Cloaking Problems Research Articles

Численная оптимизация в задачах проектирования многослойных магнитных маскировочных оболочек

A multiparameter optimization problem of magnetic cloaking is solved for a two-dimensional model of magnetostatics. This problem arises when designing cylindrical multilayer cloaking devices filled with anisotropic or isotropic magnetic media with tensor, in the general case, magnetic permeabilities. Applying the optimization method for solving inverse problems the considered problem of magnetic cloaking is reduced to a finite-dimensional extremum problem. The results of applying the developed numerical algorithm based on the particle swarm optimization method to solve the extremum problem confirm its high efficiency and make it possible to establish the important property of the obtained optimal solutions.

Fast active thermal cloaking through PDE-constrained optimization and reduced-order modelling

In this paper, we show how to efficiently achieve thermal cloaking from a computational standpoint in several virtual scenarios by controlling a distribution of active heat sources. We frame this problem in the setting of PDE-constrained optimization, where the reference field is the solution of the time-dependent heat equation in the absence of the object to cloak. The optimal control problem then aims at actuating the space–time control field so that the thermal field outside the obstacle is indistinguishable from the reference field. In particular, we consider multiple scenarios where material’s thermal diffusivity, source intensity and obstacle’s temperature are allowed to vary within a user-defined range. To tackle the thermal cloaking problem in a rapid and reliable way, we rely on a parametrized reduced order model built through the reduced basis method, thus entailing huge computational speedups compared with a high-fidelity, full-order model exploiting the finite-element method while dealing both with complex target shapes and disconnected control domains.

Feasible Optimal Solutions of Electromagnetic Cloaking Problems by Chaotic Accelerated Particle Swarm Optimization

The optimization problem of cloaking a perfectly electric conducting or dielectric spherical core is investigated. The primary excitation is due to an external magnetic dipole. The chaotic accelerated particle swarm optimization (CAPSO) algorithm is adjusted and applied to this optimization problem. The optimization variables are the radii, the permittivities and the permeabilities of a small number of spherical shells covering the core. Several feasible optimal designs are obtained, which exhibit perfect or almost perfect cloaking performance for all angles of observation. These optimal designs correspond to two, three or four spherical coating layers composed of ordinary materials. Detailed parametric investigations of the cloaking mechanism with respect to the type and radius of the core and the location of the primary dipole are carried out. The presented optimization procedure and the reported results are expected to be useful in applications like scattering and characterization of optical particles as well as in designing low-profile receiving antennas.

Optimization-Based Numerical Analysis of Three-Dimensional Magnetic Cloaking Problems

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.

Optimal design of acoustic metamaterial cloaks under uncertainty

In this work, we consider the problem of optimal design of an acoustic cloak under uncertainty and develop scalable approximation and optimization methods to solve this problem. The design variable is taken as an infinite-dimensional spatially-varying field that represents the material property, while an additive infinite-dimensional random field represents, e.g., the variability of the material property or the manufacturing error. Discretization of this optimal design problem results in high-dimensional design variables and uncertain parameters. To solve this problem, we develop a computational approach based on a Taylor approximation and an approximate Newton method for optimization, which is based on a Hessian derived at the mean of the random field. We show our approach is scalable with respect to the dimension of both the design variables and uncertain parameters, in the sense that the necessary number of acoustic wave propagations is essentially independent of these dimensions, for numerical experiments with up to one million design variables and half a million uncertain parameters. We demonstrate that, using our computational approach, an optimal design of the acoustic cloak that is robust to material uncertainty is achieved in a tractable manner. The optimal design under uncertainty problem is posed and solved for the classical circular obstacle surrounded by a ring-shaped cloaking region, subjected to both a single-direction single-frequency incident wave and multiple-direction multiple-frequency incident waves. Finally, we apply the method to a deterministic large-scale optimal cloaking problem with complex geometry, to demonstrate that the approximate Newton method's Hessian computation is viable for large, complex problems.

Optimization analysis of electrostatic cloaking problems

Optimization analysis of electrostatic cloaking problems

Экономичный метод решения двумерной задачи электростатической маскировки

An economical numerical algorithm for solving a two-dimensional problem of electrostatic cloaking is proposed and implemented. The algorithm is based on the use of an $M$-layer shell, each layer of which is filled with one of two specified materials, while the material of the last layer is determined by minimizing a special quality functional associated with the cloaking efficiency of the desired shell.

