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     

Contamination of the Accuracy of the Combined-Field Integral Equation With the Discretization Error of the Magnetic-Field Integral Equation

We investigate the accuracy of the combined-field integral equation (CFIE) discretized with the Rao-Wilton-Glisson (RWG) basis functions for the solution of scattering and radiation problems involving three-dimensional conducting objects. Such a low-order discretization with the RWG functions renders the two components of CFIE, i.e., the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE), incompatible, mainly because of the excessive discretization error of MFIE. Solutions obtained with CFIE are contaminated with the MFIE inaccuracy, and CFIE is also incompatible with EFIE and MFIE. We show that, in an iterative solution, the minimization of the residual error for CFIE involves a breakpoint, where a further reduction of the residual error does not improve the solution in terms of compatibility with EFIE, which provides a more accurate reference solution. This breakpoint corresponds to the last useful iteration, where the accuracy of CFIE is saturated and a further reduction of the residual error is practically unnecessary.     

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.     

混合ACA和MLFMA方法计算复合目标电磁散射

采用矩量法(MoM)计算电大尺寸的复合目标的电磁散射。为了能够高效快速地计算电大尺寸三维复合目标的电磁散射,提出一种新的混合方法,将自适应交叉近似(ACA)算法和多层快速多级子(MLFMA)算法相结合,共同加速矩量法的计算。其中,MLFMA用于加速目标与自身的作用,ACA用于加速目标与其他目标的相互作用。提出的混合算法在计算复合目标电磁散射时,可降低运算存储,缩短阻抗矩阵填充时间,并且能够加快矩阵矢量乘,且不影响计算精确度。数值算例表明,所提快速算法能够在保证电磁散射计算精确度前提下,比传统方法更高效。     

曲面共形阵列结构快速数值分析方法

针对曲面共形阵列结构电磁散射特性的高效、精确仿真分析需求,提出了一种并行综合函数矩量法处理方案.该方法是传统电磁经典数值算法——矩量法的一种改进形式,通过几何区域分解处理和综合基函数的方式极大降低了算法的内存消耗,使得单机分析电大尺寸问题和大规模阵列问题成为可能.更为重要的是,针对周期阵列结构,该方法具备综合函数复用特性和多区域并行处理特性,能够大大提高算法的综合处理效率.一个6×11的柱面共形贴片阵列被用于验证所提方法的性能,仿真结果表明,对于周期阵列结构,该方法的计算精度与多层快速多极子算法相当,虽然计算效率略低于多层快速多极子方法,但内存消耗比多层快速多极子方法低一个数量级.     

An Efficient Surface Integral Equation Solution to EM Scattering by Chiral Objects Above a Lossy Half Space

In this paper, an extension of the electric-magnetic current combined-field integral equation (JMCFIE) is presented for the efficient analysis of EM scattering by an arbitrary shaped homogenous chiral object located above a lossy half space. This formulation can lead to a well tested equation system using the method of moments (MoM) solution with the vector triangular basis functions and Galerkin's method. The multilevel fast multipole algorithm (MLFMA) is then employed to reduce the memory requirement and computational complexity of the MoM solution. The inner-outer flexible generalized minimal residual (FGMRES) method is used to further speed up the convergence. The accuracy and efficiency are confirmed with a couple of numerical examples.      

基于MT-SIE法求解电大介质目标电磁散射

介绍了一种用于均匀介质目标电磁散射求解的新型多区域表面积分方程(MT-SIE)方法。不同于传统的用于介质目标散射求解的积分方法,该方法将均匀介质目标分解为内、外2个独立的子区,通过在介质表面强加Robin传输条件来保证电流和磁流的连续性。由于介质目标被分解为内外2个独立的子区,不同的子区允许非共形剖分。相较于传统方法,该方法可以更高效地与多层快速多级子(MLFMA)相结合求解电大尺寸目标。为进一步加速矩阵的迭代求解,提出了一种高斯-赛德尔型预条件技术,可以有效改善矩阵的收敛,加快迭代求解速度。     

适用于电磁混合源散射求解的新型积分方程数值方法

含腔导电目标电磁散射的混合场积分方程求解方法中,将出现电场积分方程算子和磁场积分方程算子同时作用于待求混合源的复杂情况,使计算复杂度大为提高.本文导出"均衡混合场积分方程"及其数值方法,使作用于电流和磁流的积分算子完全相同,大大简化了计算.均衡混合场积分方程与多层快速多极子方法(MLFMA)结合使用,可以方便地求解含腔导体目标的电磁散射.本文给出的数值实例充分证明了这一方法的高精度和高效率.     

