Homogeneous Dielectric Scatterers Research Articles

Tensor Train Accelerated Solution of Volume Integral Equation for 2-D Scattering Problems and Magneto-Quasi-Static Characterization of Multiconductor Transmission Lines

The method of moments (MoM) discretization of volume integral equation (VIE) results in dense $N \times N$ matrix, $N$ being the number of MoM basis functions. Naive solution of such a system of linear algebraic equations (SLAE) is expensive when problems become large scale. Recently, tensor decomposition has been introduced for solving the SLAE by folding its matrix and its vectors into high-dimensional tensors. In this paper, we present detailed explanations for tensor train (TT) decomposition of the SLAE matrices and vectors resulting from MoM discretization of VIE for scalar 2-D scattering problems under TM-polarization and magneto-quasi-static characterization of multiconductor transmission lines. For Toeplitz matrices resulted from MoM discretization on structured meshes, the extraordinary performance of TT with scaling of $\log (N)$ in CPU time and memory is shown to directly solve SLAE with millions of unknowns within a few minutes and few megabytes of memory. Such $\log (N)$ performance is limited, however, to the SLAE with purely Toeplitz matrices corresponding to the scattering problems on homogeneous dielectric scatterers of the rectangular cross section. To overcome this limitation and solve the problems with arbitrarily shaped inhomogeneous objects, we propose an iterative conjugate gradient-TT (CG-TT) scheme for solving MoM discretized VIE, which utilizes TT decomposition for the fast evaluation of the matrix-vector products. The CG-TT shows CPU and memory scaling in the low- and high-frequency regimes with $O(N)$ and $O(r^{2}~N \log N)$ , respectively, $r$ being the highest rank in the TT carriages. Detailed analysis of memory and CPU time scaling with respect to a number of unknowns and time-harmonic frequency is presented.

A Hybrid Quantitative Method for Inverse Scattering of Multiple Dielectric Objects

A hybrid method is proposed to locate the boundaries and determine the dielectric constants of multiple homogeneous scatterers by postprocessing the results of the linear sampling method (LSM). The hybrid method is composed of three steps combining the capabilities of the LSM, the multiple signal classification (MUSIC) algorithm, and the generalized multipole technique (GMT). Initially, a new measure of the noise level of the multistatic response (MSR) matrix is proposed. Subsequently, the LSM is applied using the Tikhonov regularization. Then, the scattered field is back-propagated into the investigation domain using the GMT, where in order to ensure the stability, a MUSIC-based approach is exploited to determine the optimum reconstructing multipole sources. The case of limited scattered field data is also addressed by introducing a modification of the Papoulis–Gerchberg algorithm (PGA). Finally, the boundary and the dielectric constant of each scatterer are obtained simultaneously by solving an inverse boundary value problem through an optimization. The proposed hybrid approach is capable of efficiently approximating the dielectric constant of multiple homogeneous dielectric scatterers in a parallel manner. The effectiveness of the proposed method and the accuracy of the solutions are demonstrated by applying the method to both numerical and measured data.

High-order Nyström discretizations for the solution of integral equation formulations of two-dimensional Helmholtz transmission problems

We present and analyze fully discrete Nystr\"om methods for the solution of three classes of well conditioned boundary integral equations for the solution of two dimensional scattering problems by homogeneous dielectric scatterers. Specifically, we perform the stability analysis of Nystr\"om discretizations of (1) the classical second kind integral equations for transmission problems [KressRoach, 1978], (2) the single integral equation formulations [Kleinman & Martin,1988], and (3) recently introduced Generalized Combined Source Integral Equations [Boubendir,Bruno, Levadoux and. Turc,2013]. The Nystr\"om method that we use for the discretization of the various integral equations under consideration are based on global trigonometric approximations, splitting of the kernels of integral operators into singular and smooth components, and explicit quadratures of products of singular parts (logarithms) and trigonometric polynomials. The discretization of the integral equations (2) and (3) above requires special care as these formulations feature compositions of boundary integral operators that are pseudodifferential operators of positive and negative orders respectively. We deal with these compositions through Calderon's calculus and we establish the convergence of fully discrete Nystr\"om methods in appropriate Sobolev spaces which implies pointwise convergence of the discrete solutions. In the case of analytic boundaries, we establish superalgebraic convergence of the method.

