基于RWG基函数的伽略金法中奇异性积分 的精确快速计算

利用电磁场积分方程的伽略金法求解理想导体电磁散射问题时需要计算奇异性的二重面积分(即4维积分).伽略金法的基函数和检验函数广泛采用RWG(Rao-Wilton-Glisson)矢量基函数.传统上采用奇异值提取技术和Duffy坐标变换法处理该奇异性积分,本文提出了一种更为精确和高效的计算方法,该新方法通过参数坐标变换、相对坐标变换、积分区域分解和广义Duffy坐标变换相结合的技术消除了被积函数的奇异性并降低了原4维奇异性积分的数值积分维数.通过计算实例证明该方法的精确性和高收敛特性.     

Improved testing of the magnetic-Field integral equation

An improved implementation of the magnetic-field integral equation (MFIE) is presented in order to eliminate some of the restrictions on the testing integral due to the singularities. Galerkin solution of the MFIE by the method of moments employing piecewise linear Rao-Wilton-Glisson basis and testing functions on planar triangulations of arbitrary surfaces is considered. In addition to demonstrating the ability to sample the testing integrals on the singular edges, a key integral is rederived not only to obtain accurate results, but to manifest the implicit solid-angle dependence of the MFIE as well.     

导体线面连接问题中奇异函数积分的计算

在线面连接问题中，电流展开函数包含体展开函数、线展开函数和连接点展开函数三类。求解电场积分方程的积分项是电流基函数及其散度分别与自由空间格林函数的乘积，由于连接点展开函数含有一个奇异点，所以被积函数中含有两个奇异点。本文通过积分变换消除了奇异点，并将二重积分化为一重积分，使计算精度得到提高。计算实例验证了本文方法的正确性。     

Boundary integral equation technique for the singular behavior ofelectromagnetic fields at the common tip of metallic and bi-isotropiccones with arbitrary cross section

A boundary integral equation technique is developed to determine the singular field behavior at the common tip of lossless bi-isotropic and perfectly electrically and/or magnetically conducting cones with arbitrary cross section. The kernel of the set of boundary integral equations is a Green's function defined on a spherical surface. This Green's function is the associated Legendre function of the first kind. The integral equations are solved with the Galerkin method of moments. The theory is illustrated with a number of examples that show the effects of bi-isotropy on the singular field behavior     

A numerical implementation of a modified form of the electric field Integral equation

The details of a Galerkin discretization scheme for a modified form of the electric field integral equation are outlined for smooth, three-dimensional, perfectly conducting scatterers. Limitations of the divergence conforming finite-element bases in preserving the self-stabilizing properties of the electric field integral equation operator are indicated. A numerically efficient alternative is outlined which relies on an operator-based Helmholtz decomposition. The condition number of the resulting matrix equation is demonstrated to be frequency independent for scattering from a perfectly conducting sphere at various frequencies.     

Time Domain Integral Equation Analysis of Scattering From Composite Bodies via Exact Evaluation of Radiation Fields

A novel marching on in time and Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT)-based time domain integral equation solver for analyzing transient scattering from composite multiregion structures comprising homogeneous penetrable volumes and perfectly conducting surfaces is presented. To render the marching on in time procedure stable, observer and source integrals are approximated via multipoint collocation and evaluated in closed form. The proposed scheme avoids the costly evaluation of five dimensional integrals required in full space-time Galerkin schemes, and proves to be stable for all targets investigated, including coated perfect electrically conducting surfaces and multiregion penetrable volumes.      

Development and numerical solution of integral equations forelectromagnetic scattering from a trough in a ground plane

We develop a set of scalar integral equations that govern the electromagnetic scattering from a two-dimensional (2-D) trough in an infinite perfectly conducting ground plane. We obtain accurate and efficient numerical solution to these equations via the method of moments (MoM). Our numerical implementation compares favorably to popular methods such as the finite element/boundary integral (FE/BI) method, generalized network formulation (GNF), and electric field integral equation (EFIE) techniques     

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.     

