Asymptotic analysis of solutions to transmission problems in solids with many inclusions

TL;DR

This paper develops an asymptotic method to approximate solutions for transmission problems in solids with many small inclusions, providing rigorous error estimates and demonstrating efficiency through numerical comparisons.

Contribution

The paper introduces a new asymptotic approximation for transmission problems with multiple inclusions, including rigorous error bounds and numerical validation.

Findings

Asymptotic approximation closely matches finite element solutions.

Error estimates are rigorously justified.

Numerical results confirm computational efficiency.

Abstract

We construct an asymptotic approximation to the solution of a transmission problem for a body containing a region occupied by many small inclusions. The cluster of inclusions is characterised by two small parameters that determine the nominal diameter of individual inclusions and their separation within the cluster. These small parameters can be comparable to each other. Remainder estimates of the asymptotic approximation are rigorously justified. Numerical illustrations demonstrate the efficiency of the asymptotic approach when compared with benchmark finite element algorithms.