A. Heldring

Fast Iterative Solution of Integral Equations With Method of Moments and Matrix Decomposition Algorithm – Singular Value Decomposition

The multilevel matrix decomposition algorithm (MLMDA) was originally developed by Michielsen and Boag for 2D TMz scattering problems and later implemented in 3D by Rius et al. The 3D MLMDA was particularly efficient and accurate for piece-wise planar objects such as printed antennas. However, for arbitrary 3D problems it was not as efficient as the multilevel fast multipole algorithm (MLFMA) and the matrix compression error was too large for practical applications. This paper will introduce some improvements in 3D MLMDA, like new placement of equivalent functions and SVD postcompression. The first is crucial to have a matrix compression error that converges to zero as the compressed matrix size increases. As a result, the new MDA-SVD algorithm is comparable with the MLFMA and the adaptive cross approximation (ACA) in terms of computation time and memory requirements. Remarkably, in high-accuracy computations the MDA-SVD approach obtains a matrix compression error one order of magnitude smaller than ACA or MLFMA in less computation time. Like the ACA, the MDA-SVD algorithm can be implemented on top of an existing MoM code with most commonly used Greens functions, but the MDA-SVD is much more efficient in the analysis of planar or piece-wise planar objects, like printed antennas.

Multilevel Adaptive Cross Approximation (MLACA)

The Multilevel Adaptive Cross Approximation (MLACA) is proposed as a fast method to accelerate the matrix-vector products in the iterative solution of the linear system that results from the discretization of electromagnetic Integral Equations (IE) with the Method of Moments (MoM). The single level ACA, already described in the literature, is extended with a multilevel recursive algorithm in order to improve the asymptotical complexity of both the computational cost and the memory requirements. The main qualities of ACA are maintained: it is purely algebraic and independent of the integral equation kernel Greens function as long as it produces pseudo-rank-deficient matrix blocks. The algorithm is presented in such a way that it can be easily implemented on top of an existing MoM code with most commonly used Greens functions.

Compressed Block-Decomposition Algorithm for Fast Capacitance Extraction

A novel algorithm, the compressed block decomposition, is presented for highly accelerated direct (noniterative) method-of-moment capacitance extraction. The algorithm is based on a blockwise subdivision of the method-of-moment potential coefficient matrix. Matrix subblocks corresponding to distant subregions of the problem geometry are not calculated directly but approximated in a compressed form. Subsequently, the matrix is decomposed using an algorithm that preserves the compression. The efficiency of the method is demonstrated on a common benchmark problem - a 6 times 6 bus crossing. The numerical cost of the algorithm is shown, both theoretically and numerically, to scale with N log3 N and the storage space with N log2 N.

Accelerated Direct Solution of the Method-of-Moments Linear System

This paper addresses the direct (noniterative) solution of the method-of-moments (MoM) linear system, accelerated through block-wise compression of the MoM impedance matrix. Efficient matrix block compression is achieved using the adaptive cross-approximation (ACA) algorithm and the truncated singular value decomposition (SVD) postcompression. Subsequently, a matrix decomposition is applied that preserves the compression and allows for fast solution by backsubstitution. Although not as fast as some iterative methods for very large problems, accelerated direct solution has several desirable features, including: few problem-dependent parameters; fixed time solution avoiding convergence problems; and high efficiency for multiple excitation problems [e.g., monostatic radar cross section (RCS)]. Emphasis in this paper is on the multiscale compressed block decomposition (MS-CBD) algorithm, introduced by Heldring , which is numerically compared to alternative fast direct methods. A new concise proof is given for the N2 computational complexity of the MS-CBD. Some numerical results are presented, in particular, a monostatic RCS computation involving 1 043 577 unknowns and 1000 incident field directions, and an application of the MS-CBD to the volume integral equation (VIE) for inhomogeneous dielectrics.

