∈ [0, Qmax ]K ⊂ RK , where Qm = Q(τm ), m = 0, . . . , K − 1. This discretization leads to a time discretization of the cost function J and the state system (1)–(3). In order to solve accurately this state system we discretize the problem. Thus, for given N ∈ N (usually a multiple of K), we define Δ t = T /N and take tn = nΔ t, for n = 0, . . . , N. First and third equations of system (1) are discretized in an implicit way, but for the second one we use the method of characteristics that stems from considering the following identity:

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D(V q) ∂q ∂ (uq) (x,t) = (x,t) + (x,t), Dt ∂t ∂x

(10)

q) ∂ for the total derivative D(V Dt (x,t) = ∂ τ [V (x,t; τ )q(X(x,t; τ ), τ )]|τ =t , where X(x,t; τ ) is the characteristic line (providing the position at time τ of the particle that occupied the position x at time t), which is the unique solution of the following ordinary differential equation: ⎧ ⎨ dX (x,t; τ ) = u(X(x,t; τ ), τ ), (11) dτ ⎩ X(x,t;t) = x,

and V (x,t; τ ) is the evolution of the element of volume, which is given by the solution of the following ordinary differential equation: ⎧ ∂u ⎨ dV (X(x,t; τ ), τ )V (x,t; τ ), (x,t; τ ) = (12) dτ ∂x ⎩ V (x,t;t) = 1. 2

Expression (10) and the equalities uq = Aq q = qA change (7) into 

D(V q) ∂ζ  (x,t) + g (x,t) − F(A(x,t), x) + A(x,t)b (x) Dt ∂x = Q(t)W (t) cos(γ )δ (x − p) + g2 (x,t). Then, if we denote X n (x) = X(x,tn+1 ;tn ) and V n (x) = V (x,tn+1 ;tn ), the state system (1) can be approximated by the following semi-discrete system: for n = 0, . . . , N − 1 find functions An+1 (x), qn+1 (x), cn+1 (x), defined in (0, L), such that 

∂ qn+1 n+1 n (x) , (13) = A (x) + Δ t Q(tn+1 )δ (x − p) + g1 (x,tn+1 ) − A ∂x

 qn+1 (x) ∂ ζ n+1 +g (x) − F(An+1 (x), x) + An+1 (x)b (x) Δt ∂x qn (X n (x))V n (x) + Q(tn+1 )W (tn+1 ) cos(γ )δ (x − p) + g2 (x,tn+1 ), (14) = Δt ˜ n+1 (x), x), ζ n+1 (x) ∈ G(A (15) 

∂ qn+1 cn+1 cn+1 (x) − cn (x) (x) + k(x,tn+1 )cn+1 (x) + Δt ∂x An+1 = g3 (x,tn+1 ). (16) Δ τ , the set of controls in U which The admissible set Uad is approached by Uad ad are piecewise-constant in the partition of the time interval [0, T ] given by the time Δ τ , we use the following discrete step Δ τ . Finally, for any given control QΔ τ ∈ Uad approximation of the cost function J:

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K−1 ε (Qm )2 + (Qm+1 )2 J Δ t (QΔ τ ) = Δ τ ∑ 2 m=0 2

μ N−1 + Δt ∑ 2 n=0

 L n (c (x) − cmax )2+ + (cn+1 (x) − cmax )2+

2

p

dx.

(17)

To evaluate this cost function we need to solve the semi-discrete system (13)–(16) for n = 0, . . . , N − 1. Since variable cn+1 is not coupled with the first three equations of the system, we will proceed to solve it sequentially.

3.1 Computation of (An+1 , qn+1 ) We obtain a weak formulation of (14) by a classical procedure: for V = {z ∈ W 1,p (0, L) : z(0) = 0}, p ∈ [1, +∞], we multiply (14) by a test function z ∈ V and integrate in [0, L]. By using an integration by parts formula, and taking into account boundary condition A(L,t) = AL (t), we get  L n+1 q (x) 0

Δt −g

 L 0

 L

+g

z(x) dx

ζ n+1 (x)

∂z (x) dx − g ∂x

 L 0

An+1 (x)b (x) z(x) dx =

 L n n q (X (x))V n (x)

0

 L

+ 0

  F An+1 (x), x z(x) dx

0

Δt

z(x) dx

(Q(tn+1 )W (tn+1 ) cos(γ )δ (x − p) + g2 (x,tn+1 )) z(x) dx

−gG(AL (t n+1 ), L)z(L),

∀z ∈ V .

It has been proved in [Be06] that this variational problem has, at least, one solution. In order to obtain it, we choose Λh = {x0 = 0, x1 , . . . , xM = L} a partition of interval [0, L] in M subintervals Ik = [xk−1 , xk ], k = 1, . . . , M, such that there exists P ∈ {1, . . . , M − 1} satisfying xP = p. In association with this, we consider the following finite dimensional vector spaces: Wh = {Ah ∈ L2 (0, L) : Ah|I ∈ P0 , ∀k = 1, . . . , M}, k

Vh = {qh ∈ C([0, L]) : qh|I ∈ P1 , ∀k = 1, . . . , M}, k

where Pj , j = 0, 1, denotes the space of polynomials of degree j. Then, we take A0h ∈ Wh and q0h ∈ Vh as approximations of A0 and q0 and, for n = 0, . . . , N − 1, we look for An+1 ∈ Wh and qn+1 ∈ Vh satisfying: h h   ∂ qn+1 n+1 n h (x) , Ah = Ah + Δ t Q(tn+1 )δ (x − p) + g1 (x,tn+1 ) − ∂x

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 L n+1 qh (x) 0

Δt

−g =

 L 0

zh (x) dx − g

 L 0

ζhn+1 (x)

 L  n+1   F Ah (x), x zh (x) dx + g An+1 h (x)b (x) zh (x) dx 0

 L n qh (x)(Xhn (x))Vhn (x)

Δt

0

 L

+ 0

∂z (x) dx ∂x

zh (x) dx

(Q(tn+1 )W (tn+1 ) cos(γ )δ (x − p) + g2 (x,tn+1 )) zh (x) dx

−gG(AL (t n+1 ), L)zh (L),

∀zh ∈ Vh ,

˜ n+1 ), ζhn+1 ∈ G(A h qn+1 h (0) = q0 (tn+1 ), where Xhn and Vhn are the numerical solutions of (11) and (12), respectively. We resolve this nonlinear discretized system doing an implicit discretization of the operator F, and using the Bermúdez–Moreno iterative algorithm [Be81] for dealing with operator G.