含复杂细节结构的金属目标电磁散射ACE-MLFMA 分析

多层快速多极子算法(MLFMA)在计算含复杂细节结构目标的散射问题时,求解效率会迅速下降。本文介绍了快 速笛卡尔展开(ACE)算法及其与MLFMA 的结合,使得原先MLFMA 的最细层能够再局部细分,加速了阻抗矩阵的填充 和迭代求解。本文将该混合算法应用于求解含复杂细节结构目标的电磁散射问题,包括具有尖端的杏仁核和由复杂带线 结构构成的频率选择表面,计算实例验证了该方法求解效率的提高和内存开销的减少,以及算法的可靠性和高效性。     

An Efficient Implementation of the Combined Wideband MLFMA/LF-FIPWA

An efficient implementation of the low frequency fast inhomogeneous plane wave algorithm (LF-FIPWA) combined with the multilevel fast multipole algorithm (MLFMA) is presented. The seamless combination of the LF-FIPWA with the MLFMA renders a powerful method applicable for a wide frequency range. As the MLFMA is already well established we will describe possibilities for an efficient formulation of the LF-FIPWA: An implicit filtering reduces the bandwidth of the integrand. An interpolation of the excitation vectors is introduced which avoids dealing with the in general complex angles. An adaptive quadrature rule on the path in the complex plane further increases the efficiency. We present a simple but accurate interpolation and anterpolation method on this path. In addition, for symmetric objects or for perfectly conducting ground the incorporation of a symmetry wall is described.      

Comparison of Integral-Equation Formulations for the Fast and Accurate Solution of Scattering Problems Involving Dielectric Objects with the Multilevel Fast Multipole Algorithm

We consider fast and accurate solutions of scattering problems involving increasingly large dielectric objects formulated by surface integral equations. We compare various formulations when the objects are discretized with Rao-Wilton-Glisson functions, and the resulting matrix equations are solved iteratively by employing the multilevel fast multipole algorithm (MLFMA). For large problems, we show that a combined-field formulation, namely, the electric and magnetic current combined-field integral equation (JMCFIE), requires fewer iterations than other formulations within the context of MLFMA. In addition to its efficiency, JMCFIE is also more accurate than the normal formulations and becomes preferable, especially when the problems cannot be solved easily with the tangential formulations.      

基于时域混合场积分方程求解目标瞬态散射特性

当入射平面波的频谱包含目标的谐振频点时,时域电场积分方程和时域磁场积分方程求解的表面电流不稳定,会出现后期震荡现象。通过线性组合时域电场积分方程和时域磁场积分方程,可以获得一种混合场积分方程。数值结果显示,这种混合场积分方程消除了因内部谐振引起的后期震荡,得到了稳定的表面电流分布和远区散射场。     

Integral equation solution of Maxwell's equations from zerofrequency to microwave frequencies

We develop a new method to precondition the matrix equation resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE). This preconditioning method is based on first applying the loop-tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MoM matrix thus obtained still cannot be solved efficiently by iterative solvers due to the large number of iterations required. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MoM matrix that can be solved efficiently by iterative solvers. Consequently, dramatic reduction in iteration count has been observed. The various steps can be regarded as a rearrangement of the basis functions to arrive at the MoM matrix. Therefore, they are related to the original MoM matrix by matrix transformation, where the transformation requires the inverse of the connection matrix. We have also developed a fast method to invert the connection matrix so that the complexity of the preconditioning procedure is of O(N) and, hence, can be used in fast solvers such as the low-frequency multilevel fast multipole algorithm (LP-MLFMA). This procedure also makes viable the use of fast solvers such as MLFMA to seek the iterative solutions of Maxwell's equations from zero frequency to microwave frequencies     

并行多层快速多极子的高效预条件技术

多层快速多极子法是基于矩量法的快速算法,具有较低的计算复杂度和存储复杂度,被广泛应用于目标电磁散射特性分析。对于复杂结构电磁目标,由于矩阵条件数较差,往往存在迭代收敛慢甚至不收敛的问题。针对这一情况,文中利用快速多极子的近区矩阵,结合稀疏矩阵方程求解构造了一种高效预条件。数值实例表明该方法相比于块对角预条件效果更好,能有效加速多层快速多极子迭代过程。     

Some scattering results computed by surface-integral-equation and hybrid finite-element - boundary-integral techniques, accelerated by the multilevel fast multipole method

Method-of-moments (MoM) solutions of surface integral equations are especially well suited for scattering computations involving metallic objects. Improved modeling flexibility for dielectric (possibly lossy) and mixed dielectric/metallic bodies is obtained by combining a surface-integral-equation formulation, involving electric and magnetic equivalent surface-current densities, with a volumetric finite-element (FE) model of the dielectric regions. This results in the well-known hybrid FEBI (finite-element-boundary-integral) technique. For many years, hybrid FEBI techniques, as well as stand-alone Bl (surface-integral equation, often just termed MoM) techniques, were restricted to relatively small (with respect to a wavelength) geometries. However, with the development of powerful multilevel fast multipole methods/algorithms (MLFMM/MLFMA), it has become possible to compute a larger variety of practical scattering and radiation problems with the hybrid FEBI-MLFMM technique. In this contribution, we give a short review of our hybrid FEBI-MLFMM approach, with a focus on mixed dielectric/metallic geometries and multiple Bl domains. We then present a variety of scattering results for metallic and mixed dielectric/metallic objects, together with comparisons with measured RCS (radar cross section) data. Broadband computations are used to derive high-resolution range (HRR) profiles of several configurations.     

