A Practical Guide to Boundary-Element Methods
with the Software Library BEMLIB

C Pozrikidis

Chapman & Hall/CRC, 2002

ISBN: 1584883235, 440 pages,

Publisher | Amazon | Barnes & Noble | Errata

In the last twenty years, the boundary-element method (BEM) has been established as a powerful numerical technique for tackling a variety of problems in science and engineering involving elliptic partial differential equations. Examples can be drawn from elasticity, geomechanics, structural mechanics, electromagnetism, acoustics, hydraulics, low-Reynolds-number hydrodynamics, and biomechanics. The strength of the method derives from its ability to solve with remarkable efficiency and accuracy problems in domains with complex and possibly evolving geometry where traditional methods can be inefficient, cumbersome, or unreliable.

Why this book?

The purpose of this text is to provide a concise introduction to the theory and implementation of the boundary-element method, while simultaneously offering hands-on experience based on the software library BEMLIB. Expected audience includes professionals, researchers, and students in various branches of computational science and engineering. The material is suitable for self-study, and the text is appropriate for instruction in an introductory course on boundary integral and boundary element methods, or a more general course in computational science and engineering.

BEMLIB Library

The software library BEMLIB accompanying this book consists of a collection of Fortran 77 and matlab programs related to Green's functions, boundary integral, and boundary element methods for Laplace, Helmholtz, and Stokes flow problems.

The main directories contain subdirectories that include main programs, assisting subroutines, and utility subroutines. Linked with drivers, the utility subroutines become stand-alone modules. The codes of BEMLIB explicitly illustrate how procedures and concepts discussed in the text translate into code instructions, and demonstrate the mathematical formulation and structure of boundary element codes for a variety of applications. The output of the codes is recorded in tabular form so that it can be displayed using independent graphics, visualization, and animation applications. The codes of BEMLIB can be used as building blocks, and may serve as a point of departure for developing further codes.

General information on BEMLIB, the directory contents, and instructions on how to download and compile the codes are given in Chapter 8.

Contents

Consistent with the dual nature of this book as an introductory text and a software user guide, the material is divided into two parts. The first part, Chapters 1-7, discusses the theory and implementation of boundary element methods. The material includes classical topics and recent development such as the dual reciprocity method (DRM) for solving inhomogeneous, nonlinear, and time-dependent equations.

The second part, Chapters 8-12, contains the user guide of BEMLIB. The user guide explains the problem formulation and numerical method of the particular problems considered, and discusses the function of the individual subprograms and codes with emphasis on implementation.

Table of contents

Frequently Asked Questions

xi
1

Laplace's equation in one dimension

1
1.1 Green's first and second identities and the reciprocal relation 1
1.2 Green's functions 2
1.2.1 Green's function dipole 5
1.3 Boundary-value representation 7
1.4 Boundary-value equation 8
2

Laplace's equation in two dimensions

9
2.1 Green's first and second identities and the reciprocal relation 9
2.1.1 Integral form of the reciprocal relation 10
2.2 Green's functions 12
2.2.1 Green's functions of the first kind and Neumann functions 13
2.2.2 Free-space Green's function 14
2.2.3 Green's functions in bounded domains 15
2.2.4 BEMLIB directory lgf_2d 15
2.2.5 Integral properties of Green's functions 16
2.2.6 Green's function dipole 16
2.2.7 Green's function quadruple 17
2.3 Integral representation 19
2.3.1 Green's third identity 21
2.3.2 Choice of Green's functions 21
2.3.3 Integral representation of the gradient 21
2.3.4 Representation in the presence of an interface 23
2.4 Integral equations 28
2.4.1 Boundary corners 31
2.4.2 Shrinking arcs 32
2.5 Hypersingular integrals 36
2.5.1 Local analysis 37
2.5.2 Shrinking arcs 41
2.5.3 Hypersingular integral equations 42
2.5.4 Regularization of hypersingular integral equations 43
2.6 Irrotational flow 44
2.6.1 Infinite flow past a body 45
2.7 Generalized single- and double-layer representations 47
2.7.1 Interior fields 47
2.7.2 Exterior fields 48
2.7.3 Gradient of the single-layer potential 49
2.7.4 Gradient of the double-layer potential 50
3

Boundary-element methods for Laplace's equation in two dimensions

53
3.1 Boundary element discretization 53
3.1.1 Straight elements 53
3.1.2 Circular arcs 55
3.1.3 Cubic splines 56
3.1.4 BEMLIB discretization 61
3.1.5 Thomas' algorithm 61
3.2 Discretization of the integral representation 63
3.2.1 Uniform elements 64
3.2.2 Non-singular elements 65
3.2.3 Singular elements 69
3.3 The boundary-element collocation method 74
3.3.1 BEMLIB directory laplace 76
3.4 Isoparametric cubic-splines discretization 77
3.4.1 Cardinal cubic-splines functions 77
3.4.2 Boundary element discretization 78
3.5 High-order collocation methods 80
3.5.1 Spectral node distribution 82
3.5.2 Corner elements 83
3.5.3 Assembly 84
3.5.4 Grid refinement 85
3.6 Galerkin and global expansion methods 86
3.6.1 Galerkin boundary-element method 86
3.6.2 Global expansion method 89
4

