Alain Bossavit

• Research Director at Supélec

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...

Many alternatives to the classical Poynting vector P have been proposed, but they lack a property of locality that makes P special and makes it the right choice to represent the energy flux. We give a geometrical proof of this uniqueness. Similar considerations apply to the Maxwell stress tensor.

We prove the 50/50 rule of Fig. 1 by a Lagrangian technique (all fields pulled back to the reference configuration at time 0) which exposes the necessity of two circumstances for the result to hold: 1) time-invariance of currents, when expressed in “material form” and 2) equality of (magnetic) coenergy and energy, which excludes nonlinear B-H laws....

When two magnets are stuck together, where do magnetic forces operate and which formulas should one apply to compute them? Such frequently asked questions do not find immediate answers in the literature on forces, mainly because the force field is obtained, by the Virtual Power Principle (VPP), as a (mathematical, vector-valued) distribution, not a...

Electrets and piezoelectric components of various actuating devices require to take displacement currents into account, hence a less familiar modelling than in the eddy-currents case. We review basic formulations and how to discretize them. Copyright © 2013 John Wiley & Sons, Ltd.

In magnetoelastic interactions, the virtual power principle is powerful enough to solve all force problems, but only if the coupled constitutive laws are completely known. We use it to derive force densities in magnetized matter for a few such laws, insisting on the fact that a full knowledge of the magnetic field may not be enough to determine the...

Optics Express a publié en mars 2012 un article concernant la transposition au domaine de la thermique (équation de la chaleur) du principe de la cape d'invisibilité optique. Les auteurs y présentent en particulier la théorie bidimensionnelle d'un métamatériau en forme de disque, aux propriétés thermiques remarquables en conduction pure (équation d...

A plea for the introduction, in advanced electromagnetics courses, of some basic differential geometric notions: covectors, differential forms, Hodge operators. The main advantages of this evolution should be felt in computational electromagnetism. It may also shed some new light on the concept of material isotropy.

Purpose – The purpose of this paper is to clarify the status of Maxwell's tensor with respect to the virtual power principle (VPP). Design/methodology/approach – Mathematical analysis is employed. Findings – The VPP, logically stronger, is more fundamental. Maxwell's tensor derives from it, under further restrictive assumptions, and hence, its ra...

Different analytical and numerical methods for calculating the surface impedance of a high impedance surface (HIS) structure have been studied and compared. In this article, two new approaches are proposed for calculating the surface impedance for general 2D-spatial periodic HIS structures, which include both the symmetric and asymmetric planar str...

Electrets and piezolectric components of various actuating devices require not to neglect displacement currents, hence a less familiar modelling than in the eddy-currents case. We review basic formulations and how to discretize them.

We show that magnetism and elasticity have very similar mathematical structures when fields are considered as differential forms of adequate nature. The same discretization principles and techniques that succeeded in electromagnetism, notably the use of edge elements, then lead to a manageable form of coupled elasto-magnetic problems.

A method for constructing Whitney forms is proposed, which applies to tetrahe- dra, hexahedra, triangular prisms, and pyramids in a similar way, and proceeds from a unique generating principle, thus unifying their presentation. The principle automatically enforces conformity (i.e., "tangential" or "normal continuity" of the elementary proxy fields)...

In this paper, we develop a method to homogenize split-ring arrays in the frequency domain. The expected resonance and negative permeability are obtained via numerical simulations with the finite elements method. A subtle modelization of the split-ring with a closed ring pemits us to avoid meshing the small split, while maintaining the resonant beh...

Low-order Whitney elements are widely used for electromagnetic field problems. Higher-order approximations are receiving increasing interest, but their definition remains unduly complex. In this paper we propose a new simple construction for Whitney p-elements of polynomial degree higher than one that use only degrees of freedom associated to p-cha...

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...

Homogenization, which reduces the cost of numerical simulations in materials with repetitive structure, is a promising approach to the design o metamaterials. This cost reduction stems from the possibility to compute effectiv permeability and permittivity of an equivalent homogenized material by solving an auxiliary “cell problem” on the generating...

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...

