An integral equation formulation of the direct scattering problem for an inhomogeneous slab

A numerical technique based on an integral equation scheme is developed for solving the direct scattering problem for an inhomogeneous slab. The integral equation is derived, using the induced current concept and the Green's function technique. The numerical method of solving the integral equation is presented. The method is proved to be numerically satisfactory and is applied to the slab of specific profiles. Numerical results for several cases are also included.     

TM scattering by an impedance sheet extension of a paraboliccylinder

An integral equation and method of moments (MM) solution are presented for the two-dimensional (2-D) problem of transverse magnetic (TM) scattering by an impedance-sheet extension of a perfectly conducting parabolic cylinder. An integral equation is formulated for a dielectric cylinder of general cross section in the presence of a perfectly conducting parabolic cylinder. It is then shown that the solution for a general dielectric cylinder considerably simplifies for the special case of TM scattering by a thin multilayered dielectric strip that can be represented as an impedance sheet. The solution is termed an MM/Green's function solution, where the unknowns in the integral equation are the electric surface currents flowing in the impedance sheet; the presence of the parabolic cylinder is accounted for by including its Green's function in the kernel of the integral equation. The MM solution is briefly reviewed, and expressions for the elements in the matrix equation and the scattered fields are given. Sample numerical results are provided     

Combined field solution for TM scattering from multiple conducting and dielectric cylinders of arbitrary cross section

A simple moment solution is given for the problem of electromagnetic scattering from multiple conducting and dielectric cylinders of arbitrary cross section. The system of conducting and dielectric cylinders is excited by a plane-wave polarized transverse magnetic to the axis of the cylinders. The equivalence principle is used to obtain three coupled integral equations for the induced electric current on the conducting cylinders and the equivalent electric and magnetic currents on the surface of dielectric cylinders. The combined field integral equation (CFIE) formulation is used. Sample numerical results are presented. The agreement with available published data is excellent.     

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     

Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering

We introduce a new version of the combined field integral equation (CFIE) for the solution of electromagnetic scattering problems in three dimensions. Unlike the conventional CFIE, the new CFIE is well-conditioned, meaning that it is a second kind integral equation that does not suffer from spurious resonances and does not become ill conditioned for fine discretizations (the so-called "low-frequency problem"). The new CFIE combines the standard magnetic field integral operator with an analytically preconditioned electric field integral operator. We also report numerical results showing that the new formulation stabilizes the number of iterations needed to solve the CFIE on closed surfaces. This is in contrast to the conventional CFIE, where the number of iterations grows as the discretization is refined.     

Application of wavelet transforms for solving integral equationsthat arise in rough-surface scattering

We consider the problem of scattering a plane wave from a periodic rough surface. The scattered field is evaluated once the field on the boundary is calculated. The latter is the solution of an integral equation. In fact, different integral equation formulations are available in both coordinate and spectral space. We solve these equations using standard numerical techniques, and compare the results to corresponding solutions of the equations using wavelet transform methods for "sparsification" of the impedance matrix. Using an energy check, the methods are shown to be highly accurate. We limit the discussion to the Dirichlet problem (scalar), or the TE-polarized case for a one-dimensional surface. The boundary unknown is thus the normal derivative of the total (scalar) field or, equivalently, the surface current. We illustrate two conclusions. First, sparsification (using "thresholded" wavelet transforms) can significantly reduce accuracy. Second, the wavelet transform did not speed up the overall solution. For our examples, the solution time was considerably increased when thresholded wavelet transforms were used     

The three dimensional weak form of the conjugate gradient FFTmethod for solving scattering problems

The problem of electromagnetic scattering by a three-dimensional dielectric object can be formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a spatial convolution of the free space Green's function and the contrast source over the domain of interest. A weak form of the integral equation for the relevant unknown quantity is obtained by testing it with appropriate testing functions. The vector potential is then expanded in a sequence of the appropriate expansion functions and the grad-div operator is integrated analytically over the scattering object domain only. A weak form of the singular Green's function has been used by introducing its spherical mean. As a result, the spatial convolution can be carried out numerically using a trapezoidal integration rule. This method shows excellent numerical performance     

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.     

