We present an efficient method for the computation of homogenized coefficients of divergence-form operators with random coefficients. The approach is based on a multiscale representation of the homogenized coefficients. We then implement the method numerically using a finite-element method with hierarchical hybrid grids, which is a semi-implicit method allowing for significant gains in memory usage and execution time. Finally, we demonstrate the efficiency of our approach on two- and three-dimensional examples, for piecewise-constant coefficients with corner discontinuities. For moderate ellipticity contrast and for a precision of a few percentage points, our method allows to compute the homogenized coefficients on a laptop computer in a few seconds, in two dimensions, or in a few minutes, in three dimensions.
We consider a large number of randomly dispersed spherical, identical, inclusions in a bounded domain, with conductivity different than that of the host medium. In the dilute limit, with some mild assumptions on the first few marginal probability densities (no periodicity or stationarity is assumed), we prove convergence in the H 1 norm of the expectation of the solution of the steady state heat equation to the solution of an effective medium problem, where the conductivity is given by the Clausius–Mossotti formula. Error estimates are provided as well. CONTENTS Introduction ??1. Asymptotic expansion of Laplace's variational integrals ??2. Computation of dispersive media ??3. Extremal property of the hexagonal distribution of discs ??4. Random chess structure ??5. Asymptotic expansion of the effective conductivity for a small concentration of the non-conducting cellsConcluding remarks References   We consider a large number of randomly dispersed spherical, identical, perfectly conducting inclusions (of infinite conductivity) in a bounded domain. The host medium's conductivity is finite and can be inhomogeneous. In the dilute limit, with some boundedness assumption on a large number (proportional to the global volume fraction raised to the power of @math ) of marginal probability densities, we prove convergence in @math norm of the expectation of the solution of the steady state heat equation, to the solution of an effective medium problem, where the conductivity is given by the Clausius--Mossotti formula. Error estimates are provided as well.   This work is a follow-up to our previous work "A numerical approach related to defect-type theories for some weakly random problems in homogenization" (preprint available on this archive). 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 our previous work cited above. We prove the validity of an asymptotic expansion in a certain class of settings. We also extend the formal approach introduced in our former work. Our perturbative approach shares common features with a defect-type theory of solid state physics. The computational efficiency of the approach is demonstrated. We consider a medium composed of randomly dispersed spherical, identical inclusions in a bounded domain, with conductivity different than that of the host medium. We study the limit where the number inclusions tends to infinity but their volume fraction remains fixed. For small volume fractions, we prove convergence, in the @math norm ( @math ), of the expectation of the solution of the steady state heat equation to the solution of an effective medium problem, where the conductivity is given by the Clausius--Mossotti formula. This improves a previous result which required that the volume fraction tend to zero as the inclusion number goes to infinity.   