Multilevel fast multipole method solution of volume integral equations using parametric geometry modeling

We present a multilevel fast multipole method (MLFMM) solution for volume integral equations dealing with scattering from arbitrarily shaped inhomogeneous dielectrics. The solution accuracy, convergence, computer time and memory savings of the method are demonstrated. Previous works have employed the MLFMM for impenetrable targets. In this paper, we integrate the MLFMM with the volume integral equation method for scattering by inhomogeneous targets. Of particular importance is the use of curvilinear elements for better volume representation and the use of simple basis functions for ease of parallelization.     

Surface Integral Equation Solutions by Hierarchical Vector Basis Functions and Spherical Harmonics Based Multilevel Fast Multipole Method

The method of moments solution of electromagnetic surface integral equation formulations for perfect electric conductors or impedance boundary objects is obtained using hierarchical vector basis functions. The singular and hypersingular integrals involved in the near field coupling matrix are computed fully numerically using adaptive singularity cancellation technique. Multilevel fast multipole method with spherical harmonics expansion of the $k$-space representations of the basis vectors and the incoming waves at the finest level is utilized for its memory and computation time efficient implementation. Improved performance of the proposed techniques in terms of accuracy, computational time, and memory requirements is demonstrated by comparing various results with some of the existing implementations available in the literature.      

Simplified approach to probe-corrected spherical near-field scanning

A multipole representation for the response of an arbitrary receiving antenna is derived that allows the formulation of probe-corrected spherical near-field scanning simply in terms of conventional vector spherical harmonics. Both the representation and formulation are free of rotational and translational addition theorems. A sampling theorem derived for Legendre functions is used to evaluate the resulting orthogonality integrals by direct summation in a computer time proportional to (ka)3.     

Broadband Müller-MLFMA for Electromagnetic Scattering by Dielectric Objects

The multilevel fast multipole algorithm (MLFMA) is very efficient for solving large-scale electromagnetic scattering problems. However, at low frequencies, or when the discretization is small compared with the wavelength, both the MLFMA and the underlying integral equation formulation typically suffer from a subwavelength breakdown. For the electromagnetic scattering from a homogeneous dielectric object, we obtain a stable and well-conditioned surface integral formulation using a variant of the classical Muumlller formulation and linear basis functions. To overcome the subwavelength breakdown of the MLFMA, we use both propagating and evanescent plane waves to represent the fields. The implementation is based on a combination of the spectral representation of the Green's function and Rokhlin's translation formula. We also present a new interpolation scheme for the evanescent part, which significantly improves the error-controllability of the MLFMA-implementation. Several numerical results verify both the error-controllability and scalability of the proposed algorithm     

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     

Generalized Sommerfeld integrals and field expansions in two-medium half-spaces

Time harmonic modal electromagnetic fields in two-medium half-spaces are investigated. For practical and numerical considerations, the primary sources of the modal fields are chosen to be the spherical multipoles, and the potential vectors are z-directed. It is shown that modal fields of such combination are not able to represent a conventional spherical modal field. The horizontal rotating potentials are added to ensure proper representation and fast convergence. The recurrence relations which transform the spherical Hankel-Legendre functions into the Fourier-Bessel integrals are derived. The secondary fields of the Sommerfeld type are obtained for all spherical multipole sources and the added horizontally rotating potentials. The combination of the modal fields are capable of representing arbitrary electromagnetic fields resulting from radiation and scattering problems.     

Wavelet packets for fast solution of electromagnetic integralequations

This paper considers the problem of wavelet sparsification of matrices arising in the numerical solution of electromagnetic integral equations by the method of moments. Scattering of plane waves from two-dimensional (2-D) cylinders is computed numerically using a constant number of test functions per wavelength. Discrete wavelet packet (DWP) similarity transformations and thresholding are applied to system matrices to obtain sparsity. If thresholds are selected to keep the relative residual error constant the matrix sparsity is of order O(N^P) with p<2. This stands in contrast with O(N² ) sparsities obtained with standard wavelet transformations. Numerical tests also show that the DWP method yields faster matrix-vector multiplication than some fast multipole algorithms     

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.     

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.      

