Random composites and the generalized method of Schwarz I. Conductivity problems

TL;DR

This paper introduces a generalized Schwarz method for estimating effective properties of random 2D composites with inclusions, avoiding correlation function computations and providing exact symbolic formulas.

Contribution

It develops a novel approach that computes effective composite properties without using correlation functions, applicable to arbitrary multiply connected domains.

Findings

Method converges for complex domains and contrast parameters.

Provides exact symbolic formulas for effective properties.

Avoids direct computation of correlation functions.

Abstract

Two-phase composites with non-overlapping inclusions randomly embedded in matrix are investigated. A straight forward approach is applied to estimate the effective properties of random 2D composites. First, deterministic boundary value problems are solved for all locations of inclusions, i.e., for all events of the considered probabilistic space $\mathcal C$ by the generalized method of Schwarz. Second, the effective properties are calculated in analytical form and averaged over $\mathcal C$. This method is related to the classic method based on the average probabilistic values involving the $n$-point correlation functions. However, we avoid computation of the correlation functions and compute their weighted moments of high orders by an indirect method which does not address to the correlation functions. The effective properties are exactly expressed through these moments. It is proved that the generalized method of Schwarz converges for an arbitrary multiply connected doubly periodic domain and for an arbitrary contrast parameter. The proposed method yields effective in symbolic-numeric computations formulae of high order in concentration. In particular, the Torquato--Milton parameter $\zeta_1$ is exactly written for circular inclusions.