Symmetric Galerkin Boundary Element Method

Alok Sutradhar · Glaucio H. Paulino ·Leonard J. Gray

Symmetric Galerkin BoundaryElement Method

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Alok SutradharOhio State UniversityDepartment of Surgery410 W 10th Ave, Columbus OHUSAalok.sutradhar@osumc.edu

Leonard J. GrayOak Ridge National LaboratoryPOB 2008, Bldg 6012 MS 6367Oak RidgeTN 37831-6367graylj1@ornl.gov

Prof. Glaucio H. PaulinoUniversity of Illinois atUrbana-ChampaignNewmark Laboratory205 North Mathews AvenueUrbana IL 61801-2352USApaulino@uiuc.edu

ISBN: 978-3-540-68770-2 e-ISBN: 978-3-540-68772-6

Library of Congress Control Number: 2008928276

c© 2008 Springer-Verlag Berlin Heidelberg

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CONTENTS

Foreword xiii

Preface xv

1 Introduction 1

1.1 Boundary Element Method 11.1.1 Approximations and Solution 21.1.2 The Green’s function G(P,Q) 41.1.3 Singular and Hypersingular Integrals 51.1.4 Numerical Solution: Collocation and Galerkin 61.1.5 Symmetric Galerkin BEM 7

1.2 An Application Example: Automotive Electrocoating 81.2.1 Engineering Optimization 91.2.2 Electrocoating Simulation 10

1.3 Visualization 111.3.1 Virtual Reality 121.3.2 CAVE: Cave Automatic Virtual Environment 141.3.3 The MechVR 14

1.4 Other Boundary Techniques 151.4.1 Singular Integration 161.4.2 Meshless and Mesh-Reduction Methods 16

1.5 A Brief History of Galerkin BEM 19

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2 Boundary Integral Equations 23

2.1 Boundary Potential Equation 232.2 Boundary Flux Equation 282.3 Elasticity 302.4 Numerical Approximation 33

2.4.1 Approximations 332.4.2 Collocation 352.4.3 Galerkin Approximation 372.4.4 Symmetric-Galerkin 39

2.5 Hypersingular Integration: an example 392.5.1 Collocation: C1 Condition 402.5.2 Galerkin: C0 42

3 Two Dimensional Analysis 45

3.1 Introduction 453.2 Singular Integrals: Linear Element 48

3.2.1 Coincident Integration 483.2.2 Coincident: Symbolic Computation 513.2.3 Adjacent Integration 533.2.4 Cancellation of log(ε2) 563.2.5 Adjacent: shape function expansion 573.2.6 Numerical Tests 58

3.3 Higher Order Interpolation 603.3.1 Integral of G 613.3.2 Integral of ∂G/∂n and ∂G/∂N 623.3.3 Integral of ∂2G/∂N∂n 63

3.4 Other Green’s functions 633.5 Corners 643.6 Nonlinear Boundary Conditions 653.7 Concluding Remarks 67

4 Three Dimensional Analysis 69

4.1 Preliminaries 694.2 Linear Element Analysis 73

4.2.1 Nonsingular Integration 744.2.2 Coincident Integration 754.2.3 Coincident CPV integral 824.2.4 Edge Adjacent Integration 834.2.5 Vertex Adjacent Integration 884.2.6 Proof of Cancellation 90

4.3 Higher Order Interpolation 92

CONTENTS ix

4.4 Hypersingular Boundary Integral: Quadratic Element 934.4.1 Coincident Integration 944.4.2 r Expansion 944.4.3 First Integration 954.4.4 Edge Integration 96

4.5 Corners 974.6 Anisotropic Elasticity 97

4.6.1 Anisotropic Elasticity Boundary Integral Formulation 994.6.2 T Kernel: Coincident Integration 1004.6.3 Spherical Coordinates 1054.6.4 Second integration 1064.6.5 Edge Adjacent Integration 106

5 Surface Gradient 109

5.1 Introduction 1095.2 Gradient Equations 112

5.2.1 Limit Evaluation in two dimensions 1145.2.2 Example: Surface Stress 1185.2.3 Limit Evaluation in three dimensions 121

5.3 Hermite Interpolation in Two Dimensions 1235.3.1 Introduction 1235.3.2 Hermite Interpolation 1245.3.3 Iterative Solution 125