Time domain inverse scattering

The inverse scattering problem is addressed directly in the time domain using an exact space-time integral equation as the basis for solution. A time domain approach to the solution is presented and demonstrated for the case of conducting rotationally symmetric targets with both axial incidence and broadside incidence. Both an iterative and a direct solution technique are illustrated and compared.     

Calculation of electromagnetic scattering from a dielectric cylinder using the conjugate gradient method and FFT

The scattering problem of an axially uniform dielectric cylinder is formulated in terms of the electric field integral equation, where the cylinder is of general cross-sectional shape, inhomogeneity, and anisotropy, and the incident field is arbitrary. Using the pulse-function expansion and the point-matching technique, the integral equation is reduced to a system of simultaneous equations. Then, a published procedure for solving the system using the conjugate gradient method and the fast Fourier transform (FFT) is generalized to the case of oblique-incidence scattering.     

Regularized integral equations and curvilinear boundary elementsfor electromagnetic wave scattering in three dimensions

The boundary integral equations (BIEs), in their original forms, which govern the electromagnetic (EM) wave scattering in three-dimensional space contain at least a hypersingularity (1/R³ ) or a Cauchy-singularity (1/R²), usually both. Thus, obtaining reliable numerical solutions using such equations requires considerable care, especially when developing systematic numerical integration procedures for realistic problems. Regularized BIEs for the numerical computation of time-harmonic EM scattering fields due to arbitrarily-shaped scatterers are introduced. Two regularization approaches utilizing an isolation method plus a mapping are presented to remove all singularities prior to numerical integration. Both approaches differ from all existing approaches to EM scattering problems. Both work for integral equations initially containing either hypersingularities or Cauchy-singularities, without the need to introduce surface divergences or other derivatives of the EM fields on the boundary. Also, neither approach is limited to flat surfaces nor flat-element models of curved surfaces. The Muller linear combination of the electric- and magnetic-field integral equations (EFIE) and (MFIE) is used to avoid the resonance difficulty that is usually associated with integral equation-based formulations. Some preliminary numerical results for EM scattering due to single and multiple dielectric spheres are presented and compared with analytical solutions     

A weak form of the conjugate gradient FFT method fortwo-dimensional TE scattering problems

The problem of two-dimensional scattering of a transversal electric polarized wave, by a dielectric object is formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a spatial convolution of the free-space Green's function and the contrast source over the domain of interest. A weak form of the integral equation for the unknown electric flux density is obtained by testing it with rooftop functions. The vector potential is expanded in a sequence of the rooftop functions and the grad-div operator is integrated analytically over the dielectric object domain only. The method shows excellent numerical performance     

Application of the integral equation method of smoothing to random surface scattering

The integral equation method of smoothing (IEMS) is applied to the magnetic field integral equation (MFIE) weighted by the exponentialexp (jk_{1}zeta)wherezetais the stochastic surface height. An integral equation in coordinate space for the average of the product of the surface current and the exponential factor is developed. The exact closed-form solution of this integral equation is obtained based on the specularity of the average scattered field. The complex amplitude of the average scattered field is thus determined by an algebraic equation which clearly shows the effects of multiple scattering on the surface. In addition, it is shown how the incoherent scattered power can be obtained using this method. Comparisons with the Kirchhoff approximation and the dishonest approach are presented, and the first-order smoothing result is shown to be superior to both.     

A combined steepest descent-fast multipole algorithm for the fastanalysis of three-dimensional scattering by rough surfaces

A new technique, the steepest descent-fast multipole method (SDFMM), is developed to efficiently analyze scattering from perfectly conducting random rough surfaces. Unlike other prevailing methods, this algorithm has linear computational complexity and memory requirements, making it a suitable candidate for analyzing scattering from large rough surfaces as well as for carrying out Monte Carlo simulations. The method exploits the quasiplanar nature of rough surfaces to efficiently evaluate the dyadic Green's function for multiple source and observation points. This is achieved through a combination of a Sommerfeld steepest descent integral and a multilevel fast multipole-like algorithm based on inhomogeneous plane wave expansions. The fast evaluation of the dyadic Green's function dramatically speeds up the iterative solution of the integral equation for rough surface scattering. Several numerical examples are presented to demonstrate the efficacy and accuracy of the method in analyzing scattering from extremely large finite rough surfaces     

Analysis of transient scattering from composite arbitrarily shapedcomplex structures

