Alain Bossavit

The "TRIFOU" Code: Solving the 3-D eddy-currents problem by using H as state variable

In [1] and [2], a variational formulation of the eddy-currents problem using the magnetic field h as state variable was introduced. We describe here two applications of the method: computation of the impedance of a probe in non-destructive testing and prediction of the characteristics of a new design of an iron-free machine. The basic idea is first exposed, leaving aside difficulties like multiple connectedness. These are treated more thoroughly in Part 3, after the description of applications in Part 2.

Some realizations of a discrete Hodge operator: a reinterpretation of finite element techniques [for EM field analysis]

In this paper, some structures which underlie the numerical treatment of second-order boundary value problems are studied using magnetostatics as an example. The authors show that the construction of a discrete Hodge is a central problem. In this light, they interpret finite element techniques as a realization of the discrete Hodge operator in the Whitney complex. This enables one to view the Galerkin method as a way to set up circuit equations, the metric of space being encoded in the values of branch impedances.

How weak is the "weak solution" in finite element methods?

The standard Galerkin method in magnetostatics guarantees conservation of the flux about some surfaces, associated with the dual mesh, which we describe. Similar balance properties hold for edge-element approximations of Maxwells equations.

Numerical modelling of superconductors in three dimensions: a model and a finite element method

We propose a new macroscopic representation for the behavior of type-II superconductors, that is an extension of Beans model, and allows straightforward computation with finite elements in three dimensions, when superconductors as well as ordinary conductors are simultaneously present. >

Generating Whitney forms of polynomial degree one and higher

A rationale for Whitney forms is proposed; they are seen as a device to approximate manifolds, with approximation of differential forms as a byproduct. A recursive generating formula is derived. A natural way to build higher-degree forms then follows.

A new hybrid model using electric field formulation for 3-D eddy current problems

A hybrid finite-element-boundary integral method using an electric variational formulation (3-D code Trifou-e) is presented. Whitneys edge elements are used in conducting regions and the boundary element technique is used for exterior regions. The electric field is taken as the state variable for both of the regions, whether modeled by the finite-element or boundary-integral techniques, so that the problem of multiply connected regions can be treated in a convenient way. >

FORMULATION OF THE EDDY CURRENT PROBLEM IN MULTIPLY CONNECTED REGIONS IN TERMS OF H

In this paper various formulations for the eddy current problem are presented. The formulations are based on solving directly for the magnetic field h, and they differ from each other mainly by how the field on the boundary is treated. The electromagnetic problem is studied in connection with the fivefold decomposition of the space of square integrable vector fields within a bounded region. This provides us with numerical approaches with clear signposts about how to solve the eddy current problem in multiply connected domains. Besides the fivefold decomposition, another essential tool in our approach is Whitney elements, as they provide the structure needed to retain consistency between the continuous and discrete problems. The paper demonstrates the usefulness of these mathematical tools in solving electromagnetic field problems.

Simplicial finite elements for scattering problems in electromagnetism

Abstract Scattering problems in electromagnetism imply the approximate computation of vector-valued fields. To do this with finite elements which were originally designed for scalar-fields, that is, nodal elements, is not correct, for such vectorial finite elements have serious drawbacks. They impose on fields which they are meant to approximate (electric field e and magnetic field h ) a continuity of all components across inter-element boundaries which is not a necessary property of the fields. Simplicial finite elements, and more especially edge-elements, a brand of vector finite elements with degrees of freedom associated with the edges of the mesh are, as shown here, free of such drawbacks. First used in connection with eddy-current problems, they are shown to fit the scattering problem just as well, with no penalty in terms of computational time and with the expected additional advantage of not generating spurious modes when applied to the problem of the eigenmodes of resonating cavities.

Gauging in Whitney spaces

In this paper gauging is approached as a problem of selecting a representative in classes of equivalent representations. In this light we interpret how different gauging techniques are related to each other, and examine how they can be imposed on the discrete level using Whitney elements.

Discrete spaces for div and curl-free fields

In this paper we construct discrete spaces for fields whose curl or div vanishes but for which one cannot find potentials. In electromagnetism such fields appear when the problem domain is not topologically trivial. The discrete spaces are constructed with Whitney elements in simplicial meshes.