THE BOUNDARY ELEMENT METHOD SOLID MECHANICS AND ITS APPLICATIONS Volume 27 Series Editor: G.M.L. GLADWELL Solid Mechanics Division, Faculty ofE ngineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative research ers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the fIrst year graduate student. Some texts are monographs defining the current state of the fIeld; others are accessible to fInal year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages. The Boundary Element Method by W.S.HALL University ofTeesside, School of Computing and Mathematics, Middlesborough, Cleveland, U.K. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for thls book is available from the Library of Congress. ISBN 978-94-010-4336-6 ISBN 978-94-011-0784-6 (eBook) DOI 10.1007/978-94-011-0784-6 Printed on acid-free paper AU Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint ofthe hardcover Ist edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permis sion from the copyright owner. Contents Preface ............................................................................................................................. ix Chapter 1 Ordinary Integral Equations ............................................................... 1 1.1 Introduction ...................................................................................................... 1 1.2 Ordinary Integral Equations and their Applications ........................................ 1 Applications ................................................................................................. 1 Classification 0/ Integral Equations ........................................................... 8 1.3 Equivalence between Ordinary Integral and Ordinary Differential Equations ...................................................................... 9 First Order Equations ................................................................................. 9 Second Order Equations. Initial Value Problems ..................................... 10 Second Order Equations. Boundary Value Problems ............................... 14 1.4 Analytical Methods of Solution ..................................................................... 16 Fredholm Equations With Separable Kernels ........................................... 16 Iterative Methods For Second Kind Equations ......................................... 21 1.5 Numerical Methods of Solution ..................................................................... 24 Multistep Method ....................................................................................... 24 Constant Function Numerical Treatment .................................................. 28 1.6 Concluding Remarks ...................................................................................... 33 Exercises ........................................................................................................ 34 Chapter 2 Two Dimensional Potential Problems ................................................ 39 2.1 Introduction .................................................................................................... 39 2.2 Applications of Potential Formulations ......................................................... 39 Heat Conduction ....................................................................................... 40 Fluid Flow ................................................................................................. 41 2.3 Boundary Integral Equation Derivation for Interior Problems ..................... .41 Derivation/rom Green's Identity .............................................................. 42 Extension to the boundary ......................................................................... 45 2.4 Boundary Integral Equation Derivation for Exterior Problems ..................... 48 Extension to the boundary ......................................................................... 51 2.5 Treatment of Boundary Conditions ............................................................... 53 Potential boundary conditions .................................................................. 54 Flux boundary conditions .......................................................................... 55 Mixed boundary conditions ....................................................................... 56 2.6 Concluding Remarks ...................................................................................... 58 Exercises ........................................................................................................ 59 v vi Contents Chapter 3 Boundary Element Method ................................................................ 61 3.1 Introduction .................................................................................................... 61 3.2 Numerical Foundation ................................................................................... 61 3.3 Linear Approximation .................................................................................... 62 3.4 Integration on a Curve ................................................................................... 64 3.5 Constant Function Solution for Exterior Heat Conduction ............................ 69 Heat flow from a deeply buried pipe ......................................................... 69 Discretisation into elements ...................................................................... 70 Collocation ................................................................................................ 72 3.6 Evaluation of Logarithmic Integral Coefficients ........................................... 73 Case (a) Singular Element ........................................................................ 74 Case (b) Non-singular element ................................................................. 78 3.7 Concluding Remarks ...................................................................................... 80 Exercises ........................................................................................................ 81 Chapter 4 Linear Isoparametric Solution ........................................................... 85 4.1 Introduction .................................................................................................... 