3.2 Computation of cn+1 Equation (16) can be now solved by using an implicit upwind finite difference scheme. In order to do it, because of the Dirac measures characterizing the sources, we consider the following approximations: for each k = 0, . . . , M, we define δhk : [0, L] → [0, +∞) by ⎧ b − xk−1 ⎪ ⎪ if b ∈ [xk−1 , xk ], ⎪ ⎪ (x ⎨ k − xk−1 )2 xk+1 − b δhk (b) = if b ∈ [xk , xk+1 ], ⎪ ⎪ (x − xk )2 ⎪ ⎪ ⎩ k+1 0 otherwise, where δhk (b) ≈ δ (xk − b) for a generic point b ∈ [0, L]. Taking  i   0  c0 = c0 (ti ), i = 0, . . . , N and c j = c0 (x j ), j = 0, . . . , M from as data, for each n = 0, . . . , N − 1 and for each k = 1, . . . , M we compute cn+1 k the following expression: − cnk cn+1 k

Δt

+

qn+1 h (xk ) n+1 ck − An+1 h (xk )

qn+1 h (xk−1 ) n+1 ck−1 An+1 h (xk−1 )

xk − xk−1

+ k(xk ,tn+1 )cn+1 k

Optimizing Water Quality in a River

=

399

E

V

j=1

j=1

∑ n j (tn+1 )δhk (e j ) + ∑ m j (tn+1)δhk (v j ).

Finally, for each n = 0, . . . , N − 1, we approach cn+1 (x) by the unique continuous n+1 n+1 function cn+1 h (x) ∈ Vh satisfying ch (xk ) = ck , for all k = 0, . . . , M.

3.3 The Discrete Problem According to the previous discretization, the continuous problem (P) is finally approached by (PhΔ t )

min JhΔ t (QΔ τ )

Δτ QΔ τ ∈Uad

where K−1 ε (Qm )2 + (Qm+1 )2 JhΔ t (QΔ τ ) = Δ τ ∑ 2 m=0 2

μ N−1 M−1 + Δt ∑ ∑ 2 n=0 k=P

 xk+1 n 2 (ch (x) − cmax )2+ + (cn+1 h (x) − cmax )+ dx.

2

xk

(18)

Problem (PhΔ t ) is a bound constrained minimization problem that is non-convex (and even non-differentiable), which leads us to the choice of a derivative-free algorithm in order to solve it.

4 Numerical Optimization To solve the minimization problem (PhΔ t ) we propose the use of a derivative-free algorithm that has already given very good results for several environmental control problems previously studied by the authors (see, for instance, [Al02]). To do this, we need to change our discretized problem (PhΔ t ) into an unconstrained optimization problem by introducing a penalty function involving the constraints appearing in the definition of the set of admissible controls (8), that is, Q ≥ 0 and Q − Qmax ≤ 0. Thus, we define the penalty function J˜ by ˜ Δ τ ) = JhΔ t (QΔ τ ) + β J(Q

K−1

∑ max{−Qm , Qm − Qmax , 0}

(19)

m=0

where the parameter β > 0 determines the relative contribution of the objective function and the penalty terms. Function J˜ is an exact penalty function in the sense

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that, for sufficiently large β , the solutions of our constrained problem (PhΔ t ) are equivalent to the minimizers of function J˜ in RK . For computing the minimum of this penalty function J˜ we use a direct search algorithm: the Nelder–Mead simplex method [Ne65]. This is a gradient-free method, which merely compares function values where the values of the objective function are taken from a set of sample points (simplex). For the interested reader, a detailed description of the above algorithm can be found, for instance, in the paper of the authors [Al02]. Although the Nelder–Mead algorithm is not guaranteed to converge in the general case, it presents good convergence properties in low dimensions (which is our case).

Fig. 2 River data for the numerical example

5 Computational Experiments In this section we present numerical results obtained by using above method to determine the optimal inflow flux in a river which is L = 2000 m in length, and where we consider O = 3 tributaries, V = 2 domestic waste-water discharges, and one clear water discharge from a reservoir (diagram and data can be seen in Fig. 2). Moreover, we consider a parabolic river bed with a non-constant bottom in such a way that  √ 500 − x 4 H3 if 0 ≤ x ≤ 500, A = S(H, x) = , b(x) = 200 3 0 if 500 ≤ x ≤ 2000. Both initial and boundary conditions were taken as constant, particularly, AL (t) = √ 4 125 = 3 m2 , q0 (t) = q0 (x) = 1 m3 s−1 , c0 (t) = c0 (x) = 0 um−1 . The time interval to control the pollution was T = 3600 s. Moreover, the loss rate for pollutant was considered constant (k(x,t) = 10−4 s−1 ), and the bottom friction stress was neglected (S f = 0). A0 (x)

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Out of the several numerical experiments developed by the authors, we present here only one corresponding to the case of K = 4 time subintervals (remember that dimension K must be low for the efficiency of Nelder–Mead algorithm). For the objective function we chose the threshold cmax = 4.5 um−1 , the bound Qmax = 25 m3 s−1 , and the weight parameters ε = 10−3 , μ = 10 and β = 105 . For the time discretization we took N = 6000 (that is, a time step of Δ t = 0.6 s), and for the space discretization we tried a regular partition of [0, L] in M = 2000 subintervals (consequently, the clear water inflow point p = 700 m corresponds to the node xP = x700 ). Then, applying the Nelder–Mead algorithm, after 123 function evaluations we have passed from the initial random cost J˜ = 1.807 to the minimum cost J˜ = 0.696, corresponding to the optimal flow rate Q0 = 9.984 m3 s−1 , Q1 = 6.909 m3 s−1 , Q2 = 5.445 m3 s−1 , Q3 = 3.993 m3 s−1 . The effectiveness of this remedial strategy can be seen in Fig. 3, where we can observe in detail the differences between no injecting clear water (uncontrolled case) and using the optimal strategy to inject water in point p = 700 m (controlled case): in the first case the quantity of pollutant c remains over cmax up to the mouth of the last tributary e3 = 850 m at the three shown times (t = 1000, t = 2000, t = 3000); however, we can see how it turns under threshold cmax from injection point p, when the optimal discharge of clear water is considered.

Fig. 3 Uncontrolled (left) and controlled (right) quantity of pollutant at three significant times around points p = 700 and e3 = 850

Acknowledgements This work was supported by Project MTM2009-07749 of Ministerio de Ciencia e Innovación (Spain), and Project INCITE-09PXJB291083PR of Xunta de Galicia.