Laplace's equation in three dimensions

91
4.1 Green's first and second identities and the reciprocal relation 91
4.1.1 Integral form of the reciprocal relation 92
4.2 Green's functions 93
4.2.1 Green's and Neumann functions 94
4.2.2 The free-space Green's function 95
4.2.3 Green's functions in bounded domains 95
4.2.4 BEMLIB directory lgf_3d 96
4.2.5 Integral properties of Green's functions 96
4.2.6 Green's function dipole 97
4.2.7 Green's function quadruple 98
4.3 Integral representation Green's functions 100
4.3.1 Green's third identity 101
4.3.2 Integral representation of the gradient 101
4.4 Integral equations 102
4.4.1 Boundary corners 104
4.4.2 Hypersingular equations 104
4.5 Axisymmetric fields in axisymmetric domains 106
4.5.1 Computation of the free-space axisymmetric Green's function 108
4.5.2 BEMLIB} directory lgf_ax 109
4.5.3 Boundary-element methods 109
5

Boundary-element methods for Laplace's equation in three dimensions

111
5.1 Discretization 111
5.1.1 Triangulation 112
5.1.2 Successive subdivision 114
5.1.3 Local element approximation 114
5.1.4 Point collocation 116
5.1.5 High-order methods 116
5.2 Three-node flat triangles 117
5.2.1 Non-singular triangles 118
5.2.2 Single-layer integral over singular triangles 119
5.2.3 Double-layer integral over singular triangles 120
5.3 Six-node curved triangles 122
5.3.1 Non-singular triangles 124
5.3.2 Single-layer integral over singular triangles 125
5.3.3 Double-layer integral over singular triangles 126
5.4 High-order expansions 127
5.4.1 Isoparametric linear expansion 127
5.4.2 Isoparametric quadratic expansion 128
5.4.3 Spectral-element methods 128
6

Inhomogeneous, nonlinear, and time-dependent problems

131
6.1 Distributed source and domain integrals 132
6.2 Particular solutions and dual reciprocity in one dimension 134
6.2.1 The Newtonian potential 135
6.2.2 Integral representation involving a domain integral 135
6.2.3 MAPS with cardinal interpolation functions 136
6.2.4 DRM with cardinal interpolation functions 137
6.2.5 MAPS and DRM with influence functions 139
6.2.6 Cardinal functions from influence functions 140
6.3 Particular solutions and dual reciprocity in two and three dimensions 141
6.3.1 The Newtonian potential 141
6.3.2 Integral representation 142
6.3.3 MAPS and DRM with cardinal interpolation functions 142
6.3.4 MAPS and DRM with influence functions 144
6.3.5 Cardinal functions from influence functions 145
6.3.6 Radial basis functions (RBF) 145
6.3.7 General implementation of the DRM 146
6.4 Convection -- diffusion equation 150
6.4.1 Linear convection 150
6.4.2 Nonlinear convection 153
6.5 Time-dependent problems 154
6.5.1 Unsteady Green's functions method 155
6.5.2 Laplace transform method 158
6.5.3 Time-discretization methods 159
7

Viscous flow

161
7.1 Governing equations 161
7.1.1 Non-dimensionalization 163
7.1.2 Unsteady Stokes flow 164
7.1.3 Stokes flow 164
7.1.4 Modified pressure 164
7.2 Stokes flow 165
7.2.1 The Lorentz reciprocal relation 165
7.2.2 Green's functions 166
7.2.3 Free-space Green's functions 168
7.2.4 Green's functions in bounded domains 168
7.2.5 Properties of the Green's functions 169
7.3 Boundary integral equations in two dimensions 171
7.3.1 Integral equations 172
7.3.2 Integral representation for the pressure 173
7.3.3 Integral representation for the stress 174
7.3.4 Hypersingular integrals 174
7.4 Boundary-integral equations in three dimensions 175
7.4.1 Integral representation for the pressure 176
7.4.2 Integral representation for the stress 177
7.4.3 Hypersingular integrals 177
7.4.4 Axisymmetric flows in axisymmetric domains 177
7.5 Boundary-element methods 178
7.5.1 Computation of the single-layer potential 178
7.5.2 Computation of the double-layer potential 179
7.5.3 Corner singularities 179
7.6 Interfacial dynamics 180
7.7 Unsteady, Navier-Stokes, and non-Newtonian flow 182
7.7.1 Time discretization of an unsteady flow 182
7.7.2 Particular solutions and dual reciprocity 185
7.7.3 Alternative formulations 186
7.7.4 Non-Newtonian flow 187
8

BEMLIB user guide

191
8.1 General information 191
8.2 Terms and conditions 192
8.3 Directory contents 193
9

Directory: grids

197
grid_2d 198
trgl 203
10

Directory: laplace

207
lgf_2d 209
lgf_3d 230
lgf_ax 245
flow_1d 254
flow_1d_1p 261
flow_2d 265
body_2d 270
body_ax 276
tank_2d 281
ldr_3d 287
lnm_3d 291
11

Directory: helmholtz

295
flow_1d_osc 296
11

Directory: stokes

301
sgf_2d 302
sgf_3d 329
sgf_ax 349
flow_2d 364
prtcl_sw 371
prtcl_2d 376
prtcl_ax 385
prtcl_3d 390
A

Mathematical supplement

397
B

Gauss elimination and linear solvers

403
C

Elastostatics

407

References

413

Index

418