Finite Element Methods with B-Splines

Citation preview

Finite Element Methods with B-Splines

This page intentionally left blank

F R O N T I E R S IN APPLIED MATHEMATICS The SIAM series on Frontiers in Applied Mathematics publishes monographs dealing with creative work in a substantive field involving applied mathematics or scientific computation. All works focus on emerging or rapidly developing research areas that report on new techniques to solve mainstream problems in science or engineering. The goal of the series is to promote, through short, inexpensive, expertly written monographs, cutting edge research poised to have a substantial impact on the solutions of problems that advance science and technology. The volumes encompass a broad spectrum of topics important to the applied mathematical areas of education, government, and industry.

EDITORIAL BOARD H.T. Banks, Editor-in-Chief, North Carolina State University Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB Carlos Castillo-Chavez, Cornell University Doina Cioranescu, Universite Pierre et Marie Curie (Paris VI) Lisa Fauci,Tulane University Pat Hagan, Bear Stearns and Co., Inc. Belinda King,Virginia Polytechnic Institute and State University Jeffrey Sachs, Merck Research Laboratories, Merck and Co., Inc. Ralph Smith, North Carolina State University AnnaTsao, AlgoTek, Inc.

BOOKS PUBLISHED IN FRONTIERS IN APPLIED MATHEMATICS Hollig, Klaus, Finite Element Methods with B-Splines Stanley, Lisa G. and Stewart, Dawn L, Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods Vogel, Curtis R., Computational Methods for Inverse Problems Lewis, F. L.; Campos, j.; and Selmic, R., Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities Bao, Gang; Cowsar, Lawrence; and Masters, Wen, editors, Mathematical Modeling in Optical Science Banks, H.T.; Buksas, M.W.; and Lin,T, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts Oostveen, Job, Strongly Stabilizable Distributed Parameter Systems Griewank, Andreas, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation Kelley, C.T., Iterative Methods for Optimization Greenbaum.Anne, Iterative Methods for Solving Linear Systems Kelley, C.T., Iterative Methods for Linear and Nonlinear Equations Bank, Randolph E., PLTMG:A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 7.0 More, Jorge J. and Wright, Stephen J., Optimization Software Guide Rude, Ulrich, Mathematical and Computational Techniques for Multilevel Adaptive Methods Cook, L. Pamela, Transonic Aerodynamics: Problems in Asymptotic Theory Banks, H.T., Control and Estimation in Distributed Parameter Systems Van Loan, Charles, Computational Frameworks for the Fast Fourier Transform Van Huffel, Sabine andVandewalle.Joos, TheTotal Least Squares Problem:Computational Aspects and Analysis Castillo, Jose E., Mathematical Aspects of Numerical Grid Generation Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Users' Guide 6.0 McCormick, Stephen F., Multilevel Adaptive Methods for Partial Differential Equations Grossman, Robert, Symbolic Computation: Applications to Scientific Computing Coleman,Thomas F. and Van Loan, Charles, Handbook for Matrix Computations McCormick, Stephen F, Multigrid Methods Buckmasterjohn D., The Mathemat/cs of Combustion Ewing, Richard E., The Mathematics of Reservoir Simulation

Finite Element Methods with B-Splines

Klaus Hollig Universitat Stuttgart Stuttgart, Germany

Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2003 by the Society for Industrial and Applied Mathematics. 109876543 2 I All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Hollig, K. (Klaus) Finite element methods with B-splines / Klaus Hollig. p. cm. — (Frontiers in applied mathematics) Includes bibliographical references and index. ISBN 0-89871-533-4 I. Structural analysis (Engineering) 2. Finite element method. 3. Spline theory. I. Title. II. Series. TA645 .H63 2003 620'.OOI'5I535—dc2l 2002036509

is a registered trademark.

Contents Preface

ix

1

Introduction

1

2

Basic Finite Element Concepts 2.1 Model Problem 2.2 Mesh-Based Elements 2.3 Sobolev Spaces 2.4 Abstract Variational Problems 2.5 Approximation Error

7 7 9 12 16 19

3

B-Splines 3.1 The Concept of Splines 3.2 Definition and Basic Properties 3.3 Recurrence Relation 3.4 Representation of Polynomials 3.5 Subdivision 3.6 Scalar Products

23 23 25 27 29 32 35

4

Finite 4.1 4.2 4.3 4.4 4.5

37 37 40 43 46 50

5

Approximation with Weighted Splines 5.1 Dual Functions 5.2 Stability 5.3 Polynomial Approximation 5.4 Quasi-Interpolation 5.5 Boundary Regularity 5.6 Error Estimates for Standard Weight Functions

Element Bases Multivariate B-Splines Splines on Bounded Domains Weight Functions Web-Splines Hierarchical Bases

VII

53 53 56 60 63 65 67

viii

Contents

6

Boundary Value Problems 6.1 Essential Boundary Conditions 6.2 Natural Boundary Conditions 6.3 Mixed Problems with Variable Coefficients 6.4 Biharmonic Equation 6.5 Linear Elasticity 6.6 Plane Strain and Plane Stress

71 71 75 79 82 86 89

7

Multigrid Methods 7.1 Multigrid Idea 7.2 Grid Transfer 7.3 Basic Algorithm 7.4 Smoothing and Coarse Grid Approximation 7.5 Convergence

95 95 99 103 105 108

8

Implementation 8.1 Boundary Representation 8.2 Classification of Grid Cells 8.3 Evaluation of Weight Functions 8.4 Numerical Integration 8.5 Matrix Assembly

