FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
Theoretical analysis of magnetic cloaking problems using elliptical metamaterials |
Alekseev G.V. |
2023, issue 2, P. 152-160 DOI: https://doi.org/10.47910/FEMJ202313 |
Abstract |
| The conjugation problem and control problems for the 3D model of magnetostatics are considered. These problems are related to the design of three-dimensional magnetic cloaking shells. An elliptical metamaterial is chosen as a cloaking medium that fills a region which is topologically equivalent to a spherical layer. The solvability of boundary and control problems is proved, an optimality system is derived that describes the necessary conditions for an extremum, some properties of optimal solutions are established. |
Keywords: 3D model of magnetostatics, conjugation problem, elliptical metamaterial, invisibility, cloaking, control problem, solvability. |
Download the article (PDF-file) |
References |
| [1] L. S. Dolin, “O vozmozhnosti sopostavleniia trekhmernykh elektromagnitnykh sistem s neodnorodnym anizotropnym zapolneniem”, Izv. vuzov. Radiofizika, 4 (1961), 964–967. [2] J. B. Pendry, D. Shurig, D. R. Smith, “Controlling electromagnetic fields”, Science, 312 (2006), 1780–1782. [3] U. Leonhardt, “Optical conformal mapping”, Science, 312 (2006), 1777–1780. [4] B. Wood, J. B. Pendry, “Metamaterials at zero frequency”, J. Phys.: Condens. Matter., 19 (2007), 076208. [5] A. Sanchez, C. Navau, J. Prat-Camps, D.-X. Chen, “Antimagnets: controlling magnetic fields with superconductormetamaterial hybrids”, New J. Phys., 13 (2011), 093034. [6] F. G?om?ory, M. Solovyov, J. Souc, C. Navau, J. Prat-Camps, A. Sanchez, “Experimental realization of a magnetic cloak”, Science, 335 (2012), 1466–1468. [7] G. V. Alekseev, Problema nevidimosti v akustike, optike i teploperenose, Dal'nauka, V., 2016. [8] G. V. Alekseev, V. A. Levin, D. A. Tereshko, Analiz i optimizatsiia v zadachakh dizaina ustroistv nevidimosti material'nykh tel, FIZMATLIT, M., 2021. [9] G. V. Alekseev, Iu. E. Spivak, “Teoreticheskii analiz zadachi magnitnoi maskirovki na osnove optimizatsionnogo metoda”, Differentsial'nye uravneniia, 54:9 (2018), 1155– 1166. [10] G. V. Alekseev, Iu. E. Spivak, “Chislennyi analiz dvumernykh zadach magnitnoi maskirovki na osnove optimizatsionnogo metoda”, Differentsial'nye uravneniia, 56:9 (2020), 1252–1262. [11] G. V. Alekseev, Iu. E. Spivak, “Chislennyi analiz trekhmernykh zadach magnitnoi maskirovki na osnove optimizatsionnogo metoda”, Zhurn. vychislit. matematiki i mat. fiziki, 61:2 (2021), 224–238. [12] G. V. Alekseev, A. V. Lobanov, “Optimization method for solving cloaking and shielding problems for a 3D model of electrostatics”, Mathematics, 11:6 (2023), 1395. [13] H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, V. M. Menon, “Topological transitions in metamaterials”, Science, 336:6078 (2012), 205–209. [14] P. Shekhar, J. Atkinson, Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications”, Nano Convergence, 1:14 (2014), 1–17. [15] J. M. Melenk, “Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions”, Math. Comp., 79 (2010), 1871–1914. |