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References [Al02]

[Al06]

[Al09]

[Be81] [Be06]

[Br73] [Ne65]

Alvarez-Vázquez, L.J., Martínez, A., Rodríguez, C., Vázquez-Méndez, M.E.: Numerical optimization for the location of wastewater outfalls. Comput. Optim. Appl., 22, 399–417 (2002). Alvarez-Vázquez, L.J., Martínez, A., Vázquez-Méndez, M.E., Vilar, M.A.: Optimal location of sampling points for river pollution control. Math. Comput. Simul., 71, 149–160 (2006). Alvarez-Vázquez, L.J., Martínez, A., Vázquez-Méndez, M.E., Vilar, M.A.: An application of optimal control theory to river pollution remediation. Appl. Numer. Math., 59, 845–858 (2009). Bermúdez, A., Moreno, C.: Duality methods for solving variational inequalities. Comput. Math. Appl., 7, 43–58 (1981). Bermúdez, A., Muñoz-Sola, R., Rodríguez, C., Vilar, M.A.: Theoretical and numerical study of an implicit discretization of a 1D inviscid model for river flows. Math. Models Meth. Appl. Sci., 16, 375–395 (2006). Brezis, H.: Operateurs Maximaux Monotones el Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam (1973). Nelder, J.A., Mead, R.: A simplex method for function minimization. Computer J., 7, 308–313 (1965).

Boundary Integral Equations for Arbitrary Geometry Shells V.V. Zozulya

1 Introduction Theory of shells has numerous applications in engineering fields such as civil, aerospace, chemical, mechanical, naval, nuclear, and microelectronics. Different aspects of the theory of shells have been discussed in numerous publications of theoretical and applied nature [GuBL78, Ki63, Ve82]. However, there are still some problems that have not been solved or even properly understood. A fundamental problem of the theory of shells is reduction of the 3-D equations of the solid mechanics to 2-D equations of shells. It is necessary for the 2-D equations to be as simple as possible, and their solution should make it possible to reconstruct the three-dimensional stress–strain state as accurately as possible. The existing theory of shells appeared by way of a compromise between these mutually exclusive requirements. One of the unsolved problems is an application of the boundary integral equation (BIE) method for the analysis of shells with arbitrary geometry. The BIE is a very efficient method applicable for the solution of different engineering problems [BrTW84]. For an effective application of the BIE, analytical expressions of fundamental solutions for appropriate differential operators are needed. Unfortunately only the fundamental solutions for some differential operators of special form [Ho83-85], mainly with constant coefficients, are known. For differential operators with variable coefficients there does not exist an effective analytical method which permits us to obtain such fundamental solutions. The differential operators that appear in the theory elastic shells of an arbitrary geometry contain components of the first and second metric tensors of their middle surface [Ki63, Ve82] which are functions of Gaussian coordinates. Therefore, analytical expressions of the funda-

V.V. Zozulya Centro de Investigacion Cientifica de Yucatan A.C., Mérida, Mexico, e-mail: [email protected] C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5_37, © Springer Science+Business Media, LLC 2011

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mental solutions are available only for some special cases of shell’s geometry and in particular for plates [BrTW84]. The application of the BIE method to the analysis of shells with arbitrary geometry has been studied by many researchers [ArGr02, Be91, Pa91]. In some of those publications, integral equations are applied without the construction of the fundamental solution for the shell’s differential operator. In spite of the difference in the approaches to the problem, all of these publications have a common disadvantage. The integral equations obtained in those publications contain unknown functions not only on the contour of the shell but also in the surface integrals. A theory of shells based on expansion of the 3-D equations of elasticity into a series of Legendre polynomials has been developed in [Ve82]. The equations obtained are based on a strong mathematical foundation and have many applications in engineering practice [Ho83-85, Zo89, Zo06b, Zo07]. In [Zo97] integral identities of Somigliana type and also fundamental solutions for the coefficients of a Legendre polynomial expansion have been obtained based on the approach developed in [Ve82]. In this paper we extended and generalize our previous result related to the BIE application for analysis of the arbitrary geometry shells earlier published in [Zo97, Zo98]. An integral representations of Somigliana type which allow us to transform the problem of analysis of the shells with arbitrary geometry to a 1-D BIE over the contour of the shell is obtained. Thus an approach which transforms analysis of the 3-D problem of elasticity to analysis of the 1-D BIE for arbitrary geometry shells is developed. We also show here that all the kernels presented in the BIE for shells of Vekua type [Ve82] can be calculated analytically.

2 The 3-D Equations of Elasticity in Curvilinear Coordinates Let an elastic homogeneous isotropic shell of arbitrary geometry with 2h thickness occupy the domain V = Ω × [−h, h] in Euclidean space R3 . Boundary of the domain is ∂ V = S ∪ Ω+ ∪ Ω− . Here Ω is the middle surface of the shell, ∂ Ω is its boundary, Ω+ and Ω− are the outer sides and S = Ω × [−h, h] is a sheer side. Material points in the domain V will be denoted by letters X, Y, . . . . The position of the points is described by Cartesian z(z1 , z2 , z3 ) coordinates and curvilinear coordinates x(x1 , x2 , x3 ) that related to the middle surface, respectively. The coordinates x1 , x2 parameterize the points on the middle surface Ω of shell and the coordinatex3 directed along the vector normal to Ω. Here, we will only consider regular coordinate systems with such properties: for every point in the domain V there exist bijective and smooth direct and inverse mapping (isomorphism) z = z(x1 , x2 , x3 ),

x = x(z1 , z2 , z3 )

and Jacobean matrices which are nonsingular i.e.

(1)

Boundary Integral Equations for Arbitrary Geometry Shells

 Aij = ∂ xi ∂ z j ,

 Bij = ∂ zi ∂ x j ,

and

405

    det Aij  = 0, det Bij  = 0,

(2)

where the summation convention over repeated indices has been used (and will use from now on). More detailed information related to the tensor notations used here, and its application to shell theory, may be found in [Ve78]. The mapping (1) and (2) are also called coordinate transformations which relate differentials dzi = Aij dx j and dxi = Bij dz j . Denote the basis vectors in Cartesian coordinates by ei and in curvilinear by Ri , they are related in the following way: j

j

ei = Bi R j , Ri = Ai e j . Covariant, contravariant and mixed components of the metric tensor are scalar products of the basic vectors Ri · R j = gi j ,

Ri · R j = gi j ,

j

Ri · R j = gi .

(3)

Isomorphism of coordinate transformations is a group of the domain transformations. The equations of elasticity are invariant to the group of transformations. For our purpose it is convenient to write them in invariant tensor form. The stress–strain state of an elastic body is defined by stress σ and strain ε tensors and displacements u, traction p, and body forces b vectors:

σ = σ i j Ri ⊗ R j ,

ε = ε i j Ri ⊗ R j ,

u = ui Ri ,

p = pi Ri ,

b = bi Ri .

