Computational Electromagnetism : Variational Formulations, Complementarity, Edge Elements

Computational Electromagnetism refers to the modern concept of computer-aided analysis, and design, of virtually all electric devices such as motors, machines, transformers, etc., as well as of the equipment inthe currently booming field of telecommunications, such as antennas, radars, etc. The present book is uniquely written to enable the reader-- be it a student, a scientist, or a practitioner-- to successfully perform important simulation techniques and to design efficient computer software for electromagnetic device analysis. Numerous illustrations, solved exercises, original ideas, and an extensive and up-to-date bibliography make it a valuable reference for both experts and beginners in the field. A researcher and practitioner will find in it information rarely available in other sources, such as on symmetry, bilateral error bounds by complimentarity, edge and face elements, treatment of infinite domains, etc. At the same time, the book is a useful teaching tool for courses in computational techniques in certain fields of physics and electrical engineering. As a self-contained text, it presents an extensive coverage of the most important concepts from Maxwells equations to computer-solvable algebraic systems-- for both static, quasi-static, and harmonic high-frequency problems. Benefits To the Engineer A sound background necessary not only to understand the principles behind variational methods and finite elements, but also to design pertinent and well-structured software. To the Specialist in Numerical Modeling The book offers new perspectives of practical importance on classical issues: the underlying symmetry of Maxwell equations, their interaction with other fields of physics in real-life modeling, the benefits of edge and face elements, approaches to error analysis, and "complementarity." To the Teacher An expository strategy that will allow you to guide the student along a safe and easy route through otherwise difficult concepts: weak formulations and their relation to fundamental conservation principles of physics, functional spaces, Hilbert spaces, approximation principles, finite elements, and algorithms for solving linear systems. At a higher level, the book provides a concise and self-contained introduction to edge elements and their application to mathematical modeling of the basic electromagnetic phenomena, and static problems, such as eddy-current problems and microwaves in cavities. To the Student Solved exercises, with "hint" and "full solution" sections, will both test and enhance the understanding of the material. Numerous illustrations will help in grasping difficult mathematical concepts.
Chapter One Introduction: Maxwell Equations

1.1 FIELD EQUATIONS

Computational electromagnetism is concerned with the numerical study of Maxwell equations,

(1) -∂t;d + rot h = j, (2) ∂tb + rot e = 0, (3) d = ε0e + p, (4) b = μ0 (h + m),

completed by constitutive laws, in order to account for the presence of matter and for the field–matter interaction. This introductory chapter will explain the symbols, discuss constitutive laws, and indicate how a variety of mathematical models derive from this basic one.

The vector fields e, h, d, b are called electric field, magnetic field, magnetic induction, and electric induction, respectively. These four vector fields, taken together, should be construed as the mathematical representation of a physical phenomenon, that we shall call the electromagnetic field. The distinction thus made between the physical reality one wants to model, on the one hand, and the mathematical structure thanks to which this modelling is done, on the other hand, is essential. We define a model as such a mathematical structure, able to account, within some reasonably definite limits, for a class of concrete physical situations. To get a quick start, no attempt is made here either to justify the present model, on physical grounds, or to evaluate it, in comparison with others. (In time, we'll have to pay for this haste.)

The current density j, polarization p, and magnetization m are the source-terms in the equations. Each contributes its own part, as we shall see, to the description of electric charges, at rest or in motion, whose presence is the physical cause of the field. Given j, p, and m, as well as initial values (at time t = 0, for instance) for e and h, Eqs. (1–4) determine e, h, d, b for t ≥ 0. (This is no trivial statement, but we shall accept it without proof.) Maxwell's model (1–4) thus accounts for situations where j, p, and m are known in advance and independent of the field. This is not always so, obviously, and (1–4) is only the head of a series of models, derived from it by adding features and making specific simplifications, some of which will be described at the end of this chapter.

You may be intrigued, if not put off, by the notation. The choice of symbols goes against recommendations of the committees in charge of such matters, which promote the use of E, H, D, B, capital and boldface. Using e, h, d, b instead is the result of a compromise between the desire to keep the (spoken) names of the symbols as close as possible to accepted practice and the notational habits of mathematics, capitals for functional spaces and lower case for their elements, according to a hierarchy which reflects the functional point of view adopted in this book. (Explanations on this fundamental point will recur.) Boldface, still employed in the Preface for 3D vectors, according to the standard convention due to Heaviside [Sp], will from now on be reserved for another use (see p. 71). I should also perhaps call attention to the use of the ∂ symbol: If b is a time-dependent vector field, ∂tb is the field obtained by differentiating b with respect to time. Having thus ∂tb instead of ∂b/∂t is more than a mere ink-saving device: It's a way to establish the status of ∂t as an operator, on the same footing as grad, div, and rot (this will denote the curl operator) which all, similarly, yield a field (scalar- or vector-valued, as the case may be) when acting on a field—the functional viewpoint, again. Other idiosyncrasies include the use of constructs such as exp(iωt) for eiωt and, as seen here, of i for the square root of -1, instead of j.