113 114 116 119 124 127

Appendix

131

Notation and Symbols

135

Bibliography

137

Index

143

Preface Piecewise polynomial approximations are fundamental to geometric modeling, computer graphics, data fitting, and finite element methods. For most applications, B-splines have become a widely accepted standard because of their flexibility and computational efficiency. So far, finite element simulations are the sole exception. The majority of schemes use low-degree polynomial elements on meshes conforming to the geometry of the parameter domains. Boundary conditions and stability requirements seem to prevent the use of splines on uniform grids in a straightforward fashion. However, these difficulties can be overcome. The resulting methods combine the advantages of standard finite elements and B-spline representations. In particular, no mesh generation is required, which eliminates a difficult and often very time-consuming preprocessing step. We describe in this book the construction of several types of B-spline bases for approximating boundary value problems. Moreover, we discuss the implementation of spline-based finite element schemes and derive error estimates. The performance of these methods is illustrated for typical applications in field theory, fluid dynamics, and elasticity. The book is essentially self-contained. Some basic facts from functional analysis and about partial differential equations, which are required, are listed in an appendix. Hence, the material is easily accessible not only for a reader with some background in geometric modeling and finite element methods, but also for beginning graduate students in numerical analysis and engineering. I gratefully acknowledge the support by our Numerical Analysis and Geometric Design group (NAGD) in Stuttgart. Above all, I thank Ulrich Reif and Joachim Wipper for a continuing excellent cooperation, which led to the discovery of web-splines (see http://www.web-spline.de). Extensive software has been developed by Joachim Wipper (2D) and Jorg Horner (3D)—many thanks for great enthusiasm despite thousands of lines of C and MATLAB code. In addition, several students have contributed to our finite element programming projects: Alexander Fuchs [36] and Dietrich Nowottny [66] (mesh generation), Christian Apprich (penalty methods and web software), Ronald Halmen (Lagrange multiplier techniques), Anja Streit and Andreas Kopf (elasticity), Marco Bossle, Winfried Geis, and Bernhard Russig (weight functions). The University of Stuttgart and the state of Baden-Wiirttemberg have provided excellent research support. In particular, I would like to acknowledge the funding by the Technologie-Lizenz-Biiro (TLB) for our web-spline project.

IX

x

Preface

I have very much enjoyed writing this book, especially because I could concentrate on the pleasant part. Very special thanks to my daughter Natalie for helping by patiently typing various versions of my manuscript, never mentioning that KlgX seems much harder than Microsoft Word!

Klaus Hollig Zavelstein, December 2001

Chapter 1

Introduction

The finite element method (FEM) has become the most widely accepted general purpose technique for numerical simulations in engineering and applied mathematics. Principal applications arise in continuum mechanics, fluid flow, thermodynamics, and field theory (cf., e.g., [95, 52, 25, 55]). In these areas computational methods are essential and benefit strongly from the enormous advances in computer technology. While the label "finite element method" was first used by Clough [26], the key ideas date back much further [94] (see Figure 1.1). Ritz [74] solved variational problems with finite-dimensional approximations, an approach already employed by Rayleigh [70, 71]. Following an observation of Bubnow [21], Galerkin [38] approximated solutions of boundary value problems directly, without resorting to the variational formulation. Courant [29] used hat-functions, the standard finite element for analyzing the St. Vernant torsion problem. Hence, the fundamental mathematical concepts had been introduced long before their practical potential was recognized. The systematic use of variational approximations in engineering applications began with the work of Turner, Clough, Martin, and Topp [90] and Argyris [3, 4]. Their results initiated intensive research, and finite elements soon became the method of choice in structural analysis. When Strang and Fix wrote the first textbook on the mathematical theory of finite element methods [87], more than 200 papers related to this subject had been published (cf. [62] for a comprehensive listing). Now, almost 30 years later, various aspects of finite element techniques remain a central area of engineering research and development. Splines play an important role in approximation and geometric modeling. They are used in data fitting, computer-aided design (CAD), automated manufacturing (CAM), and computer graphics (cf., e.g., [44, 32]). Extensive software is available, and algorithms are almost as efficient as for polynomials [16, 51]. The early work of Schoenberg [79] revealed that splines possess powerful approximation properties. Subsequently, many approximation schemes have been proposed [2]. In particular, after de Boor's results about B-splines [14], spline techniques became popular for a broad range of applications. His algorithms [15] are still the basic building blocks for almost any spline software. Another fundamental contribution is due to Bezier [9, 10, 11],

1

2

Chapter 1. Introduction

Figure 1.1. History of finite elements and splines. who introduced modern piecewise polynomial modeling techniques to CAD/CAM. He recognized that the Bernstein basis [8] yields a very intuitive geometric description of free form curves and surfaces. Similar results were obtained earlier by de Casteljau [23], whose work is, however, much less known.'Bezier's and de Casteljau's new concepts were soon generalized to splines, an important step being the knot insertion algorithms of Bohm [12] and Riesenfeld et al. [61, 27]. This led to a revival of research on splines in the 1980s and considerably enriched the existing theory. As a consequence, B-splines became a standard not only for numerical approximation schemes, but also for free form design and geometry processing. With geometric modeling and numerical simulation closely linked in engineering applications, the use of B-splines as finite element basis functions suggests itself. But, as is illustrated in Figure 1.2, at first sight this seems infeasible for two reasons: (i) Essential boundary conditions cannot be modeled easily. For example, if a linear combination of B-splines, p = ^k u^bk, is required to vanish on the boundary 3D of a domain D, then, in general, all coefficients M* of B-splines with support intersecting 3D must be 0. Hence, p equals 0 outside the light gray region in the figure, which results in very poor approximation order for solutions of differential equations with Dirichlet boundary conditions. 'Bezier and de Casteljau were scientists working for different French car manufacturers. While Bezier's work was published, de Casteljau's research was kept confidential for many years.

Chapter 1. Introduction

3

Figure 1.2. Grid cells of biquadratic B-splines with support intersecting a bounded domain. (ii) The restricted B-spline basis is not uniformly stable. As shown in the figure, the basis may contain B-splines bj with very small support in D consisting only of portions of the dark gray boundary grid cells. This leads to excessively large condition numbers of finite element systems and can cause extremely slow convergence of iterative methods. As is implied by the title of this book, both difficulties can be overcome. The resulting methods combine the advantages of B-spline approximations on regular grids and standard mesh-based finite elements. They provide a natural link between geometric modeling and finite element methods. In hindsight, the solution to the first of the above-mentioned problems is simple. Homogeneous essential boundary conditions can be modeled via weight functions, a technique already employed by Kantorowitsch and Krylow [54]. For example, solutions which vanish on 3D are approximated with linear combinations of weighted B-splines:

where K denotes indices of B-splines with some support in D and w is a smooth positive function on D with w\dD = 0. The construction of such weight functions was extensively studied by Rvachev et al. (cf., e.g., [76, 77]). His R-function method (RFM) provides efficient algorithms for domains defined as Boolean combinations of elementary sets. Resolving the stability problem, caused by the outer B-splines bj, is slightly more subtle. A well-conditioned basis can be obtained by forming appropriate linear combinations

as described in [46]. The inner indices i e / correspond to B-splines with at least one grid cell of their support contained in D, and /(/) c K\I are small sets of neighboring outer indices j. These extended B-splines inherit all basic features of the standard B-splines &,. In particular, their linear span maintains full approximation order.

