On optimal cloaking-by-mapping transformations

Yves Capdeboscq,Michael S Vogelius

doi:10.1051/m2an/2022004

Abstract

A central ingredient of cloaking-by-mapping is the diffeomorphism which transforms an annulus with a small hole into an annulus with a finite size hole, while being the identity on the outer boundary of the annulus. The resulting meta-material is anisotropic, which makes it difficult to manufacture. The problem of minimizing anisotropy among radial transformations has been studied in Griesmaier and Vogelius [Inverse Prob. 30 (2014) 17]. In this work, as in Griesmaier and Vogelius [Inverse Prob. 30 (2014) 17], we formulate the problem of minimizing anisotropy as an energy minimization problem. Our main goal is to provide strong evidence for the conjecture that for cloaks with circular boundaries, non-radial transformations do not lead to lower degree of anisotropy. In the final section, we consider cloaks with non-circular boundaries and show that in this case, non-radial cloaks may be advantageous, when it comes to minimizing anisotropy.

Highlights

A central ingredient in the construction of cloaks by the passive cloaking technique, known as “cloaking by mapping”, is the diffeomorphism, which transforms an annulus with a small hole into an annulus with a finite size hole, and which is the identity on the outer boundary of the annulus
The push-forward of the background coefficient with the diffeomorphism represents the meta-material needed for the cloak, and the finite size hole is the area that may be used as a “hiding place” [8]
The fact that the diffeomorphism is the identity on the outer boundary ensures that the perturbation in the “far

Summary

Introduction

A central ingredient in the construction of (approximate) cloaks by the passive cloaking technique, known as “cloaking by mapping”, is the diffeomorphism, which transforms an annulus with a small hole into an annulus with a finite size hole, and which is the identity on the outer boundary of the annulus. The fact that the diffeomorphism is the identity on the outer boundary ensures that the perturbation in the “far. This provisional PDF is the accepted version. The focus of this note is to produce very strong evidence for the conjecture that when the cloak takes the shape of a classical annulus, non-radial transformations do not help in reducing the degree of anisotropy. This radial transformation has smaller energy than all other transformations with “directional field”. In the final section of this note we consider the case when the outer (and inner) boundary of the cloak are not circles, and we illustrate how the optimal radial transformation for the circular case translates into a non-radial (optimal) transformation for a non-circular cloak

Preliminaries

The radial transformation case

Optimality of radial transforms

Optimal cloaks for simply connected domains