Exact evaluation of retarded-time potential integrals for the RWG bases

A new analytical approach for obtaining the time samples of the retarded-time scalar and vector potentials due to an impulsively excited Rao-Wilton-Glisson (RWG) basis function is presented. The approach is formulated directly in the time-domain without any assumptions regarding the temporal behavior of the currents represented by the RWG bases. To the best knowledge of the authors, analytical evaluation of the potential integrals due to the RWG bases have not been formulated prior to the present work either in the time domain or the frequency domain. It is shown that the aforementioned potentials are related to the arc segments formed by the intersection of the triangular supports of the RWG basis and the sphere that is centered at the observation point and that has a radius R=ct, where c is the speed of light. In particular, the scalar potential is directly proportional to the total arc length and the vector potential is a function of the bisectors of these arc segments. A simple algorithm to evaluate these quantities is also presented. The validity of the obtained time-domain formulae is demonstrated through comparison of results to those obtained in the frequency domain by using numerical quadrature and transformed into time domain.     

导体目标瞬态电磁散射的时空矩量法研究

时域积分方程的矩量法是求解瞬态电磁散射的方法之一。研究了基于加权Lagurre函数和RWG基函数分别作为时间、空间基函数的时空矩量法,给出了时域磁场积分方程时空矩量法的全部计算公式,编制了相应串行和并行计算程序。计算结果表明：该方法具有很好的时域稳定特性,为宽带电磁散射分析提供了可能,同时也指出了其应用的局限性,为改进其方法提供了参考。     

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.     

Magnetic field integral equation at very low frequencies

It is known that there is a low-frequency breakdown problem when the method of moments (MOM) with Rao-Wilton-Glisson (RWG) basis is used in the electric field integral equation (EFIE); it can be solved through the loop and tree basis decomposition. The behavior of the magnetic field integral equation (MFIE) at very low frequencies is investigated using MOM, where two approaches are presented based on the RWG basis and loop and tree bases. The study shows that MFIE can be solved by the conventional MOM with the RWG basis at arbitrarily low frequencies, but there exists an accuracy problem in the real part of the electric current. Although the error in the current distribution is small, it results in a large error in the far-field computation. This is because a big cancellation occurs during the far field computation. The source of error in the current distribution is easily detected through the MOM analysis using the loop and tree basis decomposition. To eliminate the error, a perturbation method is proposed, from which a very accurate real part of the tree current has been obtained. Using the perturbation method, the error in the far-field computation is also removed. Numerical examples show that both the current distribution and the far field can be accurately computed at extremely low frequencies by the proposed method.     

利用时域电场积分方程分析天线辐射问题

研究了基于矩量法和RWG三角基函数的隐式电场积分方程的时域算法,引入了一种激励源的时域设置方法.利用直接在时域加激励源的方法来分析天线辐射问题,所得时域数据经傅立叶变换可得到很宽频带的频域数据,与在频域逐个频点求解相比大大节省了计算时间.通过对几种典型天线的分析计算,验证了方法的正确性和算法的稳定性.     

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.     

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.     

Hybrid MFIE/FDTD method for analysis of metallic enclosures with slots

A method for the analysis of the electromagnetic field inside metallic enclosures with apertures is presented. The model is based on a time domain hybrid approach method, characterised by the combination of the finite difference in time domain (FDTD), and the method of moments in time domain (MoMTD) that solves a magnetic field integral equation (MFIE).     

Efficient Implementation for Spherical Flux Computation and Its Application to Vascular Segmentation

Spherical flux is the flux inside a spherical region, and it is very useful in the analysis of tubular structures in magnetic resonance angiography and computed tomographic angiography. The conventional approach is to estimate the spherical flux in the spatial domain. Its running time depends on the sphere radius quadratically, which leads to very slow spherical flux computation when the sphere size is large. This paper proposes a more efficient implementation for spherical flux computation in the Fourier domain. Our implementation is based on the reformulation of the spherical flux calculation using the divergence theorem, spherical step function, and the convolution operation. With this reformulation, most of the calculations are performed in the Fourier domain. We show how to select the frequency subband so that the computation accuracy can be maintained. It is experimentally demonstrated that, using the synthetic and clinical phase contrast magnetic resonance angiographic volumes, our implementation is more computationally efficient than the conventional spatial implementation. The accuracies of our implementation and that of the conventional spatial implementation are comparable. Finally, the proposed implementation can definitely benefit the computation of the multiscale spherical flux with a set of radii because, unlike the conventional spatial implementation, the time complexity of the proposed implementation does not depend on the sphere radius.      

A multiresolution method of moments for triangular meshes

This paper presents the construction, use, and properties of a multiresolution (wavelet) basis for the method of moments (MoM) analysis of metal antennas, scatterers, and microwave circuits discretized by triangular meshes. Several application examples show fast convergence of iterative solvers and accurate solutions with highly sparse MoM matrices. The proposed basis is organized in hierarchical levels, and keeps the different scales of the problem directly into the basis functions representation; the current is divided into a solenoidal and a quasi-irrotational part, which allows mapping these two vector parts onto fully scalar quantities, where the wavelets are defined. As a byproduct, this paper also presents a way to construct hierarchical sets of Rao-Wilton-Glisson (RWG) functions on a family of meshes obtained by subsequent refinement, i.e., with the RWG of coarser meshes expressed as linear combinations of those of finer meshes.     

