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.