6 Axisymmetry 129

6.1 Introduction 1296.2 Axisymmetric Formulation 1316.3 Singular Integration 134

6.3.1 Adjacent Integration 1356.3.2 Coincident Integration 1356.3.3 Axis singularity 1366.3.4 Log Integral Transformation 1366.3.5 Analytic integration formulas 138

6.4 Gradient Evaluation 1396.4.1 Gradient Equations 1396.4.2 Coincident Integration 140

6.5 Numerical Results 142

7 Interface and Multizone 145

7.1 Introduction 1457.2 Symmetric Galerkin Formulation 147

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7.3 Interface and Symmetry 1497.3.1 Multiple Interfaces 1517.3.2 Corners 1527.3.3 Free interface 1527.3.4 Computational Aspects 152

7.4 Numerical Examples 1537.5 Remarks 155

8 Error Estimation and Adaptivity 157

8.1 Introduction 1578.2 Boundary Integral Equations 1588.3 Galerkin Residuals and Error Estimates 1608.4 Self Adaptive Strategy 161

8.4.1 Local Error Estimation 1628.4.2 Element Refinement Criterion 1628.4.3 Global Error Estimation 1638.4.4 Solution Algorithm for Adaptive Meshing 164

8.5 Numerical Example 1648.6 BEAN Code 167

9 Fracture Mechanics 171

9.1 Introduction 1719.2 Fracture parameters: Stress intensity factors (SIFs) and T-stress 1729.3 SGBEM Formulation 173

9.3.1 Basic SGBEM formulation for 2D elasticity 1739.3.2 Fracture analysis with the SGBEM 175

9.4 On Computational Methods for Evaluating Fracture Parameters 1789.5 The Two-state Interaction Integral: M-integral 179

9.5.1 Basic Formulation 1799.5.2 Auxiliary Fields for T-stress 1809.5.3 Determination of T-stress 1829.5.4 Auxiliary Fields for SIFs 1839.5.5 Determination of SIFs 1849.5.6 Crack-tip elements 1859.5.7 Numerical implementation of the M-integral 186

9.6 Numerical Examples 1879.6.1 Infinite plate with an interior inclined crack 1879.6.2 Slanted edge crack in a finite plate 1919.6.3 Multiple interacting cracks 1929.6.4 Various fracture specimen configurations 193

CONTENTS xi

10 Nonhomogenous media 197

10.1 Introduction 19710.2 Steady State Heat Conduction 198

10.2.1 On the FGM Green’s function 19910.2.2 Symmetric Galerkin Formulation 19910.2.3 Treatment of Singular and Hypersingular Integrals 202

10.3 Evaluation of singular double integrals 20310.3.1 Coincident Integration 20410.3.2 Edge Adjacent Integration 21010.3.3 Vertex Adjacent Integration 21210.3.4 Numerical Example 214

10.4 Transient heat conduction in FGMs 21610.4.1 Basic Equations 21710.4.2 Green’s Function 21810.4.3 Laplace Transform BEM (LTBEM) Formulation 21910.4.4 Numerical Implementation of the 3D Galerkin BEM 22110.4.5 Numerical Inversion of the Laplace Transform 22210.4.6 Numerical Examples 223

10.5 Concluding Remarks 224

11 BEAN: Boundary Element ANalysis Program 227

11.1 Introduction 22711.2 Main Control Window: BEAN 228

11.2.1 Menu 22811.2.2 File 22811.2.3 Geometry 22811.2.4 Boundary Conditions (BCs) 22911.2.5 Analysis 23011.2.6 Results 230

11.3 BEANPlot 23111.3.1 Menus 23111.3.2 Curves 232

11.4 BEANContour 23211.4.1 Menus 232

11.5 General Instructions 23411.6 Troubleshooting 235

11.6.1 Error Message Meanings 23611.7 Sample Problems 237

Appendix A: Mathematical Preliminaries and Notations 241

A.1 Dirac Delta function 241

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A.2 Kronecker Delta function 242A.3 Derivative, Gradient, Divergence and Laplacian 242A.4 Divergence theorem 243A.5 Stokes theorem 243A.6 Green’s Identities 243A.7 Fourier and Laplace transform 244A.8 Free Space Green’s function 244

Appendix B: Gaussian Integration 247

B.1 Gaussian rule for logarithmic singularities 247B.2 Gaussian rule for One-dimensional non-singular integration 247

Appendix C: Maple Codes for treatment of hypersingular integral 251

C.1 Maple Script: Coincident 251C.2 Maple Script: Edge Adjacent 253C.3 Maple Script: Vertex Adjacent 254

References 257

Topic Index 275

FOREWORD

The boundary element methodology (BEM) may be regarded as one of the maindevelopments in computational mechanics, fostered by the diffusion of computers,in the second half of the 20th century. If a sort of competition is attributed toresearch streams oriented to engineering applications, it can be said that triumphwas achieved by the finite element methodology, primarily because of its versatility(which, among other effects, led to a dissemination of commercial general-purposesoftware in the community of practitioners, an effect not generated by BEM).

