Parallel Hierarchical Solvers and Preconditioners for Boundary Element Methods

Acceleration of the capacitance extraction of planar structures on a single‐layered medium is reported in this article, using the forward‐backward (FB) method with a spectral acceleration (SA) algorithm. The conventional spectral‐layered medium Green's function is modified appropriately so that it can be effectively incorporated into this method. The SA algorithm substantially speeds up the computational time involving in performing matrix‐vector multiplications in the iterative method of moments (MM). However, the memory requirement for the SA algorithm is comparable to that of the FB method. The computational complexity and the memory requirement are shown to be O(N) for large capacitance extraction problems, where N is the total number of unknowns in the MM for increasing one of the surface dimensions of planar structures and fixed neighborhood distance. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:694–700, 2014

We introduce a method to compress the impedance matrix of the method of moments (MoM) for the modeling of high-fidelity multiscale structures at low- to moderate-frequencies. We start from a method recently proposed to compress static (scalar) problems, proved to have O(N) computational complexity. We represent far coupling between groups through equivalent source distributions on properly defined, automatically generated equivalence surfaces. The equivalent sources are obtained via an inverse-source process that enforces equivalence of radiated fields on testing surfaces within a prescribed accuracy, and are intrinsically multiscale. This results in an O(N) complexity scaling of matrix-vector products, but reduces multiplicative constants (i.e., reduces time and memory requirements) for all structures. It affords an improved representation of couplings in multiscale structures, resulting in a fast convergence of iterative solvers. Our approach is Green's function independent and easy to be implemented in existing MoM codes; the present version is based on the electric field integral equation (EFIE). Numerical results prove the effectiveness of the proposed algorithm for complex multiscale structures.

The electromagnetic radiation and scattering from a body of revolution (BOR) of arbitrary shape have been widely discussed during several decades. The objects can contain perfect electric conductor (PEC), homogeneous dielectric bodies, coated conducting bodies and combined dielectric and conducting bodies [1][2]. The traditional method used to solve BOR with integral equations is the Method of Moments (MoM). And the computational time consumed in solving the integral equation of the BOR problem depends on the evaluation of modal Green's function (MGF), which is a time consuming process. Some research has been done to reduce this computational complexity. For example, Abdelmageed used spherical Bessel function to expansion to evaluate the MGF [3]. In this work, we extend the fast inhomogeneous plane wave algorithm (FIPWA) [4][5] to accelerate the computation of the MoM for composite homogeneous dielectric and conducting bodies of revolution. PMCHW [6] (Poggio-Miller-Chang-Harrington-Wu) equations and electric field integral equation are used for solving the homogeneous dielectric and conducting objects, respectively. Both the memory requirement and CPU time are reduced for large-scale BOR problems. Numerical results are given to demonstrate the validity and efficiency of the FIPWA.

A three-dimensional pre-corrected fast Fourier transform (PFFT) algorithm for the rapid solution of the full-dyadic Michalski-Zheng's mixed potential integral equation is presented. The integral equation is discretized with the Rao-Wilton-Glisson (RWG) method of moments. Handling the method of moments interactions with the dyadic kernel is simplified via representation of the RWG functions in terms of barycentric shape functions. The proposed three-dimensional precorrected FFT method distributes two-dimensional FFT grids nonuniformly along the direction of stratification according to conductor locations within the layers. For <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> two-dimensional FFT grids each with an average of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Np</i> associated triangular elements the method exhibits <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Np</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Np</i> ) computational complexity and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P Np</i> ) memory usage. The low-frequency breakdown of the integral equation is eliminated via loop-tree decomposition. A unique combination of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) computational complexity, fully three-dimensional boundary-element modeling in layered substrates, and full-wave modeling from dc to multi-gigahertz frequencies makes the algorithm particularly useful for characterizing large interconnect networks embedded in multilayered substrates. The method is implemented as the electromagnetic solver in Cadence's Virtuoso RF Designer software.

Integral equation based techniques, such as the method of moments (MOM), have been widely used to analyze time harmonic surface scattering phenomena. Unfortunately, time domain counterparts of the MOM, often referred to as marching-on-in-time (MOT) schemes, have historically received scant attention. Classical MOT schemes (i) have long been conceived as intrinsically unstable, and (ii) suffer from a high computational complexity. It is well-known that the computational complexity of the MoM can be reduced by the fast multipole method. A similar time domain algorithm, namely the plane wave time domain (PWTD) algorithm, has been introduced. The PWTD algorithm is designed to be used in tandem with the conventional MOT scheme in either a two-level or a multilevel setting. This paper describes the practical implementation of the two-level scheme and demonstrates the reduction in computational complexity achieved by using the PWTD algorithm. Although the PWTD scheme is applicable to a variety of integral equations arising from the analysis of wave phenomena, this paper illustrates its application in the context of scalar wave scattering from surfaces on which Neumann boundary conditions are enforced-a problem of interest in many disciplines.

