基于高阶叠层矢量基函数的时域电磁场 积分方程方法

本文将准正交高阶叠层矢量基函数用于时域电磁场积分方程(TDIE),求解了三维金属目标的时域电磁散射问题.准正交高阶叠层矢量基函数定义在曲面四边形单元上,并且不要求网格为规范网格,给复杂目标的几何建模和电磁建模带来很大方便.在空间上利用伽略金方法、时间上采用点匹配法求解时域电磁场积分方程,并采用隐式时间步进算法,数值计算结果表明了该方法求解时域积分方程的精确性、高效性与稳定性.     

一种稳定的时域电场积分方程求解方法

提出了一种求解任意形状理想导体目标时域散射的稳定时域电场积分方程(TDIE)方法,此方法能非常有效地消除后期振荡。利用显式MOT方法求解二阶时间微分的电场积分方程,矢量位随时间的变化关系采用边缘中心近似,标量位随时间的变化关系则采用三角面元重心近似,然后运用先五步后三步的平均方法。数值结果表明,本文方法能够有效地提高TDIE方法的收敛性和稳定性。     

A new temporal basis function for the time-domain integral equationmethod

A new temporal basis function that has all-order continuous derivative has been constructed using a nonlinear optimization scheme. This new basis function provides a much more stable explicit marching-on-in-time (MOT) solution, based on the time-domain integral equation (TDIE) method, than that presently available. Two examples are presented to illustrate the superior stability of the proposed temporal basis function     

基于三次样条时域基函数的MOT算法精确求解细直线瞬时电磁辐射和散射问题

提出了采用三次样条函数作为时域基函数求解良导体细线时域积分方程(TD-EFIE)的MOT(Marching on in time)算法,由于三次样条函数是非因果的,算法中采用有限带宽信号的一步外推方法,估计将来值。三次样条函数作为时间基函数,具有插值精度高、支撑区域小的特点,在保证计算精度的情况下降低了信号采样率和计算的复杂性,提高了MOT算法的后时稳定性。计算结果表明了算法的有效性。     

Application of the Characteristic Basis Function Method Utilizing a Class of Basis and Testing Functions Defined on NURBS Patches

A novel implementation of the characteristic basis function method (CBFM) is given, in which the high-level basis functions, called characteristic basis functions (CBFs), are represented in terms of curved rooftops generated from nonuniform rational B-splines (NURBS) in the parametric (u,v) domain. The associated macro-testing functions are defined by using curved razor-blade functions corresponding to each rooftop. The underlying objective of the CBFM is the reduction of the number of unknowns that arise from the discretization process when applying the conventional method of moments (MoM). The result is, therefore, an approach which can handle many complex cases via direct solvers, without suffering from convergence problems as many of the iterative techniques are known to do when dealing with ill-conditioned matrices. As a result of the combination of the CBFM with the special class of low-level basis and testing functions directly located over NURBS surfaces, complex and realistic geometries can be efficiently analyzed to yield accurate results while reducing the CPU time as well as required memory resources.     

一种改进的时间步进稳定性平均方法

介绍了时域积分方程方法（TDIE）的基本原理,提出一种抑制时域电场积分方程（TDIE）时间步进（MOT）后期振荡的新平均算法,提高了MOT算法后期的稳定性。计算了高斯脉冲波照射到金属立方体时目标表面的电流响应,数值结果表明,该方法简单有效地推迟了TDIE后期震荡时间。     

TD‐CFIE Formulation for Transient Electromagnetic Scattering from 3‐D Dielectric Objects

In this paper, we present a time domain combined field integral equation formulation (TD‐CFIE) to analyze the transient electromagnetic response from dielectric objects. The solution method is based on the method of moments which involves separate spatial and temporal testing procedures. A set of the RWG functions is used for spatial expansion of the equivalent electric and magnetic current densities, and a combination of RWG and its orthogonal component is used for spatial testing. The time domain unknowns are approximated by a set of orthonormal basis functions derived from the Laguerre polynomials. These basis functions are also used for temporal testing. Use of this temporal expansion function characterizing the time variable makes it possible to handle the time derivative terms in the integral equation and decouples the space‐time continuum in an analytic fashion. Numerical results computed by the proposed formulation are compared with the solutions of the frequency domain combined field integral equation.     

Transient electromagnetic scattering from dielectric objects using the electric field Integral equation with Laguerre polynomials as temporal basis functions

In this paper, we propose a time-domain electric field integral equation (TD-EFIE) formulation for analyzing the transient electromagnetic response from three-dimensional (3-D) dielectric bodies. The solution method in this paper is based on the Galerkin's method that involves separate spatial and temporal testing procedures. Triangular patch basis functions are used for spatial expansion and testing functions for arbitrarily shaped 3-D dielectric structures. The time-domain unknown coefficients of the equivalent electric and magnetic currents are approximated using a set of orthonormal basis function that is derived from the Laguerre functions. These basis functions are also used as the temporal testing functions. Use of the Laguerre polynomials as expansion functions for the transient portion of response enables one not only to handle the time derivative terms in the integral equation in an analytic fashion but also completely separates the space and the time variables. Thus, the time variable along with the Courant condition can be eliminated in a Galerkin formulation using this procedure. We also propose an alternative formulation using a different expansion of the magnetic current. The total computational cost for this new method is similar to that of an implicit marching-on in time (MOT)-EFIE scheme, even though at each step this procedure requires more computations. Numerical results involving equivalent currents and far fields computed by the two proposed methods are presented and compared.     