This being said, let's return to our description. Equation (1) is Ampère's theorem. Equation (2) is Faraday's law. The term ∂td, whose introduction by Maxwell was the crowning achievement of electromagnetic theory, is called displacement current. One defines electric charge (expressed in coulombs per cubic meter) by

(5) q = div d,

a scalar field. According to (1), one has thus

(6) ∂tq + div j = 0,

with j expressed in ampères per square meter. Notice that if j is given, from the origin of times to the present, one gets the charge by integration with respect to time: assuming j and q were both null before time 0, then q(t, x) = -∫t0 (div j)(s, x) ds.

If the local differential relation (6) is integrated by applying the Ostrogradskii (Gauss) theorem to a regular spatial domain D bounded by some surface S (Fig. 1.2), one finds that

(7) d/dt ∫D q + ∫s n . j = 0,

where n denotes the field of normal vectors, of length I and outwardly directed with respect to D, on surface S. The first term in this equality is the increase, per unit of time, of the charge contained in D, whereas the second term is the outgoing flux of charge. They balance, after (7), so (6) is the local expression of charge conservation.

Quite similarly, Eqs. (1) and (2) can be integrated by using the Stokes theorem, hence global (integral) expressions which express flux and current conservation. For instance, the integral form of Faraday's law is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where S is a surface, ∂S its boundary, and τ a field of unitary tangent vectors on ∂S (inset), oriented with respect to n as prescribed by Ampère's rule. (Units are volt per meter for e and weber(Wb) per square meter for b.) Historically, such integral formulations came first and are arguably more germane to physics. Indeed, we shall have to spend some time on correcting some drawbacks of the local differential formulation (or rather, of a too literal interpretation of this formulation).

Treatises on electromagnetism often add two equations to (1–4), namely (5) and div b = 0. But the latter stems from Faraday's law (2), if one assumes a null b (or even just a null div b) before initial time, and (5) is here a definition. So there would be little justification in according to these relations the same status as (1) and (2).

A (rightful) concern for formal symmetry might suggest writing (2) as ∂tb + rot e = –k, where k would be a given field, the magnetic current, and defining magnetic charge, expressed in webers per cubic meter, as qm = div b (electric charge q would then be denoted by qe), hence the equation ∂tqm + div k = 0, which would express magnetic charge conservation. But since k and qm are null in all known physical situations, this generalization seems pointless.

Now, let us address Eqs. (3) and (4). As the next Section will make clear, the (mathematical) fields e and b suffice to describe the effect of the (physical) electromagnetic field on the rest of the world, in particular on charged particles, whose motion is described by j, p, m, and which in turn constitute the source of the field. The electromagnetic field is thus kinematically characterized by the pair {e, b}, and fields d and h are auxiliaries in its dynamic description. Moreover, there is some leeway in the very definition of d and h, because the bookkeeping on charge motion can be shared between j, p, and m in different ways.

Exercise 1.1. Rewrite (1–4) by eliminating d and h. Discuss the interchangeability of j, p, and m.

Equations (3) and (4) thus seem to define redundant entities, and indeed, many classical presentations of electromagnetism make do with two vector fields instead of four. The main advantage of their presence, which explains why this formalism is popular in the computational electromagnetics community, is the possibility this offers to express material properties in a simple way, via "constitutive laws" which relate j, p, and m to the electromagnetic field they generate.

The vacuum, in particular, and more generally, matter that does not react to the field, is characterized by p = 0 and m = 0, and thus by the coefficients ε0 and μ0. In the MKSA system, μ0 = 4π 10-7H/m and ε0 = 1/(μ0c2) F/m, where c is the speed of light (H for henry and F for farad). These values reflect two things: A fundamental one, which is the very existence of this constant c, and a more contingent one, which is the body of conventions by which historically established units for electric and magnetic fields and forces have been harmonized, once the unity of electromagnetic phenomena was established.

Let us now review these constitutive laws, which we will see are a condensed account ot the laws of charge–matter interaction in specific cases.

1.2 CONSTITUTIVE LAWS

In all concrete problems, one deals with composite systems, analyzable into subsystems, or compartments: electromagnetical, mechanical, thermal, chemical, etc. Where to put the boundaries between such subsystems is a modelling decision, open to some arbitrariness: elastic forces, for instance, can sensibly be described as electromagnetic forces, at a small enough scale. Each compartment is subject to its own equations (partial differential equations, most often), whose right-hand sides are obtained by solving equations relative to other compartments. For instance, Eqs. (1–4) govern the electromagnetic compartment, and we'll soon see how j, p, and m are provided by others. If one had to deal with all compartments at once, and thus with coupled systems of partial differential equations of considerable complexity, numerical simulation would be very difficult. Constitutive laws, in general, are the device that helps bypass this necessity: They are an approximate but simple summary of a very complex interaction between the compartment of main interest and secondary ones, detailed modelling of which can then be avoided.