A newly developed iterative method, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is applied to simulate electromagnetic scattering from large inhomogeneous objects embedded in a planarly layered medium. In this fast solver, the weak-form formulation is applied to obtain a less singular discretization of the volume electric field integral equation. Several techniques are utilized to speed up the dyadic function evaluation. To accelerate the operation of the dyadic function on an induced current (i.e., the Green's operation), the function is split into convolutional and correlational components so that FFT can be applied. The CPU time and memory cost of this BCGS-FFT method is O(NlogN) and O(N), respectively, where N is the number of unknowns, significantly more efficient than the method of moments (MoM). As a result, this method is capable of solving large-scale electromagnetic scattering problems in a planarly layered background. A large-scale scattering problem in a layered medium with more than three million unknowns has been solved on a Sun Ultra 60 workstation with 1.2 GBytes memory.

본 논문에서는 전자파를 이용하여 인체 내 암 조직을 찾아내기 위해 역산란 방법인 BIM(Born Iterative Method)을 이용한 알고리즘으로 구현하였다. Born 반복법을 이용하여 2차원 유방암 진단에 적용하였으며, 전자파산란 해석에는 적분방정식과 2차원 그린함수를 이용한 MoM(Method of Moment)을 이용하였다. 또한, 유방암 진단에 있어 적용된 역산란 알고리즘의 계산 결과를 확인하고, 암진단 의료기기로의 적용가능성과 알고리즘 사용에 제한을 분석하였다. In this paper, we used MOM-based BIM(Born Iterative Method) algorithm to implement the inverse scattering for the detection of cancer. We adopted two-dimensional breast structure, integral equations and two-dimensional Green's function is solved with MoM(Method of Moment) to analyzing electromagnetic scattering phenomena. In addition, verifying the calculation of developed inverse scattering algorithm and analyzing medical applicability and limitations of the algorithm.

In this paper, an efficient numerical technique is presented for analyzing the reflection and transmission characteristics of planar finite frequency selective surfaces (FSS) structures with multilayered dielectric substrate. For ideal infinite size FSS structures, only one single unit cell is of interest with the help of Floquet's theorem and periodic Green's functions. However, for FSS structures that are of finite size, full-wave numerical techniques must be employed to determine the electromagnetic characteristics. In this paper, the mixed potential integral equation (MPIE) for a general structure in multilayer media is first presented by using the spatial domain multilayered Green's functions. In order to simulate large-size FSS structures, the subentire- domain (SED) basis function method is then utilized to decrease memory requirement and computational complexity of method of moments (MoM). Different types of SED basis functions can be generated according to the locations of the cells in the entire structure, and then the original problem can be simplified into two smaller problems. In this way, a reduced system can be constructed and solved instead of the original large-size problem. This method enables a significant reduction to the matrix size and storage, and efficient analysis of finite FSS can be performed. To further decrease memory requirement and computational complexity, the presented numerical technique can be applied together with the fast fourier transform (FFT) algorithm for the matrix multiplication. The reflection and transmission coefficients of FSS are calculated using the far field of the FSS array and of the metallic plate with the same size. Several finite FSS structures are presented and analyzed to demonstrate the validate and efficiency of this method.

An integral equation is formulated in the spectral domain for a two slot microstrip filter using the exact Green's function for the grounded dielectric substrate. Using a moment method (MM) procedure, the integral equation has been discretized. The elements of the impedance matrix and the excitation vector are given by two dimensional Sommerfeld-type integrals in closed form. An efficient and accurate numerical integration scheme for computation of the elements is presented. The S parameters obtained from the MM procedure have been found to be in excellent agreement with measurements. >

