Preface
Computational electromagnetism begins where electromagnetic
theory stops, and stops where engineering takes over. Its purpose is not
to establish Maxwell equations and other essential physical theories, but
to use them in
mathematical modelling,
with concrete problems in view.
Modelling is the activity by which questions about a physical situation
are translated into a mathematical problem~an
equation,
if this word is
understood in a general enough sense~that will be solved, in order to
answer these questions. "Solving", nowadays, means using a computer,
in most cases. The equations one aims at, therefore, can very well be
huge systems of linear equations (solved as part of some iterative process,
in nonlinear situations, or in the simulation of transient phenomena).
Complex shapes, non-uniform physical characteristics, changing configu-
rations, can and should be taken into account. Adequate methods~not
necessarily the exclusivity of electromagnetics~have been developed for
this: finite elements, boundary integral methods, method of moments...
Strengthening their foundations, clarifying their presentation, enhancing
their efficiency, is the concern of computational electromagnetism.
The contribution of this book to such a large subject will necessarily
be limited. Three main topics are treated:
• variational formulations,
understood from a functional viewpoint,
• edge elements
and related finite elements, and
• complementarity.
These are not, by any means, the definitive pillars on which the whole
theory should be erected. Rather they are three posts, or stakes, on
which I believe some platform can be built, with a good view on the
whole subject, provided the foundations are steady. A relatively thick
appendix, entitled
• mathematical background
has been included with this in mind, which should either provide the
prerequisites or give directions to locate and study them.
xiii
xiv PREFACE
Such emphasis on foundations does not imply disregard for such
issues as implementation, algorithmic efficiency, and relevance of
numerical results, which are all important in modelling. There is rich
and fast-growing literature on all this, and lots of opportunities for
beginners to get firsthand information at conferences and specialized
gatherings (such as, for example, the TEAM Workshop, frequently
mentioned in this book). Analyses of the
conceptual bases
of modern
methods, on the other hand, are much rarer, and there is a dearth of
courses from which rapidly to learn the basic notions gathered in the
present book. Yet, these notions are needed by all those who
do
conceive,
implement, test, and run electromagnetic software.
I speak here not as a mathematics teacher but as a programmer who
had to work on numerical simulations of electrical heating processes,
and strongly felt the urge to understand what he was doing at a time
when the ideals of "structured programming" were gaining universal
favor, and while the current understanding of the finite element method
seemed unable to provide the required guidelines.
That was twenty years ago. The finite-element treatment of the Laplace
equation (Aq0 = 0, or rather, div(~t grad q0) = 0, in the context of magneto-
statics) was well understood, and its application to major problems in
electromagnetism, such as computing the pattern of magnetic lines in the
cross-section of a rotating machine, was vigorously promoted by energetic
leaders, the late P.P. Silvester among them. Tremendous successes were
obtained in 2D simulations, and the first commercial codes reached the
market. But the passage to three dimensions proved very difficultman
intriguing situation, since it didn't seem to be encountered in neighboring
fields such as heat transfer or fluid dynamics.
With hindsight, we now understand why it was so. The Laplace
equation is not
the
paradigmatic equations in electromagnetism. There
are two of them. The other one is rot(~t -1 rot a) = j, which governs the
vector potential in magnetostatics. In dimension 2, where a = {0, 0, a},
with a unique nonzero scalar component a, this vector equation reduces
to - div(~ ~ grad a) = j. Hence the easy transposition, and the illusion
that one standard model would be enough. But in dimension 3,
the
div-grad
and the
curl-curl
equations are deeply different, and require different
treatment.
This was not obvious in the 1970s. Since rot rot a = grad div a- Aa,
it could seem possible to transpose the 2D methods to 3D situations by
first imposing the "gauge condition" div a = 0mhence -Aa = jmand
then working on the three scalar components of a separately. 1 Natural
as it was, the idea led to a blind alley, and it took years to realize this.
PREFACE xv
We were fortunate, J.C. V6rit6 and myself, to enter the field at precisely
this period of unrest, with a different idea about how to deal with the
curl-curl term in the eddy-current equation 3t(~h ) + rot(~-lrot h) = 0,
and an early implementation (the "Trifou" code, some salient features of
which will be described in this book).
Our idea was inspired by the belief that
network methods
had the best
prospects for 3D generalization. In such methods (see [TL], for instance),
it was customary to take as basic unknown the electromotive force (emf)
along the branch of a circuit and to apply Kirchhoff-like laws to set up
the equations. For reasons which will be explained in Chapter 8, the
magnetomotive force (mmf) along branches seemed preferable to us, and
it was clear that such mmf's, or edge circulations of the magnetic field,
had
to be the degrees of freedom in an eddy-current computation. As we
had devised variational formulations 2 for the eddy-current equation,
the problem was to be able to
interpolate from edge mmf's:
Knowing the
circulation of h along edges of a tetrahedral finite element mesh, by
which interpolating formula to express the magnetic field inside each
tetrahedron? The help of J.C. Nedelec, who was consultant at EdF at the
time (1979), was decisive in providing such a formula: h(x) =~ x x + ~,
where x is the position, and ~, ~, two ordinary vectors, tetrahedron-
dependent. "Trifou" was coded with this vector-valued shape function,
applied to a test problem [BV], and the idea of "solving directly for h
with edge elements" gradually gained acceptance over the years.