Purpose – The paper aims at justifying the operational rule “a voltmeter's reading is the electromotive force, as it existed before branching it, along the path γ traced out by the connectors between the contact points”. Design/methodology/approach – A simple application of Faraday's law is enough to make the result plausible. Then it is shown tha...

Purpose – It is well known that complementarity can provide bilateral bounds on energy, in numerical approximations of nonlinear magnetostatics. Questions whether like force is, up to sign, the derivative of energy, can such bounds also apply to forces, torques, and other concepts? Design/methodology/approach – This is a discussion paper exploring...

Low-order Whitney forms are widely used for electromagnetic field problems. Higher-order ones are increasingly applied, but their development is hampered by the complexity of the generation of element basis functions and of the localisation of the corresponding degrees of freedom on the mesh volumes. The paper aims to give a geometrical localisatio...

In this paper we propose a novel homogenization technique for computing the quasi-static effective parameters of a three-dimensional (3D) lossless lattice of dielectric particles with complex shape. This technique offers the possibility to evaluate, in an accurate manner, the behavior of the electromagnetic field within the microstructure when the...

An efficient technique is proposed for analyzing two- and three-dimensional lossy periodic composite materials, combining an asymptotic multiscale method with the unfolding method. The computed effective conductivities for square cylinders and cubes suspended in a host isotropic medium are compared to the Maxwell-Garnett mixing formula predictions....

A novel methodology to evaluate the effective parameters of a three-dimensional lattice of chiral inclusions is presented. The homogenization is based upon mathematical arguments. The finite element technique is used to compute the constitutive parameters.

Metamaterials are crystal-like composites, derived by spatial translations from a generating cell, where highfrequency electromagnetic fields propagate in surprising ways at some frequencies: Negative refraction, "bending light the wrong way", refocusing of divergent beams into tightly parallel ones, are observed, and attributed to the fact that, a...

En nous astreignant dfinir de faon suffisamment complte le type abstrait le plus proche possible de l'objet d'tude, nous avons d en particulier dvelopper une algbre des relations entre les oprateurs du type qui se rvle trs utile la fois pour spcifier les problmes rsoudre et pour mener les calculs. Il s'avre en fin de compte que les algorithmes para...

This chapter separates the various layers present in the structure of E3 for each subpart of the Maxwell system of equations. The chapter considers four dimensions p = 0, . . . , 3 of manifolds in E3, four kinds of integrals that are constantly encountered in Physics. Such integrals can be defined on cells first, then extended by linearity to chain...

We present a novel method for analyzing 2D and 3D lossy periodic composites materials, combining an asymptotic multiscale method combined with the unfolding method. The computed effective conductivity for square cylinders (2D), and cubes (3D) suspended in a host isotropic medium are compared with the Maxwell-Garnett mixing formula predictions. The...

We study the behavior of the electromagnetic field of a medium presenting periodic mi- crostructures made of bianisotropic material. We reconsider the classical multi-scale homoge- nization technique by giving a new approach based upon the periodic unfolding method . The limiting homogeneous constitutive law is thus rigorously justified both in the...

When fields that describe a physical situation extend over the whole space, but with complex behavior (nonlinearity, coupling, etc.) only in a bounded region and simple behavior in the rest of space, it may be worthwhile to treat the inner field by a finite elements procedure and the outer field by boundary elements. We address this “marriage” prob...

In this presentation the classical multi-scale homogenization technique is associated to a new approach for the computation of the effective constitutive parameters of three-dimensional metallo-dielectric lattices. This approach is the periodic unfolding method (1, 2). It is based on the decomposition of the fields in a main macroscopic part withou...

RESUMEN RESUMEN The paper is mainly focused on the construction of two transfer operators between nested grids in the case of Whitney finite elements ( node -, edge -, face -, or volume - based ). These transfer operators , instances of what is called " chain map " in homology , have duals acting on cochains , that is to say , arrays of degrees o...

We present a novel method for analyzing 2-D and 3-D lossy periodic composites materials, combining the asymptotic multiscale method with the periodic unfolding method. The computed effective permittivity and conductivity for square cylinders (2-D), and cubes (3-D) suspended in a host isotropic medium are compared with the Maxwell-Garnett mixing for...

