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Homogenization, which reduces the cost of numerical simulations in materials with repetitive structure, is a promising approach to the design of metamaterials. We justify the procedure by a convergence result and apply it to the case of an array of split rings, where a negative effective permeability can be expected near some resonant frequency. A simplified model confirms this. Index Terms--Homogenization, metamaterials, symmetry.

How to deal with motion and forces, at the discrete level, is still an open problem, if we want the coupled discrete dynamical system that results from discretization to behave consistently (energy and momentum conservation, etc.). If discretization is conceived as putting together appropriate pieces of a toolkit, this raises the question of which pieces should correspond to items like Lorentz force, Laplace force, Poynting flux, etc. A few such objects (templates for software parts, to some extent) are described, and an approach to the discrete analogue of the Maxwell tensor is outlined. Index Terms--Discrete exterior calculus, forces, Laplace force, Lorentz force, Maxwell tensor.

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. Index Terms--Convergence, edge elements, Whitney forms.

Some field problems can be solved in two different, symmetrical ways, which are 'complementary' in the sense that each one corrects some deficiencies of the other one. In particular, one obtains error bounds this way. Taking as example the static conduction problem, we propose a geometrical presentation of this theory, which generalizes and symmetrizes the classical but a bit forgotten nowadays 'hypercircle' principle. The two complementary problems are independent, but we show how having solved one of them allows one to parallelize the solution of the other, by reducing it to a family of local problems (one for each node of the mesh, to be solved in its immediate neighborhood). Index terms - Complementarity, dual mesh, error bounds.

In this paper some structures which underlie the numerical treatment of second-order boundary value problems are studied using magnetostatics as an example. We show that the construction of a discrete Hodge is a central problem. In this light we interpret finite element techniques as a realization of the discrete Hodge operator in the Whitney complex. This enables us 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. Index terms - Finite element methods, differential forms, Hodge operator

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. Index terms - Gauge condition, Finite element method, Whitney elements

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. Index terms - de Rham cohomology groups, Whitney elements, cycles, bounding cycles

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 Maxwell's equations.

A simple finite difference method is validated for the computation of superconductor internal fields using two-dimensional simulations in transverse field. The low-cost method yields H-M curves, where M is the sample's magnetization and H is the applied field. It also yields field and current patterns, is robust and simple to use. An extended Bean model describes the fields inside the superconductors.

Perturbation approaches in terms of a small dimensionless parameter are standard and very useful in field computations. For instance, if the width of an airgap can be considered 'small', no mesh inside the gap will be required. However, it often happens that two such small parameters intervene in the modelling, and 'compete' in some way. In such situations, two, very different, limit problems are candidates as simplified formulations, depending on which of the two small parameters is taken as dominant. A justified concern for ease of computation may then involuntarily lead to the selection of a wrong model, so this raises delicate modelling issues. Some methodological guidelines are offered here in this respect. The case of the small airgap is treated in detail, showing how the penetration depth intrudes in the modelling as a second small parameter, besides the airgap width. Implications as regards numerical schemes, in both main variants where either e or h is the primary unknown, are discussed.

We show how to pass from the local Bean's model, assumed to be valid as a behavior law for a homogeneous superconductor, to a model of similar form, valid on a larger space scale. The process, which can be iterated to higher and higher space scales, consists in solving for the fields e and j over a 'periodicity cell' with periodic boundary conditions.

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

Given the shape of a magnet and its magnetization, point by point, which force does it exert on itself, also point by point? We explain what 'force' means in such a context and how to define it by using the Virtual Power Principle. Mathematically speaking, this force is a vector-valued distribution, with Dirac-like concentrations on surfaces across which the magnetization is discontinuous, i.e., material interfaces. To find these concentrations, we express the force as the divergence of a (symmetric) 2-tensor which generalizes a little the classical Maxwell tensor. [ABSTRACT FROM AUTHOR]