A sparse-matrix/canonical grid method for analyzing densely packedinterconnects

In this paper, a fast numerical method called the sparse-matrix/canonical-grid (SM/CG) method is employed to analyze densely packed microstrip interconnects that involve a large number of unknowns. The mixed-potential integral equation is solved by using the method of moments in the spatial domain. The closed-form expressions of the spatial Green's functions of microstrip structures are obtained from the combination of the fast Hankel transform and the matrix pencil method. The Rao-Wilton-Glisson triangular basis functions are used to convert the integral equation into a matrix equation. The matrix equation is then solved by using the SM/CG method, in which the far-interaction portion of the matrix-vector multiplication in the iterative solution is performed by the fast Fourier transforms (FFTs). This is achieved by the Taylor series expansions of the spatial Green's functions about the uniformly spaced canonical grid points overlaying the triangular discretization. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed method. The SM/CG method has computational complexity of O(NlogN). Furthermore, being FFT-based facilitates the implementation for parallel computation     

Time Domain CalderÓn Identities and Their Application to the Integral Equation Analysis of Scattering by PEC Objects Part I: Preconditioning

Time domain electric field integral equations often are used to analyze transient scattering from perfect electrically conducting objects. When discretized using marching-on-in-time recipes they give rise to linear systems of equations that can be solved for the induced currents for all time steps. Unfortunately, when the scatterer is approximated by increasingly dense meshes, the condition number of these systems grows rapidly, slowing down the convergence of iterative solvers. Here, time domain CalderÓn identities are derived and subsequently used to construct a CalderÓn-preconditioned time domain electric field integral equation that can be discretized even with dense meshes using Buffa-Christiansen basis functions. Numerical results that demonstrate the effectiveness and accuracy of the proposed method are presented.      

Efficient electromagnetic modeling of three-dimensional multilayermicrostrip antennas and circuits

An efficient method-of-moments (MoM) solution is presented for analysis of multilayer microstrip antennas and circuits. The required multilayer Green's functions are evaluated by the discrete complex image method (DCIM), with the guided-mode contribution extracted recursively using a multilevel contour integral in the complex _ρ-plane. An interpolation scheme is employed to further reduce the computer time for calculating the Green's functions in the three-dimensional (3-D) space. Higher order interpolatory basis functions defined on curvilinear triangular patches are used to provide necessary flexibility and accuracy for the discretization of arbitrary shapes and to offer a better convergence than lower order basis functions. The combination of the improved DCIM and the higher order basis functions results in an efficient and accurate MoM analysis for 3-D multilayer microstrip structures     

Novel monopolar MFIE MoM-discretization for the scattering analysis of small objects

We present a novel method of moments (MoM)-magnetic field integral equation (MFIE) discretization that performs closely to the MoM-EFIE in the electromagnetic analysis of moderately small objects. This new MoM-MFIE discretization makes use of a new set of basis functions that we name monopolar Rao-Wilton-Glisson (RWG) and are derived from the RWG basis functions. We show for a wide variety of small objects -curved and sharp-edged-that the new monopolar MoM-MFIE formulation outperforms the conventional MoM-MFIE with RWG basis functions.     

Accuracy of the method of moments for scattering by a cylinder

We study the accuracy and convergence of the method of moments for numerical scattering computations for an important benchmark geometry: the finite circular cylinder. From the spectral decomposition of the electric-field integral equation for this scatterer, we determine the condition number of the moment matrix and the dependence of solution error on the choice of basis functions, discretization density, polarization of the incident field, and the numerical quadrature rule used to evaluate moment-matrix elements. The analysis is carried out for both the TM polarization (weakly singular kernel) and TE polarization (hypersingular kernel). These results provide insights into empirical observations of the convergence behavior of numerical methods in computational electromagnetics     

A general approach for the stability analysis of the time-domain finite-element method for electromagnetic simulations

This paper presents a general approach for the stability analysis of the time-domain finite-element method (TDFEM) for electromagnetic simulations. Derived from the discrete system analysis, the approach determines the stability by analyzing the root-locus map of a characteristic equation and evaluating the spectral radius of the finite element system matrix. The approach is applicable to the TDFEM simulation involving dispersive media and to various temporal discretization schemes such as the central difference, forward difference, backward difference, and Newmark methods. It is shown that the stability of the TDFEM is determined by the material property and by the temporal and spatial discretization schemes. The proposed approach is applied to a variety of TDFEM schemes, which include: (1) time-domain finite-element modeling of dispersive media; (2) time-domain finite element-boundary integral method; (3) higher order TDFEM; and (4) orthogonal TDFEM. Numerical results demonstrate the validity of the proposed approach for stability analysis.     

