Eigenvalue problem in a solid with many inclusions: asymptotic analysis

TL;DR

This paper develops an asymptotic method to approximate the first eigenvalue and eigenfunction of Laplace's operator in solids with many small inclusions, providing efficient alternatives to numerical methods.

Contribution

It introduces a novel asymptotic analysis for eigenvalue problems in solids with numerous inclusions, including error estimates and numerical validation.

Findings

Asymptotic approximations are accurate and efficient.

Numerical results outperform finite element methods in complex geometries.

The approach is applicable to three-dimensional solids with clusters of inclusions.

Abstract

We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of the small inclusions is characterised by a small parameter which is much larger compared with the nominal size of inclusions. Remainder estimates for the approximations to the first eigenvalue and associated eigenfield are presented. Numerical illustrations are given to demonstrate the efficiency of the asymptotic approach compared to conventional numerical techniques, such as the finite element method, for three-dimensional solids containing clusters of small inclusions.