Abstract
Inverse problems for a three-dimensional magnetostatic model arising in the design of axisymmetric multilayered shielding and cloaking devices are stated. Assuming that the designed device consists of a finite number of spherical layers, each filled with a homogeneous isotropic medium, an optimization-based numerical algorithm is proposed for solving the problems. As a result, the considered inverse problems are reduced to finite-dimensional optimization ones in which the role of controls is played by the magnetic permeabilities of each elementary layer. The desired controls are found by applying particle swarm optimization. By analyzing numerical results, it is shown that the obtained optimal solutions correspond to cloaking devices having the highest efficiency in the considered class of devices and provide the simplicity of technical implementation.
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REFERENCES
L. S. Dolin, “On the possibility of comparing three-dimensional electromagnetic systems with an inhomogeneous anisotropic filling,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 4, 964–967 (1961).
J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterial,” Appl. Phys. Lett. 91 (183518), 1–3 (2007).
T. Han and C.-W. Qiu, “Transformation Laplacian metamaterials: Recent advances in manipulating thermal and DC fields,” J. Opt. 18 (044003), 1–13 (2016).
F. Yang, Z. L. Zhong Mei, T. Y. Jin, and T. J. Cui, “DC electric invisibility cloak,” Phys. Rev. Lett. 109 (053902), 1–5 (2012).
F. Gomory, M. Solovyov, J. Souc, C. Navau, J. Prat-Camps, and A. Sanchez, “Experimental realization of a magnetic cloak,” Science 335, 1466–1468 (2012).
S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: Cloaking and concentrating heat flux,” Opt. Express 20, 8207–8218 (2012).
Yu. V. Shestopalov and Yu. G. Smirnov, “Determination of permittivity of an inhomogeneous dielectric body in a waveguide,” Inverse Probl. 27 (9), 095010 (2011).
Y. G. Smirnov, M. Y. Medvedik, and M. A. Moskaleva, “Two-step method for permittivity determination of an inhomogeneous body placed in a rectangular waveguide,” Lobachevskii J. Math. 39, 1114–1147 (2018).
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977; Nauka, Moscow, 1986).
I. Peralta and V. D. Fachinotti, “Optimization-based design of heat flux manipulation devices with emphasis on fabricability,” Sci. Rep. 7 (6261), 1–8 (2017).
V. D. Fachinotti, A. A. Ciarbonetti, I. Peralta, and I. Rintoul, “Optimization-based design to easy-to-make devices for heat flux manipulation,” Int. J. Therm. Sci. 128, 38–48 (2018).
G. V. Alekseev, V. A. Levin, and D. A. Tereshko, “Optimization analysis of the thermal cloaking problem for a cylindrical body,” Dokl. Phys. 62 (2), 71–75 (2017).
G. V. Alekseev, V. A. Levin, and D. A. Tereshko, “The optimization method in design problems of spherical layered thermal shells,” Dokl. Phys. 62 (10), 465–469 (2017).
G. V. Alekseev and D. A. Tereshko, “Optimization method for axisymmetric problems of electric cloaking of material bodies,” Comput. Math. Math. Phys. 59 (2), 207–223 (2019).
G. V. Alekseev and D. A. Tereshko, “Particle swarm optimization-based algorithms for solving inverse problems of designing thermal cloaking and shielding devices,” Int. J. Heat Mass Transfer 135, 1269–1277 (2019).
A. V. Lobanov and Yu. E. Spivak, “Numerical analysis of problem of designing magnetic bilayer cloak,” in 2017 Progress in Electromagnetics Research Symposium Spring (PIERS) (St. Petersburg, 2017), pp. 1362–1366.
G. V. Alekseev, D. A. Tereshko, and Yu. V. Shestopalov, “Optimization approach for axisymmetric electric field cloaking and shielding,” Inverse Probl. Sci. Eng. 28, 1–16 (2020).
G. V. Alekseev, “Stability estimates in the problem of cloaking material bodies for Maxwell’s equations,” Comput. Math. Math. Phys. 54 (12), 1788–1803 (2014).
G. V. Alekseev, “Analysis and optimization in problems of cloaking of material bodies for the Maxwell equations,” Differ. Equations 52 (3), 361–372 (2016).
G. V. Alekseev and Yu. E. Spivak, “Theoretical analysis of the magnetic cloaking problem based on an optimization method,” Differ. Equations 54 (9), 1125–1136 (2018).
Yu. E. Spivak, “Optimization method in two-dimensional magnetic cloaking problems,” Sib. Elektron. Mat. Izv. 16, 812–825 (2019).
R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization: An overview,” Swarm Intel. 1, 33–57 (2007).
G. V. Alekseev, Problem of Cloaking in Acoustics, Optics, and Heat Transfer (Dal’nauka, Vladivostok, 2016) [in Russian].
J. Kennedy and R. Eberhart, “Particle swarm optimization,” Proceedings of IEEE International Conference on Neural Networks IV (1995), pp. 1942–1948.
A. C. Chiang, Elements of Dynamic Optimization (McGraw-Hill, New York, 1992).
M. Solovyov, F. Gomory, J. Souc, E. Mikulasova, M. Usakova, and E. Usak, “Force acting on a magnetic cloak placed in magnetic field,” The 13th Biennial European Conference on Applied Superconductivity (2017), Poster No. 3LP4-03.
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This work was performed within the state assignment of the Institute of Applied Mathematics of the Far Eastern Branch of the Russian Academy of Sciences. Spivak acknowledges the support of the Russian Foundation for Basic Research, project no. 19-31-90039.
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Translated by I. Ruzanova
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Alekseev, G.V., Spivak, Y.E. Optimization-Based Numerical Analysis of Three-Dimensional Magnetic Cloaking Problems. Comput. Math. and Math. Phys. 61, 212–225 (2021). https://doi.org/10.1134/S0965542521020032
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DOI: https://doi.org/10.1134/S0965542521020032