Chapter 1. Introduction

4

Combining the above ideas gives rise to the definition of weighted extended B-splines (web-splines) [46]. These basis functions possess the usual properties of standard finite elements. In addition, there are a number of algorithmic advantages of B-spline bases which lead to very efficient simulation techniques [47, 49]. • No mesh generation is required. • The uniform grid is ideally suited for parallelization and multigrid techniques. • Accurate approximations are possible with relatively low-dimensional subspaces. • Smoothness and approximation order can be chosen arbitrarily. • Hierarchical bases permit adaptive refinement. Given the difficulty of constructing finite element meshes (cf., e.g., [50, 67, 37]), the first property is probably the most important one. Utilizing a regular grid not only eliminates a difficult preprocessing step, but also permits very efficient implementations of algorithms. Moreover, the use of B-splines reduces the dimension of finite element systems, in particular when high accuracy is required. Regardless of the degree, there is only one parameter for each grid cell. Outline of the Text The description of the new spline-based methods involves a combination of finite element analysis and approximation theory. Therefore, we review in the first two chapters some of the most relevant definitions and results from both fields. In chapter 2 we describe the RitzGalerkin approximation, the classical finite element scheme. We consider the variational problem where a is an elliptic bilinear form and X a continuous linear functional on a Hilbert space H (cf., e.g., [24, 87]). Despite its simplicity, this abstract formulation includes a broad range of applications. Moreover, it permits very elegant derivations of the standard error estimates. Chapter 3 is devoted to univariate uniform B-splines. We derive their basic properties and describe algorithms which are particularly important for finite element methods. Having briefly discussed necessary background material, we turn to the description of spline-based finite element methods. Chapter 4 introduces three types of finite element bases: weighted B-splines, web-splines, and hierarchical B-splines. These basis functions have similar properties as standard finite elements. We discuss this in detail for web-splines in chapter 5. In particular, we prove uniform stability and construct projection operators with optimal approximation order. With the aid of these results, the analysis of RitzGalerkin methods with weighted splines is routine. We consider in chapter 6 several typical variational problems of the form (1.1), including Poisson's equation as a simple model problem, incompressible flow, the plate problem, and the Lame-Navier system in linear elasticity. The last two chapters are devoted to the implementation of finite element schemes. In chapter 7 we describe the basic components of multigrid algorithms and establish the

Chapter 1. Introduction

5

grid-independent convergence rate for the w-cycle. Chapter 8 provides algorithmic details for the construction of the web-basis and the assembly of Ritz-Galerkin matrices. As is to be expected, the regular structure of the spline spaces offers many computational advantages. Most of the topics in this book do not require advanced mathematical techniques. Merely for existence theorems and error estimates some background in functional analysis is helpful. The relevant results are listed in an appendix. Notation Important definitions and results are highlighted in light gray boxes. They are labeled with c.n, where c is the chapter and n the statement number. The labels A.n refer to statements in the appendix. As usual, equations are labeled by (c.n) and [n] refers to the list of references. Dependencies on parameters are not always indicated if they are clear from the context. For example, we write for the m-variate tensor product B-spline of coordinate degree n, grid width n, and support In estimates, the dependence of constants on parameters pv is indicated in the form const(pi, p2, ...). If the dependence is clear from the context, or not particularly relevant, we use the symbols , and x. For example,

characterizes all points jc with distance > const h from the boundary of D. A spline approximation M/, with grid width n of a function u is usually written in the form

and U = {uk\keK is the vector of coefficients. The terms vector and matrix are used in a broader sense. Their subscripts need not be integers. For example, for the Ritz-Galerkin matrices

the indices are often integer vectors from a subset / of TLm. With this more general interpretation it is convenient not to distinguish between row and column vectors; i.e., a scalar product is written as U V without transposing one of the vectors. We say that a function is smooth if it possesses the regularity required by the current context. Using this convention, we do not have to keep track of the minimal number of required derivatives, which sometimes distracts from the essential arguments. Slightly more restrictive than standard terminology, the term domain is used for a bounded open set D c W with Lipschitz boundary (cf. A.6 for details). Partial derivatives are denoted by

6

Chapter 1.

Introduction

For example, is the gradient of an w-variate function M, and A = Y^™=\ ^v l& me Laplace operator. The 2-norm for vectors and matrices is denoted by || • ||. Finally, ||u||i is the norm of a function u in the Sobolev space Hi(D}, corresponding to the scalar product {•, -}t (cf. section 2.3).

Chapter 2

Basic Finite Element Concepts This chapter introduces fundamental finite element principles. The basic theory is described in a very general context, which is independent of the specifics of the differential equations, boundary conditions, and approximating subspaces. Despite this conceptual simplicity, the methods apply to a broad range of applications. In section 2.1 we explain the classical Ritz-Galerkin scheme for Poisson's equation, which serves as a typical model problem. Section 2.2 gives examples of basis functions, defined on triangular or quadrilateral meshes. These standard finite elements are discussed only very briefly, merely to point out essential differences to the weighted spline bases, introduced in chapter 4. After briefly introducing Sobolev spaces in section 2.3, we formulate in section 2.4 the finite element method in an abstract setting. In particular, we define the concept of ellipticity and prove the Lax-Milgram existence theorem for variational problems. Finally, in section 2.5 we derive two basic error estimates, Cea's inequality, and the duality principle of Aubin and Nitsche.

