Table Of ContentC
OMPUTATIONAL
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LECTROMAGNETISM
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ARIATIONAL ORMULATIONS
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OMPLEMENTARITY DGE LEMENTS
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A l a i n B o s s a v i t
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COMPUTATIONAL
ELECTROMAGNETISM
Variational Formulations,
Complementarity, Edge Elements
ACADEMIC PRESS SERIES IN ELECTROMAGNETISM
Electromagnetism is a classical area of physics and engineering which
still plays a very important role in the development of new technology.
Electromagnetism often serves as a link between electrical engineers,
material scientists, and applied physicists. This series presents volumes
on these aspects of theoretical electromagnetism that are becoming
increasingly imporant in modern and rapidly developing technology. Its
objective is to meet the needs of researchers, students, and practicing
engineers.
This is a volume in
ELECTROMAGNETISM
ISAAK MAYERGOYZ, SERIES EDITOR
UNIVERSITY OF MARYLAND, COLLEGE PARK, MARYLAND
COMPUTATIONAL
ELECTROMAGNETISM
Variational Formulations,
Complementarity, Edge Elements
Alain Bossavit
Électricité de France
ACADEMIC PRESS
San Diego London Boston
New York Sydney Tokyo Toronton
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AB, 20 11 03
Foreword
This book is the second volume in the Academic Press Electromagnetism
Series, written by Professor Alain Bossavit, one of the most active
researchers in the area of electromagnetic field calculations. Professor
Bossavit is well known and highly regarded in the electromagnetic
community for his seminal contributions to the field of computational
electromagnetics. In particular, he has pioneered and strongly advocated
the use of edge elements in field calculations. These elements, which are
now widely accepted by engineers, have become indispensable tools in
numerical analysis of electromagnetic fields. His work on the use of
symmetry in numerical calculations, computational implementation of
complementarity, and evaluation of electromagnetic forces have also been
extremely important for the development of the field.
This book reflects the unique expertise and extensive experience of the
author. It is written with a strong emphasis on comprehensive and critical
analysis of the foundations of numerical techniques used in field
calculations. As a result, the book provides many valuable insights into
the nature of these techniques. It contains information hardly available
in other sources and no doubt will enrich the reader with new ideas and a
better conceptual understanding of computational electromagnetics. The
material presented in the book can be expected to contribute to the
development of new and more sophisticated software for electromagnetic
field analysis.
The book is distinctly unique in its original style of exposition, its
emphasis, and its conceptual depth. For this reason, it will be a valuable
reference for both experts and beginners in the field. Researchers as well
as practitioners will find this book challenging, stimulating, and
rewarding.
Isaak Mayergoyz, Series Editor
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vi
Contents
Preface xiii
CHAPTER 1 Introduction: Maxwell Equations 1
1.1 Field Equations 1
1.2 Constitutive Laws 6
1.2.1 Dynamics of free charges: the Vlasov–Maxwell model 7
1.2.2 Dynamics of conduction charges: Ohm’s law 9
1.2.3 Dynamics of bound charges: dielectric polarization 11
1.2.4 Magnetization 13
1.2.5 Summing up: linear materials 14
1.3 Macroscopic Interactions 15
1.3.1 Energy balance 16
1.3.2 Momentum balance 18
1.4 Derived Models 20
Exercises 24
Solutions 26
References 28
CHAPTER 2 Magnetostatics: “Scalar Potential” Approach 31
2.1 Introduction: A Model Problem 31
2.2 Honing Our Tools 33
2.2.1 Regularity and discontinuity of fields 33
2.2.2 Jumps 37
2.2.3 Alternatives to the standard formalism 39
2.3 Weak Formulations 41
2.3.1 The “divergence” side 42
2.3.2 The “curl” side 45
2.3.3 The uniqueness issue 47
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viii CONTENTS
2.4 Modelling: The Scalar Potential Formulation 48
2.4.1 Restriction to a bounded domain 48
2.4.2 Introduction of a magnetic potential 51
2.4.3 Uniqueness 53
2.4.4 Laplace, Poisson, Dirichlet, and Neumann 55
Exercises 56
Solutions 57
References 59
CHAPTER 3 Solving for the Scalar Magnetic Potential 61
3.1 The “Variational” Formulation 61
3.2 Existence of a Solution 65
3.2.1 Trying to find one 65
3.2.2 Φ* is too small 67
3.2.3 Completing Φ* 68
3.3 Discretization 70
3.3.1 The Ritz–Galerkin method 71
3.3.2 Finite elements 74
3.3.3 The linear system 78
3.3.4 “Assembly”, matrix properties 81
Exercises 84
Solutions 87
References 93
CHAPTER 4 The Approximate Scalar Potential: Properties 95
and Shortcomings
4.1 The “m-weak” Properties 95
4.1.1 Flux losses 96
4.1.2 The dual mesh, and which fluxes are conserved 99
4.1.3 The flux through Sh 103
4.2 The Maximum Principle 105
4.2.1 Discrete maximum principle 105
4.2.2 Voronoi–Delaunay tessellations and meshes 107
4.2.3 VD meshes and Stieltjes matrices 109
4.3 Convergence and Error Analysis 111
4.3.1 Interpolation error and approximation error 113
4.3.2 Taming the interpolation error: Zlamal's condition 114
4.3.3 Taming the interpolation error: flatness 116