Third order differential subordination and superordination results for analytic functions involving the Hohlov operator

The problems of third order differential subordination as well as superordination for functions analytic in the open unit disk seem to be new and have been of interest after the seminal work of Antonino and Miller, and Tang. In the present paper by considering suitable classes of admissible functions associated with Hohlov operator, various third order differential subordination as well as differential superordination results are obtained. As a consequence, the dual problems which gives the sandwich-type relations are presented. Upon suitable choice of the parameters, the results obtained in this paper include some classical as well as recently studied results. An attempt has also been made to illustrate the applications of the new results in the context of electromagnetic cloaking problem.

Optimization-based method of solving 2D thermal cloaking problems

We consider inverse problems for 2D heat conduction model concerning with designing cylindrical thermal cloaking shells. The shells are assumed consisting of a finite number of layers every of which is filled with homogeneous isotropic medium. Using the optimization method these inverse problems are reduced to corresponding control problems. Thermal conductivities of the shell layers play the role of passive controls. A numerical algorithm based on the particle swarm optimization is proposed and the results of numerical experiments are discussed. Rigorous optimization analysis shows that high cloaking efficiency of the shell can be achieved using a multilayer shell composed of only three isotropic nature materials with optimally selected thermal conductivities.

Optimization method in 2D magnetic cloaking problems

Optimization method in 2D magnetic cloaking problems

Optimization Method for Axisymmetric Problems of Electric Cloaking of Material Bodies

Inverse problems of designing spherical shells intended for cloaking material bodies in a stationary electric field are studied. By applying an optimization method, these problems are reduced to optimization ones in which the role of controls is played by components of the diagonal electrical conductivity tensor of the shell’s anisotropic material. The solvability of the direct and optimization problems is proved. A numerical algorithm based on particle swarm optimization is proposed for solving optimization problems, and numerical results are discussed.

Theoretical Analysis of the Magnetic Cloaking Problem Based on an Optimization Method

Control problems are considered for a model of magnetic scattering on a permeable anisotropic obstacle shaped as a spherical layer. Such problems arise in developing technologies for designing magnetic cloaking devices when the corresponding inverse problems are solved by an optimization method. The solvability of the direct and extremal problems for the model in question is proved and the optimality system is derived. Its analysis permits obtaining sufficient conditions on the initial data which ensure the local uniqueness and stability of the optimal solutions.

Analysis of a Two-Dimensional Thermal Cloaking Problem on the Basis of Optimization

For a two-dimensional model of thermal scattering, inverse problems arising in the development of tools for cloaking material bodies on the basis of a mixed thermal cloaking strategy are considered. By applying the optimization approach, these problems are reduced to optimization ones in which the role of controls is played by variable parameters of the medium occupying the cloaking shell and by the heat flux through a boundary segment of the basic domain. The solvability of the direct and optimization problems is proved, and an optimality system is derived. Based on its analysis, sufficient conditions on the input data are established that ensure the uniqueness and stability of optimal solutions.

Solving approximate cloaking problems using finite element methods

Motivated by the approximate cloaking problem, we consider a variable coefficient Helmholtz equation with a fixed wave number. We use finite element methods to discretize the equation. Numerical results are shown to exhibit cloaking behaviour. References A. Alu and N. Engheta. Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72 :016623, 2005. doi:10.1103/PhysRevE.72.016623 W. Cai, U. K. Chettiar, A. V. Kildishev and V. M. Shalaev. Optical cloaking with metamaterials. Nature Photonics 1 :224–227, 2007. doi:10.1038/nphoton.2007.28 A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann. Cloaking devices, electromagnetic wormholes and transformation optics. SIAM Rev. 51 :3–33, 2009. doi:10.1137/080716827 R. V. Kohn, D. Onofrei, M. S. Vogelius and M. I. Weinstein. Cloaking via change of variables for the Helmholtz equation. Comm. Pure Appl. Math. 63 :973–1016, 2010. doi:10.1002/cpa.20341 R. V. Kohn, H. Shen, M. S. Vogelius and M. I. Weinstein. Cloaking via change of variables in electric impedance tomography. Inverse Problems 24 :015016, 2008. doi:10.1088/0266-5611/24/1/015016 U. Leonhardt. Optical conformal mapping. Science 312 :1777–1780, 2006. doi:10.1126/science.1126493 M. Maischak. Book of Numerical Experiments (B.O.N.E) . http://www.ifam.uni-hannover.de/ maiprogs/ J. B. Pendry, D. Schurig and D. R. Smith. Controlling electromagnetic fields. Science 312 :1780–1782, 2006. doi:10.1126/science.1125907 D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr and D. Smith. Metamaterial electromagnetic cloak at microwave frequencies. Science 314 :977–980, 2006. doi:10.1126/science.1133628

