Mathematical Methods in Electromagnetism: Linear Theory and Applications

396 Pages·1996·65.407 MB·English

by  Michel Cessenat

Series on Advances in Mathematics for Applied Sciences - Vol. 41
MATHEMATICAL METHODS
IN ELECTROMAGNETS
Linear Theory and Applications
Michel Cessenat
CEA/DAM
Centre d'Etudes de Bruydre$-le-Chdtel
France
World Scientific
Singapore »New Jersey'London* Hong Kong
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Library of Congress Cataloguig-in-Piiblicatioii Data
Cessenat, Michel.
Mathematical methods in etectromagnetism: linear theory and
applications / Michel Cessenat.
396 p. 22.5 cm. - (Series on advances in mathematics for applied sciences; vol. 41)
Includes bibliographical references and index.
ISBN 9810224672
1. Electromagnetism — Mathematics. I. Title. II. Series.
QC760.C43 1996
537\01'51-dc20 96-11628
CIP
British Library Cataloguing-in-Publicatkm Data
A catalogue record for this book is available from the British Library.
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MfiTHEMflTICfiL METHODS
III ELECTROMflGNETISM
Linear Theory and Applications
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Acknowledgements
I wish to thank Professor R. Dautray for drawing my attention to certain
problems of electromagnetism. I also wish to thank Professor J.L. Lions for
suggesting to me, one day at Orly airport, to write this book. Applying his ideas
on asymptotic analysis to electromagnetism is an exciting challenge! I wish to
express my gratitude to Professor A. Bossavit for constructive remarks on ways
to improve this book. I am grateful to Professors M. Artola, P. Benilan, R.
Petit and also to Dr. A. Gervat for useful comments. Dr. G. Zerah must be
thanked for fruitful discussions on ferromagnetism.
I am also gratefully indebted to Dr. J.-M. Clarisse, and O. Cessenat, and also
to C. Averseng and C. Mares for their help.
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Preface
Electromagnetism has numerous applications whether in energy transport or
signal transmission in devices as diverse as antennas, waveguides, optical
fibers, gratings, electrical circuits, light in lasers and plasmas. Thus, much
research has been devoted to problems in electromagnetism essentially from a
physical point of view. One aim of this book is to present a more global analysis
encompassing both the physical and mathematical aspects of
electromagnetism. In particular, powerful modern mathematical methods are
emphasized, which lead to numerical applications.
Phenomena under consideration are modelled as "distributed system" by the
"electromagnetic field" which must satisfy Maxwell's equations. In general,
media are described using the "continuum assumption" by introducing specific
constants and thus, without entering their fine structures. This description
leads to constitutive relations. Various conditions must also be treated such as:
initial, boundary or transmission conditions, conditions at infinity, finite (or
locally finite) energy conditions.
In the absence of any coupling with other phenomena such as fluid motion,
determining the electromagnetic field which satisfies all the necessary
conditions is already a difficult problem.
The student or the engineer may feel quite uneasy in face o fthese problems
and think that usual mathematical books are helpless. Indeed, problems in
electromagnetism can be very diverse, and differ from classical problems in
partial differential equations in the vectoriai nature of the electromagnetic
field. Thus, differential geometry is present at all stages whether when
modelling, choosing a mathematical framework, or computing theoretical or
numerical solutions. Another peculiarity is inherent to the constitutive
relations and to the specific "constants" of the media. These constants may be
complex or take the form of matrices for usual dissipative and anisotropic
media, or be positive real numbers for free space (or ideal conservative media).
Therefore, different methods must be applied depending on the media. It must
also be the case that these specific constants strongly depend on frequency,
thus leading to evolution problem with time delay, or depend on the
electromagnetic field itself, giving rise to nonlinear effects.
Last but not least, electromagnetism problems are wave propagation
problems, therefore presenting classical features of such problems. Hence
stationary problems in the frequency domain are often ill-posed, and become
well-posed only by imposing radiation conditions at infinity or on the direction
of propagation. This results in difficulties when treating scattering by infinite
obstacles.
VII
VIII PREFACE
One aim of this book is to provide the reader with the basic tools and the