85 4.2 Linear Function Approximation for Exterior Heat Conduction .................... 85 4.3 Assembly of Left Hand Side Coefficients ..................................................... 89 4.4 Singular and Nonsingular Elements ............................................................... 93 4.5 Evaluation of Right Hand Side Terms ........................................................... 97 4.6 Exterior Neumann Problem for Velocity Potential ........................................ 99 Illustration of Non-singular Integration ................................................. 101 4.7 Singularity Elimination for the Derivative Kernel... .................................... 107 4.8 Interior Mixed Boundary Value Problem .................................................... 108 4.9 Concluding Remarks .................................................................................... 117 Exercises ...................................................................................................... 118 Chapter 5 Quadratic Isoparametric Solution ................................................... 121 5.1 Introduction .................................................................................................. 121 5.2 Interior Mixed Boundary Value Problems ................................................... 121 5.3 Treatment of Singular Integrals ................................................................... 126 Row sum elimination ............................................................................... 126 Exact integration ..................................................................................... 127 Weighted Gaussian Integration ............................................................... 12 7 5.4 Subtraction and Series Expansion Method for Singular Integration ........... 129 Expansion of the shapefunction .............................................................. 131 Expansion of the logarithm ..................................................................... 132 Expansion of the Jacobian ...................................................................... 134 Expansion of the complete integrand ...................................................... 135 Treatment of the remainder integrals ...................................................... 137 5.5 Concluding Remarks .................................................................................... 13 7 Exercises ...................................................................................................... 138 Chapter 6 Three Dimensional Potential Problems ........................................... 141 6.1 Introduction .................................................................................................. 141 6.2 Boundary Integral Equation Formulation .................................................... 142 6.3 Electrostatics Application ............................................................................ 144 6.4 Shape functions and boundary elements ...................................................... 146 6.5 The Boundary Element Method ................................................................... 151 6.6 Surface Jacobian .......................................................................................... 152 Contents vii 6.7 Assembly of Coefficients ............................................................................. 154 6.8 Generation of a System of Equations ........................................................... 157 6.9 Summary of the Three Dimensional Boundary Element Method ............... 157 6.10 Concluding Remarks .................................................................................... 158 Exercises ...................................................................................................... 158 Chapter 7 Numerical Integration for Three Dimensional Problems ............. 161 7.1 Introduction .................................................................................................. 161 7.2 Integration in the Local Coordinate Plane ................................................... 161 7.3 Singular Integration ..................................................................................... 166 Integration by Regularization ................................................................. 166 Subtraction and Series Expansion ........................................................... 170 7.4 Concluding Remarks .................................................................................... 174 Exercises ................................................................................................................... 174 Chapter 8 Two-Dimensional Elastostatics ........................................................ 177 8.1 Introduction .................................................................................................. 177 8.2 Review of Linear Elasticity ......................................................................... 177 Equilibriwn Equation .............................................................................. 179 Plane Stress ............................................................................................. 180 Traction vector ........................................................................................ 182 Deformations and Strains ........................................................................ 183 Generalised Hooke's Law ....................................................................... 184 Kelvin's Solution ..................................................................................... 187 8.3 Derivation of the Boundary Integral Equation ............................................. 190 Betti's theorem and Somigliana's identity .............................................. 190 Displacement and Stress at an Internal Point .........................................1 94 8.4 Boundary Element Solution ......................................................................... 195 Five Element Illustration ......................................................................... 201 Singular integration using rigid body displacement solution ................. 