Covariant and mixed components of the related tensors and vectors can be found in the standard way by raising and lowering indexed using metric tensors (3). These quantities are not independent as they are related by the equations of linear elasticity. If the displacements and their gradients are small, the Cauchy relations can be used:

ε=

 1 (∇u)T + ∇u , 2

1 1 εi j = (∇i ui + ∇ j ui ) = (∂i ui + ∂ j ui ) − Γi kj uk . 2 2

(4)

Here ∇i are the covariant derivatives, Γi kj are the Christoffel symbols and ∂i = ∂ ∂ xi are partial derivatives with respect to the space variables xi . The equations of equilibrium have the form 

∇ · σ + b = 0,

∇ j σ i j + bi = 0,

∇ j σ i j = ∂ j σ i j + Γjki σ k j + Γjk σ ik . j

(5)

The stress and strain tensors are related by the generalized Hook’s law

σ = λ θ E + 2μ ε ,

σ i j = (λ gi j gkl + 2μ gil g jk )εkl ,

(6)

where λ and μ are Lame constants, E is a unit tensor. Substituting Cauchy relations (4) and Hook’s law (6) in the equations of equilibrium (5), we obtain the differential equations of equilibrium in displacements in the

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vector form A = (λ + μ )∇∇ + μ ∇ · ∇ = Ai j Ri ⊗ R j ,

A · u = b,

where the differential operators ∇∇ and ∇ · ∇ have the form ∇ · ∇u = Δ u = Ri ∇k ∇k ui = gk j Ri ∇ j ∇k ui .

∇∇ u = gk j Ri ∇i ∇k u j ,

The differential equations of equilibrium for components of the displacement vector in curvilinear coordinates may be presented in the form Ai j u j (x) + bi (x) = 0, Ai j

= [μ gi j gkl

+ μ )gil g jk ]∇

+ (λ

∀x ∈ V,

k ∇l ,

∇i u j = ∂i u j − Γi kj uk .

(7)

Boundary conditions in shell theory are usually stated in the following form. On the outer surfaces Ω+ and Ω− the traction vectors are prescribed P · u = p+ ,

∀x ∈ Ω+ ,

P · u = p− ,

Pi j u j (x) = pi+ (x) ,

∀x ∈ Ω+ ,

Pi j u j (x) = pi− (x) ,

or

∀x ∈ Ω− ∀x ∈ Ω− .

(8)

On the sheer side S = S p ∪ Su , S p ∩ Su = 0/ the mixed boundary conditions P·u = ψ, or

Pi j u j (x) = ψ i (x) ,

∀x ∈ S p u = ϕ , ∀x ∈ S p ;

∀x ∈ Su

ui (x) = ϕi (x) ,

∀x ∈ Su

(9)

are prescribed. The traction operator in (8) and (9) in curvilinear coordinates has the form   P = Pi j Ri ⊗ R j = λ n · ∇ + μ n · (∇ )T + ∇ , (10)   Pi j = λ ni g jk + μ (nk gi j + n j gik ) ∇k .

3 The 3-D Somigliana Identity and Fundamental Solutions The boundary value problem (7)–(9) may be transformed to a system of BIE using the Somigliana’s identity. For this purpose the work that the body forces and the surface tractions perform on the corresponding displacements is considered and is given by   1 1 bi (x)ui (x)dV + pi (x)ui (x)dS. (11) L(ui , pi , bi ) = 2 2 V

, pi , bi

∂V

let us consider another system of forces, tractions and disTogether with ui placements u¯i , p¯i , b¯ i . From the reciprocal theorem of Betty [Ki63] it follows that

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407

L(ui , pi , bi ) = L(u¯i , p¯i , b¯ i ).

(12)

Utilizing the reciprocal theorem, Somigliana’s integral representation for displacements vector may be obtained. For this purpose the fundamental solutions for the differential equations (7) in an infinite region caused by unique concentrated body force, which apply at the point y and act in the direction of axis y j have to be considered. Thus a differential equation for the fundamental solution has the form [Ho83-85] (13) Ai j (∇x )U jk (x − y) + δki δ (x − y) = 0 , x, y ∈ R3 . Here δ (x − y) is the Dirac delta distribution and U jk (x − y) is a symmetric secondorder tensor, which is a displacement of the point x in the direction of the axis xk . Because the differential equations of elasticity (13) are invariant to coordinate transformations, we can solve it in Cartesian coordinates and then transform solution to the curvilinear coordinates using (1), (2). In Cartesian coordinates the fundamental solution of (13) has the form [BrTW84] Ui j (z(x) − z(y)) =

(3 − 4ν )δi j + ∂i R∂ j R , 16π μ (1 − ν )R

(14)

 where R = (z1 (x) − z1 (y))2 + (z2 (x) − z2 (y))2 + (z3 (x) − z3 (y))2 is the Euclidean distance between points x and y. The fundamental tensor of traction Wi j (z(x), z(y)) in Cartesian coordinates may be found by applying the operator Pi j from (10) in the form Pi j = λ ni ∂k + μ (δik ∂n + nk ∂i ) to the displacement tensor Ui j (z(x) − z(y)) [BrTW84] giving Wi j (z(x), z(y)) =

(1 − 2ν )(n j ∂i R − ni ∂ j R) − ((1 − 2ν )δi j + ∂i R∂ j R)∂n R . 8π μ (1 − ν )R2

(15)

In (14)  and (15) zi (x) and zi (y) are the Cartesian coordinates of the points x and y, R = (z1 (x) − z1 (y))2 + (z2 (x) − z2 (y))2 + (z3 (x) − z3 (y))2 is the Euclidean distance between points x and y and υ is Poisson’s ratio. From (11) and (12) and taking into account that b¯ i = gij δ (x − y), p¯i = W ji (x, y) and

u¯i = Ui j (x − y),

the Somigliana integral representation for the displacements in curvilinear coordinates may be written in the form 

ui (y) = ∂V

j

(p j (x)U ji (x − y) − u j (x)Wi (x − y))dS +



p j (x)U ji (x − y)dV.

(16)

V

The fundamental tensors of the displacements and tractions (14), (15) in curvilinear coordinates using coordinates transformations (1) and (2) may be presented in the

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form

Ui j (x − y) = Aki (x)Alj (y)Ukl (z(x) − z(y)), W ji (x, y) = Bik (x)Alj (y)Wlk (z(x), z(y)).

The integral representation (16) may be used for further development of the BIE for 3-D elasticity in arbitrary curvilinear coordinates.

