THE BOUNDARY ELEMENT METHOD

Transcription

1 THE BOUNDARY ELEMENT METHOD

2 SOLID MECHANICS AND ITS APPLICATIONS Volume 27 Series Editor: G.M.L. GLADWELL Solid Mechanics Division, Faculty of Engineering 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 researchers 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.

3 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.

4 A C.I.P. Catalogue record for thls book is available from the Library of Congress. ISBN DOI / ISBN (ebook) 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.

5 Contents Preface... ix Chapter 1 Ordinary Integral Equations Introduction Ordinary Integral Equations and their Applications... 1 Applications... 1 Classification 0/ Integral Equations Equivalence between Ordinary Integral and Ordinary Differential Equations... 9 First Order Equations... 9 Second Order Equations. Initial Value Problems Second Order Equations. Boundary Value Problems Analytical Methods of Solution Fredholm Equations With Separable Kernels Iterative Methods For Second Kind Equations Numerical Methods of Solution Multistep Method Constant Function Numerical Treatment Concluding Remarks Exercises Chapter 2 Two Dimensional Potential Problems Introduction Applications of Potential Formulations Heat Conduction Fluid Flow Boundary Integral Equation Derivation for Interior Problems Derivation/rom Green's Identity Extension to the boundary Boundary Integral Equation Derivation for Exterior Problems Extension to the boundary Treatment of Boundary Conditions Potential boundary conditions Flux boundary conditions Mixed boundary conditions Concluding Remarks Exercises v

6 vi Contents Chapter 3 Boundary Element Method Introduction Numerical Foundation Linear Approximation Integration on a Curve Constant Function Solution for Exterior Heat Conduction Heat flow from a deeply buried pipe Discretisation into elements Collocation Evaluation of Logarithmic Integral Coefficients Case (a) Singular Element Case (b) Non-singular element Concluding Remarks Exercises Chapter 4 Linear Isoparametric Solution Introduction Linear Function Approximation for Exterior Heat Conduction Assembly of Left Hand Side Coefficients Singular and Nonsingular Elements Evaluation of Right Hand Side Terms Exterior Neumann Problem for Velocity Potential Illustration of Non-singular Integration Singularity Elimination for the Derivative Kernel Interior Mixed Boundary Value Problem Concluding Remarks Exercises Chapter 5 Quadratic Isoparametric Solution Introduction Interior Mixed Boundary Value Problems Treatment of Singular Integrals Row sum elimination Exact integration Weighted Gaussian Integration Subtraction and Series Expansion Method for Singular Integration Expansion of the shapefunction Expansion of the logarithm Expansion of the Jacobian Expansion of the complete integrand Treatment of the remainder integrals Concluding Remarks Exercises Chapter 6 Three Dimensional Potential Problems Introduction Boundary Integral Equation Formulation Electrostatics Application Shape functions and boundary elements The Boundary Element Method Surface Jacobian

7 Contents vii 6.7 Assembly of Coefficients Generation of a System of Equations Summary of the Three Dimensional Boundary Element Method Concluding Remarks Exercises Chapter 7 Numerical Integration for Three Dimensional Problems Introduction Integration in the Local Coordinate Plane Singular Integration Integration by Regularization Subtraction and Series Expansion Concluding Remarks Exercises Chapter 8 Two-Dimensional Elastostatics Introduction Review of Linear Elasticity Equilibriwn Equation Plane Stress Traction vector Deformations and Strains Generalised Hooke's Law Kelvin's Solution Derivation of the Boundary Integral Equation Betti's theorem and Somigliana's identity Displacement and Stress at an Internal Point Boundary Element Solution Five Element Illustration Singular integration using rigid body displacement solution Concluding remarks Exercises Appendix A Integration and Differentiation Formulae Appendix B Matrix Partitioning for the Mixed Boundary Value Problem Appendix C Answers to Selected Exercises Bibliography In.dex

8 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

9 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.