Analysis of scattering by a large array of waveguide-fed wide-slot millimeter wave antennas using precorrected-FFT algorithm

An effective numerical technique employing the precorrected-fast Fourier transform (P-FFT) technique based on the method-of-moments (MoM) is presented to study the scattering from wide-slot antennas fed by waveguides with arbitrary terminations. As the Rao-Wilton-Glisson (RWG) functions are used as the basis and testing functions, the magnetic currents with arbitrary slot directions are considered. In addition, the P-FFT technique is employed to accelerate the MoM procedure for a large slot array. The numerical results are obtained and compared with both the traditional MoM results obtained using the entire-domain basis functions and the experiments results, to demonstrate better effectiveness and accuracy of this technique for millimeter wave scattering from arbitrary wide-slot antennas.     

The use of curl-conforming basis functions for the magnetic-field integral equation

Divergence-conforming Rao-Wilton-Glisson (RWG) functions are commonly used in integral-equation formulations to model the surface current distributions on planar triangulations. In this paper, a novel implementation of the magnetic-field integral equation (MFIE) employing the curl-conforming n~/spl times/RWG basis and testing functions is introduced for improved current modelling. Implementation details are outlined in the contexts of the method of moments, the fast multipole method, and the multilevel fast multipole algorithm. Based on the examples of electromagnetic modelling of conducting scatterers, it is demonstrated that significant improvement in the accuracy of the MFIE can be obtained by using the curl-conforming n~/spl times/RWG functions.     

考虑热松弛时间的半无限粘弹性压电杆的热冲击问题

基于具有一个松弛时间的广义热粘弹性理论,研究了半无限压电杆热冲击问题。利用现代控制理论中状态空间法求解了偏微分控制方程,获得了拉普拉斯变换域内的精确解。利用数值反变换方法,得到了位移、温度和应力的分布曲线,同时将广义理论和传统耦合理论做了对比。从这些结果来看,当所考察时间的量级短到与材料热松弛时间量级相同时,热松弛影响着所有场变量的分布。材料粘性性质不影响温度分布,但在时间很短时对应力分布影响显著,而对位移分布的影响为时间愈长愈明显。     

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.     

Magnetostatic image current and its application to an analytic identification of a current dipole inside a conducting sphere

The image solution for the static magnetic field outside a conducting sphere with an internal current dipole is considered. The image current, which is a linear distribution of magnetic dipoles on the line segment between the dipole point and the center of the sphere, is derived by using the fact that the induced current does not have any contribution to the radial component of the magnetic field outside the sphere. The image is then used to obtain some explicit formulas for identifying the location and tangential moment of the primary current dipole. This explicit identification method is also tested with a real model for a patient's brain.     

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.     

A comparison of performance of three orthogonal polynomials in extraction of wide-band response using early time and low frequency data

The objective of this paper is to generate a wideband and temporal response of three-dimensional composite structures by using a hybrid method that involves generation of early time and low-frequency information. The data in these two separate time and frequency domains are mutually complementary and contain all the necessary information for a sufficient record length. Utilizing a set of orthogonal polynomials, the time domain signal (be it the electric or the magnetic currents or the near/far scattered electromagnetic field) could be expressed in an efficient way as well as the corresponding frequency domain responses. The available data is simultaneously extrapolated in both domains. Computational load for electromagnetic analysis in either domain, time or frequency, can be thus significantly reduced. Three orthogonal polynomial representations including Hermite polynomial, Laguerre function and Bessel function are used in this approach. However, the performance of this new method is sensitive to two important parameters-the scaling factor l/sub 1/ and the expansion order N. It is therefore important to find the optimal parameters to achieve the best performance. A comparison is presented to illustrate that for the classes of problems dealt with, the choice of the Laguerre polynomials has the best performance as illustrated by a typical scattering example from a dielectric hemisphere.     

各向异性材料涂覆目标电磁散射特性仿真

该文基于阻抗边界条件(IBC),提出了3维各向异性材料涂覆目标电磁散射特性的矩量法(MoM)解决方案。根据表面等效原理,采用感应电磁流以3维RWG(Rao-Wilton-Glisson)矢量基函数展开的伽略金法。以表面阻抗矩阵表征电磁参数,实现各向异性材料涂覆目标的电磁仿真,算例结果与Mie级数解等精确结果吻合良好。对各向异性材料涂覆复杂目标的电磁散射特性进行分析,为目标的雷达隐身和反隐身提供理论支持。     

An accurate approach for the calculation of MoM matrix elements

The method of moments solution on integral equations for electromagnetic scattering and radiation problems with Rao-Wilton-Glisson (RWG) and n/spl times/RWG basis functions requires the calculation of singular double surface integrals over triangular subdomains. In this paper, an approach is presented to deal with the singularity and almost singularity in which the triangle subdomain is divided into several subtriangles and then the singular or almost singular integral is calculated over each of the subtriangles. This approach is mainly based on the projection process and the local polar frame. It can be used to deal with the singular double surface integrals of order 1/R and 1/R/sup 2/, in which vector functions may be of zeroth-order and higher order. Numerical examples are provided to verify the validity of the method presented in this paper for dealing with the singularity and almost singularity.     

Parametric analysis of slot-loaded trapezoidal patch antennas

Slot-loaded trapezoidal patch antennas are analyzed in order to achieve a multifrequency operation. A moment method formulation based on new analytical entire domain basis functions is developed and compared to the one involving the Rao-Wilton-Glisson (1982) RWG triangular discretization. The numerical results obtained through the new efficient implemented code are compared to the measurements showing a good agreement. A parametric analysis of such antennas is also presented leading to useful design graphs showing the main antenna properties (resonance frequencies, cross-polarization levels, matching, etc.) as a function of the patch geometrical dimensions, symmetric collocation of the slots on the patch, and slot dimensions.