An IE-ODDM-MLFMA Scheme With DILU Preconditioner for Analysis of Electromagnetic Scattering From Large Complex Objects

For electrically large complex electromagnetic (EM) scattering problems, huge memory is often required for most EM solvers, which is too difficult to be handled by a personal computer (PC) even a workstation. Although the multilevel fast multipole algorithm (MLFMA) effectively deals with electrically large problems to some extent, it is still time and memory consuming for very large objects. In order to further reduce the CPU time and the memory requirement, a hybrid algorithm, based on the overlapped domain decomposition method for integral equations (IE-ODDM), MLFMA and block-diagonal, incomplete lower and upper triangular matrices (DILU) preconditioner, is proposed for the analysis of electrically large problems. The dominant memory requirement for plane wave expansions in the three processes of aggregation, translation and disaggregation in the MLFMA is drastically reduced by the first two techniques. The iterative procedure for each overlapped subdomain solved by the MLFMA is effectively sped up by the DILU preconditioner. After integrating these techniques, the proposed hybrid algorithm is more efficient in computing time and memory requirement compared to the conventional MLFMA and is more suitable for analyzing very large EM scattering problems. Enough accurate solution can be obtained within quite a few outer iterations, where an outer iteration means a complete sweep for all the subdomains. Some numerical examples are presented to demonstrate its validity and efficiency.     

电大均匀介质目标三维散射的并行多层快速多极子计算

实现了计算电大均匀介质体散射问题的高效混合并行混合场积分方程(Electric and Magnetic Current Combined-Field Integral Equation, JMCFIE)求解, 在单纯消息传递接口(Message Passing Interface, MPI)并行基础上采用共享存储并行编程(Open Multi-Processing, OpenMP)进一步提升性能.该混合MPI与OpenMP的并行多层快速多极子技术通过灵活的进程和线程策略, 提升了负载平衡和可扩展性.数值实验展示了此混合MPI与OpenMP的并行多层快速多极子技术的计算能力, 计算了不同尺寸的电大目标体(包含一个半径120 m、1.1亿未知数目的介质球).     

On the formulation of hybrid finite-element and boundary-integralmethods for 3-D scattering

This paper studies, in detail, a variety of formulations for the hybrid finite-element and boundary-integral (FE-BI) method for three-dimensional (3-D) electromagnetic scattering by inhomogeneous objects. It is shown that the efficiency and accuracy of the FE-BI method depends highly on the formulation and discretization of the boundary-integral equation (BIE) used. A simple analysis of the matrix condition number identifies the efficiency of the different FE-BI formulations and an analysis of weighting functions shows that the traditional FE-BI formulations cannot produce accurate solutions. A new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance. Finally, the multilevel fast multipole algorithm (MLFMA) is employed to significantly reduce the memory requirement and computational complexity of the proposed FE-BI method     

Radiation and scattering from infinite periodic printed antennaswith inhomogeneous media

The hybrid method of moments (MoM)/Green's function method technique is applied to infinite periodic printed antenna arrays containing dielectric inhomogeneities. The solution uses an integral equation for an infinite periodic printed array on or over a homogeneous dielectric substrate, coupled with equivalent volume polarization currents for dielectric inhomogeneities on top of the homogeneous substrate. Volume pulse-basis functions were used to expand the volume polarization currents. A hybrid MoM/Green's function method solution was then obtained through the matrix form of the problem. The two-dimensional (2-D) solution of plane wave scattering from a grounded dielectric slab was used to validate the reaction impedance of the dielectric inhomogeneity. Several infinite periodic printed dipole arrays with dielectric supports and overlays were studied with this solution and good agreement was observed between the hybrid MoM/Green's function method and waveguide simulator experiments     

二维组合目标RCS的MoM-PO混合法分析

利用矩量法求解二维目标结构的电磁散射问题,并在电流基混合法的基础上,对矩量法和物理光学混合法进行了研究,推导出了一种矩阵方程表达式,并通过仿真实例分析了以物理光学混合法计算多组合复杂目标散射时,物理光学区域与矩量法区域的划分方式,仿真结果证明,此区域划分结论的准确性、可行性。