Table Of ContentThis is the third edition of the principal text on the finite element method, which
has become a major solution technique for engineering electromagnetics. It
presents the method in a mathematically undemanding style, accessible to
undergraduates who may be encountering it for the first time. Like the earlier
editions, it begins by deriving finite elements for the simplest familiar potential
fields, then builds on these to formulate finite elements for a wide range of applied
electromagnetics problems. These include wave propagation, diffusion, and static
fields; open-boundary problems and nonlinear materials; axisymmetric, planar
and fully three-dimensional geometries; scalar and vector fields.
This new edition is more than half as long again as its predecessor and includes
much new material. Exterior problems of wave propagation, edge elements, and
many other recent developments are now included. The original material has been
extensively revised and in many cases entirely new treatments have been
substituted for the old. A wide selection of fully documented demonstration
programs is included with the text, allowing the reader to see how the methods are
put to work in practice.
This graduate-level textbook is easily accessible to the senior undergraduate for
it requires neither previous experience with finite elements, nor any advanced
mathematical background. It will also be a valuable reference text for
professional engineers and research students to make full use of finite elements in
electrical engineering.
From reviews of the second edition:
Highly recommended for senior electrical engineering students and professionals in the field
of modern electromagnetics.
Engineering
This book provides a clear, detailed introduction to finite element analysis ... it is suitable
for use as a text in a graduate electrical engineering course. It is also an excellent reference
for all electrical engineers who want to understand finite-element analysis well enough to
write or modify their own finite-element codes.
IEEE Antennas and Propagation Magazine
Highly recommended for engineering libraries serving senior and graduate electrical
engineering students, as well as for professionals in the field of electromagnetics.
American Library Association
Finite elements for electrical engineers
Finite elements for
electrical engineers
Third edition
PETER P. SILVESTER
University of British Columbia and McGill University
RONALD L. FERRARI
University of Cambridge
CAMBRIDGE
UNIVERSITY PRESS
Published by the Press Syndicate of the University of Cambridge
The Pitt Building, Trumpington Street, Cambridge CB2 1RP
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Melbourne 3166, Australia
© Cambridge University Press 1983, 1990, 1996
First published 1983
Reprinted 1986
Second edition 1990
Third edition 1996
A catalogue record for this book is available from the British Library
Library of Congress cataloguing in publication data
Silvester, P.P. (Peter Peet)
Finite elements for electrical engineers/Peter P. Silvester,
Ronald L. Ferrari.-3rd ed.
p. cm.
Includes bibliographical references.
ISBN 0 521 44505 1 (he) - ISBN 0 521 44953 7 (pb)
1. Electric engineering-Mathematics. 2. Finite element method.
I. Ferrari, R.L. (Ronald L.) II. Title.
TK153.S53 1996
621.3/01/515353-dc20 95-32756 CIP
ISBN 0 521 44505 1 hardback
ISBN 0 521 44953 7 paperback
Transferred to digital printing 2003
KW
Contents
Preface to the third edition xvii
Preface to the first edition xviii
1 Finite elements in one dimension 1
1. Introduction 1
2. A direct-current transmission line 1
3. The finite element approach 4
3.1 Power redistribution and minimization 4
3.2 Explicit expressions for power 5
3.3 Piecewise approximation 6
3.4 Finite element matrices 9
4. Global minimization 11
4.1 Joining the elements 11
4.2 Total power minimization 14
5. A Fortran program 15
5.1 Program structure 15
5.2 Solutions and errors 17
6. Further reading 20
7. Bibliography 21
8. Programs 21
First-order triangular elements for potential problems 28
1. Introduction 28
2. Laplace's equation 29
2.1 Energy minimization 29
2.2 First-order elements 32
2.3 Continuity between elements 34
2.4 Solution of the connected problem 37
viii Contents
3. Inhomogeneous sources and materials 39
3.1 Poisson's equation 39
3.2 Modelling the source term 42
3.3 Material inhomogeneities 44
3.4 Natural interface conditions 45
