The scattering of electromagnetic waves by a perfectly conducting infinite cylinder

We consider the scattering of a plane time-harmonic electromagnetic wave by a perfectly conducting infinite cylinder with axis in the direction k , where k is the unit vector along the z axis. Suppose the incident wave propagates in a direction perpendicular to the cylinder. For a given observation angle θ, let F_D(θ, α) k be the far-field pattern of the electric field corresponding to an incident wave with direction angle α polarized perpendicular to the axis and let F_N(θ; α) k be the far-field pattern of the magnetic field corresponding to an incident wave with direction angle α polarized parallel to the z axis. Let {α_n}_n=1^∞ be a distinct set of angles in [ ? π, π] and μ a complex number. Then, necessary and sufficient conditions are given for the set {(1 ? μ)F_D(θ;α_n) + μF_N(θ;α_n)}_{n = 1}^∞ to be complete in L²[ ? π, π]. Applications, together with numerical examples, are given to the inverse scattering problem of determining the shape of the cylinder from a knowledge of the far-field data.     

An adaptive anisotropic perfectly matched layer method for 3-D time harmonic electromagnetic scattering problems

We develop an anisotropic perfectly matched layer (PML) method for solving the time harmonic electromagnetic scattering problems in which the PML coordinate stretching is performed only in one direction outside a cuboid domain. The PML parameters such as the thickness of the layer and the absorbing medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the proposed adaptive anisotropic PML method provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the choice of the thickness of the PML layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.     

ANALYSIS OF TWO-DIMENSIONAL ELECTROMAGNETIC SCATTERING BY A PERFECTLY CONDUCTING OBSTACLE IN A HOMOGENEOUS CHIRAL ENVIRONMENT

The time-harmonic electromagnetic plane waves incident on a perfectly conducting obstacle in a homogeneous chiral environment are considered.A two-dimensional direct scat- tering model is established and the existence and uniqueness of solutions to the problem are discussed by an integral equation approach.The inverse scattering problem to find the shape of scatterer with the given far-field data is formulated.Result on the uniqueness of the inverse problem is proved.     

An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate

This article is concerned with uniqueness for reconstructing a periodic inhomogeneous medium sitting on a perfectly conducting plate. We deal with the problem in the framework of time-harmonic Maxwell systems without TE or TM polarization. An orthogonal relation is obtained for two refractive indices and then used to prove that the refractive index can be uniquely identified from a knowledge of the incident fields and the total tangential electric field on a plane above the inhomogeneous medium, utilizing the eigenvalues and eigenfunctions of a quasi-periodic Sturm–Liouville eigenvalue problem.     

Discrete singular convolution method with perfectly matched absorbing layers for the wave scattering by periodic structures

A new computational algorithm is introduced for solving scattering problem in periodic structure. The PML technique is used to deal with the difficulty on truncating the unbounded domain while the DSC algorithm is utilized for the spatial discretization. The present study reveals that the method is efficient for solving the problem.     

An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems

An adaptive perfectly matched layer (PML) technique for solving the time harmonic electromagnetic scattering problems is developed. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the adaptive PML technique provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorbing layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.

     

Volume singular integral equations for problems of scattering on three-dimensional dielectric structures

We consider volume singular integral equations describing problems of scattering of electromagnetic waves in bounded three-dimensional dielectric structures. We analyze the equations mathematically. Existence and uniqueness theorems are presented, and the spectrum of the integral operators is studied.     

Numerical solution of 3D problems of electromagnetic wave diffraction on a system of ideally conducting surfaces by the method of hypersingular integral equations

We construct a numerical method for solving problems of electromagnetic wave diffraction on a system of solid and thin objects based on the reduction of the problem to a boundary integral equation treated in the sense of the Hadamard finite value. For the construction of such an equation, we construct a numerical scheme on the basis of the method of piecewise continuous approximations and collocations. Unlike earlier known schemes, by using the below-suggested scheme, we have found approximate analytic expressions for the coefficients of the arising system of linear equations by isolating the leading part of the kernel of the integral operator. We present examples of solution of a number of model problems of the diffraction of electromagnetic waves by the suggested method.     

A finite element method for approximating electromagnetic scattering from a conducting object

We provide an error analysis of a fully discrete finite element – Fourier series method for approximating Maxwell's equations. The problem is to approximate the electromagnetic field scattered by a bounded, inhomogeneous and anisotropic body. The method is to truncate the domain of the calculation using a series solution of the field away from this domain. We first prove a decomposition for the Poincaré-Steklov operator on this boundary into an isomorphism and a compact perturbation. This is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem. Using this decomposition, and edge elements to discretize the interior problem, we prove an optimal error estimate for the overall problem.     