However, as this book evidences, there are several meaningful problems of prac-tical interest, for the numerical solutions of which BE approaches are significantlyadvantageous. As an example, additional to the variety of such situations dealtwith by the Authors, I would like to mention here the customary assessment of“added masses” for traditional dynamic analysis of a dam, by solving preliminar-ily a (linear, tridimensional) potential problem over the reservoir with results ofpractical interest confined on part of the boundary only.

Within the BE area, the “symmetric Galerkin (SG) method”, systematicallytreated in this book, does deserve more attention and further efforts, both in re-search institutions and in industrial environments. Although it implies some pecu-liar conceptual and mathematical difficulties, the SGBEM turns out to be a timely,attractive and promising subject, as the Authors, who are internationally acknowl-edged leading researchers in the field, show in this book.

It is well known that the BE methodology has remote deep roots in the his-tory of mechanics. The central integral equation was established in 1886 by theyoung descendent of Alessandro Volta, Carlo Somigliana (then 26 year old, at thebeginning of his scientific career, which ended when he was 95, after his last twenty

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years in Milan). Somigliana investigated consequences of the reciprocity theoremin elasticity due to his master Enrico Betti and, in particular, the involvement init of the fundamental solution found in 1848 by William Thomson, when he was24 (not yet Lord Kelvin, Baron of Largs), at the beginning of his professorshipin Glasgow (which lasted 53 years!). A few years later (1891) Michele Gebbia,in Palermo, established his two-point fundamental solution, more intriguing thanKelvin’s kernel due to its hyper-singularity, and later he related it to the works ofthe mathematicians Green and Fredholm (with some mild polemical hints to thelatter).

As for the SGBEM it is worth noticing first that Boris Galerkin (1871-1945)personified great, fruitful synergy between mathematics and engineering by meansof his crucial roles both in Saint Petersburg Mathematical Society and in design ofdams and hydro-and thermo- electrical power-plants (and his one-and-half years inprison as a student under the Czars witness his passion for social justice).

The concept of symmetry has been recurrent in science, from Pythagoras’ canonsof music and Vitruvius’ theory of architecture up to present particle physics, throughemphasis on it in works of great mathematicians like Abel and Galois.

The synergistic combination of the symmetric integral equation couple in elas-ticity with the solution approximations by Galerkin’s weighted residual approachgave rise to the SGBEM, now extensively expounded in this book.

The Authors discuss variants and alternative approaches within, and also out-side, the BE methodology and elucidate potentialities and limitations of the SGBEM.Among the pros, it may be mentioned here the fact that, when symmetry reflectsessential features of the physical problem, its preservation in passing from con-tinuum to discrete problem formulations sometimes permits to achieve theoreticalresults in a simpler algebraic context.

Readers are likely to particularly appreciate in the book various parts devotedto recent or current developments, desirable but unexpected in a textbook, namelye.g.: a fairly detailed survey of visualization techniques and relevant software;a whole chapter on error estimation and adaptivity; another chapter on non-homogeneous media like functionally graded materials, nowadays fashionable sub-ject of research in various technologies and especially in micro-technologies.

As a conclusion of this brief foreword, it can be stated that this is a timely book:it probably fills a niche in computationally mechanics and might promote a revivalof research on BE methods and on their fruitful engineering applications.

GIULIO MAIER

Professor Emeritus of Structural EngineeringTechnical University (Politecnico) of Milan, ItalyJanuary 2007

PREFACE

As a number of good books covering the boundary element method have appearedrelatively recently, we are perhaps obligated to justify the appearance of this vol-ume. In this preface we would therefore like to delineate how this text differs fromits predecessors and what we hope that it can add to the field.

