Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking
نویسنده
چکیده
In this paper, we study approximate cloaking of active devices for the Helmholtz equation in the whole space of dimension 2 or 3 for the scheme introduced by Kohn, Shen, Vogelius, and Weinstein in [11]. More precisely, we assess the degree of invisibility, determine the limit of the field inside the cloaked and cloaking regions, and show that the scheme is unstable with respect to the material parameters. As a consequence, we obtain some feasible properties of “perfect” cloaking.
منابع مشابه
Approximate Cloaking for the Full Wave Equation via Change of Variables
We study, in the context of the full wave equation, an approximate cloaking scheme, that was previously considered for the Helmholtz equation [8], [17]. This cloaking scheme consists in a combination of an absorbing layer with an anisotropic layer, obtained by so-called transformation optics. We give optimal bounds for the visibility that tend to zero as a certain regularization parameter appro...
متن کاملApproximate Cloaking for the Full Wave Equation via Change of Variables | SIAM Journal on Mathematical Analysis | Vol. 44, No. 3 | Society for Industrial and Applied Mathematics
We study, in the context of the full wave equation, an approximate cloaking scheme that was previously considered for the Helmholtz equation [R. V. Kohn, D. Onofrei, M. S. Vogelius, and M. I. Weinstein, Comm. Pure Appl. Math., 63 (2010), pp. 973–1016, H.-M. Nguyen and M. S. Vogelius, Arch. Ration. Mech. Anal., 203 (2011), pp. 769–807]. This cloaking scheme consists in a combination of an absorb...
متن کاملCloaking via change of variables for the Helmholtz equation
The transformation optics approach to cloaking uses a singular change of coordinates, which blows up a point to the region being cloaked. This paper examines a natural regularization, obtained by (i) blowing up a ball of radius ρ rather than a point, and (ii) including a well-chosen lossy layer at the inner edge of the cloak. We assess the performance of the resulting near-cloak as the regulari...
متن کاملAnalysis of an approximate cloaking for acoustic scattering problems in R3
In this paper, we consider the acoustic cloaking based on singular transformations and its approximation in the context of scattering problems in R. The cloaking based on singular transformations can not avoid a singularity of the resulting Laplace–Beltrami operator because singular transformations blow up one point to a sphere. In order to treat the singularity properly, we use the variational...
متن کاملResearch Summary
1. Overview 2 2. Electromagnetic waves 3 2.1. Approximate cloaking via change of variables 3 2.1.1. Cloaking for the Helmholtz equation 3 2.1.2. Full range estimates for the degree of invisibility 4 2.1.3. Perfect cloaking for the Helmholtz equation 4 2.2. Electrical impedance tomography 5 2.3. Generalized impedance boundary conditions 5 2.3.1. The Helmholtz equation 6 2.3.1. The time harmonic ...
متن کامل