Optimization Method for Solving Cloaking and Shielding Problems for a 3D Model of Electrostatics

Abstract

Inverse problems for a 3D model of electrostatics, which arise when developing technologies for designing electric cloaking and shielding devices, are studied. It is assumed that the devices being designed to consist of a finite number of concentric spherical layers filled with homogeneous anisotropic or isotropic media. A mathematical technique for solving these problems has been developed. It is based on the formulation of cloaking or shielding problems in the form of inverse problems for the electrostatic model under consideration, reducing the latter problems to finite-dimensional extremum problems, and finding their solutions using one of the global minimization methods. Using the developed technology, the inverse problems are replaced by control problems, in which the role of controls is played by the permittivities of separate layers composing the device being designed. To solve them, a numerical algorithm based on the particle swarm optimization method is proposed. Important properties of optimal solutions are established, one of which is the bang-bang property. It is shown on the base of the computational experiments that cloaking and shielding devices designed using the developed algorithm have the simplicity of technical implementation and the highest performance in the class of devices under consideration.