The rapid development of communication systems and the impending traffic congestion of RF spectrum have led to an expansion in frequency usage towards the millimeter wave and sub-millimeter wave bands. In dealing with high frequency spectra in these bands, the conventional fabrication technology for microwave integrated circuit (MWIC) such as microstrip technology no longer applies, primarily because of high transmission loss. Although low-loss transmission media such as dielectric waveguide structures can be used as alternatives, they are difficult to fabricate and integrate with passive or active devices. This results in a growing interest in combining the advantages of microstrip and dielectric waveguide technologies. One of the most common implementations of interconnecting these two dissimilar structures is offered by aperture coupling. The theme of this dissertation is to introduce a rigorous formulation for modeling the coupling between microstrip and open-dielectric related structures (rectangular dielectric resonator antenna and image guide) via an aperture. The model is the first major rigorous method published for analyzing such types of coupling structures. The method is based on a surface integral equation approach by which the dielectric waveguide is modeled as equivalent surface electric and magnetic currents and subsequently treated via a set of combined field integral equations (CFIE). The microstrip structure and the associated coupling mechanism are formulated using the mixed potential integral equation (MPIE) approach. The coupled integral equations are then solved by the method of moments (MoM) in the spatial domain using Galerkin's procedure. Closed-form Green's functions are employed in the calculation to enhance numerical efficiency. The model is demonstrated to be a very versatile and is a practical design tool with the ability to perform accurate full-wave analysis very efficiently. The key to formulating 3-D open-dielectric waveguide structures is the introduction of the hybrid basis functions technique for modeling propagating and scattered fields associated with a guiding structure. It is based on the use of a combination of sub-domain (rooftop) and travelling wave basis functions to represent the irregular and modal surface currents on the dielectric. The technique was first applied to a 2-D planar guided structure (microstrip leaky wave antenna) before it was expanded to the 3-D geometry. The consequence of this technique is far-reaching as it has made possible a MoM analysis of 3-D finite open-dielectric waveguides that has not been reported.

Various methods are available at the disposal for computing the solution of electromagnetic problems expeditiously. Electromagnetic scattering problems can be classified into two broader categories named as surface and volume problems. These problems can be solved using differential and integral equations. Differential equations based techniques use Finite Element Method (FEM) and Finite Difference Time Domain (FDTD) methods and major integral formulation based technique is Method of Moment (MOM). Any set of linear equations for a given system can be solved using direct or iterative methods. Direct solvers confront many computational complexities and use large amount of memory while iterative solvers reduce computational complexities and use less amount of memory. In this research work some integral equations based solver schemes such ass Finite Element Method (FEM), Finite Difference Time Domain Method (FDTD) and Method of Moments (MOM) for surface and volume scatters are discussed and results are compared. The comparison is with respect to complexity and memory requirements for different solution schemes.

In this paper, a new method based on the complex images technique is presented for analysis of 2-D periodic devices such as waveguides, directional couplers, and coupled cavity waveguides (CCW). The main idea of this method lies in representing the infinite summation of the periodic Green's functions of such kind of structures in terms of finite terms of complex images. The electric field integral equation has been used to formulate the problem and the method of moments has been applied afterward for solving the equation. The method has been used successfully in the analysis of different waveguides, directional couplers, and CCWs in both rectangular and triangular lattices. Fast convergence, simple formulations, and flexibility of the method in analyzing different structures are the main advantages of the proposed method.

The closed-form Green's functions of the vector and scalar potentials in the spatial domain are presented for the sources of horizontal electric, magnetic, and vertical electric, magnetic dipoles embedded in general, multilayer, planar media. First, the spectral domain Green's functions in an arbitrary layer are derived analytically from the Green's functions in the source layer by using a recursive algorithm. Then, the spatial domain Green's functions are obtained by adding the contributions of the direct terms, surface waves, and complex images approximated by the Generalized Pencil of Functions Method (GPOF). In the derivations, the main emphasis is to put these closed-form representations in a suitable form for the solution of the mixed potential integral equation (MPIE) by the method of moments in a general three-dimensional geometry. The contributions of this paper are: 1) providing the complete set of closed-form Green's functions in spectral and spatial domains for general stratified media; 2) using the GPOF method, which is more robust and less noise sensitive, in the derivation of the closed-form spatial domain Green's functions; and 3) casting the closed-form Green's functions in a form to provide efficient applications of the method of moments.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In this letter, we present the analysis of large multiscale wire-surface structures through the method of moments (MoM) discretization of the electric field integral equation (EFIE). The MoM system is solved via the Green's function interpolation with fast Fourier transform (GIFFT) fast solver, extended here to geometries with wires and wires connected to surfaces. The implemented solver is entirely integrated with a hierarchical hybrid preconditioner, to allow an efficient analysis, in terms of computational time and memory requirements, of realistic and topologically complex geometries.

Solving electromagnetic (EM) problems by integral equation methods relies on the accurate evaluation of singular integrals related to the Green's function. In the method of moments (MoM) with the Rao-Wilton-Glisson (RWG) basis function for solving surface integral equations (SIEs), the gradient operator on the scalar Green's function can be moved onto the basic function and testing function, resulting in a 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> weak singularity in the integral kernel, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> is the distance between an observation point and a source point. The weakly singular integral can be evaluated with the well-known Duffy's method, but it requires a two-fold numerical integration. In this work, we develop a novel approach to evaluate the singular integral by using a local polar coordinate system. The approach can automatically cancel the singularity and reduce the integral to a one-fold numerical integration by deriving a closed-form expression for the integral over the polar coordinate. Numerical examples are presented to demonstrate the effectiveness of the approach.