Integral equation solution for analyzing scattering fromone-dimensional periodic coated strips

A set of integral equations based on the surface/surface formulation are developed for analyzing electromagnetic scattering by one-dimensional periodic structures. To compare the accuracy, efficiency, and robustness of the formulation, the electric field integral equation (EFIE), magnetic field integral equation (MFIE), and combined field integral equation (CFIE) are developed for analyzing the same structure for different excitations. Due to the periodicity of the structure, the integral equations are formulated in the spectral domain using the Fourier transform of the integrodifferential operators. The generalized-biconjugate-gradient-fast Fourier transform method with subdomain basis functions is used to solve the matrix equation     

On filtered binary processes

The problem of calculating the probability density function of the output of anRCfilter driven by a binary random process with intervals generated by an equilibrium renewal process is studied. New integral equations, closely related to McFadden's original integral equations, are derived and solved by a matrix approximation method and by iteration. Transformations of the integral equations into differential equations are investigated and a new closed-form solution is obtained in one special case. Some numerical results that compare the matrix and iteration solutions with both exact solutions and approximate solutions based upon the Fokker-Planck equation are presented.     

混合场积分方程结合MLFMA分析导体介质复合目标电磁散射问题

针对组合目标电磁散射问题,采用一类新的混合场积分方程进行分析.通过合理选择比例系数组合表面电场和磁场积分方程,构造出具有良好收敛性的阻抗矩阵.MLFMA的迭代求解采用广义最小残差方法(GMRES),结合预条件技术进一步减少迭代次数,加速计算并提高处理电大尺寸导体介质复合目标的能力.研究了几类典型目标电磁散射特性并比较了计算效率,数值算例验证了该方法的准确性和高效性.     

Integral equation based analysis of scattering from 3-D inhomogeneous anisotropic bodies

This paper presents an integral equation based scheme to analyze scattering from inhomogeneous bodies with anisotropic electromagnetic properties. Both the permittivity and permeability are assumed to be generalized tensors. Requisite integral equations are derived using volume equivalence theorem with the electric and magnetic flux densities being the unknown quantities. Matrix equations are derived by discretizing these unknowns using three dimensional Rao-Wilton-Glisson basis functions. Reduction of the integral equation to a corresponding matrix equation is considerably more involved due to the presence of anisotropy and the use of vector basis function; methods for evaluation of the integrals involved in the construction of this matrix is elucidated in detail. The method of moments technique is augmented with the fast multipole method and a compression scheme. The latter two enable large scale analysis. Finally, several numerical results are presented and compared against analytical solutions to validate the proposed scheme. An appendix provides analytical derivations for the formulae that are used to validate numerical method, and the necessary formulae that extends the approach presented herein to the analysis of scattering bianisotropic bodies.     

Integral equation formulations for imperfectly conducting scatterers

Integral equation formulations are presented for characterizing the electromagnetic (EM) scattering interaction for nonmetallic surfaced bodies. Three different boundary conditions are considered for the surfaces: namely, the impedance (Leontovich), the resistive sheet, and its dual, the magnetically conducting sheet boundary. The integral equation formulations presented for a general geometry are specialized for bodies of revolution and solved with the method of moments (MM). The current expansion functions, which are chosen, result in a symmetric system of equations. This system is expressed in terms of two Galerkin matrix operators that have special properties. The solutions of the integral equation for the impedance boundary at internal resonances of the associated perfectly conducting scatterer are examined. The results are compared with the Mie solution for impedance-coated spheres and with the MM solutions of the electric, magnetic, and combined field formulations for impedance-coated bodies.     

A method for computing scattering by large arrays of narrow strips

The electromagnetic scattering characteristics of an array of narrow, conducting strips can he developed readily by extending the work of Butler and Wilton who show that Chebyshev polynomials augmented with the edge condition can be used to solve the narrow-strip/narrow-slot integral equations. The strips reside in a homogeneous medium of infinite extent and are considered narrow relative to wavelength in the medium at the frequency of excitation. The unknown current distributions on the strips are represented as linear combinations of certain basis functions that are exact solutions to the approximate equation for an isolated narrow strip subject to a special excitation. The resulting power-series treatment allows easy calculation of the coupling terms among the strips in the array in a simple matrix equation by which the unknown coefficients in the current distribution expansions may be readily computed. With these coefficients, one can obtain the distribution of current on each strip and the total scattered field. The method is particularly well suited for handling large arrays with more strips than could be accommodated by the usual moment method. Numerical data-currents and scattered fields-are presented for various cases of interest.     

On combined source solutions for bodies with impedance boundary conditions

Numerical solutions to the impedance boundary condition (IBC) combined source integral equation (CSIE) for scattering from impedance spheres are presented. The CSIE formulation is a well-posed alternative to the IBC electric and magnetic field integral equations which can be contaminated by spurious resonant modes. Compared with the IBC combined field integral equation (CFIE), CSIE solutions have the same accuracy when the combined source coupling admittance is chosen to be the same value as the combined field coupling admittance. However, the CSIE formulation is better suited than the CFIE for creating a general purpose computer code capable of handling aperture radiation problems and/or a scatterer which has a spatially varying surface impedance.     

