Volume Singular Integral Equation Research Articles

Numerical Method for Reconstruction of Inhomogeneity Parameters of a Body Placed in a Semi-Infinite Rectangular Waveguide

The article is devoted to an inverse problem of reconstruction of inhomogeneity parameters of a body in semi-infinite rectangular waveguide. We suggest a new numerical method for solving this one. The inverse problem is reduced to the solving of volume singular integral equation of first kind and additional procedure of calculation the parameters of inhomogeneity. The numerical results are presented.

A modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann series

This work addresses the solvability and solution of volume integrodifferential equations (VIEs) associated with 3D free-space transmission problems (FSTPs) involving elastic or conductive inhomogeneities. A modified version of the singular volume integral equation (SVIE) associated with the VIE is introduced and shown to be of second kind involving a contraction operator, i.e. solvable by Neumann series, implying the well-posedness of the initial VIE. Then, the solvability of VIEs for frequency-domain FSTPs (modelling the scattering of waves by compactly-supported inhomogeneities) follows by a compact perturbation argument. This approach extends work by Potthast (1999) on 2D electromagnetic problems (transverse-electric polarization conditions) involving orthotropic inhomogeneities in a isotropic background, and contains recent results on the solvability of Eshelby's equivalent inclusion problem as special cases. The proposed modified SVIE is also useful for iterative solution methods, as Neumannn series converge (i) unconditionally for static problems and (ii) on some inhomogeneity configurations for which divergence occurs with the usual SVIE for wave scattering problems.

Volume integral equations for electromagnetic scattering in two dimensions

We study the strongly singular volume integral equation that describes the scattering of time-harmonic electromagnetic waves by a penetrable obstacle. We consider the case of a cylindrical obstacle and fields invariant along the axis of the cylinder, which allows the reduction to two-dimensional problems. With this simplification, we can refine the analysis of the essential spectrum of the volume integral operator started in a previous paper (Costabel et al., 2012) and obtain results for non-smooth domains that were previously available only for smooth domains. It turns out that in the transverse electric (TE) case, the magnetic contrast has no influence on the Fredholm properties of the problem. As a byproduct of the choice that exists between a vectorial and a scalar volume integral equation, we discover new results about the symmetry of the spectrum of the double layer boundary integral operator on Lipschitz domains.

A volume integral equation method for periodic scattering problems for anisotropic Maxwell's equations

This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell's equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green's functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.

Determination of the effective permittivity of a body in a waveguide from the reflection coefficient

The inverse problem of diffraction of the electromagnetic field by an inhomogeneous body placed in a rectangular waveguide with perfectly conducting walls is considered. The problem is reduced to a nonlinear volume singular integral equation. The integral equation is solved with the help of the iteration method. The permittivity is determined with the use of the reflection coefficient. Computation results for a figure of a complex shape are presented.

Singular electromagnetic modes in an anisotropic medium

We construct the singular mode corresponding to a spatial essential spectrum of the integral operator for the scattering of electromagnetic waves by a three-dimensional body of finite size with inhomogeneous and anisotropic dielectric permittivity tensor. The permittivity tensor field is assumed to be Hölder continuous throughout the whole space. The singular volume integral equation, transformed from Maxwell’s equations, makes it feasible to deduce explicit forms of both the continuous essential spectrum and the corresponding singular modes. The obtained singular mode is a natural extension of the previously obtained one for the isotropic case and is applicable to a much wider class of dielectric scattering bodies. A discussion is made of the possibility for realizability of electromagnetic waves, with finite energy, concentrated at a point in the body.

Volume integral equations for scattering from anisotropic diffraction gratings

We analyze electromagnetic scattering of transverse magnetic polarized waves from a diffraction grating consisting of a periodic, anisotropic, and possibly negative index dielectric material. Such scattering problems are important for the modelization of, for example, light propagation in nano‐optical components and metamaterials. The periodic scattering problem can be reformulated as a strongly singular volume integral equation, a technique that attracts continuous interest in the engineering community but has rarely received rigorous theoretic treatment. In this paper, we prove new (generalized) Gårding inequalities in weighted and unweighted Sobolev spaces for the strongly singular integral equation. These inequalities also hold for materials for which the real part of the material parameter takes negative values inside the diffraction grating, independently of the value of the imaginary part. Copyright © 2012 John Wiley & Sons, Ltd.

