HOMOGENIZATION, SYMMETRY, AND PERIODIZATION IN DIFFUSIVE RANDOM MEDIA

doi:10.1016/S0252-9602(12)60008-3

Abstract

Abstract:

We present a systematic study of homogenization of diffusion in random me-dia with emphasis on tile-based random microstructures. We give detailed examples of several such media starting from their physical descriptions, then construct the associated probability spaces and verify their ergodicity. After a discussion of material symmetries of random media, we derive criteria for the isotropy of the homogenized limits in tile-based
structures. Furthermore, we study the periodization algorithm for the numerical approxi-mation of the homogenized diffusion tensor and study the algorithm’s rate of convergence. For one dimensional tile-based media, we prove a central limit result, giving a concrete rate of convergence for periodization. We also provide numerical evidence for a similar central limit behavior in the case of two dimensional tile-based structures.

Key words: homogenization, periodization, random media, ergodic dynamical systems, material symmetry, isotropy

CLC Number:

37A05

Alen Alexanderian, Muruhan Rathinam, Rouben Rostamian. HOMOGENIZATION, SYMMETRY, AND PERIODIZATION IN DIFFUSIVE RANDOM MEDIA[J].Acta mathematica scientia,Series B, 2012, 32(1): 129-154.

Trendmd

References

[1] Alexanderian A, Rathinam M, Rostamian R. Irreducibility of a symmetry group implies isotropy. J Elasticity, 2011, 102(2): 151–174

[2] Alicandro R, Cicalese M, Gloria A. Integral representation results for energies defined on stochastic lattices
and application to nonlinear elasticity. Arch Ration Mech Anal, 2011, 200(3): 881–943

[3] Bal G. Central limits and homogenization in random media. Multiscale Modeling & Simulation, 2008, 7(2): 677–702

[4] Bensoussan A, Lions J L, Papanicolaou G. Asymptotic Analysis for Periodic Structures. Volume 5 of Studies in Mathematics and its Applications. Amsterdam: North-Holland Publishing Co, 1978

[5] Berdichevsky V, Jikov V, Papanicolaou G. Homogenization. Volume 50 of Series on Advances in Mathe-matics for Applied Sciences. River Edge, NJ: World Scientific Publishing Co Inc, 1999

[6] Bourgeat A,PiatnitskiA.Estimatesinprobabilityofthe residualbetween the random andthe homogenized
solutions of one-dimensional second-order operator. Asym Anal, 1999, 21(3/4): 303–315

[7] Bourgeat A, Piatnitski A. Approximations of e?ective coe?cients in stochastic homogenization. Ann Inst
H Poincar′e Probab Statist, 2004, 40(2): 153–165

[8] Brockwell P J, Davis R A. Time Series: Theory and Methods. Springer Series in Statistics. 2nd ed. New
York: Springer-Verlag, 1991

[9] Bystr¨om J, Dasht J, Wall P. A numerical study of the convergence in stochastic homogenization. J Anal
Appl, 2004, 2(3): 159–171

[10] Bystr¨om J, Engstr¨om J, Wall P. Periodic approximation of elastic properties of random media. Adv
Algebra Anal, 2006, 1(2): 103–113

[11] Chechkin G A, Piatnitski A L, Shamaev A S. Homogenization. Volume 234 of Translations of Mathe-matical Monographs. Methods and Applications, Translated from the 2007 Russian Original by Tamara Rozhkovskaya. Providence, RI: American Mathematical Society, 2007

[12] Cioranescu D, Donato P. An Introduction to Homogenization. Volume 17 of Oxford Lecture Series in
Mathematics and its Applications. New York: The Clarendon Press Oxford University Press, 1999

[13] Cornfeld I P, Fomin S V, Sinaffi Ergodic Theory. Volume 245 of Grundlehren der Mathematischen Wis-
senschaften [Fundamental Principles of Mathematical Sciences]. New York: Springer-Verlag, 1982

[14] Efendiev Y, Pankov A. Numerical homogenization and correctors for nonlinear elliptic equations. SIAM J Appl Math, 2004, 65(1): 43–68 (electronic)

[15] Efendiev Y, Pankov A. Numerical homogenization of nonlinear random parabolic operators. Multiscale
Model Simul, 2004, 2(2): 237–268 (electronic)

[16] Fulton W, Harris J. Representation Theory. Volume 129 of Graduate Texts in Mathematics. New York:
Springer-Verlag, 1991