Invisibility problem in acoustics, electromagnetism and heat transfer. Inverse design method

Two approaches (direct design and inverse design methods) for solving problems of designing devices providing invisibility of material bodies of detection using different physical fields – electromagnetic, acoustic and static are discussed. The second method is applied for solving problems of designing cloaking devices for the 3D stationary thermal scattering model. Based on this method the design problems under study are reduced to respective control problems. The material parameters (radial and tangential heat conductivities) of the inhomogeneous anisotropic medium filling the thermal cloak and the density of auxiliary heat sources play the role of controls. A unique solvability of direct thermal scattering problem in the Sobolev space is proved and the new estimates of solutions are established. Using these results, the solvability of control problem is proved and the optimality system is derived. Based on analysis of optimality system, the stability estimates of optimal solutions are established and numerical algorithms for solving particular thermal cloaking problem are proposed.

Optimization method in problems of acoustic cloaking of material bodies

Optimization problems for a three-dimensional model of acoustic scattering are formulated and studied. These problems arise in designing tools for cloaking material bodies by applying the wave flow method. The cloaking effect is achieved due to an optimal choice of variable parameters of the inhomogeneous isotropic medium occupying the sought shell. The solvability of direct and optimization problems for the acoustic scattering model is proved, and sufficient conditions ensuring the uniqueness and stability of optimal solutions are established.

Computational material design for acoustic cloaking

A topology optimization technique based on the topological derivative and the level set function is utilized to design/synthesize the micro-structure of a pentamode material for an acoustic cloaking device. The technique provides a micro-structure consisting of a honeycomb lattice composed of needle-like and joint members. The resulting metamaterial shows a highly anisotropic elastic response with effective properties displaying a ratio between bulk and shear moduli of almost 3 orders of magnitude. Furthermore, in accordance with previous works in the literature, it can be asserted that this kind of micro-structure can be realistically fabricated. The adoption of a topology optimization technique as a tool for the inverse design of metamaterials with applications to acoustic cloaking problems is one contribution of this paper. However, the most important achievement refers to the analysis and discussion revealing the key role of the external shape of the prescribed domain where the optimization problem is posed. The efficiency of the designed micro-structure is measured by comparing the scattering wave fields generated by acoustic plane waves impinging on bare and cloaked bodies.

Surface-admittance equivalence principle for nonradiating and cloaking problems

In this paper, we address non-radiating and cloaking problems exploiting the surface equivalence principle, by imposing at any arbitrary boundary the control of the admittance discontinuity between the overall object (with or without cloak) and the background. After a rigorous demonstration, we apply this model to a non-radiating problem, appealing for anapole modes and metamolecules modeling, and to a cloaking problem, appealing for non-Foster metasurface design. A straightforward analytical condition is obtained for controlling the scattering of a dielectric object over a surface boundary of interest. Previous quasi-static results are confirmed and a general closed-form solution beyond the subwavelength regime is provided. In addition, this formulation can be extended to other wave phenomena once the proper admittance function is defined (thermal, acoustics, elastomechanics, etc.).

Optimization analysis of the thermal cloaking problem for a cylindrical body

The inverse heat transfer problem associated with constructing multilayer material shells cloaking the presence of a cylindrical body in an externally applied temperature field is studied. As the original mathematical model, the steady-state heat equation for an anisotropic shell is used. With the help of the optimization method, this inverse problem is reduced to the corresponding control problem. A numerical algorithm of its solution based on the particle swarm optimization is proposed, and the results of numerical experiments are discussed.

On the Use of the Analogue Transformation Acoustics in Aeroacoustics

The objective of the paper is the assessment of the Analogue Transformation Acoustics (ATA) in the design of acoustic metamaterial for aeronautical applications. The work focuses on the consistency of the background flow resulting from the application of the ATA with the equations governing the potential aerodynamics. Indeed, in case of acoustic perturbations propagating within moving media, the convective terms in the governing equations are responsible for the failure of formal invariance under the action of conformal mappings. The ATA approach overcomes this limitation, introducing the possibility of handling the convective form of the wave equation in a straightforward and elegant way. It is based on the concept of analogue space-time and fully relies on the analytical tools of Lorentzian differential geometry. The present paper analyses the relationship between the analogue velocity field with a realistic potential flow. The method is validated through numerical simulations using two widely assessed acoustic cloaking problems. The preliminary results obtained show that the use of numerical, quasi-conformal mappings can lead to transformed streamlines negligibly deviating from those of the potential velocity field satisfying the fluid-dynamic conservation laws, but with incompatible intensity of the local velocity.