一种电大尺寸组合体散射的快速计算方法

采用自适应多层快速多极子算法分析电大尺寸组合体的雷达散射截面。推导了组合体表面的积分方程,通过将基函数和权函数分别用不同空间位置上的点源函数展开,自适应多层快速多极子算法实现了阻抗积分的快速计算,通过采用射线传播法,远场近似和对称性计算法则使转移因子的计算效率大大提高,所有转移过程可由快速傅里叶变换计算完成。这种算法计算组合体散射时所需的计算时间和内存显著降低,且数值计算结果和实际测试结果吻合良好。     

Broadband MLFMA With Plane Wave Expansions and Optimal Memory Demand

A broadband multilevel fast multipole algorithm (MLFMA) in 3D is presented based on plane wave expansions and diagonal translations on all division levels. The radiation and incoming wave patterns on all levels are presented by trigonometric polynomials, and optimal sampling rates and translator degrees are found and tabulated. On the super-wavelength levels the memory need and the computational cost are lowered more than by half compared to the traditional approach. On the sub-wavelength levels also a novel saving method for the field patterns is presented which is as efficient as saving the multipole series coefficients while all translations with the proposed method can still be carried out in the fast diagonal form. Also the direction dependence of the radiation and incoming wave patterns is lowered by half. The proposed broadband MLFMA has a very good error control on all division levels which we demonstrate by numerical testing.     

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     

一种基于矢量有限元与多层快速多极子技术的 电磁散射快速并行算法

在混合矢量有限元/多层快速多极子算法的基础上提出了一种快速算法及其并行算法,该算法中,其有限元部分的计算可在单元级上完成,无须生成总体系数矩阵,因此可大大节省内存及计算时间;对多层快速多极子部分,将基函数和权函数分别用不同空间位置上的点源函数展开,使阻抗积分计算得到大大简化,所有转移过程可由快速傅立叶变换计算完成,同时还给出了一些其他的改进措施.数值结果说明了算法的有效性.     

A Spectral Multigrid Method Combined With MLFMM for Solving Electromagnetic Wave Scattering Problems

A new spectral multigrid method (SMG) combined with the multilevel fast multipole method (MLFMM) is proposed for solving electromagnetic wave scattering problems. The MLFMM is used to speed up the matrix-vector product operations and the SMG is employed to accelerate the convergence rate of the Krylov iteration. Unlike traditional algebraic multigrid methods (AMG), the spectral multigrid method is an algebraic two-grid cycle built on a preconditioned Krylov iterative method that is used as the smoother, and the grid transfer operators are defined using the spectral information of the preconditioned matrix. Numerical experiments indicate that this class of multigrid method is very effective with the MLFMM and can reduce both the iteration number and the overall simulation time significantly.     

Ray Optical Electromagnetic Far-Field Scattering Computations Using Planar Near-Field Scanning Techniques

For accurate scattering computations in the far-field of flat finite objects, field based ray optical methods cannot be used directly, since the finiteness of the objects is not considered in the formulations. In this paper, planar near-field scanning techniques are used to overcome this problem. In particular, scattered ray optical fields are first computed in a scanning plane in the near-field region of the involved objects and are transformed into the far-field afterwards using field expansions in terms of spectrum density functions of outgoing waves. Since evanescent waves are avoided in the scanning plane, sampling rates less than lambda₀/2 can be used for restricted angle range around the normal direction to the scanning plane. Reduced accuracy at grazing directions of observation is overcome by combining solutions provided by several scanning planes. The proposed approach is applied in the postprocessing stage of the recently developed hybrid method combining the uniform geometrical theory of diffraction with the finite element boundary integral technique and with the multilevel fast multipole method.     

一种多层快速多极子的高效并行方案

基于多层快速多极子中不同层间数据分布和计算时间不同的特点,提出一种按不同方式并行处理不同层平面波和转移矩阵的高效并行方案.从理论分析和数值试验出发,给出了在不同层间高效使用不同并行方式的原则及具体公式.在北京理工大学信息科学技术学院电磁仿真中心的高性能计算机集群上成功计算了直径为144个波长、超过1000万未知数模拟的金属球散射;与美国伊利诺伊大学计算电磁中心的报道进行了比较,展示了本方案的高效精确.还计算了飞机模型的散射,显示本方案对电大复杂目标散射问题的求解能力.     

双位积分方程在分析地面上线天线中的应用

导出了任意取向线天线在基于sommerfeld积分的反射系数近似条件下的双位积分方程,采用多项式基函数和点匹配的矩量法对该方程求解,给出阻抗元素计算的一般表达式.计算结果表明了该方法的有效性.     

An alternative formulation for the fast multipole method

This paper describes an alternative formulation for the fast multipole method based on spherical waves decomposition. It is somewhat simpler to implement than the standard fast multipole method and also better suited at low frequencies. The new formulation is mainly based on a technique for the interpolation of the bistatic radar cross section derived from the Wacker's method for antenna measurements.     

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.     

A weak form of the conjugate gradient FFT method for plate problems

A number of electromagnetic field problems for planar structures 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 convolution of the free-space Green's function and some surface current density over the domain of interest. A weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only. As a next step, the vector potential is expanded in a sequence of subdomain basis functions and the grad-div operator is integrated analytically over the plate domain only. For the problem of electromagnetic scattering by a plate, the method shows excellent numerical performance. The numerical difficulties encountered in some previous conjugate gradient fast Fourier transform (CGFFT) methods have been eliminated