A time-domain surface integral equation approach based on the electric field formulation is utilized to calculate the transient scattering from both conducting and dielectric bodies consisting of arbitrarily shaped complex structures. The solution method is based on the method of moments (MoM) and involves the modeling of an arbitrarily shaped structure in conjunction with the triangular patch basis functions. An implicit method is described to solve the coupled integral equations derived utilizing the equivalence principle directly in the time domain. The usual late-time instabilities associated with the time-domain integral equations are avoided by using an implicit scheme. Detailed mathematical steps are included along with representative numerical results     

一种分析多目标散射问题的区域分解方法

提出了一种用于分析复杂多目标散射问题的区域分解方法. 在该方法中,每个目标作为一个独立的计算区域采用矢量有限元方法进行分析;各个区域之间通过基于格林函数的边界积分方程进行耦合;所得到的耦合矩阵方程采用基于Foldy-Lax多径散射方程的特征基函数方法进行求解. 由于矢量有限元方法的灵活性,该区域分解方法特别适合于求解多个具有相同结构复杂目标的散射问题. 数值算例验证了该方法的准确性和处理复杂多目标散射问题的能力.     

Investigation of millimeter-wave scattering from frequencyselective surfaces

A comparative numerical and experimental analysis of scattering from dielectric-backed frequency-selective surfaces in W-band (75-110 GHz) was carried out. The examples studied include metal (aluminium), resistive (bismuth), and bismuth-loaded I-pole or linearized Jerusalem cross arrays on fused silica, all of which exhibit a band-stop resonance in W-band as a general feature. The arrays were fabricated using standard photolithographic techniques. The numerical analysis involves the solution of an electric field integral equation using subdomain rooftop basis and testing functions within the framework of the Galerkin testing procedure. The lossy nature of the materials was fully accounted for. A comparative analysis of doubly stacked aluminium I-pole arrays was also performed. The numerical analysis exploits a variant of the cascade method in that the immediately adjacent dielectric layers are included in the construction of the scattering matrix for the frequency selective surface. This allows the higher-order evanescent Floquet modes to decay sufficiently at the dielectric boundaries so they can be ignored in the scattering matrix     

Protecting EFIE-based scattering computations from effects ofinterior resonances

A numerical method, based on the standard matrix formulation of the electric field integral equation (EFIE) for calculating the scattering from conducting bodies near resonant frequencies, is given for both stabilizing the numerical calculations and finding the form of the resonant fields. When this method is used along with a direct solution to the original matrix equation, it requires insignificant additional computation. An additional advantage of this approach is that it can easily be implemented in existing computer codes by using a single, standard but general, plug-in module     

Two-dimensional scattering by a homogeneous anisotropic rod

An integral equation based on a plane wave representation of the fields in a simply connected and anisotropic medium has been derived in order to handle the problem of two-dimensional scattering by a homogeneous anisotropic cylinder. Its simplicity: one-dimensional, finite range of integration, one unknown with no derivatives of it involved, and a nonsingular kernel, results in an efficient and straightforward numerical implementation. Both polarizations are considered and computer results are presented, discussed, and used to check our scheme thoroughly.     

Low-grazing angle scattering from rough surfaces in a duct formedby a linear-square refractive index profile

The problem of rough surface scattering and propagation over rough terrain in a ducting environment has been receiving considerable attention in the literature. One popular method of modeling this problem is the parabolic wave equation (PWE) method. An alternative method is the boundary integral equation (BIE) method. The implementation of the BIE in inhomogeneous media (ducting environments) is not straightforward, however, since the Green's function for such a medium is not usually known. In this paper, a closed-form approximation of the Green's function for a two-dimensional (2-D) ducting environment formed by a linear-square refractive index profile is derived using asymptotic techniques. This Green's function greatly facilitates the use of the BIE approach to study low-grazing angle (LGA) rough surface scattering and propagation over rough surfaces in the aforementioned ducting environment. This paper demonstrates how the BIE method can model the combined effects of surface roughness and medium inhomogeneity in a very rigorous fashion. Furthermore, it illustrates its capability of accurately predicting scattering in all directions including backscattering. The boundary integral equation of interest is solved via the method of ordered multiple interactions (MOMI), which eliminates the requirements of matrix storage and inversion and, hence, allows the application of the BIE method to very long rough surfaces