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Bounds on the effective behavior of a square conducting lattice
, 2004
"... A collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behavior of a square two–dimensional lattice of such resistors when each type is present in fixed proportion in the lattice. The macroscopic behavior is that of an anisotropic ..."
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A collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behavior of a square two–dimensional lattice of such resistors when each type is present in fixed proportion in the lattice. The macroscopic behavior is that of an anisotropic conductor at the continuum level and the goal of the paper is to describe the set of all possible such conductors. This is thus a problem of bounds in the footstep of an abundant literature on the topic in the continuum case. The originality of the paper is that the investigation focusses on the interplay between homogenization and the passage from a discrete network to a continuum. A set of bounds is proposed and its optimality is shown when the proportion of each resistor on the discrete lattice is 1 2. We conjecture that the derived bounds are optimal for all proportions. Keywords: Γ–convergence, lattice, resistor network, homogenization, bounds, optimality. 1
REDUCTION OF THE RESONANCE ERROR PART 1: APPROXIMATION OF HOMOGENIZED COEFFICIENTS
, 2010
"... Abstract. This paper is concerned with the approximation of effective coefficients in homogenization of linear elliptic equations. One common drawback among numerical homogenization methods is the presence of the socalled resonance error, which roughly speaking is a function of the ratio ε/η, where ..."
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Abstract. This paper is concerned with the approximation of effective coefficients in homogenization of linear elliptic equations. One common drawback among numerical homogenization methods is the presence of the socalled resonance error, which roughly speaking is a function of the ratio ε/η, where η is a typical macroscopic lengthscale and ε is the typical size of the heterogeneities. In the present work, we propose an alternative for the computation of homogenized coefficients (or more generally a modified cellproblem), which is a first brick in the design of effective numerical homogenization methods. We show that this approach drastically reduces the resonance error in some standard cases.
HOMOGENIZATION ON LATTICES: SMALL PARAMETER LIMITS, HMEASURES, AND DISCRETE WIGNER MEASURES
"... Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of m ..."
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Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of magnetic dipoles on a Bravais lattice, (letting the lattice parameter tend to zero). In order to describe the smallparameter limit, we use discrete Wigner transforms to transform the storedenergy which is given by the double convolution of a sequence of (dipole) functions on a Bravais lattice with a kernel, homogeneous of degree with N with the cancellation property, as the lattice parameter tends to zero. By rescaling and using Fourier methods, discrete Wigner transforms in particular, to transform the problem to one on the torus, we are able to characterize the smallparameter limit of the energy depending on whether the dipoles oscillate on the scale of the lattice, oscillate on a much longer lengthscale, or converge strongly. In the case where> N, the result is simple and can be characterized by anintegral with respect to the Wigner measure limit on the torus. In the case where = N, oscillations essentially on the scale of the lattice must be separated from oscillations essentially onamuch longer lengthscale in order to characterize the energy in terms of the Wigner measure limit on the torus, an Hmeasure limit, and the limiting magnetization. We show that the classical
par
, 2013
"... Sujet: Qualitative and quantitative results in stochastic homogenization Soutenance le 24 février 2012 devant le jury composé de: ..."
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Sujet: Qualitative and quantitative results in stochastic homogenization Soutenance le 24 février 2012 devant le jury composé de:
E. Cancès and S. Labbé, Editors NUMERICAL HOMOGENIZATION: SURVEY, NEW RESULTS, AND PERSPECTIVES
"... Abstract. These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate ..."
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Abstract. These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which Hconverges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar’s correctors. Hence the convergence analysis of these methods relies on the Hconvergence theory. When one is interested in convergence rates, the story is different. In particular one first needs to make additional structure assumptions on the heterogeneities (say periodicity for instance). In that case, a crucial tool is the spectral interpretation of the corrector equation by Papanicolaou and Varadhan. Spectral analysis does not only allow to obtain convergence rates, but also to devise efficient new approximation methods. For both qualitative and quantitative properties, the development and the analysis of numerical homogenization methods rely on seminal concepts of the homogenization theory. These notes contain some new results. Résumé. Ces notes de cours dressent un état de l’art des méthodes d’homogénéisation numérique pour les équations elliptiques linéaires. Le fil conducteur choisi est l’analyse. La plupart des méthodes d’homogénéisation numérique s’interprète comme des discrétisations (plus ou moins différentes) d’une