A new well-conditioned Integral formulation for Maxwell equations in three dimensions

We present a new boundary integral equation dedicated to the solution of the boundary problem of a perfectly electrically conducting surface for the harmonic Maxwell equations in unbounded domains. Any solution of the harmonic Maxwell equations is represented as the electromagnetic field generated by a combination of electric and magnetic potentials. These potentials are those appearing in the classical combined field integral equation (CFIE), but their coupling is realized by an operator Y/spl tilde//sup +/ instead of a coefficient. Therefore, the integral equation obtained can be viewed as a generalization of the CFIE. In this paper, we propose an explicit construction of the coupling operator Y/spl tilde//sup +/ which is designed to approximate the exterior admittance operator of the scattering obstacle. A local approximation by the admittance operator of the tangential plane seems to be relevant thanks to the localization effects related to high-frequency phenomena. The provided numerical simulations show that this formulation leads to linear systems that are better conditioned compared to more classical integral equations, which speeds up the resolution when solved with iterative techniques.     

A higher order parallelized multilevel fast multipole algorithm for3-D scattering

A higher order multilevel fast multipole algorithm (MLFMA) is presented for solving integral equations of electromagnetic wave scattering by three-dimensional (3-D) conducting objects. This method employs higher order parametric elements to provide accurate modeling of the scatterer's geometry and higher order interpolatory vector basis functions for an accurate representation of the electric current density on the scatterer's surface. This higher order scheme leads to a significant reduction in the mesh density, thus the number of unknowns, without compromising the accuracy of geometry modeling. It is applied to the electric field integral equation (EFIE), the magnetic field integral equation (MFIE), and the combined field integral equation (CFIE), using Galerkin's testing approach. The resultant numerical system of equations is then solved using the MLFMA. Appropriate preconditioning techniques are employed to speedup the MLFMA solution. The proposed method is further implemented on distributed-memory parallel computers to harness the maximum power from presently available machines. Numerical examples are given to demonstrate the accuracy and efficiency of the method as well as the convergence of the higher order scheme     

Iterative determination of permittivity and conductivity profiles of a dielectric slab in the time domain

A numerical method is presented to compute one unknown constitutive parameter of an inhomogeoeous lossy dielectric slab from the reflected field in the time domain. The method is based upon a space-time discretization of the integral equation for the reflected field. In the inversion, especially those space-time points where the numerical computation of the electric-field strength in the slab is most accurate are taken into account. This is achieved by computing the unknown parameter iteratively. Alternately solving equations for an approximate direct-scattering problem and an approximate inverse-scattering problem yields successive approximations for the electric field in the slab and the unknown constitutive coefficient. Both problems lead to an infinite system of linear equations from which a finite subsystem is selected. General criteria for this selection are presented. Various profiles have been reconstructed numerically from the reflected field due to a sine-squared incident pulse.     

一种求解内谐振条件下导体散射特性的有效方法

众所周知,在内谐振条件下,用积分方程法分析导体的散射特性时,不论是电场积分方程还是磁场积分方程,所求得的解都是不唯一或者不稳定的。本文提出了一种新的方案,通过引入一个微小的复频率,并结合逼近理论求得导体表面的真实电流密度,从而得到正确的导体散射特性。此方法具有实现简单和概念清晰的优点。文中分别以无限长理想导体正方柱和两个理想导体球为例,并将计算结果与混合场积分方程法所得的结果进行比较,它们之间良好的一致性说明了本文所提方法的正确性和有效性。     

Levinson-like and Schur-like fast algorithms for solvingblock-slanted Toeplitz systems of equations arising in wavelet-basedsolution of integral equations

The Wiener-Hopf integral equation of linear least-squares estimation of a wide-sense stationary random process and the Krein integral equation of one-dimensional (1-D) inverse scattering are Fredholm equations with symmetric Toeplitz kernels. They are transformed using a wavelet-based Galerkin method into a symmetric “block-slanted Toeplitz (BST)” system of equations. Levinson-like and Schur-like fast algorithms are developed for solving the symmetric BST system of equations. The significance of these algorithms is as follows. If the kernel of the integral equation is not a Calderon-Zygmund operator, the wavelet transform may not sparsify it. The kernel of the Krein and Wiener-Hopf integral equations does not, in general, satisfy the Calderon-Zygmund conditions. As a result, application of the wavelet transform to the integral equation does not yield a sparse system matrix. There is, therefore, a need for fast algorithms that directly exploit the (symmetric block-slanted Toeplitz) structure of the system matrix and do not rely on sparsity. The first such O(n²) algorithms, viz., a Levinson-like algorithm and a Schur (1917) like algorithm, are presented. These algorithms are also applied to the factorization of the BST system matrix. The Levinson-like algorithm also yields a test for positive definiteness of the BST system matrix. The results obtained are directly applicable to the problem of constrained deconvolution of a nonstationary signal, where the locations of the smooth regions of the signal being deconvolved are known a priori     

Application of FD-MEI to electromagnetic scattering fromtransversally anisotropic inhomogeneous cylinders