2.1

Model Problem

To explain the basic finite element idea, we consider Poisson's equation with homogeneous boundary conditions, for a domain D c Mm as a model problem. This boundary value problem describes a number of physical phenomema (cf., e.g., [30]). A simple two-dimensional example is shown in Figure 2.1. An elastic membrane is fixed at its boundary 3D and subjected to a vertical force with density /. If the resulting displacement u(x\ , Jt2) is small, it can be accurately modeled by Poisson's equation. Multiplying the differential equation — Aw = / by a smooth function u, which vanishes on the boundary, and integrating by parts (cf. A.9), it follows that

This weak form of Poisson's problem suggests a natural discretization. We approximate the solution u by a linear combination 7

Chapter 2. Basic Finite Element Concepts

8

Figure 2.1. Displacement (magnified) of an elastic membrane.

of basis functions fi, which satisfy the boundary condition Bj\3D = 0. Usually, the "finite elements" Bf are piecewise polynomials with small support on a mesh of the domain D, and h denotes the maximum diameter of the mesh cells (examples will be given in the next section). Replacing « by w/, in (2.2) and choosing v = B^, we obtain a linear system

for the coefficients U — {«/}. We summarize this basic finite element scheme, which was first proposed by Ritz [74] and Galerkin [38] at the beginning of the 20th century, as follows. 2.1 Ritz-Galerkin Approximation of Poisson's Problem The coefficients of a standard finite element approximation

for the boundary value problem

are determined from the linear system GU = F with

2.2. Mesh-Based Elements

9

The Ritz-Galerkin method can also be derived via a variational approach. To this end we note that a smooth solution u of Poisson's problem minimizes the energy functional

over all smooth functions which vanish on 3D. The characterization of a minimum,

where t € R. is arbitrary, again leads to (2.2). Since the right-hand side is a parabola in t, the expression in square brackets must vanish for all admissible v. We can define the finite element approximation by minimizing Q over the linear span

of the basis functions fi,. Expanding Q(^(- «/#/) yields the quadratic form

which is minimal if GU = F. There is a subtle point hidden in this rather formal argumentation. We have to choose the appropriate class of functions H to ensure that inf ueH Q(u) is attained. Of course, from a numerical point of view, it is legitimate to assume the existence of a smooth solution and to focus entirely on its approximation. However, as we will see in section 2.4, finite element methods and the analysis of variational problems are intimately related. We can establish the existence of weak solutions for very general boundary value problems and, at the same time, prove the solvability of Ritz-Galerkin systems. This will justify the somewhat heuristic arguments for the model problem.

2.2

Mesh-Based Elements

We give in this section a brief overview of some typical classical finite elements. These well-known basis functions will not be used in the remainder of the book, which is devoted to meshless spline approximations. We sketch the standard constructions primarily to compare them with different techniques. They also serve as convenient examples for the abstract variational approach discussed in this chapter. Most commonly used finite elements are defined on a mesh, i.e., a partition of the domain D into triangles, quadrilaterals, tetrahedra, hexahedra, or other polygonal cells. Triangles and tetrahedra are preferred for most applications since they can be adapted more easily to complicated boundaries. In particular, generating hexahedral meshes in three dimensions is rather difficult. Often one has to resort to mixed partitions in order to overcome the geometric difficulties. Figure 2.2 shows a triangulation of a two-dimensional domain with the hat-function, the basic piecewise linear finite element. A hat-function B{ equals 1 at an interior vertex jc,- and vanishes on all triangles T not containing jc, . Hence, the graph of #, is a pyramid

10

Chapter 2. Basic Finite Element Concepts

Figure 2.2. Hat-function on a triangulation. with star-shaped support. For this very simple basis function, the coefficients M, of an approximation

coincide with the values M /,(*/). The Ritz-Galerkin approximation 2.1 of Poisson's problem is easily computed. The linear system GU = F is assembled by adding the contributions from each triangle r of the triangulation, i.e.,

The gradients in the first integral are constant and can be determined by transforming the hat-functions to a standard reference triangle. For the entries of the right-hand side F numerical integration is used. Because of the small support of the hat-functions, the matrix G is sparse, and the Ritz-Galerkin system can be solved efficiently with iterative methods. While simple to implement, piecewise linear finite element methods are not very accurate. In general, the error is at best of order 0(/z 2 ), where h is the maximum diameter of the triangles. Moreover, if standard triangulation algorithms are used, homogeneous Dirichlet boundary conditions are fulfilled exactly essentially only for convex domains. Strictly speaking, #/ is not an admissible basis function if an edge of its support lies outside the domain D (cf. [87] for a discussion of such variational crimes). Better approximations can be obtained with polynomials of higher degree. Figure 2.3 gives a few examples of commonly used constructions. It is customary to describe these finite elements by the parameters which determine the polynomials on each triangle. Dots indicate values, circles gradients and second-order derivatives, and small marks on edges normal derivatives. For example, the data for the Argyris element consist of derivatives

2.2. Mesh-Based Elements

11

Figure 2.3. Examples of bivariate finite elements. up to second order at each vertex and normal derivatives on the midpoints of each edge. The total number of parameters per triangle is 3(1 +2 + 3) + 3 = 21, which matches the dimension of quintic bivariate polynomials. For each of the finite elements in Figure 2.3, the data shared by adjacent triangles are chosen to yield at least continuity across edges. This is necessary to permit the computation of gradients. For example, the approximations constructed with the quartic Lagrange element on the left are univariate quartic polynomials along each edge, which are uniquely determined by the five interpolated values. Hence, polynomials on adjacent triangles match, so that the basis functions are continuous (fi, 6 C°). Continuity of the gradient is more difficult to achieve. Either one has to resort to high polynomial degree or auxiliary subdivision, as is illustrated by the two examples on the right of the figure. For the Argyris triangle as well as for the Clough-Tocher element, the data associated with an edge determine not only the polynomial but also its normal derivative along the edge uniquely. The corresponding basis functions have continuous first-order partial derivatives; i.e., they belong to C 1 . As for hat-functions, the data for finite element basis functions fi, have exactly one nonzero entry. However, in general there are different types, depending on whether /?, is associated with an interpolated value or a derivative. For example, the Clough-Tocher finite element subspace is spanned by three basis functions B" for each vertex jc/ and one basis function B^ for each edge e^ • The functions B" have support on the macro triangles sharing the vertex X(. For a = (0, 0) they equal 1 at *,• and grad B?(x{) = (0, 0); for a equal to (1, 0) or (0, 1) they vanish at Jt, with grad #,(•*/) = oc. The function Bk has support on the macro triangles sharing the edge e^ and its normal derivative equals 1 at the midpoint of this edge. There exists an extensive theory of mesh-based elements (cf., e.g., [24]). The examples in Figure 2.3 are not even representative of the broad spectrum of possibilities. However, they illustrate the following typical properties of basis functions on unstructured meshes.

