Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T22:18:05.644Z Has data issue: false hasContentIssue false
  • English
  • Français

Elements of Mathematical Foundations for Numerical Approaches for Weakly Random Homogenization Problems

Published online by Cambridge University Press:  20 August 2015

A. Anantharaman  and
C. Le Bris
A. Anantharaman*
Affiliation:
Université Paris-Est, CERMICS, Project-team Micmac, INRIA-Ecole des Ponts, 6 & 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France
C. Le Bris*
Affiliation:
Université Paris-Est, CERMICS, Project-team Micmac, INRIA-Ecole des Ponts, 6 & 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France
*
Email:ananthaa@cermics.enpc.fr
Corresponding author.Email:lebris@cermics.enpc.fr
Get access

Abstract

This work is a follow-up to our previous work. It extends and complements, both theoretically and experimentally, the results presented there. Under consideration is the homogenization of a model of a weakly random heterogeneous material. The material consists of a reference periodic material randomly perturbed by another periodic material, so that its homogenized behavior is close to that of the reference material. We consider laws for the random perturbations more general than in. We prove the validity of an asymptotic expansion in a certain class of settings. We also extend the formal approach introduced in. Our perturbative approach shares common features with a defect-type theory of solid state physics. The computational efficiency of the approach is demonstrated.

Keywords

35B2735J1535R6082D30Homogenizationrandom mediadefects
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 4 , April 2012 , pp. 1103 - 1143
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Anantharaman, A., Mathematical Analysis of Some Models in Electronic Structure Calculations and Homogenization, PhD Thesis, Thèse de l’Université Paris-Est, available at http://tel.archives-ouvertes.fr/tel-00558618/fr.Google Scholar
[2]Anantharaman, A. and Le Bris, C., A numerical approach related to defect-type theories for some weakly random problems in homogenization, SIAM Multiscale Model. Sim., 9 (2011), 513–544.CrossRefGoogle Scholar
[3]Anantharaman, A. and LeBris, C., Homogenization of a weakly randomly perturbed periodic material, C. R. Acad. Sci. Paris Sèrie I, 348(9-10) (2010), 529–534.Google Scholar
[4]Blanc, X., Le Bris, C. and Lions, P.-L., Stochastic homogenization and random lattices, J. Math. Pures Appl., 88 (2007), 34–63.Google Scholar
[5]Bourgeat, A. and Piatnitski, A., Approximations of effective coefficients in stochastic homog-enization, Annales de l’Institut Henri Poincaré(B) Probabilités et Statistiques, 40(2) (2004), 153–165.CrossRefGoogle Scholar
[6]Hömander, L., The analysis of linear partial differential operators I: distribution theory and fourier analysis, Grundl. Math. Wissenschaft., 256, Springer, 1983.Google Scholar
[7]Jikov, V. V., Kozlov, S. M. and Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals, Springer Verlag, 1994.Google Scholar