Time domain inverse scattering for a buried homogeneous cylinder in a slab medium using NU-SSGA

A time-domain inverse scattering technique for reconstructing a buried homogeneous cylinder with arbitrary cross section in a slab medium is proposed. For the forward scattering, the FDTD method is employed to calculate the scattered E fields. Base on the scattering fields, these inverse scattering problems are transformed into optimization problems. The non-uniform steady state genetic algorithm (NU-SSGA) is applied to reconstruct the location shape and permittivity of the two-dimensional homogeneous dielectric cylinder. The NU-SSGA is a population-based optimization approach that aims to minimize the objective function between measurements and computer-simulated data. A set of representative numerical results is presented for demonstrating that the proposed approach is able to efficiently reconstruct the electromagnetic properties of homogeneous dielectric scatterer even when the initial guess is far away from the exact one. In addition, the effects of Gaussian noises on the image reconstruction are also investigated.

Time Domain Inverse Scattering for a Homogenous Dielectric Cylinder by Asynchronous Particle Swarm Optimization

Abstract In this paper, we propose a time domain inverse scattering technique for reconstructing the electromagnetic properties of a homogeneous dielectric cylinder based on the finite difference time domain method and the asynchronous particle swarm optimization (APSO). The homogeneous dielectric cylinder with unknown electromagnetic properties is illuminated by transverse magnetic pulse and the scattered field is recorded outside. By minimizing the discrepancy between the measured and estimated scattered field data, the location, shape, and permittivity of the dielectric cylinder are reconstructed. The inverse problem is resolved by an optimization approach and the global searching scheme APSO is then employed to search the parameter space. A set of representative numerical results is presented for demonstrating that the proposed approach is able to efficiently reconstruct the electromagnetic properties of the homogeneous dielectric scatterer even when the initial guess is far away from the exact one. In addition, the effects of Gaussian noises on imaging reconstruction are also investigated.

Time Domain Image Reconstruction for a Buried 2D Homogeneous Dielectric Cylinder Using NU-SSGA

This article presents an image reconstruction approach for a buried homogeneous cylinder with arbitrary cross-section in half space. The computational method combines the finite difference time domain (FDTD) method and non-uniform steady state genetic algorithm (NU-SSGA) to determine the shape and location of the subsurface scatterer with arbitrary shape. The FDTD-subgirdding technique is implemented for modeling the shape of the cylinder more closely. The inverse problem is reformulated into an optimization problem and the global searching scheme NU-SSGA with closed cubic-spline is then employed to search the parameter space. A set of representative numerical results is presented for demonstrating that the proposed approach is able to efficiently reconstruct the electromagnetic properties of homogeneous dielectric scatterer even when the initial guess is far from the exact one. In addition, the effects of Gaussian noises on imaging reconstruction are also investigated.

A novel semi-analytic method for the analysis of scattering by dielectric objects immersed in uniform media

The method for the solution of scattering problems with homogeneous dielectric scatterers based on a single coordinate multipole expansion, the Stratton–Chu integral and Maxwell's equations in the integral form is proposed in this paper. Its convergence is proved. The sources of ill-conditionality of constitutive algebraic systems are established. The method of their regularization based on the use of a scalar control parameter is suggested. For the sake of validation, the numerical analysis is performed for different testing objects.

Image reconstruction for 2D homogeneous dielectric cylinder using FDTD method and SSGA

This paper presents an image reconstruction approach for a 2-D homogeneous cylinder with arbitrary cross section in free space. The computational method combines the finite difference time domain (FDTD) method and non-uniform steady state genetic algorithm (NU-SSGA) to determine the shape and location of the scatterer with arbitrary shape. The subgirdding technique is implemented for modeling the shape of the cylinder more closely. The inverse problem is reformulated into an optimization problem and the global searching scheme NU-SSGA with closed cubic-spline is then employed to search the parameter space. A set of representative numerical results is presented for demonstrating that the proposed approach is able to efficiently reconstruct the electromagnetic properties of homogeneous dielectric scatterer even when the initial guess is far away from the exact one. In addition, the effects of Gaussian noises on imaging reconstruction are also investigated.

Effective response and scattering cross section of spherical inclusions in a medium

The Maxwell-Garnett theory for a right-handed homogeneous system is extended in order to investigate the effective response of a medium consisting of low density distributed 3-dimensional inclusions. The polarisability factor is modified to account for inclusions with binary layered volumes and it is shown that such a configuration can yield doubly negative effective permittivity and permeability. Terms representing second-order scattering interactions between binary inclusions in the medium are derived and are used to reformulate conventional effective medium theory. In the appropriate limit, the one-body theory of Maxwell-Garnett is recovered. The scattering cross section of the spherical inclusions is determined and comparison is made to homogeneous dielectric scatterers in the Rayleigh limit. It is found that the scattering resonances can be manipulated using the inclusion parameters. Furthermore, the effect that two-interacting spherical inclusions in a medium have on the scattering cross section is investigated via higher order dipole moments while the issue of reducing the scattering cross section to zero is also examined.