12

Chapter 2. Basic Finite Element Concepts

12 Properties of Finite Elements The basis functions fi, of standard mesh-based finite element subspaces axe piecewise polynomials of degree < n with support on few neighboring mesh cells. They are at least continuous and compatible with homogeneous boundary conditions. The first property implies that smooth functions can be approximated with order O(h"+l) under certain mild additional assumptions. The second condition is necessary for the standard Ritz-Galerkin scheme. We will not go into details since a comprehensive discussion will be given for the weighted spline-based methods described in this book. However, we note a few disadvantages of mesh-based finite elements, apparent from the above examples. Generating good meshes can be very difficult, particularly in three dimensions (cf., e.g., [50, 67]). The meshing algorithms often require the major portion of the computing time in finite element simulations. While the assembly of Ritz-Galerkin matrices is straightforward, using quadratic or higher degree polynomials leads to excessively large systems. For example, the dimension of cubic polynomials is (^) = 20 in three dimensions. Therefore, only moderately accurate approximations are possible. Finally, the boundary conditions are merely approximated for domains with curved boundaries. To maintain the overall approximation order, curved isoparametric elements have to be used. Weighted spline-based finite elements, introduced in chapter 4, overcome the above difficulties. No mesh generation is required, accurate smooth approximations are possible with relatively low-dimensional finite element subspaces, and boundary conditions are satisfied exactly.

2.3

Sobolev Spaces

For analyzing partial differential equations as well as numerical schemes, the choice of the appropriate function spaces is important. The classical theory of elliptic boundary value problems uses functions with Holder-continuous derivatives (cf., e.g., [59]), requiring in particular that all terms appearing in the differential equations are continuous up to the boundary. However, this approach is limited to smooth problems and not well suited for finite element approximations. The natural framework for variational techniques becomes apparent when considering Poisson's problem

on the unit square D — (0, I) 2 . For this special domain the solution is readily obtained by expanding the right-hand side into a Fourier series:

Since the sine functions —m, u is integrable. Moreover, it is not difficult to see that is an integrable weak derivative of u if p > I — m despite the singularity at the origin. Using polar coordinates, we have

Therefore, u e //'(D) if p > 1 — m/2. Since for m > 2 a negative exponent p is possible, we see that functions with square integrable derivatives need not be bounded. Weak differentiability does not imply continuity. As a second example we consider on D — (— 1, I)"7 the piecewise constant function

which is discontinuous across the hyperplane S = {x e W" : x\ = 0}. We claim that u does not have a square integrable weak derivative v — d\u, i.e., that the identity

cannot hold for arbitrary smooth functions (p with compact support in D, if we insist that v € Li(D). To see this, we choose

with V(0) > 0. Then, the right-hand side of (2.5) equals

2.3. Sobolev Spaces

15

while the left-hand side tends to zero:

by the Cauchy-Schwarz inequality (cf. A. 2) and a scaling of jci . The reason for this contradiction is the assumption v e LI. If we relax the requirement of integrability, we can give a meaningful interpretation to v = d\ u as a measure IJL supported on the lower dimensional set S:

This is a simple example of so-called generalized derivatives, a basic concept in the theory of distributions (cf., e.g., [75]). Sobolev spaces can also be defined as the closure of smooth functions with respect to the norm || • ||^. This confirms that Hl is a Hilbert space. Moreover, many identities (such as integration by parts) can be conveniently derived by a limit process. While the spaces Hl are adequate for basic finite element analysis, it should be mentioned that Sobolev spaces are defined more generally for p-integrable functions (cf. A. 8). The interplay between integrability, differentiability, and dimension gives rise to a very beautiful theory, which has become indispensable for studying partial differential equations. Famous results are Sobolev's embedding theorem A. 11 and the construction of extension operators by Calderon and Stein (cf. A. 13). For working with Sobolev spaces the regularity of the domain D is important. The commonly used hypothesis is that the boundary is Lipschitz-continuous (cf. A.6), which we assume in what follows. This minimal requirement is satisfied by almost all applications of practical relevance. The domains of interest are usually more regular. They have piecewise smooth boundaries, and cusp-like singularities do not occur. For approximating elliptic problems it is convenient to use Sobolev spaces which incorporate the boundary conditions. After reduction to homogeneous form they can be imposed as linear constraints. 2.4 Sobolev Spaces with Boundary Conditions The subspace H^(D) C He(D) consists of all functions which vanish on 3D. More precisely, HQ (D) is the closure of all smooth functions with compact support in D with respect to the norm || • ||£. There is a subtle point about boundary conditions for functions in Sobolev spaces. Since integrable functions can be modified on sets of measure zero (such as dD), offhand, boundary values are not well defined. However, we can use a limit process, as in the above definition. In this way, the restriction operator

which is well defined for smooth functions, is generalized by continuous extension to Hl (D) (cf.A.10).