On propose dans cette Note une justification rigoureuse de la loi de comportement limite d'un matériau électromagnétique bianisotrope avec mémoire présentant une structure périodique et dont tous les paramètres constitutifs dépendent du temps. Cette étude est menée sur les formulations temporelle puis fréquentielle des équations de Maxwell en appli...

The idea of modelling space as two interacting equivalent networks, one for currents, one for magnetic fluxes, pervades computational electromagnetics since its beginnings. The Yee scheme, the TLM method, can thus be interpreted. But this is also true of finite element- or finite volume-inspired more recent proposals, as we show, so the idea is not...

In this paper, we focus on the representation of a divergence-free vector field, defined, on a connected nonsimply connected domain $\Omega \subset \R^3$ with a connected boundary $\Gamma$, by its curl and its normal component on the boundary. The considered problem is discretized with H(curl)- and H(div)-conforming finite elements. In order to ens...

Solving a given problem by two "complementary" formulations gives useful bilateral bounds on some quantities. Mixed hybrid methods achieve the same result by solving only one algebraic system and by using two different methods to interpolate from the computed degrees of freedom.

Existent software does not seem to be able to cope with situations where two conductors slide on each other, with current going through the interface. Modelling a homopolar dynamo, for instance, would require this capability. We extend standard eddy current theory to cover this case, and describe the basic algorithm. The approach is Lagrangian in s...

The topological features of a given domain Ω in R3 are here analyzed by means of the homology groups of first and second order. Algebraic topology together with a particular QR type factorization in Z can be used to know whether Ω is connected and simply connected, as well as to check if a given discretization of Ω by means of simplices has been co...

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

The idea of modelling space as two interacting equivalent networks, one for currents, on for magnetic fluxes, pervades computational electromagnetics since its beginnings. The Yee scheme, the TLM method can thus be interpreted. But this is also true of finite element-or finite volume-inspired more recent proposals, as we show, so the idea is not in...

The v × × × × B term in eddy current equations for conducting fluids is an instance of contraction of a differential form by a vector field. We search for a natural way to discretize such contractions. Looking at the operation of extrusion of a manifold, which is dual to contraction, provides the main clue. Two example applications, Carpenter's gau...

Large ratios of extreme nonzero conductivities in eddy current problems cause poor conditioning, due to the kernel of the curl-curl operator. Solutions to this problem (wrongly blamed on edge elements by some) are discussed

We argue that, in static situations (magnetostatics, for instance) all materials can be considered as locally isotropic, though not homogeneous. Genuine anisotropy appears only in dynamic situations, e.g., in wave propagation problems.

The aim of upscaling is to determine equivalent homogeneous parameters at a coarse-scale from a spatially oscillating fine-scale parameter distribution. To be able to use a limited number of relatively large grid-blocks in numerical oil reservoir simulators or groundwater models, upscaling of the permeability is frequently applied. The spatial fine...

The geometrical approach to Maxwell's equations promotes a way to discretize them that can be dubbed "Generalized Finite Differences", which has been realized independently in several computing codes. The main features of this method are the use of two grids in duality, the "metric-free" formulation of the main equations (Ampère and Faraday), and t...

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We propose an analysis (discretization techniques, convergence) of numerical schemes for Maxwell equations which use two meshes (not necessarily tetrahedral), dual to each other. Schemes of this class generalize Yee's “finite difference in time domain” method (FDTD). We distinguish network equations (the discrete equivalents of Faraday's law and Am...

Hysteresis is not a property of the constitutive laws of superconductors, but an observed macroscopic property of some devices, including superconductive material, which one uses in order to access a key parameter in modelling, namely the critical current density. The paper aims at clarifying these issues, by exposing the relation between the criti...

We propose an analysis (discretization techniques, convergence) of numerical schemes for Maxwell equations which use two meshes (not necessarily tetrahedral), dual to each other. Schemes of this class generalize Yee's "finite difference in time domain" method. We distinguish network equations (the discrete equivalents of Faraday's law and Amp re's...

