On near-cloaking for linear elasticity

TL;DR

This paper investigates near-cloaking in linear elasticity, establishing theoretical foundations, numerical validation, and proposing an approximate isotropic cloaking algorithm for elastic materials.

Contribution

It provides a rigorous analysis of near-cloaking for elasticity systems using transformation media theory and introduces an approximate isotropic cloaking method.

Findings

Numerical experiments demonstrate effective near-cloaking.

Sharp estimates for solution convergence are validated.

An approximate isotropic cloak algorithm is proposed.

Abstract

We make precise some results on the cloaking of displacement fields in linear elasticity. In the spirit of transformation media theory, the transformed governing equations in Cosserat and Willis frameworks are shown to be equivalent to certain high contrast small defect problems for the usual Navier equations. We discuss near-cloaking for elasticity systems via a regularized transform and perform numerical experiments to illustrate our near-cloaking results. We also study the sharpness of the estimates from [H. Ammari, H. Kang, K. Kim and H. Lee, J. Diff. Eq. 254, 4446-4464 (2013)], wherein the convergence of the solutions to the transmission problems is investigated, when the Lam\'e parameters in the inclusion tend to extreme values. Both soft and hard inclusion limits are studied and we also touch upon the finite frequency case. Finally, we propose an approximate isotropic cloak algorithm for a symmetrized Cosserat cloak.