4 The 2-D Equations of Elasticity in Coordinates Related to the Middle Surface of the Shell As mentioned in [ArGr02] and [Be91] the application of the 3-D BIE for elasticity to problems in the theory of shells has some methodological and computational difficulties. In order to avoid those problems an approach based on the reduction of a 3-D problem to a 2-D one has been developed here. For convenience we transform above equations of elasticity taking into account that the radius vector R(x) of any point in domain V , occupied by material points of the shell, may be presented as R(x) = r(xα ) + x3 n(xα )

(17)

or in a coordinate form zi (x) = zi (xα ) + x3 ni3 (xα ), where r(xα ) is the radius vector of the points located on the middle surface of the shell, xα = (x1 , x2 ) are curvilinear coordinates associated with the middle surface, n(xα ) is a unit vector normal to the middle surface. The basis vectors in curvilinear coordinates are related to the middle surface of the shell and, taking (17) into account, are presented in the form Rα (x) = rα (xα ) + x3 nα (xα ),

R3 (x) = n(xα ).

(18)

Now we rewrite the equations of elasticity (4)–(7) in a special coordinate system associated with the middle surface. The Cauchy relations (4) and the equations of equilibrium (5) have the form 1 1 β εαβ = (∇α uβ + ∇β uα ) − bαβ u3 , εα 3 = (∇α u3 + ∂3 uα ) − bα uβ , 2 2 ε33 = ∂3 u3 ,

(19)

γ

∇β σ αβ − 2bαβ σ 3β − bγ σ 3β + ∂3 σ α 3 + bα = 0, γ

∇β σ 3β + bαβ σ αβ − bγ σ 33 + ∂3 σ 33 + b3 = 0,

(20)

respectively. The generalized Hook’s law has the same form as (6) after taking into account that the components of the metric tensor (3), along with (17) and (18) are

Boundary Integral Equations for Arbitrary Geometry Shells

gαβ = aαβ ,

gα 3 = 0,

409

g33 = 1.

(21)

Here and in (19), (20) aαβ = rα · rβ and bαβ = −rα · nβ are coefficients of first and second quadratic forms of the surface. The differential equations of equilibrium for the displacements (7) are Aαβ uβ (x) + Aα 3 u3 (x) + bα (x) = 0

∀x ∈ V,

A3β uβ (x) + A33 u3 (x) + b3 (x) = 0

where the differential operators Ai j have the form   Aαβ = μ gαβ gγυ + (λ + μ )gαυ gβ γ ∇γ ∇υ , Aα 3 = (λ + μ )gαυ ∇υ ∂3 , A3β = (λ + μ )gβ γ ∇γ ∂3 , A33 = μ gγυ ∇γ ∇υ + (λ + μ )∂32 .

(22)

(23)

The next step in the transformation of the 3-D equations of elasticity into 2-D equations of shells consists of the following. Assume that the components of stress σ αβ and strain εαβ tensors; and displacements ui , body forces bi , and surface tractions pi vectors are sufficiently smooth functions of coordinate x3 and may be expanded into a series of Legendre’s polynomials. Then using the approach developed in [Ve82], they can be expressed as

σ (x) = ij

∞

n ij

∑σ

(xα )Pn (ω ) ,

n=0

εi j (x) =

∞

∑ εi j (xα )Pn (ω ) , n

n ij

2n + 1 σ (xα ) = 2h n

εi j (xα ) =

n=0 ∞

ui (x) =

∑ ui (xα )Pn(ω ) , n

n

ui (xα ) =

n=0

∞ n pi (x

pi (x) = ∑

n=0

bi (x) =

∞ n i

α )Pn (ω ) ,

∑ b (xα )Pn(ω ) ,

n=0

2n + 1 2h 2n + 1 2h

n i

2n + 1 b (xα ) = 2h

σ i j (xα , x3 )Pn (ω )dx3 ,

−h

h

εi j (xα , x3 )Pn (ω )dx3 ,

−h

h

−h h

2n + 1 p (xα ) = 2h n i

h

−h h

ui (xα , x3 )Pn (ω )dx3 ,

(24)

pi (xα , x3 )Pn (ω )dx3 , bi (xα , x3 )Pn (ω )dx3 ,

−h

 3

where ω = x h is a dimensionless coordinate. Substituting the series expansions (24) into the equations of elasticity (19) and (20) we get the corresponding relations for the coefficients of the expansion (24). The Cauchy relations have the form n n n 1 1 n n 3 n n n β n εαβ = (∇α uβ +∇β uα ) − bαβ u3 , εα 3 = ∇α u3 + ui −bα uβ , ε33 = ui , 2 2 − −

(25)

410

V.V. Zozulya

where n

ui = −

2n + 1 n+1 n+3 ( u3 + u3 + · · · ). h

The equations of equilibrium are n

n

n σ 3β

n σ αβ

γ

n

n

n

∇β σ αβ −2bβα σ 3β −bγ σ 3β + σ α 3 + f α = 0, ∇β where

n

+bαβ

n

f i (xα ) = bi (xα ) +

− n n n γ 33 −bγ σ + σ 33 + f 3 −

(26)

= 0,

2n + 1 i3 (σ+ (xα ) − (−1)n σ−i3 (xα )), h

n

n+3 2n + 1 n+1 (σ i3 + σ i3 + · · · ). − h The generalized Hook’s law for the coefficients of the expansion (24) has the same form as (6) after considering (21). Now with taking into account (25), (26) the differential equations for coefficients of expansion (24) of the displacements has the form

σ α3 =

n

n

n

0

1

n

Aαβ uβ (xα ) + Aα 3 u3 (xα ) + Lα (ui , ui , . . . , ) + f α (xα ) = 0

∀xα ∈ Ω,

n n n 0 1 n A3β uβ (xα ) + Aα 3 u3 (xα ) + L3 (ui , ui , . . . , ) + f 3 (xα ) = 0

(27)

n

where Ai j are the differential operators defined in (22), (23), Li is the operator that depend on all members of the series expansion (24) and include first order differential operators. The boundary conditions (9) for the coefficients of expansion (24) have the form n

Pαβ uβ (xα ) = pα+ (xα ) , n

uα (xα ) =

pi− (xα ) ,

∀xα ∈ ∂ Ω p , ∀xα ∈ ∂ Ωu .

(28)

5 The 2-D Somigliana Identity and BIE for Arbitrary Geometry Shells Substituting the series expressions (24) into the equation for the work that the body forces and the surface tractions perform on the corresponding displacements (11), we have

Boundary Integral Equations for Arbitrary Geometry Shells

L(ui , pi , bi ) =

1 2



∞ n i

∑ b (xα )Pn (ω )

V n=0

+

1 2

∞

411

∑ ui (xα )Pm (ω )dV m

m=0



∞

∞ n

∑ pi (xα )Pn(ω ) ∑ ui (xα )Pm (ω )dS.