The numerical solution of the finite difference with the measured equation of invariance (FD-MEI) for transversally anisotropic cylinders is presented. It is different from the currently available methods for the anisotropic scatterers; this approach involves both the finite-difference equations at interior nodes of the mesh and the measured equation of invariance for the boundary nodes of the mesh. It has the merit of saving the computing time and computer memory needs. By introducing the novel nine-point mesh the generalized finite-difference equations for the inhomogeneous transversally anisotropic material are derived. The radar cross section (RCS) for inhomogeneous, lossy, electrically large, and arbitrarily shaped two-dimensional transversally anisotropic objects are calculated. The computing efficiency and accuracy are assessed using comparisons with two other methods, namely, the integral equation based on a plane wave representation of the fields and the combined field surface integral equations when available     

Electromagnetic scattering from composite objects using a mixtureof exact and impedance-boundary conditions

Different surface integral equations for characterizing the electromagnetic scattering from a surface impedance object partially coated with dielectric materials are presented. The impedance boundary condition (IBC) is applied on the impedance surface and the exact boundary condition is applied on the dielectric surface. The resulting integral equations are solved for bodies of revolution using the method of moments. The numerical results are compared with the exact solution for a sphere. Other geometries are considered, and their results are verified by comparing results of the numerical solutions which were obtained using different formulations. The internal resonance problem is examined. It is found that the combined field integral equation (CFIE) can be used at any frequency and with any surface impedance     

Fast spectral domain algorithm for rapid solution of integralequations

A fast spectral domain algorithm is presented for rapid solution of planar surface integral equations. The method of moments coupling integral matrix is formulated in the spectral domain but not explicitly calculated. Thus, in conjunction with an iterative equation solver, the pertinent matrix/vector products are evaluated with complexity O(n) where n is the number of unknowns. Validation and timing results are presented for an array analysis approach using a hybrid finite element (FE)-boundary integral implementation     

A hybrid equation approach for the solution of electromagneticscattering problems involving two-dimensional inhomogeneous dielectriccylinders

A numerical procedure for the solution of electromagnetic scattering problems involving inhomogeneous dielectric cylinders of arbitrary cross section is discussed. The cases of illumination by both transverse magnetic (TM) and transverse electric (TE) plane waves are considered. The scattering problems are modeled via a hybrid integral-equation/partial-differential-equation approach. The method of moments is applied to obtain a system of simultaneous equations that can be solved for the unknown surface current densities and the interior electric field. The interior region partial differential equation and the exterior region surface integral equation are coupled in such a manner that many existing surface integral equation computer codes for treating problems involving scattering by homogeneous dielectric cylinders can be modified easily to generate the block of the matrix corresponding to the surface current interactions. The overall system matrix obtained using the method of moments is largely sparse. Numerical results are presented and compared with exact solutions for homogeneous and inhomogeneous circular cylinders     

An Improved Weak-Form BCGS-FFT Combined With DCIM for Analyzing Electromagnetic Scattering by 3-D Objects in Planarly Layered Media

Fast algorithms for electrically large objects buried in layered media are mainly hindered by two time-consuming processes. One is the table filling of Green's function, and the other is the solving of the impedance matrix equation. For the first, to accelerate the evaluation of the time-consuming Sommerfeld integral in the dyadic Green's function (DGF), the discrete complex image method (DCIM) is introduced to get a closed-form DGF. To further accelerate the calculation of DGF for the volume electric field integral equation (EFIE), DGF is split before applying DCIM. For the second, the iterative solver stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) is combined with DCIM for solving the matrix equations. Meanwhile, the closed-form DGF enables the "spherical-mean" Green's function, which eliminates the singularity of Green's function. Numerical results show that the weaker singularity results in a faster and steadier convergence rate for iterative solvers     

Current induced by TE excitation on a conducting cylinder located near the planar interface between two semi-infinite half-spaces

An analysis is presented for determining the current induced by a known transverse electric excitation on a perfectly conducting cylinder located near the planar interface separating two semi-infinite, homogeneous half-spaces of different electromagnetic properties. The conducting cylinder of general cross section is of infinite extent and the excitation is transverse electric to the cylinder axis. Two types of integral equations, the magnetic field integral equation and the electric field integral equation, are formulated, and the Green's functions for the integral equations are derived in an appendix. Numerical solution methods for solving the integral and integrodifferential equations are presented. For a strip parallel or perpendicular to the interface, a circular cylinder, and a rectangular cylinder, data are presented and discussed for selected parameters, including the case of a cylinder resting on the interface.     

A survey of the near-field far-field inverse scattering inverse source integral equation

The Kirchhoff direct integration of the scalar wave equation is reviewed, and some properties of the Kirchhoff surface integral are discussed, from the perspective of the inverse scattering inverse source problem. A modified Kirchhoff surface integral is introduced, leading to a Fredholm integral equation of the first kind for the unknown sources (induced by the incident field) inside a volume in terms of the (scattered) fields on the surface enclosing this volume. The properties and physical meaning of this integral equation are discussed. A generalization of this integral equation for the vector electromagnetic wave equations is presented.