Multiscale CBD for fast direct solution of MoM linear system

This paper presents the Multiscale CBD method for accelerated direct solution of scattering and radiation problems with the Method of Moments. The method has all the advantages of a direct solution and it is essentially parameter-free (there is only a parameter tau controlling accuracy against efficiency). It has an improved efficiency and storage requirements with respect to the ordinary CBD. Two significant computations have been presented, an open pipe and the NASA almond.

Multilevel adaptive cross approximation (MLACA)

A novel algorithm based on the Adaptive Cross Approximation (ACA) [1] has been developed. It has been proved, both theoretically and numerically, to have a computational complexity scaling with N2 and memory requirement growing with NlogN.

Application of multilevel adaptive cross approximation (MLACA) to electromagnetic scattering and radiation problems

The Multilevel Adaptive Cross Approximation (MLACA), an algorithm to solve MoM electromagnetic problems with computational cost O(N2) and a storage scaling with O(NlogN), is presented here and for the first time applied to a whole electromagnetic problem and not only to the interaction between blocks whose containing spheres do not intersect each other.

New and more efficient formulation of MLMDA for arbitrary 3D antennas and scatterers

The Multilevel Matrix Decomposition Algorithm (MLMDA) was originally developed by Michielsen and Boag for 2-D TMz scattering problems and later implemented in 3-D by Rius et al. The 3-D MLMDA was particularly efficient and accurate for piece-wise planar objects such as printed antennas. However, for arbitrary 3-D problems it was not as efficient as the Multilevel Fast Multipole Algorithm (MLFMA). This paper will introduce some improvements in 3-D MLMDA, like new placement of equivalent functions and SVD post-compression, that make it comparable with MLFMA in terms of computation time and memory requirements, but more accurate.

Tangential-Normal Surface Testing for the Nonconforming Discretization of the Electric-Field Integral Equation

Nonconforming implementations of the electric-field integral equation (EFIE), based on the facet-oriented monopolar-RWG set, impose no continuity constraints in the expansion of the current between adjacent facets. These schemes become more versatile than the traditional edge-oriented schemes, based on the RWG set, because they simplify the management of junctions in composite objects and allow the analysis of nonconformal triangulations. Moreover, for closed moderately small conductors with edges and corners, they show improved accuracy with respect to the conventional RWG-discretization. However, they lead to elaborate numerical schemes because the fields are tested inside the body, near the boundary surface, over volumetric subdomains attached to the surface meshing. In this letter, we present a new nonconforming discretization of the EFIE that results from testing with RWG functions over pairs of triangles such that one triangle matches one facet of the surface triangulation and the other one is oriented perpendicularly, inside the body. This “tangential-normal” testing scheme, based on surface integrals, simplifies considerably the matrix generation when compared to the volumetrically tested approaches.

Stable Discretization of the Electric-Magnetic Field Integral Equation With the Taylor-Orthogonal Basis Functions

We present two new facet-oriented discretizations in method of moments (MoM) of the electric-magnetic field integral equation (EMFIE) with the recently proposed Taylor-orthogonal (TO) and divergence-Taylor-orthogonal (div-TO) basis functions. These new schemes, which we call stable, unlike the recently published divergence TO discretization of the EMFIE, which we call standard, result in impedance matrices with stable condition number in the very low frequency regime. More importantly, we show for sharp-edged objects of moderately small dimensions that the computed RCS with the stable EMFIE schemes show improved accuracy with respect to the standard EMFIE scheme. The computed RCS for the sharp-edged objects tested becomes much closer to the RCS computed with the RWG discretization of the electric-field integral equation (EFIE), which is well-known to provide good RCS accuracy in these cases. To provide best assessment on the relative performance of the several implementations, we have cancelled the main numerical sources of error in the RCS computation: (i) we implement the EMFIE so that the non-null static quasi-solenoidal current does not contribute in the far-field computation; (ii) we compute with machine-precision the strongly singular Kernel-contributions in the impedance elements with the direct evaluation method.