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Boundary elements: Theory and applications

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The author's ambition for this publication was to make BEM accessible to the student as well as to the professional engineer. For this reason, his main task was to organize and present the material in such a way so that the book becomes "user-friendly" and easy to comprehend, taking into account only the mathematics and mechanics to which students have been exposed during their undergraduate studies. This effort led to an innovative, in many aspects, way of presenting BEM, including the derivation of fundamental solutions, the integral representation of the solutions and the boundary integral equations for various governing differential equations in a simple way minimizing a recourse to mathematics with which the student is not familiar. The indicial and tensorial notations, though they facilitate the author's work and allow to borrow ready to use expressions from the literature, have been avoided in the present book. Nevertheless, all the necessary preliminary mathematical concepts have been included in order to make the book complete and self-sufficient.

Throughout the book, every concept is followed by example problems, which have been worked out in detail and with all the necessary clarifications. Furthermore, each chapter of the book is enriched with problems-to-solve. These problems serve a threefold purpose. Some of them are simple and aim at applying and better understanding the presented theory, some others are more difficult and aim at extending the theory to special cases requiring a deeper understanding of the concepts, and others are small projects which serve the purpose of familiarizing the student with BEM programming and the programs contained in the CD-ROM.

The latter class of problems is very important as it helps students to comprehend the usefulness and effectiveness of the method by solving real-life engineering problems. Through these problems students realize that the BEM is a powerful computational tool and not an alternative theoretical approach for dealing with physical problems. My experience in teaching BEM shows that this is the students' most favorite type of problems. They are delighted to solve them, since they integrate their knowledge and make them feel confident in mastering BEM.

The CD-ROM which accompanies the book contains the source codes of all the computer programs developed in the book, so that the student or the engineer can use them for the solution of a broad class of problems. Among them are general potential problems, problems of torsion, thermal conductivity, deflection of membranes and plates, flow of incompressible fluids, flow through porous media, in isotropic or anisotropic, homogeneous or composite bodies, as well as plane elastostatic problems in simply or multiply connected domains. As one can readily find out from the variety of the applications, the book is useful for engineers of all disciplines. The author is hopeful that the present book will introduce the reader to BEM in an easy, smooth and pleasant way and also contribute to its dissemination as a modern robust computational tool for solving engineering problems.

Author(s): J.T. Katsikadelis
Edition: 1
Publisher: Elsevier Science
Year: 2002

Language: English
Pages: 351
Tags: Математика;Вычислительная математика;

Front Cover ......Page 1
Boundary Elements: Theory and Applications......Page 4
Copyright Page ......Page 5
Table of Contents......Page 12
Preface......Page 8
1.1 Scope of the book......Page 16
1.2 Boundary Elements and Finite Elements......Page 17
1.3 Historical development of the BEM......Page 20
1.4 Structure of the book......Page 22
1.6 References......Page 24
2.2 The Gauss-Green theorem......Page 28
2.3 The divergence theorem of Gauss......Page 30
2.4 Green's second identity......Page 31
2.5 The adjoint operator......Page 32
2.6 The Dirac delta function......Page 33
Problems......Page 38
3.1 Introduction......Page 40
3.2 Fundamental solution......Page 41
3.3 The direct BEM for the Laplace equation......Page 43
3.4 The direct BEM for the Poisson equation......Page 49
3.5 Transformation of the domain integrals to boundary integrals......Page 51
3.6 The BEM for potential problems in anisotropic bodies......Page 54
3.7 References......Page 59
Problems......Page 61
4.1 Introduction......Page 62
4.2 The BEM with constant boundary elements......Page 64
4.3 Evaluation of line integrals......Page 68
4.4 Evaluation of domain integrals......Page 72
4.5 The Dual Reciprocity Method for Poisson's equation......Page 73
4.6 Program LABECON for solving the Laplace equation with constant boundary elements......Page 76
4.7 Domains with multiple boundaries......Page 100
4.8 Program LABECONMU for domains with multiple boundaries......Page 101
4.9 The method of subdomains......Page 111
4.10 References......Page 117
Problems......Page 119
5.1 Introduction......Page 120
5.2 Linear elements......Page 122
5.3 The BEM with linear boundary elements......Page 126
5.4 Evaluation of line integrals on linear elements......Page 130
5.5 Higher order elements......Page 144
5.6 Near-singular integrals......Page 151
5.7 References......Page 155
Problems......Page 157
6.2 Torsion o f non-circular bars......Page 158
6.3 Deflection o f elastic membranes......Page 189
6.4 Bending o f simply supported plates......Page 193
6.5 Heat transfer problems......Page 196
6.6 Fluid flow problems......Page 202
6.7 Conclusions......Page 208
6.8 References......Page 212
Problems......Page 213
7.2 Equations o f plane elasticity......Page 216
7.3 Betti ' s reciprocal identity......Page 226
7.4 Fundamental solution......Page 228
7.5 Stresses due to a unit concentrated force......Page 234
7.6 Boundary tractions due to a unit concentrated force......Page 235
7.7 Integral representation o f the solution......Page 236
7.8 Boundary integral equations......Page 239
7.9 Integral representation o f the stresses......Page 243
7.10 Numerical soiution of the boundary integral equations......Page 245
7.11 Body forces......Page 249
7.12 Program ELBECON for solving the plane elastostatic problem with constant boundary elements......Page 256
7.13 References......Page 294
Problems......Page 296
Appendix A: Derivatives of r......Page 300
B.1 Gauss integration of a regular function......Page 304
B.3 Double integrals of a regular function......Page 313
B.4 Double singular integrals......Page 320
B.5 References......Page 322
Appendix C: Answers to selected problems......Page 324
Author Index......Page 336
Subject Index......Page 340