Some Numerical Approaches for Weakly Random Homogenization

Conference paper

Abstract

We overview a series of recent works addressing homogenization problems for some materials seen as small random perturbations of periodic materials (in a sense made precise in the body of the text). These recent works are joint works with several collaborators: Blanc (Paris 6), Lions (Collège de France), Legoll, Anantharaman, Costaouec (Ecole Nationale des Ponts et Chaussées and INRIA). The theory, developed in [C. R. Acad. Sci. Série I, 343, 717–724 (2006), Journal de Mathématiques Pures et Appliquées, 88, 34–63 (2007)], is only outlined. Next a collection of numerical appropriate approaches introduced in [Note aux Comptes Rendus de l’Académie des Sciences (2009), Thèse de l’ Université Paris Est, C. R. Acad. Sci. Série I, 348, 99–103 (2010)] is presented. The theoretical considerations and the numerical tests provided here show that for the materials with only a small amount of randomness that are considered, a dedicated approach is far more efficient than a direct, stochastic approach.

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References

  1. 1.
    G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (6), pp 1482–1518, 1992MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Anantharaman, Thèse de l’ Université Paris Est, Ecole des PontsGoogle Scholar
  3. 3.
    A. Anantharaman, C. Le Bris, Homogénéisation d’un matériau périodique faiblement perturbé aléatoirement, [Homogenization of a weakly randomly perturbed periodic material], C. R., Math., Acad. Sci. Paris 348, No. 9–10, 529–534, 2010Google Scholar
  4. 4.
    A. Bensoussan, J. L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, 5. North-Holland, Amsterdam-New York, 1978Google Scholar
  5. 5.
    X. Blanc, C. Le Bris, P.-L. Lions, Une variante de la théorie de l’homogénéisation stochastique des opérateurs elliptiques [A variant of stochastic homogenization theory for elliptic operators], C. R. Acad. Sci. Série I, 343, pp 717–724, 2006MATHGoogle Scholar
  6. 6.
    X. Blanc, C. Le Bris, P.-L. Lions, The energy of some microscopic stochastic lattices, Arch. Rat. Mech. Anal., 184, pp 303–339, 2007MATHCrossRefGoogle Scholar
  7. 7.
    X. Blanc, C. Le Bris, P.-L. Lions, Stochastic homogenization and random lattices, Journal de Mathématiques Pures et Appliquées, 88, pp 34–63, 2007MATHCrossRefGoogle Scholar
  8. 8.
    D. Cioranescu, P. Donato, An introduction to homogenization. Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999Google Scholar
  9. 9.
    R. Costaouec, Thèse de l’ Université Paris Est, Ecole des PontsGoogle Scholar
  10. 10.
    R. Costaouec, C. Le Bris, F. Legoll, Approximation numérique d’une classe de problèmes en homogénéisation stochastique, [Numerical approximation of a class of problems in stochastic homogenization], C. R. Acad. Sci. Série I, 348, pp 99–103, 2010MATHGoogle Scholar
  11. 11.
    R. Costaouec, C. Le Bris, F. Legoll, Variance reduction in stochastic homogenization: proof of concept, using antithetic variables, Bol. Soc. Esp. Mat. Apl., 50, pp 9–27, 2010Google Scholar
  12. 12.
    V. V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals. Springer, Berlin, 1994Google Scholar
  13. 13.
    U. Krengel, Ergodic theorems, de Gruyter Studies in Mathematics, vol. 6, de Gruyter, 1985Google Scholar
  14. 14.
    F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. 5 (4) pp 485–507, 1978MathSciNetGoogle Scholar
  15. 15.
    G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (3), pp 608–623, 1989MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. N. Shiryaev, Probability, Graduate Texts in Mathematics, vol. 95, Springer, Berlin, 1984Google Scholar
  17. 17.
    L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot–Watt Symposium, Vol. IV, pp. 136–212, Res. Notes in Math., 39, Pitman, MA, 1979Google Scholar
  18. 18.
    M. Thomas, Propriétés thermiques de matériaux composites : caractérisation expérimentale et approche microstructurale, [Thermal properties of composite materials: experimental characterization and microstructural approach], Thèse de l’ Université de Nantes, Laboratoire de Thermocinétique, CNRS - UMR 6607, 2008Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.CERMICSÉcole Nationale des Ponts et ChausséesMarne-La-Vallée Cedex 2France
  2. 2.INRIA Rocquencourt, MICMAC project, Domaine de VoluceauLe Chesnay CedexFrance

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