Nearly cloaking the elastic wave fields

TL;DR

This paper develops a mathematical framework for nearly cloaking elastic waves using transformation elastodynamics, introducing a regularized approach with a lossy layer to achieve high-accuracy cloaking of arbitrary contents.

Contribution

It presents a comprehensive mathematical approach to approximate elastic wave cloaking, including regularization techniques and the design of lossy layers to overcome resonances.

Findings

Regularized cloaking avoids singularities in material tensors.

Lossy layers are essential for effective cloaking without resonance issues.

The scheme achieves high-accuracy near-invisibility for elastic waves.

Abstract

In this work, we develop a general mathematical framework on regularized approximate cloaking of elastic waves governed by the Lam\'e system via the approach of transformation elastodynamics. Our study is rather comprehensive. We first provide a rigorous justification of the transformation elastodynamics. Based on the blow-up-a-point construction, elastic material tensors for a perfect cloak are derived and shown to possess singularities. In order to avoid the singular structure, we propose to regularize the blow-up-a-point construction to be the blow-up-a-small-region construction. However, it is shown that without incorporating a suitable lossy layer, the regularized construction would fail due to resonant inclusions. In order to defeat the failure of the lossless construction, a properly designed lossy layer is introduced into the regularized cloaking construction . We derive sharp asymptotic estimates in assessing the cloaking performance. The proposed cloaking scheme is capable of nearly cloaking an arbitrary content with a high accuracy.