Geometric multipole expansion and its application to semi-neutral inclusions of general shape

TL;DR

This paper introduces a geometric multipole expansion method using Faber polynomials for 2D conductivity and elasticity problems, enabling the design of semi-neutral inclusions with minimal field disturbance.

Contribution

It develops a novel geometric multipole expansion framework and applies it to create semi-neutral inclusions of arbitrary shape with tailored material properties.

Findings

Semi-neutral inclusions effectively reduce field perturbations.

Method applicable to general-shaped inclusions.

Material parameters can be tuned via multipole expansion coefficients.

Abstract

This paper presents a new concept of geometric multipole expansion for the conductivity or anti-plane elasticity problem in two dimensions by using the Faber polynomials. As an application, we construct semi-neutral inclusions of general shape that show relatively negligible field perturbations for low-order polynomial loadings. These inclusions are of the multilayer structure whose material parameters are determined such that some coefficients of geometric multipole expansion vanish.