Electromagnetic scattering from extended wires and two- and three-dimensional surfaces

Efficient numerical solutions for the electromagnetic scattering for classes of electrically large one-, two-, and three-dimensional perfectly conducting scatterers are presented. The formulation is based on solution of the electric field integral equation (EFIE) using the method of moments (MM). An entire domain Galerkin representation is used for wires and two-dimensional surfaces and a combination of entire and subdomain representations is applied to surfaces in three dimensions. The analysis is extendable to corrugated surfaces formed from sections of surfaces of translation or rotation. Numerical results are presented for wires, infinite strips, and finite strips (or plates). The behavior of the solutions with the number of terms in the entire domain expansion is examined. The reconstruction of the traveling-wave contribution to the scattering cross section using various approximations is discussed, and representative examples are given.     

Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions

The method of moments (MoM) solution of combined field integral equation (CFIE) for electromagnetic scattering problems requires calculation of singular double surface integrals. When Galerkin's method with triangular vector basis functions, Rao-Wilton-Glisson functions, and the CFIE are applied to solve electromagnetic scattering by a dielectric object, both RWG and n/spl times/RWG functions (n is normal unit vector) should be considered as testing functions. Robust and accurate methods based on the singularity extraction technique are presented to evaluate the impedance matrix elements of the CFIE with these basis and test functions. In computing the impedance matrix elements, including the gradient of the Green's function, we can avoid the logarithmic singularity on the outer testing integral by modifying the integrand. In the developed method, all singularities are extracted and calculated in closed form and numerical integration is applied only for regular functions. In addition, we present compact iterative formulas for computing the extracted terms in closed form. By these formulas, we can extract any number of terms from the singular kernels of CFIE formulations with RWG and n/spl times/RWG functions.     

Scattering by an indentation satisfying a dyadic impedance boundarycondition

We derive a pair of boundary integral equations for the problem of scattering of an electromagnetic wave by an indentation in a perfectly conducting screen. The wall of the indentation obeys a dyadic impedance boundary condition. The unknowns are the electric current density on the wall of the indentation and the total tangential magnetic field in the opening of the indentation. We also derive integral representations for the fields everywhere in free space, including the far-field region. In all cases, the integrals involved extend over finite surfaces only     

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     

Applications of differential forms to boundary integral equations

In this paper we discuss the application of differential forms to integral equations arising in the study of electromagnetic wave propagation. The usual Stratton-Chu integral equations are derived in terms of differential forms and corresponding Galerkin formulations are constructed. All numerical schemes require the specification of basis functions and the use of differential forms provides a very general method for the construction of arbitrary order basis functions on curvilinear geometries. It is noted that the lowest order approximations on flat geometries reduce to forms essential equivalent to the standard Rao-Wilton-Glisson functions. The effect on accuracy is investigated for electric field integral equation and magnetic field integral equation formulations for a range of bases. Hierarchical classes of functions are also developed, as are transition elements useful in p-adaptive schemes where variable orders of approximation are sought.     

On the testing of the magnetic field integral equation with RWGbasis functions in method of moments

For electromagnetic analysis using method of moments (MoM), three-dimensional (3-D) arbitrary conducting surfaces are often discretized in Rao, Wilton and Glisson basis functions. The MoM Galerkin discretization of the magnetic field integral equation (MFIE) includes a factor Ω₀ equal to the solid angle external to the surface at the testing points, which is 2π everywhere on the surface of the object, except at the edges or tips that constitute a set of zero measure. However, the standard formulation of the MFIE with Ω₀=2π leads to inaccurate results for electrically small sharp-edged objects. This paper presents a correction to the Ω₀ factor that, using Galerkin testing in the MFIE, gives accuracy comparable to the electric field integral equation (EFIE), which behaves very well for small sharp-edged objects and can be taken as a reference     

金属-各向异性介质体组合目标频域体表积分方程矩量法

利用体表积分方程矩量法求解了具有任意的介电常数张量和磁导率张量的各向异性介质与金属的组合目标的电磁散射问题.给出了基于RWG面基函数和SWG体基函数的体表积分方程阻抗矩阵元素表达式并详细推导了阻抗矩阵元素所涉及的各种积分运算的计算方法;通过数值计算实例与解析解或其它数值方法的详细对比分析,证明了计算公式的正确性.     