∂V

n=0

m

(29)

m=0

The integrals in this formula may be transformed in following way: 

 ∂V

∞ n

∞



∞ n

∑ bi (xα )Pn (ω ) ∑ ui (xα )Pm (ω )dV = ∑ bi (xα ) ui (xα )dV , m

m=0 V n=0 ∞ ∞ n m pi (xα )Pn (ω ) ui (xα )Pm (ω )dS n=0 m=0

∑

∑





=

∞

∂Ω

Ω ∞ n i

n

∑ p (xα ) ui (xα )dS n

n=0

∑ [pi+ (xα ) − (−1)n pi− (xα )] ui (xα )dS.

+

(30)

n=0

n

(31)

n=0

Ω

Here, the orthogonal property of Legendre’s polynomials is used together with the fact that Pn (1) = 1 and Pn (−1) = (−1)n , and boundary conditions given in (8). Substituting expressions for the volume (30) and surface integral (31) into (29) results in n

n

n

L(ui , pi , bi ) = where



1 2

Ω

n

∞ n i

∑

n

f (xα ) ui (xα )d Ω +

n=0

1 2

 ∂Ω

∞ n

∑ pi (xα ) ui (xα )dl, n

(32)

n=0

n

f i (xα ) = bi (xα ) + (pi+ (xα ) − (−1)n pi− (xα )). From (32), similar to 3-D elasticity, a corresponding reciprocal theorem for the coefficients of the expansion into the Legendre’s polynomial series may be obtained in the form n n n n n − − − n L(ui , pi , bi ) = L(ui , pi , bi ),

(33)

which is an analog of Betty’s reciprocity theorem for (12). In [Zo97] it has been proved that if n

−

b = δi δ i

j nm

n

n

(xα − yα ),

nm ui = Ui j (xα , yα ), −

−

nm

pi = W ji (xα , yα ),

where nm Ui j (xα , yα ) =

2n + 1 2m + 1 · 2h 2h

h h −h −h

U ji (x − y)Pn (ω )Pm (ω )dx3 dy3 ,

412

V.V. Zozulya nm

W ji (xα , yα ) =

δ

2n + 1 2m + 1 · 2h 2h

(xα − yα ) =

nm

h h

h h

Wi j (x, y))Pn (ω )Pm (ω )dx3 dy3 ,

−h −h

(34)

δ (x − y))Pn (ω )Pm (ω )dx dy 3

3

−h −h

put in (33) the integral representation for the expansion of the coefficients of the displacement vector components into a Legendre polynomial series may be obtained. These integral representations are presented in the form   ∞ nm

n

nm

∑ Ui j (xα , yα ) pi (xα ) − W ji (xα , yα ) ui (xα )

m

ui (yα ) = ∂Ω



+ Ω

n=0 n ∞ nm Ui j (Xα , yα ) f i (xα )dS n=0

∑

n

dl

∀yα ∈ / ∂ Ω (m = 0, 1, 2, . . . , ∞). (35)

This equation is the analog of the Somigliana identity in the theory of elasticity [Ki63]. Somigliana’s identity (16) is used for formulation of the BIE in theory of elasticity [BrTW84]. In the same way the BIE for shells of arbitrary geometry may be obtained using the analogy of the Somigliana’s identity (35). For that purpose the point yα must be brought to a position on the boundary ∂ Ω through a limiting process. The boundary properties of the integral operator in (35) depends on the boundary properties of the elastostatic fundamental solutions (14) and (15). These properties are well known (see for example [BrTW84]). In [Pa91] it was shown that the boundary properties of the integral operators in (35) are expressed by

   n n nm nm Ui j (xα , yα ) pi (xα )dl = Ui j (xα , yα ) pi (xα )dl, lim yα →∂ Ω



lim

yα → ∂ Ω

∂Ω  nm n i W j (xα , yα ) ui (xα )dl

∂Ω

∂Ω

1 n = ∓ ui (yα ) + 2

 nm

n

(36)

W ji (xα , yα ) ui (xα )dl.

∂Ω

Now the point yα is brought to a position on the boundary ∂ Ω through a limiting process applied to the Somigliana integral representation. The boundary properties of the integral operators in (35) are expressed by (36). Taking into account all of the above, the BIE for shells of arbitrary geometry take the form   n nm ∞ nm 1m n i i ± ui (yα ) = ∑ Ui j (xα , yα ) p (xα ) − W j (xα , yα ) ui (xα ) dl 2 n=0 ∂Ω



+ Ω

∞ nm

n

∑ Ui j (xα , yα ) f i (xα )dS

n=0

have been obtained.

∀yα ∈ ∂ Ω (m = 0, 1, 2, . . . , ∞) (37)

Boundary Integral Equations for Arbitrary Geometry Shells

413

6 First Approximation Equations As mentioned earlier, we consider a deformable body which is an elastic homogeneous shell of arbitrary geometry with thickness 2h which is supposed to be small compared to other sizes. In the approach developed here the shell is substituted by its middle surface and its stress-strain state is described by the infinite system of differential equations (27) with boundary conditions (28). Using the regular approximation theorem, we use only a finite number of terms in the Legendre polynomial series (24). The order of the system of equations depends on an assumption regarding the thickness on the distribution of the stress–strain parameters. The thickness is supposed to be relatively small in comparison with other parameters of the shell. Therefore following [GuBL78] and [Ve82] we only use two members in polynomial expansion (24). In this case we will get first order approximation equations of shell, which we usually refer to as Vekua’s shell theory. The BIE in this case have nm

nm

the form (37) , where the kernels Ui j (xα , yα ) and W ji (xα , yα ) are given by (34) and n, m = 0, 1. Analysis of the expressions for fundamental solutions show that the dimensionless double integral 1 1 −1 −1

(ξ 3 − ζ 3 )a (ξ 3 )b (ζ 3 )c (r2 (x

α , yα

) + h2 (ξ 3 − ζ 3 )2 )k/2

dξ 3dζ 3,

(38)

k = 1, 3, 5; a = 0, 1, 2; b = 0, 1; c = 0, 1, where R2 (xα , yα , x3 , y3 ) = r2 (xα , yα ) + (x3 − y3 )2 , has to be calculated. Some of the integrals in (38) are equal to zero, while other integrals are nonzero and can be calculated analytically. We cannot present the results of all of these integral calculations here because it would require too much space. In order to have an idea how they look, we present only two of them: 1 1 −1 −1

1 1 −1 −1



dξ 3d ζ 3 2 r(x = , y ) − r2 (xα , yα ) + 4h2 α α h3 R2 (xα , yα , ξ 3 , ζ 3 )1/2 

2 2 2 2 + h ln(h r (xα , yα ) + 4h + 2h) − h ln(h r (xα , yα ) + 4h − 2h) , (ξ 3 − ζ 3 )2 d ξ 3 d ζ 3 R2 (xα , yα , ξ 3 , ζ 3 )5/2  r2 (xα , yα ) + 2h2 4 1  . (39) = 3 − 3h r2 (xα , yα ) r2 (xα , yα ) + 4h2 r(xα , yα )

Some integrals given here are singular and need special consideration. For these the approach developed in [Zo06a] can be used. We compare results of analytical

414

V.V. Zozulya

and numerical methods for evaluating the double integrals in (38). Calculations using analytical expressions for (39) have been almost more than ten times faster than using numerical quadrature formulas.