Singular boundary method for 3D time-harmonic electromagnetic scattering problems

A frequency domain singular boundary method is presented for solving 3D time-harmonic electromagnetic scattering problem from perfect electric conductors. To avoid solving the coupled partial differential equations with fundamental solutions involving hypersingular terms, we decompose the governing equation into a system of independent Helmholtz equations with mutually coupled boundary conditions. Then the singular boundary method employs the fundamental solutions of the Helmholtz equations to approximate the scattered electric field variables. To desingularize the source singularity in the fundamental solutions, the origin intensity factors are introduced. In the novel formulation, only the origin intensity factors for fundamental solutions of 3D Helmholtz equations and its derivatives need to be considered which have been derived in the paper. Several numerical examples involving various perfectly conducting obstacles are carried out to demonstrate the validity and accuracy of the present method.     

High-frequency projection method for plane problems of wave diffraction on dielectric bodies

A version of the Galerkin incomplete projection method is described for plane problems of wave diffraction on dielectric bodies of arbitrary shape. The proposed method generalizes the Sommerfeld method, which constructs diffraction series rapidly converging at high frequencies for circular and spherical bodies. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 58–67.     

A preconditioned GMRES for complex dense linear systems from electromagnetic wave scattering problems

We consider the numerical solution of linear systems arising from the discretization of the electric field integral equation (EFIE). For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. The electromagnetic scattering problem is here solved by means of a preconditioned GMRES in the context of the multilevel fast multipole method (MLFMM). The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM, as preconditioner for the GMRES iterations. As experience shows, the efficiency of an ILU preconditioning for such systems essentially depends on a sufficient fill-in, which apparently sacrifices the sparsity of the near-field matrix. In the light of this experience we propose a multilevel near-field matrix and its corresponding HSS representation as a hierarchical preconditioner in order to substantially reduce the number of iterations in the solution of the resulting system of equations.     

A point source method for inverse acoustic and electromagnetic obstacle scattering problems

We present the mathematical foundation for a point source methodto solve some inverse acoustic and electromagnetic obstaclescattering problems in three dimensions. We investigate theinverse acoustic scattering problem by a sound-soft and a sound-hardscatterer and the inverse electromagnetic scattering problemby a perfect conductor. Two independent approaches to the methodare presented which reflect its strong relation to basic propertiesof obstacle scattering problems.     

Matrix solution to electromagnetic scattering by a conducting cylinder with an eccentric metamaterial coating

Closed series solution to scattering by an eccentric coated cylinder is realized in matrix form. Diffracted radiation characteristics are investigated for N incident plane transverse electric (TE) waves. The solution is obtained by the boundary value analysis and the addition theorem of the Bessel's functions. Wave transformation and orthogonality of the complex exponentials are also used to find an infinite series in the solution. Numerical results are shown by reducing the infinite series to a limited number of terms and compared to previously published works.     

Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell’s equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.     

Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric

Consider the problem of scattering of electromagnetic waves by a doubly periodic Lipschitz structure. The medium above the structure is assumed to be homogenous and lossless with a positive dielectric coefficient. Below the structure there is a perfect conductor with a partially coated dielectric boundary. We first establish the well‐posedness of the direct problem in a proper function space and then obtain a uniqueness result for the inverse problem by extending Isakov's method. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust and high-order effective boundary conditions for perfectly conducting scatterers coated by a thin dielectric layer

A new approach is considered to construct effective boundaryconditions for the solution of problems related to the scatteringof electromagnetic waves by perfectly conducting cylinders coatedby a thin dielectric shell. These boundary conditions aim tobe both robust and of high order while remaining set in termsof surface differential operators involving at most second-orderderivatives. Error estimates yield a theoretical basis for theuse of these boundary conditions in practical computations.Some numerical experiments at frequencies beyond the range ofvalidity of the usual impedance boundary conditions validatethe efficiency of the approach.     

Analysis of electromagnetic wave propagation in complex dielectric materials with uncertainty

We consider the propagation of high frequency electromagnetic pulses in complex materials with nonlinear polarization. The physical problem is modeled by Maxwell’s equations in variational form, and well-posedness results are established with respect to probability distributions on the polarization parameters (in a Prohorov metric sense).