The choice of representative volumes in the approximation of effective properties of random materials

TL;DR

This paper rigorously analyzes a selection method for representative volumes in stochastic homogenization, showing it generally outperforms random selection and can significantly improve accuracy in estimating effective properties of random materials.

Contribution

The paper provides a rigorous theoretical validation of a selection approach for representative volumes, demonstrating its advantages over random sampling in stochastic homogenization.

Findings

Selection method never performs worse than random sampling.

Method can significantly improve accuracy with suitable selection criteria.

Theoretical analysis confirms empirical effectiveness of the approach.

Abstract

The effective large-scale properties of materials with random heterogeneities on a small scale are typically determined by the method of representative volumes: A sample of the random material is chosen - the representative volume - and its effective properties are computed by the cell formula. Intuitively, for a fixed sample size it should be possible to increase the accuracy of the method by choosing a material sample which captures the statistical properties of the material particularly well: For example, for a composite material consisting of two constituents, one would select a representative volume in which the volume fraction of the constituents matches closely with their volume fraction in the overall material. Inspired by similar attempts in material science, Le Bris, Legoll, and Minvielle have designed a selection approach for representative volumes which performs remarkably well in numerical examples of linear materials with moderate contrast. In the present work, we provide a rigorous analysis of this selection approach for representative volumes in the context of stochastic homogenization of linear elliptic equations. In particular, we prove that the method essentially never performs worse than a random selection of the material sample and may perform much better if the selection criterion for the material samples is chosen suitably.