204 8.5 Concluding remarks ..................................................................................... 205 Exercises ...................................................................................................... 206 Appendix A Integration and Differentiation Formulae ....................................... 208 Appendix B Matrix Partitioning for the Mixed Boundary Value Problem .•••.•• 210 Appendix C Answers to Selected Exercises. ......................................................... 213 Bibliography .................................................................................................................. 219 In.dex .............................................................................................................................. 221 Preface The Boundary Element Method is a simple, efficient and cost effective computational technique which provides numerical solutions - for objects of any shape - for a wide range of scientific and engineering problems. In dealing with the development of the mathematics of the Boundary Element Method the aim has been at every stage, only to present new material when sufficient experience and practice of simpler material has been gained. Since the usual background of many readers will be of differential equations, the connection of differential equations with integral equations is explained in Chapter 1, together with analytical and numerical methods of solution. This information on integral equations provides a base for the work of subsequent chapters. The mathematical formulation of boundary integral equations for potential problems - derived from the more familiar Laplace partial differential equation which governs many important physical problems - is set out in Chapter 2. It should be noted here that this initial formulation of the boundary integral equations reduces the dimensionality of the problem. In the key Chapter 3, the essentials of the Boundary Element Method are presented. This first presentation of the Boundary Element Method is in its simplest and most approachable form - two dimensional, with the shape of the boundary approximated by straight lines and the functions approximated by constants over each of the straight lines. The following chapters develop the method by improving the levels of approximation and by dealing with the resulting problems of, for example, the accurate integration of singular kernels. Thus Chapter 4 brings the function approximation to the same linear level as the boundary approximation. In Chapter 5 both approximations are quadratic. By the time Chapters 6 and 7 are reached, sufficient experience will have been gained of the Boundary Element Method to deal with three dimensional problems. Chapter 6 again takes partial differential equations and converts them to boundary integral equations, applies approximations to the boundary and to the functions and produces numerical solutions for three dimensional problems. The more advanced problems of performing accurate integration arising from three dimensional problems are dealt with in Chapter 7. In all previous chapters, in order to gain experience, the application of the Boundary Element Method has been to relatively simple potential problems. Chapter 8 presents the application of the Boundary Element Method to the mainstream engineering problem of elastostatics. The Boundary Element Method serves as a standard introductory reference text of the mathematics of this method and is ideal for final year undergraduate study as well as for postgraduates, scientists and engineers new to the subject. Worked examples and exercises are provided throughout the text ix x Preface In producing the text I would like to thank all of those who, over the years, have helped to generate the material of the book and who have helped in its production. In particular Ferri Aliabadi, Wilf Blackburn, Alan Cook, Ciaran Flood, Terry Hibbs, Alan Jeffrey, Xin-qiang Mao, Peter Milner, Mike Parks, Pedro Parreira, Melvin Phemister, Andrew Pullan, Bill Robertson, David Rooke, Bill Spender, Gordon Symrell and Terry Wilkinson. Finally, I would like to dedicate the book to my family and particularly to my wife, Pauline, and daughter, Charlotte, for their support and forebearance. Professor W. S. Hall School of Computing and Mathematics, The University of Teesside, Middlesbrough, Cleveland, UK. Ordinary Integral Equations 1 1.1 Introduction The Boundary Element Method is a general numerical technique which solves boundary integral equations. To understand fully the complexity of these equations it is first necessary to become familiar with simple integral equations, such as those which model one dimensional problems. This chapter introduces such simple integral equations which will be termed ordinary integral equations because of their equivalence with ordinary differential equations which is shown later in the chapter. lllustrations are given of the one-dimensional problems to which they apply. Before proceeding to show how integral equations can be analytically and numerically solved it will be shown that ordinary integral and ordinary differential equations are equivalent 1.2 Ordinary Integral Equations and their Applications In this section, the ways in which integral equations arise direcdy from applications in mathematics, mechanics, physics, engineering and control are considered. This leads to a treatment of various kinds of integral equations and their classification. Given the equivalence between an ordinary integral equation and an ordinary differential equation it will be appreciated that the multitude of problems treatable by differential equations may also be treated by integral equations. This is in addition to the problems coming from the direct formulation of integral equations from applications. Some applications which lead directly to ordinary integral equations are now considered, starting with a simple geometrical problem and proceeding to problems from mechanics, physics and control. Applications Application 1 - Fixed area under a curve (Bernoulli's Problem) A simply stated geometric problem is that of finding the shape of a curve, y(x), such that the area under it is a fixed proportion, p, of the area of the rectangle circumscribing it. The curve y(x) is shown in Figure 1.1.