Numerical Analysis of Quasioptical Multireflector Antennas in 2-D With the Method of Discrete Singularities: E-Wave Case

Numerical method for modeling the E-polarized wave scattering by electrically large quasioptical two-dimensional (2-D) reflectors is presented. Reflectors are assumed zero-thickness and perfectly electrically conducting. Efficient numerical solution is obtained from the coupled singular integral equations discretized using new quadrature formulas of interpolation type. It has controlled accuracy and deals with small-size matrices. To simulate a small-horn feeding, the incident field is taken as a beam generated by a complex-source-point (CSP) current. Presented numerical results validate empirical rule of -10 dB edge illumination needed to provide the best electromagnetic performance of reflector     

Efficient evaluation of reaction integrals in the EFIE analysis ofplanar layered structures with uniaxial anisotropy

This paper presents an efficient implementation of the electric-field integral-equation (EFIE) method to deal with planar anisotropic layered printed structures. A convenient treatment of the kernel of the integral equation gives rise to reaction integrals that only involve quasi-singularities and R^-1-type singularities. When the well-known Rao-Wilton-Glisson triangular basis functions are used in conjunction with the Galerkin's method, closed-form expressions are found for the singular parts of the self-reaction integrals, as well as for the inner convolution integrals of the remaining singular/quasi-singular reaction integrals. Thus, the present procedure sets the EFIE method as a competitive alternative to other formulations     

The generalized forward-backward method for analyzing thescattering from targets on ocean-like rough surfaces

In Holliday et al. (1995, 1996), the iterative forward-backward (FB) method has been proposed to solve the magnetic field integral equation (MFIE) for smooth one-dimensional (1-D) rough surfaces. This method has proved to be very efficient, converging in a very small number of iterations. Nevertheless, this solution becomes unstable when some obstacle, like a ship or a large breaking wave, is included in the original problem. In this paper, we propose a new method: the generalized forward-backward (GFB) method to solve such kinds of complex problems. The approach is formulated for the electric field integral equation (EFIE), which is solved using a hybrid combination of the conventional FB method and the method of moments (MoM), the latter of which is only applied over a small region around the obstacle. The GFB method is shown to provide accurate results while maintaining the efficiency and fast convergence of the conventional FB method. Some numerical results demonstrate the efficiency and accuracy of the new method even for low-grazing angle scattering problems     

Hybrid solutions at internal resonances

The use of hybrid solutions for integral equation (IE) formulations in electromagnetics is illustrated at frequencies where a perfectly conducting scatterer exhibits internal resonances. Hybrid solutions, incorporating the Fock theory and physical optics Ansatzes, and the Galerkin representation, are compared with the method of moments (MM) solutions of the electric, magnetic, and combined field formulations at such frequencies. Numerical results are presented for spheres and a right circular cylinder.     

Electromagnetic scattering from electrically large coated flat and curved strips: Entire domain Galerkin formulation

Efficient numerical solutions are presented for electromagnetic scattering for classes of electrically large, coated, perfectly conducting strips which are flat or curved. The formulation is based on the solution of a coupled system of electric- and magnetic-field integral equations using the method of moments (MM). Entire domain Galerkin representations for the currents are used on the surface of the coating and at the coating-conductor interface. The resulting symmetric matrix equation is well conditioned and admits rapid, accurate solutions. Numerical results are presented for various coating thicknesses, strip widths, and curvatures for the transverse electric (TE) and transverse magnetic (TM) cases. The convergence of the Galerkin solution is examined as a function of these parameters. The effect of the edge approximation on the choice of expansion functions is discussed. The numerical results are compared with experimental measurements.