Numerical Analysis of Two-Dimensional Magnetic Cloaking Problems Based on an Optimization Method
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RICAL METHODS
Numerical Analysis of Two-Dimensional Magnetic Cloaking Problems Based on an Optimization Method G. V. Alekseev1,2∗ and Yu. E. Spivak1,2∗∗ 1
Institute of Applied Mathematics, Far East Branch, Russian Academy of Sciences, Vladivostok, 690041 Russia 2 Far Eastern Federal University, Vladivostok, 690090 Russia e-mail: ∗ [email protected], ∗∗ [email protected] Received March 13, 2020; revised March 13, 2020; accepted May 14, 2020
Abstract—We study inverse problems of magnetostatics that arise when designing two-dimensional multilayered shielding and cloaking devices. It is assumed that the device to be designed has the form of a circular shell consisting of finitely many layers, each filled with a homogeneous isotropic medium. Using an optimization method, the inverse problems under consideration are reduced to finite-dimensional control problems in which the constant permeabilities of the layers play the role of control parameters. A numerical algorithm based on the particle swarm method is proposed for solving the control problems. Some properties of the algorithm proposed and of the resulting optimal solutions are established. Results of computational experiments are discussed. DOI: 10.1134/S0012266120090116
INTRODUCTION In recent years, a new scientific research trend has been intensively developing, which aims to create technologies for the design of invisibility devices, energy concentrators, and other special functional devices used to control physical fields. Solving design problems for these functional devices leads to the need to solve inverse problems for the corresponding physical field model. These problems include selection of the parameters of a medium that fills the designed device with a given topology, based on some additional information about the field to be created, arising from the purpose of the desired device. This trend was initiated by the papers [1–3]. They put forward the idea of cloaking concealed objects by placing them inside a volume cloaking shell filled with an inhomogeneous anisotropic medium with specially selected parameters. To find the desired parameters of the medium, the authors of these papers developed an original method called the “transformation optics” (TO-approach) in [2, 3]. Later, the TO-approach was extended to acoustic and static (thermal, electric, and magnetic) fields (see, for example, [4–7]). It should be noted that the solutions of cloaking problems obtained using this method have a significant drawback; namely, they are not feasible in practical terms, because materials with properties that correspond to the solutions obtained are lacking in nature or in modern engineering practice. One of the ways to overcome this drawback and solve cloaking problems with proper regard to the capabilities of modern technology is to consider these problems as inverse problems for the corresponding system of differential equations and to solve the latter by applying the optimization method. Starting from the fundamental works by Tikhonov [8], the optimization
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