Evaluation of the Green's function for the mixed potential integral equation (MPIE) method in the time domain for layered media

In the use of the time-domain integral equation (TDIE) method for the analysis of layered media, it is important to have the time-domain layered medium Green's function computed for many source-to-field distances /spl rho/ and time instants t. In this paper, a numerical method is used that computes the mixed potential Green's functions G/sub v/(/spl rho/,t) and G/sub A/(/spl rho/,t) for a multilayered medium for many /spl rho/'s and t's simultaneously. The method is applicable to multilayered media and for lossless or lossy dispersive media. Salient features of the method are: 1) the use of complex /spl omega/ so that the surface wave poles are lifted off the real k/sub /spl rho// axis such that pole extractions are not required; 2) the use of half-space extraction so that the integrand for the Sommerfeld integral decays exponentially along the k/sub /spl rho// axis to obtain fast convergence of the integral; and 3) the use of the fast Hankel transform so that the Green's function is calculated for many values of /spl rho/ simultaneously. For a four-layer medium, we illustrate the numerical results by a three-dimensional plot of /spl rho/G/sub v/(/spl rho/,t) versus /spl rho/ and t and demonstrate the space-time evolution of these Green's functions. For a maximum frequency range of 8 GHz, the method requires only a few CPU minutes to compute a table of 100 (points in /spl rho/) /spl times/ 168 (points in t) uniformly spaced values of G/sub v/(/spl rho/,t) on an 867-MHz Pentium PC.     

Limitations of the Cubical Block Model of Man in Calculating SAR Distributions

Block models of man which consist of a limited number of cubical cells are commonly used to predict the internal electromagnetic (EM) fields and specific absorption rate (SAR) distributions inside the human body. Numerical results, for these models, are obtained based on moment-method solutions of the electric-field integral equation (EFIE) with a pulse function being used as the basis for expanding the unknown internal field. In this paper, we first examine the adequacy of the moment-method procedure, with pulse basis functions, to determine SAR distributions in homogeneous models. Calculated results for the SAR distributions in some block models are presented, and the stability of the solutions is discussed. It is shown that, while the moment-method, using pulse basis functions, gives good values for whole-body average SAR, the convergence of the solutions for SAR distributions is questionable. A new technique for improving the spatial resolution of SAR distribution calculations using a different EFIE and Galerkin's method with linear basis functions and polyhedral mathematical cells is also described.     

A 3-D Discontinuous Spectral Element Time-Domain Method for Maxwell's Equations

A discontinuous spectral element time-domain method is proposed to analyze transient electromagnetic properties of general 3-D structures. This method is advantageous in that its mass matrices are block-diagonal due to the Gauss-Lobatto-Legendre polynomials, and it allows different orders of basis functions for each subdomain. The Riemann solver is employed in the boundary integral terms to communicate fields between adjacent subdomains. Perfectly matched layers are utilized to truncate the computational domain. Galerkin method is used for spatial discretization, and a fourth-order Runge-Kutta scheme is employed for the time integration. The validity of the proposed approach is demonstrated through several numerical examples of initial value problems and scattering problems.     

Coupled TDIE-PO method for transient scattering from electrically large conducting objects

An efficient hybrid method based on the time domain integral equation (TDIE) coupled with physical optics (PO) is proposed for the transient scattering from electrically large conducting objects. The computational complexity of the proposed hybrid method is drastically reduced compared with full TDIE, and the accuracy is improved compared with only PO. The numerical results demonstrate the validity and efficiency of the hybrid method.     

精确稳定求解时域电场积分方程的一种新方法

时域电场、磁场和混合场积分方程已被广泛用来分析散射体的时域散射响应.基于适当的空间积分方法和隐式的时间步进算(MOT)法在求解时域磁场和混合场积分方程时总是稳定的,然而在求解TDEFIE时则是不稳定的.在本文中,时域电场积分方程的非奇异性积分采用标准的高斯求积法来计算;而利用参数坐标变换和极坐标变换将其奇异性积分转换成为可以分区域精确快速计算的非奇异性积分.通过数值实验表明,利用该方法可以非常精确稳定地求解时域电场积分方程,即使是在时间迭代后期也不必采用任何求平均的过程;另外,该方法可以用于任意时间基函数并可以推广到高阶空间基函数的情形.     

Efficient hybrid spatial and spectral techniques in analyzingplanar periodic structures with nonuniform discretizations

A new efficient technique for analyzing planar periodic structures with arbitrary unit cell geometry rendered in a nonuniform discretization is proposed in this paper. The mixed potential integral equation is solved by the method of moments in conjunction with the Rao-Wilton-Glisson triangular discretization. The convergence of computing each element in the impedance matrix is accelerated using Ewald's method for contributions of quasi-dynamic and complex images and the lattice-sum method for the surface-wave contribution. Numerical efficiency and accuracy of this hybrid method are compared with the spectral-domain method