However, the analytic form of these shape functions was a puzzle
(why precisely ~ x x + ~ ?) and remained so for several years. A key
piece was provided by R. Kotiuga when he suggested a connection with
a little-known compartment of classical differential geometry, Whitney
forms [Ko]. This was the beginning of a long work of reformulation and
1Plus a fourth unknown field, the electric potential ~, in transient situations. The pair
{a, ~} fully describes the electromagnetic field, as one well knows, so such a choice of
unknown variables was entirely natural. What was wrong was not the {a, ~} approach
per se, but the
separate
treatment of Cartesian components by
scalar-valued
finite elements,
which resulted, in all but exceptional cases of uniform coefficients, in hopeless entanglement
of components' derivatives. Much ingenuity had to be lavished on the problem before
{a, ~d}-based methods eventually become operational. Some reliable modern codes do use
them.
2One of them, using the current density as basic unknown, was implemented [V6]. It
was equivalent (but we were not aware of it at the time) to a low-frequency version of
Harrington's "method of moments" [Ha].
3Trifou, now the reference code of l~lectricit6 de France for all electromagnetic problems,
continues to be developed, by a task-crew under the direction of P. Chaussecourte.
xvi PREFACE
refoundation, the result of which is given in Chapter 5, where Whitney
forms are called "Whitney elements". Although the picture thus given is
not definitive, we now can boast of understanding the whole structure of
Maxwell equations, both in "continuous" and "discrete" form, with enough
detail to firmly ground an orderly object-oriented programming approach.
This is what the book tries to convey. Its strong mathematical
component stems from the desire to help the reader track all concepts
back to their origin, in a self-contained package. The approach is descriptive
and concept-oriented, not proof-oriented. (Accordingly, the Subject Index
points only to the page where a
definition
is provided, for most entries.)
Motivations thus being given, it's time to detail the contents. After
an introductory Chapter 1, which reviews Maxwell equations and derived
models, Chapter 2 focuses on one of them, a very simple model problem
in magnetostatics, which is treated in Chapter 3 by solving for the scalar
potential. This is intended as an introduction to
variational formulations.
Variational methods have an ancient and well known history. The
solution to some field problem often happens to be the one, among a
family of a priori "eligible" fields, that minimizes some energy-related
quantity, or at least makes this quantity stationary with respect to small
variations. By restricting the search of this optimum to a well-chosen
finite
subfamily of eligible fields, one obtains the desired finite system of
equations, the solution of which provides a near-optimum. This powerful
heuristics, or
Galerkin method,
applicable to most areas of physics, leads
in a quite natural way to finite element methods, as we shall see in
Chapters 2 and 3, and suggests a method for error analysis which makes
most of Chapter 4.
At this stage, we are through with the div-grad operator, and wish
to address the curl-curl one, by treating the same problem in vector
potential formulation. The same approach is valid, provided the right
finite elements are available, and ...
Edge elements
are it; at least, if one takes a large enough view, by
considering the whole family of "Whitney elements" (as the foregoing
Whitney forms are called in this book), a family which includes the standard
node-based scalar elements, edge elements (with degrees of freedom
associated with edges of a mesh), face elements, etc. And with these, we
discover mathematical structures which, if not foreign to other branches
of physics, at least seem ready-made for electromagnetism. Reasons for
this marvelous adequacy will be suggested: It has to do with the essentially
geometric character of Maxwell equations, which revolve around different
versions of the Stokes theorem. This is the substance of Chapter 5. Although
PREFACE xvii
the "classical" language used in this work (only vectorial analysis, no
differential geometry at all) does not permit a full explanation of such
things, I hope the reader will feel some curiosity for more advanced
books, in which the differential geometric viewpoint would prevail from
the onset, and for the specialized literature. Some informal forays into
homology and the basics of topology should help show the way.
At the end of Chapter 5, we shall have a better view of the classical
scalar potential vs vector potential diptych, and be ready to apply to the
curl-curl side what we learned by close scrutiny of the div-grad panel.
But the symmetry thus revealed will suggest a new idea, which is the
essence of
complementarity.
Indeed, both approaches lead to a solution of
the model problem, but using
both
of them, in a deliberately redundant
way, brings valuable additional benefits: a posteriori error
bounds
(much
better than asymptotic
estimates),
bilateral bounds on some important
results, usable error estimators for mesh-refinement procedures. Chapter
6 implements this working program, revisiting the time-honored
"hypercircle" idea (cf. p. 171) along the way. Emphasis in this chapter,
as elsewhere, is on the structures which underlie the methods, and finer
details have been relegated to Appendices B and C.
A technical, but important, issue then arises and is covered in Chapter
7: how to deal with infinite regions with only a finite number of elements.
Integral methods
on an artificial boundary, in association with finite elements
in a bounded region, solve this problem.
It's only then that we shall go beyond magnetostatics. The strong
imbalance of this book's contents in favor of such a limited subject may
surprise the reader. But this limited scope is precisely what makes
magnetostatics such a good model with which to explain concepts of
much broader application. The last two chapters are intended to illustrate
this point by a rapid examination of a few other, more complex, models.
Chapter 8 is devoted to
eddy current
problems. This can serve as a
short theoretical introduction to the Trifou code, with in particular a
detailed account of how the hybrid method of Chapter 7 (finite elements
inside the conductor, boundary elements on the air-conductor interface)
is applied.
Chapter 9 then addresses the "microwave oven" problem, that is,
Maxwell equations in harmonic regime in a bounded region. The point
is to show how easily, once variational techniques and edge elements are
mastered, one can pass from the basic equations to solvable linear systems.
It's also another opportunity to see complementarity and the symmetry
of Maxwell equations at work.
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