A numerical approach related to defect-type theories for some weakly random problems in homogenization

TL;DR

This paper introduces a computational method to determine the effective macroscopic properties of materials with small random perturbations, using an asymptotic expansion inspired by defect theories, demonstrating efficiency in calculations.

Contribution

It develops a novel perturbative approach for homogenization of weakly random media, providing first and second-order corrections with demonstrated computational efficiency.

Findings

Effective computation of homogenized properties with small random perturbations.

Asymptotic expansion approach validated for weakly random materials.

Method shares features with defect-type theories in solid state physics.

Abstract

We present in this paper an approach for computing the homogenized behavior of a medium that is a small random perturbation of a periodic reference material. The random perturbation we consider is, in a sense made precise in our work, a rare event at the microscopic level. It however affects the macroscopic properties of the material, and we indeed provide a method to compute the first and second-order corrections. To this end, we formally establish an asymptotic expansion of the macroscopic properties. Our perturbative approach shares common features with a defect-type theory of solid state physics. The computational efficiency of the approach is demonstrated.