5.5 Energy and capacitance evaluation 47
4. Program construction issues 48
4.1 Handling of boundary conditions 49
4.2 Interconnection of multiple elements 51
4.3 Programming and data structures 53
5. A sample program 55
6. Further reading 58
7. Bibliography 59
8. Programs 60
3 Electromagnetics of finite elements 68
1. Introduction 68
2. Maxwell's equations 68
2.1 Differential relations 68
2.2 Integral relations 70
2.3 Complex phasor notation 71
2.4 Boundary rules 72
3. Waves and potentials 73
3.1 Wave equations 73
3.2 The classical potentials 75
3.3 Potential wave equations 78
3.4 Coulomb gauge 79
3.5 Diffusion gauge 80
3.6 Lorentz gauge 82
3.7 Separation of variables 83
3.8 Retarded potentials 85
4. Functionals over potentials and fields 88
4.1 The general scalar Helmholtz equation 88
4.2 A functional for the Helmholtz equation 89
4.3 General boundary conditions 92
4.4 Inhomogeneous Neumann boundaries 93
4.5 Inhomogeneous Cauchy boundaries 93
4.6 Natural continuity conditions 94
4.7 The vector Helmholtz equation 96
4.8 Electric and magnetic walls 98
5. Projective solutions 99
5.1 Functions and spaces 100
5.2 Inner product spaces 101
5.5 Projective approximation 103
5.4 Geometric interpretation 105
5.5 Galerkin's method for the Poisson equation 106
5.6 Weighted residual methods 108
Contents ix
5.7 Protective weighted residuals 109
5.8 Collocation methods 111
6. Two-dimensional field problems 112
6.1 Translational uniformity 112
6.2 Axisymmetrie regularity or uniformity 114
6.3 Modified axisymmetric potentials 116
7. Magnetic scalar potentials 117
7.1 The scalar potential 118
7.2 Carpenter-Noether potentials 119
7.3 Tozoni-Mayergoyz potentials 120
7.4 Two scalar potentials 122
8. Further reading 122
9. Bibliography 124
4 Simplex elements for the scalar Helmholtz equation 126
1. Introduction 126
2. Interpolation functions in simplex coordinates 127
2.1 Simplex coordinates 127
2.2 Cartesian-simplex conversion 129
2.3 Interpolation on 1-simplexes 129
2.4 Interpolation functions on n-simplexes 130
2.5 Differentiation in simplex coordinates 133
3. General simplex elements 136
3.1 Approximation of the variational functional 136
3.2 High-order triangular-element matrices 138
3.3 Interconnection and continuity 140
3.4 Rotations and column permutations 141
4. A high-order finite element computer program 145
4.1 Program description 145
4.2 High-order elements in waveguide analysis 146
5. Embeddings, projections, and their uses 149
5.7 Axisymmetric scalar fields 149
5.2 Embedding operators 152
5.5 Axisymmetric scalar matrices 155
5.4 Solution of coaxial-line problems 156
5.5 Axisymmetric vector fields 158
5.6 Projection operators 160
6. Tetrahedral elements 162
6.1 Interpolation and the metric T 162
6.2 Dirichlet matrices for tetrahedra 163
6.3 Permutations and rotations of tetrahedra 166
7. Further reading 168
8. Bibliography 169
9. Programs 169
t Contents
5 Differential operators in ferromagnetic materials 183
1. Introduction 183
2. Functional minimization for magnetic fields 183
2.1 Energy functionals 184
2.2 Minimization over finite elements 186
3. Solution by simple iteration 187
3.1 Simple iteration 188
3.2 A lifting magnet 190
4. Solution by Newton iteration 192
4.1 Formulation of Newton iteration 192
4.2 First-order triangular Newton elements 194
4.3 Material models 196
5. A magnetic analysis program 198
5.7 Program structure 198
5.2 Material models 199
5.5 Practical Newton iteration 201
6. Analysis of a DC machine 202
6.1 Boundary conditions 203
6.2 Performance evaluation 205
6.3 Periodicity conditions 206
7. Anisotropic materials 208
8. Further reading 210
9. Bibliography 211
10. Programs 212
6 Finite elements for integral operators 222
1. Introduction 222
2. One-dimensional finite elements for electrostatics 224
2.1 Stationary functionals for transmission lines 224
2.2 Finite element modelling of the integral equation 226
2.3 Finite element matrices 229
2.4 Numerical integration 231
3. Green's function for a dielectric slab 233
3.1 Dielectric half-space 233
3.2 The dielectric slab 234
3.3 Microstrip transmission-line analysis 236
4. A strip transmission-line program 236
5. Integral equations in magnetostatics 238
5.7 Magnetization integral equation 239
5.2 Boundary integral method 241
6. Integral operators in antenna theory 244
6.1 Pocklingtons equation 244
6.2 Projective formulation 247
6.3 Finite element matrices 249
6.4 Quasi-singular terms 250