functional analysis concepts corresponding to the usual physical hypotheses of
the modelling stage. We also present in the Appendix, certain properties of
differential geometry, the corner stone of electromagnetism. As examples of
application of these tools and concepts, we treat several fundamental problems
of electromagnetism, e.g.: scattering of an incident electromagnetic wave by a
bounded obstacle, scattering by a grating (periodically infinite obstacle), wave
propagation in a waveguide or in an optical fiber.
Several recent approaches suitable for solving electromagnetism problems
and related to numerical methods are presented, in particular:
Integral methods, resulting in solving an integral equation on a given surface
(typically the surface of the obstacle in a stationary scattering problem) thus
sparing one spacial dimension. However, the resulting matrices are full.
Semigroup methods, allowing to treat evolution problems, and also
stationary problems both with planar or cylindrical geometries, and a given
direction of wave propagation.
Variational methods (particularly suited to numerical applications), in the
case of dissipative media.
Spectral methods, when spectral decompositions (of normal operators) are
possible. However, these approaches, well known in physics as modal
decompositions, and which are the core of many textbooks on electromagnetism
are often used beyond their scope.
In many cases, hybrid approaches are preferable which combine the
respective advantage of different methods. For instance, when solving a
stationary scattering problem, a finite element method is applied to an
inhomogeneous obstacle and coupled to an integral method for the outer
domain. The determination of constants for a sample in a waveguide is another
typical example: a finite element method is used in the sample and coupled to a
spectral method for the rest of the homogeneous waveguide.
One of the main ideas developed in this book is the use of Calderon
projectors and operators (also called impedance or admittance surface
operators). These operators are surface operators containing all the
information relevant to the electromagnetic properties of the domain bounded
by the surfaces where they are defined. Such operators are especially useful
when tackling asymptotic problems in electromagnetism: e.g. the scattering by
an obstacle of high conductivity or with fine periodic inclusions.
We hope that this book is useful for students or engineers having to solve
problems in electromagnetism. We also hope that mathematical notions will
not conceal physical concepts from the reader, but rather allow a better
understanding of modelization in electromagnetism, and emphasize the
essential features related to the geometry and nature of materials.
Some prerequisites in functional analysis may be useful as for example
Dautray-Lions [1].
Contents
Chapter 1 - Mathematical Modelling of the Electromagnetic Field in
Continuous Media: MaxweU Equations and Constitutive Relations 1
I - Evolution Maxwell Equations 1
2-Stationary Maxwell Equations... 4
3 - Constitutive Relations 7
3.1 - Linear isotropic dielectric media 7
3.2- Linear anisotropic dielectric media • 20
3.3- Linear chiral media 22
3.4-Nonlinear constitutive relations 23
Chapter 2 - Mathematical Framework for Electromagnetism 26
1-Spaces for curl and div: Trace Theorems 28
2- Jump Formulas Across a Bounded Hypersurfaee T in Rn 32
3-Differential Operators on a Regular Surface T 33
4 - The spaces H ~ 1/2(div,D, H " lT2(curl,r). Trace for H(curl,Q) 35
5-TraceforWl(div,Q) 40
6-Some Complementary Results. Polar Sets 41
7-Traces on a Sheet 42
7.1 -Trace problems on sheets (the scalar case) 43
7.2 - Trace problems on sheets (the vectorial case) 46
S-Some Regularity Results 49
9 - The Hodge Decomposition 52
10 - Interpolation Results 57
II -Some Variational Frameworks 58
11.1- Variational frameworks based on Hf curl ,Q) spaces 58
11.2- Variational frameworks based on H (Q) and the Laplacian .... 60
12 - First Problems with Inhomogeneous Boundary Conditions 62
12.1 -Problems with H~1/2(div,D boundary conditions 62
12.2 - Problems with H1/2(r) boundary conditions 66
13 - Boundary Problems of Cauchy Type. Uniqueness Theorems 67
14-Whitney Elements; Numerical Treatment 68
Chapter 3 - Stationary Scattering Problems with Bounded Obstacles 71
1 - Stationary Waves due to Sources in Bounded Domains 71
1.1- The main properties for Helmholtz equation in R3 71
1.1.1 -Local regularity properties 72
1.1.2- Outgoing and incoming Sommerfeld conditions 72
1.1.3-Elementary outgoing (incoming) solution 73
1.1.4-Fundamental properties 73
1.2-The main properties for Maxwell equations in R3 75
1.2.1-Relation to the Helmholtz problem •• — •..-.
1.2.2- The Silver-Mulier conditions *
78
IX