Spectral properties of the Neumann-Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system

Ando, Kazunori; Ji, Yong-Gwan; Kang, Hyeonbae; Kim, Kyoungsun; Yu, Sanghyeon

doi:10.48550/arxiv.1510.00989

Cited by 16 publications

(46 citation statements)

References 12 publications

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“…Recently, the plasmon resonances were investigated for the linear elasticity governed by the Lamé system, where negative shear and bulk modulus are allowed to present [8,9,15,[18][19][20]. In [19,20], the plasmon resonances in both two and three dimensions were considered.…”

Section: 1mentioning

confidence: 99%

“…However, only energy blowup and dependence on the source location were shown by using the variational approach, and the localising and cloaking effects cannot be shown. In [8,9], much more accurate characterisation of the anomalous localised resonance in the linear elasticity was established based on a spectral approach using the spectral properties of the Neumann-Poincaré (N-P) operator associated with the Lamé system. However, the corresponding study was only conducted in two dimensions and the major obstacle for the extension to the three-dimensional case is the lack of the spectral properties of the N-P operator associated with the Lamé system in R 3 .…”

Section: 1mentioning

confidence: 99%

“…We first derive the full spectral properties of the Neumann-Poincaré operator for the 3D elastostatic system in the spherical geometry. The spectral result is of significant interest for its own sake, and serves as a highly nontrivial extension of the corresponding 2D study in [8]. The derivation of the spectral result in 3D involves much more complicated and subtle calculations and arguments than that for the 2D case.…”

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confidence: 99%

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On spectral properties of Neuman–Poincaré operator and plasmonic resonances in 3D elastostatics

Deng

Liu

2018

J. Spectr. Theory

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We consider plasmon resonances and cloaking for the elastostatic system in R 3 via the spectral theory of Neumann-Poincaré operator. We first derive the full spectral properties of the Neumann-Poincaré operator for the 3D elastostatic system in the spherical geometry. The spectral result is of significant interest for its own sake, and serves as a highly nontrivial extension of the corresponding 2D study in [8]. The derivation of the spectral result in 3D involves much more complicated and subtle calculations and arguments than that for the 2D case. Then we consider a 3D plasmonic structure in elastostatics which takes a general core-shell-matrix form with the metamaterial located in the shell. Using the obtained spectral result, we provide an accurate characterisation of the anomalous localised resonance and cloaking associated to such a plasmonic structure.

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Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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confidence: 99%

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On spectral properties of Neuman–Poincaré operator and plasmonic resonances in 3D elastostatics

Deng

Liu

2018

J. Spectr. Theory

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“…Since the operator S D : H * → H is invertible in three dimensions [4], from the first equation in (3.15) one can obtain…”

Section: Novel Plasmonic Structures and Resonancesmentioning

confidence: 99%

“…* is the inner product on H −1/2 (∂D) and we denote by • H * the norm induced by this inner product; see[4] for more relevant results. Furthermore, there holds…”

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On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonances

Liu

2018

Journal de Mathématiques Pures et Appliquées

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This paper is concerned with the theoretical study of plasmonic resonances for linear elasticity governed by the Lamé system in R 3 , and their application for cloaking due to anomalous localized resonances. We derive a very general and novel class of elastic structures that can induce plasmonic resonances. It is shown that if either one of the two convexity conditions on the Lamé parameters is broken, then we can construct certain plasmon structures that induce resonances. This significantly extends the relevant existing studies in the literature where the violation of both convexity conditions is required. Indeed, the existing plasmonic structures are a particular case of the general structures constructed in our study. Furthermore, we consider the plasmonic resonances within the finite frequency regime, and rigorously verify the quasi-static approximation for diametrically small plasmonic inclusions. Finally, as an application of the newly found structures, we construct a plasmonic device of the core-shellmatrix form that can induce cloaking due to anomalous localized resonance in the quasi-static regime, which also includes the existing study as a special case.

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Quantitative Characterization of Stress Concentration in the Presence of Closely Spaced Hard Inclusions in Two-Dimensional Linear Elasticity

Kang

2018

Arch Rational Mech Anal

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In the region between close-to-touching hard inclusions, the stress may be arbitrarily large as the inclusions get closer. The stress is represented by the gradient of a solution to the Lamé system of linear elasticity. We consider the problem of characterizing the gradient blow-up of the solution in the narrow region between two inclusions and estimating its magnitude. We introduce singular functions which are constructed in terms of nuclei of strain and hence are solutions of the Lamé system, and then show that the singular behavior of the gradient in the narrow region can be precisely captured by singular functions. As a consequence of the characterization, we are able to regain the existing upper bound on the blow-up rate of the gradient, namely, −1/2 where is the distance between two inclusions. We then show that it is in fact an optimal bound by showing that there are cases where −1/2 is also a lower bound. This work is the first to completely reveal the singular nature of the gradient blow-up in the context of the Lamé system with hard inclusions. The singular functions introduced in this paper play essential roles to overcome the difficulties in applying the methods of previous works. Main tools of this paper are the layer potential techniques and the variational principle. The variational principle can be applied because the singular functions of this paper are solutions of the Lamé system.

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Spectral properties of the Neumann-Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system

Cited by 16 publications

References 12 publications

On spectral properties of Neuman–Poincaré operator and plasmonic resonances in 3D elastostatics

On spectral properties of Neuman–Poincaré operator and plasmonic resonances in 3D elastostatics

On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonances

Quantitative Characterization of Stress Concentration in the Presence of Closely Spaced Hard Inclusions in Two-Dimensional Linear Elasticity

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