Construction of weakly neutral inclusions of general shape by imperfect interfaces

TL;DR

This paper presents a method to construct weakly neutral inclusions of arbitrary shape in two dimensions using imperfect interfaces, reducing field perturbations compared to perfect bonding.

Contribution

It introduces a geometric condition and an imperfect interface parameter to realize weakly neutral inclusions of general shape, extending beyond circular shapes in isotropic media.

Findings

Weakly neutral inclusions cause significantly less field perturbation than perfect interfaces.

The geometric condition is linked to the conformal mapping coefficient.

Numerical examples demonstrate the effectiveness of the proposed construction.

Abstract

Upon insertion of an inclusion into a medium with the uniform field, if the field is not perturbed at all outside the inclusion, then it is called a neutral inclusion. It is called a weakly neutral inclusion if the field is perturbed weakly. The inclusions neutral to multiple uniform fields are of circular shape if the medium is isotropic, and any other shape cannot be neutral. We consider in this paper the problem of constructing inclusions of general shape which are weakly neutral to multiple fields in two dimensions. We show that a simply connected domain satisfying a certain geometric condition can be realized as a weakly neutral inclusion to multiple fields by introducing an imperfect interface parameter on the boundary. The geometric condition on the domain and the imperfect interface parameter are determined by the first coefficient of the conformal mapping from the exterior of the unit disk onto the exterior of the domain. We provide some numerical examples to compare field perturbations by weakly neutral inclusions and perfectly bonding interfaces. They clearly show that the field perturbation by weakly neutral inclusions is much weaker.