References [ArGr02]

Artyuhin, Yu.P., Gribanov, A.P.: Solution of the Problems of Nonlinear Deformation of Plates and Shallow Shells by Boundary Element Method, Kazan University Publisher House, Kazan (2002) (Russian). [BrTW84] Brebbia, C.A., Telles, J.C.F., Wrobel, L.C.: Boundary Element Techniques. Theory and Applications in Engineering, Berlin (1984) [Be91] Beskos, D.E.: Boundary Element Analysis of Plates and Shells, Springer, Berlin, 93– 140 (1991). [GuBL78] Guliaev, V.I., Bajenov, V.A., Lizunov, P.P.: Nonclassical Shell Theory and Its Application for Engineering Problems Solution, Vyshcha shkola, Lviv (1978) (Russian). [Ho83-85] Hormander, L.: The Analysis of Linear Partial Differential Operators, vols. I–IV, Springer Verlag, Berlin (1983–1985). [Ki63] Kil’chevski, N.A.: Foundation of Analytical Theory of Shells, Publisher House of ANUkrSSR, Kiev (1963) (Russian). [Pa91] Paymushin, V.I., Sidorov, I.N.: Variant of boundary integral equation method for solution of static problems for isotropic arbitrary geometry shells. Mech. Solids, 1, 60–169 (1991). [Ve78] Vekua, I.N.: Fundamental of Tensor Analysis and Theory of Covariants, Nauka, Moskow (1978) (Russian). [Ve82] Vekua, I.N.: Some General Methods of Construction Various Variants of Shell Theory, Nauka, Moskow (1982) (Russian). [Zo89] Zozulya, V.V.: The combined problem of thermoelastic contact between two plates though a heat conducting layer. J. App. Maths. and Mech., 53, n. 5, 722–727 (1989). [Zo97] Zozulya, V.V.: Somigliano identity and fundamental solutions for arbitrary geometry shells. Doclady Akademii Nauk of Ukraine, 6, 60–65 (1997) (Russian). [Zo98] Zozulya, V.V.: The boundary integral equations for the shells of arbitrary geometry. Inter. App. Mech., 34, n. 5, 79–83 (1998). [Zo06a] Zozulya, V.V.: Regularization of the divergent integrals. I. General consideration. Electr. J. Boun. Elem., 4, n. 2, 49–57 (2006). [Zo06b] Zozulya, V.V.: Laminated shells with debonding between laminas in temperature field. Inter. App. Mech., 42, n. 7, 135–141 (2006). [Zo07] Zozulya, V.V.: Mathematical modeling of pencil-thin nuclear fuel rods, in: Structural Mechanics in Reactor Technology (Editor: A. Gupta), SMIRT, Toronto, C04–C12, (2007).

Index

A a posteriori pointwise estimator, 329 absolute minimizers, 181 absorbing boundary conditions, 236 adaptive particle filter, 47 adjoint state technique, 395 advection–diffusion equation, 25 advective terms, 276 aerodynamical applications, 259 algebraic equation, elliptic quadratic qualitative analysis of, 256 algebraic equations, elliptic quadratic, 253 qualitative analysis of, 254 almost periodicity, 141 asymptotic analysis of singularities, 379 asymptotics, 161, 386 atmospheric and ionospheric ducts, evaporative, 367 atmospheric pollutant dispersion model, 262 sources, 261 B Banach principle, 143 Bayes theorem, 48 Bayesian approach, 47 Bermúdez–Moreno iterative algorithm, 398 Bernoulli equation, 149 Bessel potential spaces, 110 bimaterial, linearly elastic, 242 Boltzmann equation, 17 Boltzmann–Gibbs–Shanon formula, 51 bootstrap filter, 55 boundary element method, 327

homogenization, 161 integral equation, 79, 403 method, 129, 147, 403 maps, 71 boundary–domain integral equations localized, 91 segregated, 109 Bragg curve, 19 Brownian random walk, 262 bubble behavior, 147 C captured shock, 358 cardinal theorem of interpolation theory, 352 carrier function, 363 Cauchy integral equations, 1 principal value, 2 relations, 405 Cauchy–Kowalewski theorem, 32, 350 change of variable methods, 207 characteristic function, 342 method, 339 Chebyshev polynomial, 377 basis, 73 Christoffel symbols, 405 co-normal derivative, 115 operators, 93 collocation, 195 method, 1 conjugate gradient methods, 80 contact problems, elastodynamic, 241 continuous waves, 364 contraction mapping, 143 control valve model, 305

C. Constanda, P.J. Harris (eds.), Integral Methods in Science and Engineering, DOI 10.1007/978-0-8176-8238-5, © Springer Science+Business Media, LLC 2011

415

416 controlled nuclear reactions, 35 convolution, 380 product, 143 Copenhagen experiment, 265 correlation decay rate, 75 function, 237 cost function, 395 Coulomb friction law, 243 crack, semi-infinite, 381 creep crack propagation, 191 critical hypersurface, 260 parabola, 257 parabolic line, 258 D density-weighted potential, 151 deterministic subsystems, 235 diffusion equation, 225 Dirac’s distribution, 95 direct boundary integral method, 328 field propagators, 235 Dirichlet boundary condition, 226 problem, 131, 327 discharge coefficient, 301 displacement vector, 129 dissipative structures, nonlinear localized, 357 distribution, 131, 380 drift coefficient, 262 dual, 132 E eikonal methods, 363 elliptic theta functions, 370 energy analysis dynamic, 69 statistical, 69 conservation equation, 279 functionals, 181 equilibrium equations, curvilinear coordinates, 406 ergodic ray dynamics, 75 error estimation, 327 Euler’s function, 382 Euler–Lagrange equation, 184 evolution problems, 313 exponential integral, 375

