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Homogenization of a weakly randomly perturbed periodic material. (Homogénéisation d’un matériau périodique faiblement perturbé aléatoirement.) (French. Abridged English version) Zbl 1193.35008

Summary: We present an approach aiming at computing the first-order homogenized behaviour of a medium consisting of a randomly perturbed periodic reference material. The approach, which proves to be very efficient from a computational point of view, is rigorously founded in a certain class of settings and has been successfully numerically tested for more general settings.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
74Q05 Homogenization in equilibrium problems of solid mechanics
65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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References:

[1] A. Anantharaman, Thèse de l’Université Paris-Est, en préparation; A. Anantharaman, Thèse de l’Université Paris-Est, en préparation
[2] A. Anantharaman, C. Le Bris, A numerical approach based on defect-type theories for some weakly random problems in homogenization, en préparation; A. Anantharaman, C. Le Bris, A numerical approach based on defect-type theories for some weakly random problems in homogenization, en préparation · Zbl 1233.35014
[3] A. Anantharaman, C. Le Bris, Mathematical foundations and genericity of a numerical approach for some weakly random problems in homogenization, en préparation; A. Anantharaman, C. Le Bris, Mathematical foundations and genericity of a numerical approach for some weakly random problems in homogenization, en préparation · Zbl 1373.35027
[4] Babuska, I.; Andersson, B.; Smith, P. J.; Levin, K., Damage analysis of fiber composites. Part I : Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg., 172, 27-77 (1999) · Zbl 0956.74048
[5] Blanc, X.; Le Bris, C.; Lions, P.-L., Stochastic homogenization and random lattices, J. Math. Pures Appl., 88, 34-63 (2007) · Zbl 1129.60055
[6] Bourgeat, A.; Piatnitski, A., Approximations of effective coefficients in stochastic homogenization, Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, 40, 2, 153-165 (2004) · Zbl 1058.35023
[7] Costaouec, R.; Le Bris, C.; Legoll, F., Approximation numérique d’une classe de problèmes en homogénéisation stochastique, C. R. Acad. Sci. Paris Série I vol., 348, 1-2, 99-103 (2010) · Zbl 1180.65166
[8] Jikov, V. V.; Kozlov, S. M.; Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals (1994), Springer-Verlag
[9] Sakata, S.; Ashida, F.; Kojima, T.; Zako, M., Three-dimensional stochastic analysis using a perturbation-based homogenization method for elastic properties of composite material considering microscopic uncertainty, International Journal of Solids and Structures, 45, 894-907 (2008) · Zbl 1167.74384
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