16

2.4

Chapter 2. Basic Finite Element Concepts

Abstract Variational Problems

The finite element method can be formulated in a very general framework. Generalizing the model problem (2.1), we consider an abstract boundary value problem

with a differential operator £ and an operator B describing the boundary conditions. Moreover, analogously to (2.2), we assume that this problem admits a variational formulation

where a is a bilinear form and A. is a linear functional on a Hilbert space H (cf. A.3, A.4). Then, for a finite element subspace B/, C //, the Ritz-Galerkin approximation Uh is defined by The coefficients w, of «/, with respect to a basis /?/ of B/, are determined via the linear system obtained by using u/, = Bk as test functions. 2.5 Ritz-Galerkin Approximation Hie Ritz-Galerkin approximation «/, = ]jT}r «,• BI € B* c H of the variational problem

is determined by the linear system

which we abbreviate as Gt/ = F. For the model problem (2.1), £ is the negative Laplace operator — A, $M = M, and

The Hilbert space H is the Sobolev space HQ (D) of functions with square integrable first derivatives and zero boundary data, defined in 2.4. A simple finite element subspace B/, consists of piecewise linear functions on a triangulation of D, as described in section 2.2. To analyze the abstract variational formulation of finite element approximations, the following property of the bilinear form a is crucial. 2.6 Ellipticity A bilinear form a on a Hilbert space H is elliptic if it is bounded and equivalent to the norm on H, i.e., if for all u, v € H

with positive constants Cb and ce.

2.4. AbstractVariational Problems

17

As we will see in chapter 6, many physical problems have natural associated elliptic bilinear forms a. Usually, the boundedness of these bilinear forms can be easily established. However, the norm equivalence is in general more subtle. For the Poisson bilinear form, the Cauchy-Schwarz inequality implies

where

is the norm on H = HQ (D) c H] (D). Hence, the constant Q, in the upper bound equals 1 in this case. The lower bound is a consequence of the Poincare-Friedrichs inequality A. 12:

Adding fD ||grad u ||2 — a(u, M) to both sides, we see that ce = (const(D) + I)" 1 . We now formulate the main existence result for elliptic variational problems. 2.7 Lax-Milgram Theorem If a is an elliptic bilinear form and A is a bounded linear functional on a Hilbert space H, then the variational problem

has a unique solution u € V for any closed subspace V of H. Moreover, if a is syaaaetric, the solution u can be characterized as the minimum of the quadratic form

onV. For finite element approximations, the subspace V = B/, has finite dimension, and the variational problem is equivalent to the Ritz-Galerkin system GU = F, defined in 2.5. In this case, the ellipticity of a implies that the matrix G is positive definite:

for Uh 7^ 0. Hence, G is invertible and the Ritz-Galerkin system uniquely solvable. This simple observation already settles the part of the Lax-Milgram theorem relevant for numerical schemes. However, the infinite-dimensional case is also of fundamental importance. In particular, choosing V = H yields the existence of weak solutions. Therefore, we include a proof, although it does not quite fall into our context.

18

Chapter 2. Basic Finite Element Concepts

With the appropriate tools from functional analysis the arguments are not difficult. We interpret the variational problem as an identity between bounded linear functionals on the Hilbert space V and denote the functional on the left-hand side by

Its boundedness follows from the boundedness of a:

By the Riesz representation theorem A.5, any bounded linear functional Q on the Hilbert space V can be written as where {•, •) is the scalar product on V (identical to the scalar product on //) and 7£ is an isometry onto V. Hence, the variational problem becomes

We rewrite this identity as the fixed point equation

and show that the linear operator S is a contraction for small positive &) (the infinitedimensional analogue of Richardson's iteration for symmetric positive definite matrices). As a consequence, the unique solvability, asserted by the Lax-Milgram theorem 2.7, follows from Banach's fixed point theorem. The norm

can be estimated using the ellipticity of a. By definition of ~R, and A, the scalar product on the right-hand side equals

which is less than 1 for CD = The characterization of a solution as the minimum of a quadratic form in the symmetric case can be verified as in section 2.1. Alternatively, we can use the fact that a(u, u) is a scalar product which induces an equivalent norm || • ||a on H:

Then, by the Riesz representation theorem A.5, the functional A can be identified with an element RX of H, i.e., Hence, we can rewrite the quadratic form as

2.5. Approximation Error

19

This shows that minimizing Q is equivalent to finding the best approximation to RX from V. We recognize the variational problem

as the well-known characterization of the solution to this basic approximation problem (cf. A.4). The simplicity of the symmetric case is deceptive. According to the above argument we have u = Rh. if V = H, an almost trivial existence proof. However, we should not forget that this approach just yields a weak solution. In the model problem, u e //0'; i.e., merely the first derivatives are square integrable. Some hard analysis is required to improve the regularity and justify a pointwise interpretation of the differential equation.

2.5

Approximation Error

The error of finite element approximations w/, can be related in a natural way to the error of the best approximation from the subspaces B/,. This allows a priori estimates without any detailed knowledge about the differential equation or the Ritz-Galerkin solution. The crucial identity is the orthogonality relation

which follows by subtracting the Ritz-Galerkin equations (2.7) from the characterization (2.6) of weak solutions, setting v = Vh — Wh- If the elliptic bilinear form a is symmetric, it represents a scalar product on H, and (2.8) implies that Uh is the best approximation to u with respect to the norm Figure 2.4 illustrates the elementary geometric argument. Since ce \\u \\2 < a(u, u) < Q, \\u ||2, the error of the best approximation in the standard norm of H is at most by a constant factor larger. This important a priori bound for RitzGalerkin approximations is also valid for nonsymmetric bilinear forms.

Figure 2.4. Best approximation Uh e B/, to u in the scalar product norm

20

Chapter 2. Basic Finite Element Concepts

2.8 Cea's Inequality The error of the Ritz-Galerkin approximation UH & u for an elliptic bilinear form a satisfies where Q, and ce are the constants in definition 2.6. The proof follows from the identity

which is a consequence of the orthogonality relation (2.8) with Wf, — UH — u/,. By the ellipticity condition 2.6, the left-hand side is > ce\\u — uh\\2, and the right-hand side is < cb\\u — uh\\\\u — Vh\\. Cancelling the common factor \\u — w/J on both sides of the resulting inequality yields the desired estimate since u/, e K/, is arbitrary. For piecewise linear Ritz-Galerkin approximations (cf. Figure 2.2) of the model problem (2.1), Cea's inequality implies

where h is the mesh width of the triangulation. In this case, || • || = || • || \ is the norm of the underlying Hilbert space H = HQ (D), and we have used without proof that

for the best piecewise linear approximation VH to u [24]. This estimate requires some mild geometric assumptions. It is sufficient that the triangulations conform to the boundary and are quasi-uniform, i.e., that the quotient of the longest and shortest edge is uniformly bounded as h —> 0. If the boundary of the polygonal domain D is convex, elliptic regularity [59] yields

and we obtain an error estimate entirely in terms of the data of the boundary value problem:

with eh = u — Uh and c\ = (cb/ce)ccr. The natural norm || • || , associated with the boundary value problem, usually involves derivatives. To obtain estimates in weaker norms, such as the L2-norm || • ||Q for the model problem, the following result is useful.