Index extension operator, 133 F fast iterative methods, 79 multipole methods, 83 Fenchel dual, 183 Fickian closure, 25 fluid flow, 7 fluid–rigid system, 339 flying configuration in subsonic flow, 259 Fokker–Plank equation, 262 Fox H-function, 226 fractional Caputo derivative, 225 Fredholm operator of index zero, 101 fundamental solution, 370, 404, 406 G Gakhov operator, 380 Gamma function, 229 Gas–Liquid Cylindrical Cyclone/Slug Damper System (GLCC-SD), 299 Gauss–Chebyshev quadrature, 73 Gaussian coordinates, 403 distribution, 51 quadrature, 29, 152 General Integral Transform Technique (GITT), 59 Generalized Integral Advection Diffusion Multilayer Technique (GIADMT), 26 Green’s function, 148, 236 representation formula, 92 Green’s function, 69 greenhouse gas, 261 H Hadron cancer therapy, 15 Hamilton–Jacobi theory, 182 harmonic loading, 241 heat flux, 130 heavy-ion transport, 17 Helmholtz equation, 79, 225, 234 problem, 79 Hilbert integral, 187 transformation, 181 holomorphic mapping, 134 Hook’s law, generalized, 405

Index hybrid method, 233 zonal solutions, 259 I image deblurring model, 83 implicit upwind difference scheme, 398 incompressible liquid flow, 301 indirect boundary integral equation method, 109 initial-boundary value problem, 130, 136 instantaneous flow rates, 299 integral equation, Abel-type, nonlinear, 191 problems, iterative methods for, 79 integro-differential equations, 287 system, 143 interface bridged crack, 293 cracks, 241 bridge zone of, 287 problems, 92 interior potential, 333 intermediate variable methods, 85 interpolation methods, 204 projection operator, 2 inverse deconvolution, 79 inverse-direct reduction method, 161 iterative algorithm, 246

417 Lévy–Gnedenko central limit theorem, 52 Liapunov–Tauber theorem, 116 likelihood function, 51 Lipschitz condition, 142 localized layer potentials, 96 Levi function, 95 parametrix, 95 volume potentials, 95 localizing functions, 105 M macroscopic-microscopic relationships, 8 mammary adenocarcinoma, 213 Markov process, 49 matched asymptotic expansions, 274 Maxwell–Boltzmann transport, 8 method of sources and sinks, 369 metric tensors, 403 minimax principle, 169 mixed boundary value problem, 112 moment–stress operator, 136 moments, bending and twisting, 129 Monin–Obukhov similarity, 282 Monte Carlo simulation, 36 validation procedure, 15 multigrid method, 82 multilayer perceptron artificial neural network (MLP-ANN), 261 N

K Kalman filter, 47 Kolmogorov time scale, 262 L Lagrangian particle model LAMBDA, 262 Lamé constants, 130 laminar regime, 7 Langevin equation, 262 Laplace transform of source functions, 369 transformation, 28, 131 Lax–Milgram theorem, 112 layer potentials, 112 Lebesgue’s theorem of dominated convergence, 229 Legendre polynomials, 404 Levi function, 91, 112 Levi-Civita symbol, 9

Navier–Stokes equations, 7 layer, 259 Nelder–Mead algorithm, 400 Neumann boundary condition, 79 problem, 136 neural networks, artificial, 261 neutron diffusion equation, 59, 349 transport, 37 Newmann product method, 369 Newton method, 196 Noether identity, 184 non-extensive entropy, 48 parameter, 53 non-smooth boundary, 379 nonnegative discrete spectrum, 171 normal boundary flux, 331

418 nuclear reactor, spectral criticality, 35 Nyström method, 195

Index Reynolds number, 7 Richardson extrapolation, 327 rigid functions, 341

O S observables, 8 operator convection–diffusion, 173 operators boundary integral, 115 optimal control problem, hyperbolic, 391 optimality criterion, 185 optimization based multilevel methods, 85 orthogonal eigenfunctions, 27 orthonormal eigenfunctions, 351 oscillatory integrals, 203 P parametrix, 91, 112 particle transport, 16 penalty function, 399 Perron–Frobenius operator, 71 perturbation techniques, 273 phase shift, 361 space operators, 71 pipeline slugging model, 306 plates thermoelastic, 129 with cracks, 129 pneumatic line model, 305 pollutant, 391 concentration, 27 potential, 8 double-layer, 226 Lennard–Jones, 8 Newton, 114 preconditioning, 79 primary separator, 299 pseudodifferential operator, 100, 115, 379 pseudospectrum, 173 Q quadratic forms of a surface, 409 quadrature methods, 204 quasimode with remainder, 313 R Rayleigh–Plesset bubble, 152 regularization procedure, 1 reservoir flow, 369

scale invariance, 181 Schwartz class, 380 semi-discretization, 343 semilinear systems, 141 separable Hilbert space, 314 shells of arbitrary geometry, 403 Signorini unilateral constraints, 243 similarity solutions, 184 slug damper model, 301 flow, 299 Sobolev–Slobodetski spaces, 110 solitary waves, nonlinear, 358 Somigliana representation, 244, 407 Sommerfeld radiation–type condition, 243 specific heat, 130 spectral Neumann problem, 159 stability, 173 stiff problems, 159 stability regimes, 278 stable distribution, 47 standing waves, 314 stationary flow problem, 72 statistical energy analysis (SEA), 233 Stehfest algorithm, 369 stochastic reverberant field, 235 stratified atmospheric boundary layer, 273 stress analysis, 287 intensity factors, 242, 292 strongly oscillating boundary, 159 Sturm–Liouville problem, 59 subdiffusion, limiting cases of, 225 T thermal conductivity, 130 expansion coefficient, 130 neutron flux, 61 thermostatistics, 52 toroidal bubble model, 154 trace operators, 132 transfer operator, 69 transformation parameter, 131 transgenic mouse model, 214 transmission problem

Index Dirichlet, 97 mixed, 101 with variable coefficients, 91 transport equation, 10 transverse shear forces, 129 trapezium quadrature rule, 196 trigonometric polynomials, 1 turbulence, 7, 26 turbulent parametrization, 30 regime, 7 V variational method, 109, 129 problem, 397 Vekua’s shell theory, 413 vibro-acoustic energy, 233 virial theorem, 237

419 viscoelastic plane, 192 viscosity, 8 Volterra integral equation, nonlinear, 193 vortex correlator, 8 vorticity, 8 confinement, 358 W water quality, optimization of, 391 wave confinement (WC), 358 energy flow, 69 equations, 357 factorization method, 380 wavelet methods, 83 wavenumbers, 79 weak solution, 134, 138, 139 Weierstrass excess function, 188 weighted Sobolev spaces, 109