Inverse scattering in guides

We present statements and a method of solution to the inverse scattering problem of reconstructing permittivity of a dielectric inclusion in a 2D or 3D waveguide from the transmission characteristics. The approach employs a volume singular integral equation (VSIE) method. The unique solvability of VSIE is established. The inverse problem is solved by the method of iterations applied to VSIE; the convergence of the method is proved

Subhierarchical method for solving problems of diffraction of electromagnetic waves by a dielectric body in a rectangular waveguide

The problem of diffraction of an external electromagnetic field by a locally inhomogeneous body placed in a rectangular waveguide with perfectly conducting walls is considered. The formulated problem is reduced to a volume singular integral equation. The problem is solved with the use of the numerical collocation method. A subhierarchical method is applied to analyze the problem for structures of a complex geometric shape. Because of a large amount of computation, the problem is solved with the use of parallel algorithms realized on a supercomputer complex.

On the existence and uniqueness of solutions of the inverse boundary value problem for determining the permittivity of materials

The problem of determining the permittivity of material samples of arbitrary shape placed in a rectangular waveguide with perfectly conducting walls is investigated. The problem is reduced to solving a nonlinear volume singular integral equation. A theorem on the existence and uniqueness of solutions to the nonlinear volume singular integral equation and of the inverse boundary value problem for determining the permittivity of the material is proved.

Existence and uniqueness of a solution to the inverse problem of the complex permittivity reconstruction of a dielectric body in a waveguide

This paper presents a statement, a proof of uniqueness, and a method of solution to the inverse problem of the determination of permittivity of a lossy dielectric inclusion in a three-dimensional waveguide of rectangular cross-section from the transmission characteristics. The approach is based on the solution to a volume singular integral equation (VSIE). The examination of this equation is based on the analysis of the corresponding boundary value problem (BVP) for the system of Maxwell's equations and the equivalence of this BVP and VSIE. The existence and uniqueness for VSIE in the space of square-integrable functions are proved. The permittivity reconstruction employs a method of iterations applied to the solution of VSIE.

Inverse Boundary Value Problem for Determination of Permittivity of Dielectric Body in a Waveguide using the Method of Volume Singular Integral Equation

The problem of diffraction of an external electromagnetic field by a locally nonhomogeneous body in a perfectly conducting waveguide is examined. This problem is reduced to solving a volume singular integral equation (VSIE). The examination of this equation is based on the analysis of the corresponding boundary value problem (BVP) for the system of Maxwell's equations and the equivalence of this BVP and VSIE. The existence and uniqueness for VSIE in L2 are proved. A numerical Galerkin method for the solution of VSIE is proposed and its convergence is proved.Inverse boundary value problem for determination of permittivity of dielectric body in a waveguide is reduced to the nonlinear volume singular integral equation.

Investigation of Electromagnetic Diffraction by a Dielectric Body in a Waveguide Using the Method of Volume Singular Integral Equation

The problem of diffraction of an electromagnetic field by a locally nonhomogeneous body in a perfectly conducting waveguide of rectangular cross section is considered. This problem is reduced to solving a volume singular integral equation (VSIE). The examination of this equation is based on the analysis of the corresponding boundary value problem (BVP) for the system of Maxwell's equations and the equivalence of this BVP and VSIE. The existence and uniqueness for VSIE in the space of square-integrable functions are proved. A numerical Galerkin method for the solution of VSIE is proposed, and its convergence is proved.

Existence and Uniqueness of a Solution of a Singular Volume Integral Equation in a Diffraction Problem

Existence and Uniqueness of a Solution of a Singular Volume Integral Equation in a Diffraction Problem

Numerical Analysis of an Unbounded Operator Arising from an Electromagnetic Interior Scattering Problem

In this paper a 1-D singular integral equation motivated by the well-known singular volume integral equation associated with electromagnetic interior scattering is considered. In the 3-D case the kernel (the dyad Green’s function) is O(R−3) and in the present 1-D case the kernel is O(R−1). The numerical solution is obtained by using a simple Nystrom method. The mapping properties of the integral operator and the numerical integral operators are studied in various (Holder) subspaces of C([a, b]). Convergence theorems for the numerical integral operators as well as for the numerical solutions are proved.