A Hybrid Approach to 3D Microwave Imaging by Using Linear Sampling and ACO

A hybrid approach is proposed to inspect three-dimensional homogeneous dielectric scatterers by using microwaves. Namely, the considered method consists in two steps: first, the supports of the scatterers are retrieved by the linear sampling method; then the permittivities of the targets are estimated by an ant colony optimization algorithm. The described strategy combines the efficiency of the linear sampling method with the global optimization capabilities of the stochastic procedure. Numerical results assess the capabilities of the approach also in the presence of noise.

Scattering From Axisymmetric Dielectric Scatterers: A Hybrid Method of Solving the Surface Integral Equations

Scattering of EM waves from homogeneous dielectric scatterers is formulated in terms of two surface integral equations, for the components of the total electric and magnetic fields that are tangential to the surface of the scatterer. Two equivalent systems of such equations may be used, corresponding to the two types of Maue's integral equations for perfectly conducting scatterers. The appearance of surface divergence terms of both components in the dielectric case, in addition to the unknowns themselves, causes serious complications when the method of solution is based on some kind of division of the scatterer surface into patches; these complications become particularly apparent in the process of evaluating the locally dominant self-patch contribution to the surface integrals. They can be effectively avoided if a third equivalent system of surface integral equations is used, arising from a proper summation of the original two systems; then, the two singular Green functions that appear in the surface divergence terms are replaced by their non-singular difference and this, followed by a further transformation of these critical terms, results in the complete elimination of the surface divergence terms. The self-patch contribution can then be evaluated analytically and this helps reduce the size of the matrix, via which the values of the tangential field components at the patch centers are calculated. Scatterers of considerable electrical size, for instance spheres with ka of the order of 10 or 15, can then be treated with moderate size matrices. Numerical results for spheres, sphere-cone-spheres and other shapes are obtained and compared with results from other methods. Apart from spheres, results for other shapes are very rare and limited to small sizes in the literature.

Full-Vectorial Three-Dimensional Microwave Imaging Through the Iterative Multiscaling Strategy—A Preliminary Assessment

In this letter, a multiscaling strategy for full-vectorial three-dimensional inverse scattering problems is presented. The approach is fully iterative, and it avoids solving any forward problem at each step. Thanks to the adaptive multiresolution model, which offers considerable flexibility for the inclusion of the a priori knowledge and of the knowledge acquired during the iterative steps of the multiscaling process, the overall computational burden as well as the dimension of the search-space is considerably reduced. This allows to balance effectively the tradeoff between computational costs and achievable resolution accuracy. The effectiveness of the proposed approach is demonstrated through a selected set of preliminary experiments using homogeneous dielectric scatterers in a noisy synthetic environment.

Matrix interpretation of the spectral iteration technique for 3D dielectric scatterers in the resonance region

The matrix interpretation of the spectral iteration technique (MSIT) has recently been shown to provide rapid convergence for the electromagnetic problems associated with 2D homogeneous dielectric scatterers in the resonance region for both TM and TE incidence. The MSIT is presented in a form suitable for use on 3D homogeneous dielectric and lossy-dielectric scatterers. Rapid convergence on test problems in the resonance region is demonstrated with significantly fewer than 3√(N) iterations, where N is the number of unknowns, needed for electrically large scatterers. The bistatic RCS computed from the MSIT for spheres with a range of permittivities and diameters up to 2λ0 are shown to be in good agreement with analytic solutions. Current distributions from some initial tests on lossy-dielectric thin plates show well-behaved and symmetric results.

Recent advances in frequency domain techniques for electromagnetic scattering problems

Some recent applications of numerical techniques for the analysis of electromagnetic scattering problems involving several general classes of geometries are reviewed. The solution procedures are all formulated directly in the frequency domain, using the method of moments or the finite-element method, in terms of an integral equation and/or a partial differential equation. Geometrical configurations discussed include three-dimensional conducting scatterers and two- and three-dimensional homogeneous and inhomogeneous dielectric scatterers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>