The first distinguishing feature is that, as indicated by the title, this book isconcerned solely with a Galerkin approximation of boundary integral equations,and more specifically, symmetric Galerkin. Most books on boundary elements dealprimarily with more traditional collocation methods, so we hope that this volumewill complement existing material. The symmetric Galerkin approximation is anaccurate and versatile numerical analysis method, possessing the attractive featureof producing a symmetric coefficient matrix. Moreover, it is based upon the ability- that Galerkin provides - to handle the hypersingular (as well as the standardsingular) boundary integral equation by means of standard continuous elements(i.e. no special elements are needed). Thus, a second noteworthy aspect of thisbook is that singular and hypersingular equations are introduced together and aretreated numerically by means of a unified approach, as elaborated below.

A primary reason that this can be accomplished, and as well an important themeof this book, is that all singular integrals - weakly singular, singular (Cauchy prin-cipal value), and hypersingular - can be handled using the same basic concept andalgorithms. These algorithms are based upon a mathematically rigorous definitionof the integrals as limits to the boundary , certainly a unique feature of this book.We hope that our readers, especially students, will find this direct approach moreintuitive than (in our prejudiced view) the somewhat sleight-of-hand removal ofdivergences in the (more or less) standard principal value and Hadamard finite

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part treatments of the singular and hypersingular integrals. Moreover, as the limitbased singular integration methods are largely independent of the particular Green’sfunction, it suffices to separately discuss the analyses for two and three-dimensionalproblems. While most texts have separate chapters devoted to specific applications(potential theory, elasticity, etc.), here we take a different approach to emphasizethat all equations can be treated by fundamentally the same methods leading to avariety of applications.

The consolidation of the basic analysis procedures into two chapters also leavesus room to include some important topics not usually covered in other texts. Theseinclude surface gradient evaluation (a key advantage of limit based singular integra-tion methods), error estimation (once again based upon the ability to easily treathypersingular equations), and non-homogeneous (e.g., graded) materials. Finally,a symmetric Galerkin Matlab c© educational program called BEAN, which standsfor Boundary Element ANalysis, is available for download, and our hope is thatthis will assist in using the book as classroom material and also in learning how toprogram the symmetric Galerkin boundary element method.

The book is arranged into 11 Chapters. A brief summary of the organization isas follows.

Chapter 1 provides an introduction to the boundary element method withspecial emphasis on the symmetric Galerkin formulation. The basic aspects of theintegral equation formulation are introduced, and the advantages of this techniqueare discussed in the context of a specific industrial application, the electrodepositionof paint in automotive manufacturing. This application is integrated with advancedvisualization techniques, including virtual reality. Other integral equation methods,such as meshless and mesh-reduction techniques, are briefly discussed, and thechapter concludes with a succinct history of the Galerkin approximation.

Chapter 2 introduces boundary integral equations and their numerical approxi-mations. For the Laplace equation, the integral equations for surface potential andfor surface flux are derived, which involve the Green’s function and its first andsecond derivatives. These functions are divergent when the source and field pointscoincide, the singularity becoming progressively stronger with higher derivatives,and thus the evaluation of (highly) singular integrals is of paramount importance.As noted above, the fundamental approach adopted in this book is to define andevaluate all singular integrals as ‘limits to the boundary’. This approach unifies thenumerical analysis of the integral equations and affects almost every aspect of thebook, most especially the evaluation of gradients examined in Chapter 5.

Chapters 3 & 4 are the core of the book, presenting the numerical imple-mentation of a Galerkin boundary integral analysis in two and three dimensions,respectively. The analysis techniques presented in these Chapters represent wellthe book philosophy outlined above. The primary task is the evaluation of singularintegrals, and for the hypersingular integral it is necessary to isolate the diver-gent terms and to prove that they cancel. The methods are first described in thesimplest possible setting, a piecewise linear solution of the Laplace equation. Subse-quently, higher order curved interpolation and more complicated Green’s functionsare considered. As noted above, the limit to the boundary approach provides aconsistent scheme for defining all singular integrals, and moreover results in directsemi-analytical evaluation algorithms. Symbolic computation is utilized to simplifythe work involved in carrying out the limit process and related analytical integra-tion, and example Maple c© codes are provided.

PREFACE xvii

Chapter 5 addresses the evaluation of surface gradients. A significant advantageof the boundary limit approach is that it leads to a highly accurate and efficientscheme for computing surface derivatives: only local singular integrals need to beevaluated, i.e. a complete boundary integration is not required. The key is toexploit the limit definition, writing the gradient equation as a difference of interiorand exterior limits. In many applications, most notably moving boundary problems,the knowledge of boundary derivatives is necessary, e.g., the potential gradient forLaplace problems or the complete stress tensor in elasticity. A specific, and inmany ways typical, example of a free boundary problem is discussed, a coupledlevel set-boundary element analysis for modeling two-dimensional breaking wavesover sloping beaches. This example demonstrates the advantage of the gradienttechniques for the general class of moving front problems.