2.5. Approximation Error

21

2.9 Aubin-Nitsche Duality Principle If H is a subspace of a Hilbert space H*, the error e^ = u — Uh of the Ritz-Galerkin approximation satisfies

where w* is the solution of the dual problem

and {•,•}* denotes the scalar product on //*. As for Cea's inequality, the proof is based on the orthogonality condition (2.8). We have for any vh e B/,, and the result follows from the boundedness of a. For the model problem (2. 1) we can deduce from theorems 2.8 and 2.9 that

for piecewise linear finite elements on quasi-uniform triangulations of a convex polygonal domain. In this case, H — //0' (D), H* = L 2 (D), || • H* = || • ||0, and the two factors in the Aubin-Nitsche estimate can be bounded by

respectively. Since a(v, H*) = fD grad w* grad u, the dual problem is the weak form of

Hence, elliptic regularity implies ||w*||2 < cr ||e/i||o, proving (2.9) with CQ = Q,CI (ccr). The error estimates in this section are completely independent of the particular type of finite elements. We will apply them in chapter 6 to spline approximations. In this case,

where u/, is the best spline approximation to u of degree < n and with grid width h. For second-order problems, Cea's inequality implies that for the //' -norm (t = 1) this optimal approximation order is retained. This is also the case for the L2-norm (t = 0) by the Aubin-Nitsche duality principle. However, here we must assume that

i.e., that the dual problem has optimal regularity.

This page intentionally left blank

Chapter 3

B-Splines

In this chapter we define B-splines and describe identities and algorithms, particularly useful for implementing finite element schemes. We consider only uniform knots since the essential advantages of arbitrary knot sequences do not persist in several variables. For local refinement we have to use hierarchical bases rather than knot insertion strategies, which would have a global effect. Hence, uniform B-splines are adequate for multivariate approximation and of course much simpler from a computational point of view. Moreover, the theory, originating from Schoenberg's classical work on cardinal splines, is particularly elegant. After a brief introduction to the concept of splines in section 3.1, we define in section 3.2 uniform B-splines and derive their basic properties. In section 3.3 we prove the fundamental recurrence relation, which allows one to evaluate B-splines efficiently and to compute their polynomial segments. The recursion is used in section 3.4 to obtain an explicit representation of polynomials as linear combinations of B-splines. This result is very important for the stabilization of finite element bases and for deriving error estimates. Finally, in sections 3.5 and 3.6 we discuss algorithms for grid refinement and computation of scalar products.

3.1

The Concept of Splines

Polynomials provide good local approximations for smooth functions. However, on large intervals, the accuracy can be low. Moreover, local changes have global influence. Therefore, it is quite natural to use piecewise polynomials, defined on a partition of the parameter interval D. If the break points are uniformly spaced, and the polynomial segments join smoothly, this leads to Schoenberg's classical definition [79]. 3,1 Splines A spline of degcee < « and grid width h is (n — l)-tiiaes continuously differeatiable and agrees with a polynomial of degree < n on each grid interval 0\ i + l}h of the parameter interval D. 23

24

Chapter3. B-Splines

Figure 3.1. Derivatives of a cubic spline p with grid width h = 2 and parameter interval D = [0, 10]. Figure 3.1 shows a cubic spline p with five polynomial segments, which, by definition, is twice continuously differentiable. The discontinuities of the third derivative at the break points (marked with circles) are hardly visible. Also the derivative of p, which is a quadratic spline, appears to be smooth. Only by knowing that the segments of p' are parabolas, we notice the abrupt changes in the curvature. Definition 3.1 is not well suited for computations since it does not reveal the free parameters. Moreover, offhand it is not clear whether the smoothness constraints at the break points interfere with the approximation order of the polynomial segments. The key to the theory and numerical treatment of splines is the construction of a local basis. The appropriate basis functions, the so-called B-splines, will be defined in the next section. While in hindsight quite simple, the construction is by no means obvious. Therefore, we consider, as a first illustration, the elementary piecewise linear case. As is apparent from the example in Figure 3.2, a linear spline p is uniquely determined by its values at the break points ji, = i h. Hence, it can be represented as a linear combination of hat-functions bt:

The basis functions bj equal 1 at jt/+i and vanish at all other break points. The range of summation depends on the parameter interval D under consideration. As in the example, all hat-functions with some support in D have to be included. The hat-function basis consists of scaled translates of a single function:

where the standard hat-function &' has support [0, 2] and grid width 1. It is remarkable that

3.2. Definition and Basic Properties

25

Figure 3.2. Linear spline p, represented as a linear combination of hat-functions bj.

this simple structure of the spline space persists in general. For arbitrary degree, splines with uniform break points can be represented in terms of a single B-spline (cf. definition 3.6). Of course, this is a considerable advantage for the implementation of algorithms and also permits a particularly elegant theory. It should be noted that splines can be defined also for nonuniform partitions and with more general smoothness constraints. The corresponding theory has reached a state of perfection [16, 80]. It provides very flexible approximation methods, in particular when adaptive refinement is necessary. However, most nonuniform techniques are limited to one variable. Therefore, for the construction of finite elements the simpler uniform spline spaces are adequate.