1.2.1 Dynamics of free charges: the Vlasov–Maxwell model

A concrete example will illustrate this point. Let's discuss the problem of a population of charged particles moving in an electromagnetic field which they significantly contribute to produce. Coupled problems of this kind occur in astrophysics, in plasma physics, in the study of electronic tubes, and so forth. To analyze such a physical system, we may consider it as made of two compartments (Fig. 1.3): the electromagnetic one (EM), and the "charge motion" compartment (CM), which both require a kinematical description, and influence each other's dynamics, in a circular way.

Let's enter this circle at, for instance, CM. A common way to describe its kinematics is to treat charge carriers as a fluid, characterized by its charge density "in configuration space", a function [??](t, x, v) of time, position, and (vector-valued) velocity. The actual charge density and current density are then obtained by summing up with respect to v:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where dv is the volume element in the three-dimensional space of velocities. CM thus influences EM by providing a source {q, j} for it. (Later we'll see that q is redundant, (6) being satisfied.)

The influence of EM on CM is via "Lorentz force". Recall that the force exerted by the field on a point charge Q passing at point x at time t with the (vector-valued) speed v is Q times the vector e(t, x) + v x b(t, x). (The part independent of celerity, that is Q e(t, x), is "Coulomb force".) Here we deal with a continuum of charge carriers, so let us introduce, and denote by [??], the density of force in configuration space: [??](t, x, v) dx dv is thus the force exerted on the packet of charges which are in volume dx around x, and whose speeds are contained in the volume dv of velocity space around v, all that at time t. So we have, in condensed notation,

(9) [??] = [??](e + v x b).

These forces produce work. We note for further use that the power density π thus communicated from EM to CM is what is obtained by integrating with respect to v:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

after (8).

(Continues...)

Excerpted from COMPUTATIONAL ELECTROMAGNETISM by Alain Bossavit Copyright © 1998 by Academic Press. Excerpted by permission of Academic Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site. Computational Electromagnetism refers to the modern concept of computer-aided analysis, and design, of virtually all electric devices such as motors, machines, transformers, etc., as well as of the equipment inthe currently booming field of telecommunications, such as antennas, radars, etc.

The present book is uniquely written to enable the reader-- be it a student, a scientist, or a practitioner-- to successfully perform important simulation techniques and to design efficient computer software for electromagnetic device analysis. Numerous illustrations, solved exercises, original ideas, and an extensive and up-to-date bibliography make it a valuable reference for both experts and beginners in the field. A researcher and practitioner will find in it information rarely available in other sources, such as on symmetry, bilateral error bounds by complimentarity, edge and face elements, treatment of infinite domains, etc.

At the same time, the book is a useful teaching tool for courses in computational techniques in certain fields of physics and electrical engineering. As a self-contained text, it presents an extensive coverage of the most important concepts from Maxwells equations to computer-solvable algebraic systems-- for both static, quasi-static, and harmonic high-frequency problems.

Benefits
To the Engineer
A sound background necessary not only to understand the principles behind variational methods and finite elements, but also to design pertinent and well-structured software.

To the Specialist in Numerical Modeling
The book offers new perspectives of practical importance on classical issues: the underlying symmetry of Maxwell equations, their interaction with other fields of physics in real-life modeling, the benefits of edge and face elements, approaches to error analysis, and "complementarity."

To the Teacher
An expository strategy that will allow you to guide the student along a safe and easy route through otherwise difficult concepts: weak formulations and their relation to fundamental conservation principles of physics, functional spaces, Hilbert spaces, approximation principles, finite elements, and algorithms for solving linear systems. At a higher level, the book provides a concise and self-contained introduction to edge elements and their application to mathematical modeling of the basic electromagnetic phenomena, and static problems, such as eddy-current problems and microwaves in cavities.

To the Student
Solved exercises, with "hint" and "full solution" sections, will both test and enhance the understanding of the material. Numerous illustrations will help in grasping difficult mathematical concepts. Theoretical in nature, this book addresses main computational techniques, especially finite elements as applied to electromagnetic calculations. The book is aimed at graduate students and researchers involved in computer simulation of electromagnetic phenomena, and originates from courses the author taught on the subject on numerical analysis of electromagnetism. Computational Electromagnetism should also be useful to practitioners in the industry working on numerical computations of electromagnetic fields and design of electromagnetic devices, in addition to applied mathematicians interested in applied PDE's and the finite element community.