The present paper is in the spirit of the latter approach. It is, however, very preliminary, as other issues than boundary conditions would need attention from the above viewpoint, which are not addressed here. We also ignore non-homogeneous local conditions, including so-called "absorbing" ones. Our main concern is with topological features: prese...

Although edge elements satisfactorily solve the eddy current problem, formulations allowing the use of standard, node-based elements, are still looked for. But “well-posed” formulations have been elusive up to now. We propose one, based on a particular gauge, div(s a)=-s 2µ v, close to the “Lorenz gauge” of several recent publications, but not iden...

We discuss a method of approximation of eddy-current equations in bounded space that relies on the use of the electric field as primary variable in the conductive parts and of the magnetic field in the air. Theoretical issues about this hybrid approach are addressed (is the problem well-posed?) and one shows that edge elements offer a straightforwa...

Some EM 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. In particular, one obtains error bounds this way. Taking as an example the static conduction problem, the authors propose a geometrical presentation of this theory, which generalizes...

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

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...

A Galerkin edge-element solution technique for Maxwell's equations in time domain is discussed. With proper diagonal lumping of one of the mass matrices, it can be seen as a generalization to a tetrahedral mesh and its barycentric dual of the staggered-grid finite difference scheme known nowadays as FDTD, or Yee's scheme. A new approach to the lump...

Voltage drops", in the 2D situation of Figs 1 & 2, are precisely defined and used in a variational formulation of the problem in terms of the electric field, allowing outside circuit conditions and (at variance with the more standard vector potential formulation) nonlinear conduction laws.

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 Hedge is a central problem. In this light we interpret finite element techniques as a realization of the discrete Hedge operator in the Whitney compl...

Some popular techniques in the numerical simulation of Eddy-Current NDT (separate computation of the "no crack field" and of the "crack current", superposition, use of differently graded meshes) are discussed and justified by considerations of symmetry and perturbation theory. An important subtask in crack-localization is the so-called "direct prob...

First Page of the Article

In this chapter, the two mirror symmetries with respect to vertical planes are taken into account. The chapter discusses the model problem of the magnetostatics version of the “Bath cube” setup. The chapter looks for the solution as (first step) an element of some predefined functional space that (second step) can be characterized as the minimizer...

This chapter solves a non-stationary model, and the starting point is Maxwell's system with Ohm's law. The chapter introduces the main simplification, often described as the “low-frequency approximation,” which consists in neglecting the term of “Maxwell displacement currents,” as well as the method—implemented under the code name Trifou. The key i...

This chapter explains symbols, discusses constitutive laws, and indicates how a variety of mathematical models derive from the basic Maxwell equations. Computational electromagnetism is concerned with the numerical study of Maxwell equations completed by constitutive laws to account for the presence of matter and the field–matter interaction. Const...

This chapter addresses a typical magnetostatics modelling, for which the computational domain is a priori in the whole space. The technique of “infinite elements and boundary elements in association,” which is essential to the treatment of all the open-space problems—static or not—is introduced in the chapter. Irrespective of the selected formulati...

This chapter contains an enlarged mathematical framework, studying the three fundamental operators—grad, rot, div—from the functional point of view, thus making visible a rich structure, which happens to be the right functional framework for Maxwell's equations. The three differential operators should be treated not only in parallel but also as an...

By using complementarity (solving for both potentials, scalar and vector), one often can provide bilateral bounds for some quantitites of interest, like for instance the reluctance of a circuit. However, the vector potential method is expensive, and does not make use of information acquired in the first phase of solving for the scalar potential. We...

This paper deals with a numerical method to study the electromagnetic behavior in a solid superconductor. The model shows that the anisotropy of hysteretic phenomena is influenced by the dependence of the critical current density on the induction field

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

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

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 integr...

Discusses some general algorithmic problems encountered when solving linear systems in electromagnetism, such as, e.g. the well-known difficulties linked with the curl-curl operator in discrete form. The problem is examined as part of a larger one: solving “mixed” (or as we prefer to say, “constrained”) linear systems, with Lagrange multipliers, fo...