3.2

Definition and Basic Properties

Uniform B-splines can be defined in several ways. We prefer the following simple averaging process, which makes all basic properties of B-splines immediately apparent. 3.2 B-Splines The uniform B-spline bn of degree n is defined by the recursion

starting from the characteristic function b° of the unit interval [0, 1). Equivalently,

with bn (0)^0. The first few B-splines are shown in Figure 3.3. One average of the characteristic function yields the piecewise linear hat-function

Chapter3. B-Splines

26

Figure 3.3. B-splines of degree n = 1,2,3.

already defined in the previous section. With the next average we obtain the quadratic B-spline b2. For example, for x e [1,2],

The shaded area under the graph of b2 represents an average f*_ yielding a value of the cubic B-spline b3. As is illustrated with the examples in the figure, each average increases the length of the support, the smoothness, and the degree by 1. We summarize these basic properties as follows. • Positivity and local support: b" is positive on (0, n + 1) and vanishes outside this interval. • Smoothness: b" is (n — l)-times continuously differentiable with discontinuities of the nth derivative at the break points 0 , . . . , n + 1. • Piecewise polynomial structure: b" is a polynomial of degree n on each interval Finally, we note two qualitative properties. 33 Symmetry and Monotonicity The B-spline of degree « is symmetric, i.e.,

and strictly monotone on [0, (n + l)/2] and [(n [(n + 1)/2, n + 1].

These properties are easily proved by induction on the degree. Assuming that both assertions are valid up to degree n — 1, it follows from definition 3.2 that

3.3. Recurrence Relation

27

which establishes the symmetry. To show the monotonicity on the left interval, we note that the derivative of bn , is positive for x e (0, n/2] by induction. This is also true for n/2 < x < (n + l)/2 since in this case bn~l (x) = bn~l(n - x) > bn-]((n - l)/2) > bn~[(x - 1).

3.3

Recurrence Relation

While definition 3.2 allows us to derive the main properties of B-splines in a straightforward manner, it is not well suited for computations. A simple algorithm for evaluating B-splines is provided by the following recursion, proved by de Boor [14] and Cox [31] more generally for arbitrary break points.

3.4 Recurrence Relation The B-spline bn is a weighted combination of B-splines of degree n — I:

This recurrence relation is proved by induction, showing the equivalence to the formula for the derivative in definition 3.2. Since both sides of the identity vanish at x = 0, it is sufficient to check that the derivatives match, i.e., that

where we used the abbreviation bnk = bn(x —k). The term in curly braces can be written in the form

Assuming by induction that the recursion is valid up to degree n — 1 , the terms in brackets are equal to bn~l (x) and bn~l (x — 1), respectively. Hence,

and both sides of (3.2) agree. The recurrence relation 3.4 is illustrated in Figure 3.4. We start with hat-functions rather than with the characteristic functions b°(- — k} to avoid assigning values at break points with discontinuities. Depending on the argument x, some branches are not needed. In the example, x e [1, 2], so that the rightmost hat-function gives no contribution.

Chapters. B-Splines

28

Figure 3.4. Recurrence relation for a cubic B- spline. With the aid of the recurrence relation we can also compute the polynomial segments of the B-splines. This has to be done separately for each grid interval, and it is convenient to use the Taylor expansion with respect to the left break point. 3.5 Taylor Coefficients

The n + I polynomial segments

of the B-spline V1 can be computed with the recursion 1

starting with a^o = 1an(i setting a% £ = 0 if either ^ or € are ^ {0,..., n}. For the proof we just have to expand the terms in the recurrence relation 3.4. Restricting x = k + t to the interval k + [0, 1] yields

Adding the expressions on the right-hand side and collecting the coefficients of ti, we obtain the summands in the recursion for akn £ . The Taylor coefficients for the first few B-splines are listed in Table 3. 1 . For example, the polynomial segments of the quadratic B-spline (n = 2) on the intervals [0, 1], [1,2], and [2, 3] are

29

3.4. Representation of Polynomials

n

2

1

£ h is the sum of these expressions over all unit vectors a. As a further application, we compute the integrals

which appear in the Ritz-Galerkin approximation of Poisson's equation. Because of the product form of multivariate B-splines, the contributions from each partial derivative factor, i.e.,

Chapter 4. Finite Element Bases

40

1 360

7 180

7 180

13 90

1 ~12 7 180 1

360

1 30 13 90

7 180

1 12 1 30 11

To 1 30 1 12

7 180

1 360

13 90

7 180

1 30

1 ~12

13 90

7 180

7 180

1 360

Figure 4.2. Ritz-Galerkin integrals g0^, \tv\ < 2, for biquadratic B-splines. Hence,

where 5" and dn are the scalar products defined in theorem 3.12 of the previous section. Figure 4.2 shows the integrals for biquadratic B-splines. Since gk,i depends only on the difference k — t, we have chosen k — 0 and associated each value with the center of the support ofblh. The supports of b\ h and b2(2^h are highlighted as well as their intersection, which is the region where the scalar product of the gradients is nonzero. For this pair of B-splines, using the values in Table 3.2.

4.2

Splines on Bounded Domains

As a first step toward the construction of spline-based finite element subspaces, we recall the standard definition of splines as linear combinations of B-splines.

43 Splines

The splines Bj> (D) on a bounded domain D c Rm consist of all linear combinations

of relevant B-splines; i.e., the set K of relevant indices contains all k with b% h (x) ^ 0 for some x e D.

4.2. Splines on Bounded Domains

41

Figure 4.3. Relevant biquadratic B-splines b^, k e K, spanning B. To simplify notation, we will omit parameters if they are fixed throughout the context. For example, we write B = BJj(D) and bk = bnkh. Definition 4.3 of the spline space B is illustrated in Figure 4.3 for quadratic splines on a two-dimensional domain D (cf. also Figure 1.2). The relevant B-splines bk, k e K, are marked with circles at the lower left corner of their support kh + [0, 3]2h. In contrast to univariate splines, B may contain B-splines with very small support in D; one example is highlighted in the figure. Moreover, depending on the shape of the domain, the set of relevant indices K can be rather irregular. To simplify computations, it is therefore convenient to work with a rectangular array of indices containing K. For analytical arguments we may even sum over k e Z7"; i.e., we interpret p e B as the restriction of an m-variate cardinal spline to D,

This is particularly useful for manipulating sums. Restricting the argument x of the B-splines is simpler than specifying the minimal summation range. To obtain accurate approximations, it is crucial that B contains polynomials. The proof is not difficult and based on Marsden's identity 3.7. We restate this univariate formula in the less precise form

where g/, is a polynomial of total degree n in t and k. Forming a product yields the

42

Chapter 4. Finite Element Bases

multivariate identity

where