Far Eastern Mathematical Journal

To content of the issue


Optimization method in two-dimensional electrical cloaking problems


Lobanov A.V., Spivak Yu.E.

2019, issue 1, Ñ. 31-42


Abstract
In this paper, we study control problems for the two-dimensional electrical cloaking model in the case when the cloaking device has shape of a ring filled with an inhomogeneous anisotropic medium. These problems arise while developing technologies of design of electrical cloaking devices using an optimization method for solving inverse problems. The solvability of the direct and control problems for considered electrical transmission model is proved. The optimality system that describes the necessary conditions for an extremum is derived, and its properties are investigated. The possibility of implementing two types of numerical algorithms that are used to find approximate solutions is discussed.

Keywords:
electric transmission problem, inverse problem, optimization method, existence, uniqueness, optimality system, invisibility

Download the article (PDF-file)

References

[1] J. B. Pendry, D. Shurig, D. R. Smith, “Controlling electromagnetic fields”, Science, 312, (2006), 1780–1782.
[2] S. A. Cummer, B. I. Popa, D. Shurig, D. R. Smith, J. Pendry, M. Rahm, A. Starr, “Scattering theory derivation of a 3D acoustic cloaking shell”, Phys. Rev. Lett., 100, (2008), 024301.
[3] A. Sanchez, C. Navau, J. Prat-Camps, D. X. Chen, “Antimagnets: controlling magnetic fields with superconductormetamaterial hybrids”, New J. Phys., 13, (2011), 093034.
[4] W. Jiang, C. Luo, H. F. Ma, Z. L. Mei, T. J. Cui, “Enhancement of current density by dc electric concentrator”, Sci. Rep., 2, (2012), 956.
[5] S. Guenneau, C. Amra, D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux”, Opt. Express., 20, (2012), 8207–8218.
[6] G. V. Alekseev, Problema nevidimosti v akustike, optike i teploperenose, Dal'nauka, Vladivostok, 2016.
[7] A. N. Tikhonov, V. Ia. Arsenin, Metody resheniia nekorrektnykh zadach, Nauka, M., 1986.
[8] R. V. Brizitskii, Zh. Iu. Saritskaia, “Obratnye koeffitsientnye zadachi dlia nelineinogo uravneniia konvektsii-diffuzii-reaktsii”, Izv. RAN (Seriia Matematicheskaia), 82:1, (2018), 17–33.
[9] R. V. Brizitskii, Zh. Iu. Saritskaia, “Zadacha granichnogo upravleniia dlia nelineinogo uravneniia konvektsii–diffuzii–reaktsii”, Zh. vychisl. matem. i matem. fiz., 58:12, (2018), 2139–2152.
[10] D.A. Tereshko, “Chislennoe vosstanovlenie granichnogo potoka tepla dlia statsionarnykh uravnenii teplovoi konvektsii”, Sib. zhurn. industr. matem., 17:4, (2014), 111–119.
[11] S. Xu, Y. Wang, B. Zhang, H. Chen, “Invisibility cloaks from forward design to inverse design”, Science China Information Sciences, 56, (2013), 120408.
[12] B.I. Popa, S. A. Cummer, “Cloaking with optimized homogeneous anisotropic layers”, Phys. Rev. A., 79, (2009), 023806.
[13] S. Xi, H. Chen, B. Zhang, B.-I. Wu, J.A. Kong, “Route to low-scattering cylindrical cloaks with finite permittivity and permeability”, Phys. Rev. B., 79, (2009), 155122.
[14] G. V. Alekseev, “Upravlenie granichnym impedansom v dvumernoi zadache maskirovki material'nykh tel metodom volnovogo obtekaniia”, Zhurn. vych. matem. i matem. fiz., 53:12, (2013), 98–115.
[15] G. V. Alekseev, A. V. Lobanov, “Otsenki ustoichivosti v dvumernoi zadache maskirovki material'nykh tel”, Dal'nevost. matem. zhurn., 14:2, (2014), 127–140.
[16] G. V. Alekseev, “Analiz i optimizatsiia v zadachakh maskirovki material'nykh tel dlia uravnenii Maksvella”, Differentsial'nye uravneniia, 52:3, (2016), 366–377.
[17] G. V. Alekseev Yu. E. Spivak, “Analysis of the 3D acoustic cloaking problems using optimization method”, J. Phys.: Conf. Ser., 722, (2016), 012002.
[18] G. V. Alekseev, A. V. Lobanov, Iu. E. Spivak, “Optimizatsionnyi metod v zadachakh akusticheskoi maskirovki material'nykh tel”, Zhurn. vych. matem. i matem. fiz., 57:9, (2017), 79–95.
[19] G. V. Alekseev, “Analiz dvumernoi zadachi teplovoi maskirovki na osnove optimizatsionnogo metoda”, Zh. vychisl. matem. i matem. fiz., 58:4, (2018), 504–519.
[20] G. V. Alekseev, Iu. E. Spivak, “Teoreticheskii analiz zadachi magnitnoi maskirovki na osnove optimizatsionnogo metoda”, Differentsial'nye uravneniia, 54:9, (2018), 1155–1166.
[21] G. V. Alekseev, V. A. Levin, D. A. Tereshko, “Optimizatsionnyi analiz zadachi teplovoi maskirovki tsilindricheskogo tela”, Dokl. AN., 472:4, (2017), 398–402.
[22] G.V. Alekseev, D. A. Tereshko, “Optimizatsionnyi metod v osesimmetrichnykh zadachakh elektricheskoi maskirovki material'nykh tel”, Zh. vychisl. matem. i matem. fiz, 59:2, (2019), 217–234.
[23] G. Alekseev, D. Tereshko, “Particle swarm optimization-based algorithms for solving inverse problems of designing thermal cloaking and shielding devices”, International Journal of Heat and Mass Transfer, 135, (2019), 1269–1277.
[24] H. Han, X. Wu, Artificial Boundary Method, Springer-Verlag (Tsinghua University Press), Berlin (Beijing), 2013.
[25] A. D. Ioffe, V. M. Tikhomirov, Teoriia ekstremal'nykh zadach, Nauka, M., 1974.

To content of the issue