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Register a new userOn optimal cloaking-by-mapping transformations
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A central ingredient of cloaking-by-mapping is the diffeomorphisn 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 [4]. In this work, as in [4], 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.
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A body of literature has developed concerning cloaking by anomalous localized resonance. The mathematical heart of the matter involves the behavior of a divergence-form elliptic equation in the plane, $ ablacdot (a(x) abla u(x)) = f(x)$. The complex-valued coefficient has a matrix-shell-core geometry, with real part equal to 1 in the matrix and the core, and -1 in the shell; one is interested in understanding the resonant behavior of the solution as the imaginary part of $a(x)$ decreases to zero (so that ellipticity is lost). Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. We introduce a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source $f$ plays a crucial role in determining whether or not resonance occurs.
The acoustic cloaking theory of Norris (2008) permits considerable freedom in choosing the transformation function f from physical to virtual space. The standard process for defining cloak materials is to first define f and then evaluate whether the materials are practically realizable. In this paper, this process is inverted by defining desirable material properties and then deriving the appropriate transformations which guarantee the cloaking effect. Transformations are derived which result in acoustic cloaks with special properties such as 1) constant density 2) constant radial stiffness 3) constant tangential stiffness 4) power-law density 5) power-law radial stiffness 6) power-law tangential stiffness 7) minimal elastic anisotropy.
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 Lame 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.
We study the approximate cloaking via transformation optics for electromagnetic waves in the time harmonic regime in which the cloaking device {it only} consists of a layer constructed by the mapping technique. Due to the fact that no-lossy layer is required, resonance might appear and the analysis is delicate. We analyse both non-resonant and resonant cases. In particular, we show that the energy can blow up inside the cloaked region in the resonant case and/whereas cloaking is {it achieved} in {it both} cases. Moreover, the degree of visibility {it depends} on the compatibility of the source inside the cloaked region and the system. These facts are new and distinct from known mathematical results in the literature.
The geometry of an admissible Backlund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal prolongation of a $|1|$--graded semi-simple Lie algebra.
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