Chapter 6 considers three-dimensional axisymmetric problems. As an example,the boundary integral equation for the axisymmetric Laplace equation is solved byemploying modified Galerkin weight functions. The alternative weights smooth outthe singularity of the Green’s function at the symmetry axis, and restore symmetryto the formulation. The modified weight functions, together with a boundary limitdefinition, also result in a relatively simple algorithm for the post-processing of thesurface gradient.

Chapter 7 presents a symmetric Galerkin boundary integral method for inter-face and multizone problems. This type of problem arises, for example, in applica-tions such as composite materials (bi-material interfaces) and geophysical simula-tions (internal boundaries). In the present formulation, the physical quantities areknown to satisfy continuity conditions across the interface, but no boundary con-ditions are specified. The algorithm described herein achieves a symmetric matrixof reduced size.

Chapter 8 addresses error estimation and adaptivity from a practical view-point. The so-called a-posteriori error estimators are used to guide the associatedadaptive mesh refinement procedure. The estimators make use of the “hypersin-gular residuals”, originally developed for error estimation in a standard collocationapproximation, and later extended to the symmetric Galerkin setting. This leads tothe formulation of Galerkin residuals, which are natural to the symmetric Galerkinboundary integral approach, and forms the basis of the present error estimationscheme. The error estimation and adaptive procedure are implemented in the edu-cational, user-friendly, symmetric Galerkin Matlab c© code BEAN mentioned above,which solves problems governed by the Laplace equation.

Chapter 9 is dedicated to one of the most important and successful applicationareas for boundary integral methods: fracture analysis. Problems involving fractureand failure arise in many critical engineering areas, and boundary integral methodshave inherent advantages for these calculations. It is therefore essential that anefficient and effective symmetric Galerkin approximation be developed for this classof problems, and this chapter demonstrates that this is indeed the case.

Chapter 10 focuses on the development of symmetric Galerkin formulationsfor nonhomogeneous problems of potential theory. Specifically, a formulation andcorresponding implementation for heat conduction in three dimensional functionallygraded materials are presented. The Green’s function of the actual problem isused to develop a boundary-only formulation without any domain discretization, inwhich the thermal conductivity varies exponentially in one coordinate. A transient

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implementation using the Laplace transform Galerkin boundary element method isalso provided.

Chapter 11 is dedicated to the educational computer code Boundary ElementANalysis, including its graphical user interface. BEAN is a user-friendly adaptivesymmetric Galerkin BEM code to solve the two-dimensional Laplace Equation. TheChapter outlines the specific procedures to set up the problems, and the steps toutilize BEAN’s post-processing capabilities. The book website contains additionalrelated material, as discussed below. In an effort to make the book as self-containedas possible, three appendices are provided, covering mathematical preliminaries,some Gaussian integration tables, and the symbolic Maple c© codes. As discussedabove, the Maple c© codes are an integral part of the book and should be used inconjunction with Chapters 2-4.

A web site at http://www.ghpaulino.com/SGBEM_book associated with thisbook will be maintained, primarily as a means of allowing access to the codesemployed in the text. It contains the BEAN code, including the graphical user in-terface. The complete source-code is provided, together with a library of practicalexamples, and a video tutorial which demonstrates step-by-step how to use thesoftware. The source code can be used for instructional purposes, as well as thebuilding block for new applications. In addition, the symbolic Maple c© codes, forsingular and hypersingular integrations, are also provided. These codes supplementthe explanations of Chapters 2 through 4. We hope that readers will also use theweb site to report misprints, errors, and other comments/suggestions about thetext.

Finally, this book would not have been possible without the patience of our wivesand children. We also owe a tremendous debt to the many colleagues and students,most of whom are now good friends, with whom we have collaborated over the pasttwenty years. They are too numerous to mention, but they know who they are,and we thank them for many enjoyable hours of struggling to get on the right path,hunting down insidious bugs, and on occasion, finding gold.

We hope that you enjoy the book.

ALOK SUTRADHAR, G. H. PAULINO, L. J. GRAY

February, 2008