Fast direct solution of volume integral equations based on skeletonization by method of moments

The skeletonization accelerated multilevel fast multipole algorithm (MLFMA) in terms of surface integral equation (SIE) is extended to the method of moments (MoM) solution of volume integral equation (VIE). Boxes at the finest MLFMA level are recursively divided into smaller ones and the associated rank-deficient submatrices are approximated by skeletonization. Numerical experiments are conducted to study the performance of the proposed algorithm.

Electromagnetic problems with inhomogeneous or anisotropic penetrable media require using volume integral equations (VIEs) to describe in the integral equation approach. The VIEs are usually solved by the method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function, but the implementation may not be convenient because conforming meshes are required to define the basis function. In this paper, we propose a novel meshless method to discretize the VIEs. The method uses the Green-Gauss theorem to transform a volume integral into a boundary integral so that discretizing volume domains can be removed, resulting in a truly meshless method. For electrically large problems, we incorporate the method with multilevel fast multipole algorithm (MLFMA) to accelerate the solving process. A numerical example is presented to demonstrate the approach and its effectiveness and robustness have been validated.

A butterfly acceleration scheme is proposed for the fast solution of volume integral equation (VIE). The proposed scheme efficiently constructs the method of moments (MoM) matrix in the compressed format and obtains an approximate inverse of the MoM matrix, used as a highly effective preconditioner. The proposed scheme enables to efficiently solve problems involving millions of unknowns. Furthermore, it requires much less computational resources compared to $\mathcal{H}$ matrix scheme for the electromagnetic (EM) scattering analysis of electrically large structures.

The Numerical Solution of Integral Equations on the Half-Line

The method of moments (MoM) discretization of the volume integral equation (VIE) is an effective tool for calculating field responses in the presence of arbitrary scatterers. This method scales poorly with problem size, however, due to the inherent difficulty of inverting the relevant matrix equations. The application of tensor train (TT) decomposition techniques has recently shown promise in alleviating the complexity of such calculations, with dramatic efficiency boosts in both computational time and memory having been proven possible for very simple scatterers. For arbitrary scatterers, previous work has shown that a conjugate gradient- TT (CG- TT) procedure was still capable of producing favorable runtime and memory complexities, particularly in quasi-constant rank regimes. In this work, we consider a direct solution of TT decomposed MoM matrix equation approach that is largely insensitive to MoM matrix condition number, while maintaining the generality of CG- TT. Preliminary results indicate that this approach is capable of matching the efficiency of CG- TT in the quasi-constant rank regime for scatterers of arbitrary geometry.

Abstract. Accurate evaluation of singular potential integrals is essential for successful method of moments (MoM) solutions of surface integral equations. In mixed potential formulations for metallic and dielectric scatterers, kernels with 1/R and r1/R singularities must be considered. Several techniques for the treatment of these singularities will be reviewed. The most common approach solves the MoM source integrals analytically for specific observation points, thus regularizing the integral. However, in the case of r1/R a logarithmic singularity remains for which numerical evaluation of the testing integral is still difficult. A recently by Yl¨a-Oijala and Taskinen proposed remedy to this issue is discussed and evaluated within a hybrid finite element – boundary integral technique. Convergence results for the MoM coupling integrals are presented where also higher-order singularity extraction is considered.

This paper present a Galerkin-type method of moments (MoM) solution of volume surface integral equation (VSIE) and Taylor series expansion (TSE). This VSIE-TSE scheme is developed for the wideband frequency analysis of composite metallic and dielectric (CMD) structures. By expanding the exponential of the Green's function into a Taylor series, the impedance matrices can be efficiently calculated and generated, when the frequency sweeping process. In comparison with the conventional VSIE method, our scheme requires much less computational time while the accuracy remains at the same level. Several numerical examples are presented to demonstrate the accuracy and efficiency of the VSIE-TSE scheme.

The first section of this chapter collects results from [240] which gives a further contribution to the analysis of the hp-version of the boundary element method (BEM) by presenting a more general result for Dirichlet and Neumann problems than [21] allowing the use of a general geometric mesh refinement on the polygonal boundary Γ. Here as in [240] we prove the exponential convergence of the hp- version of the boundary element method by exploiting only features of the solutions of the boundary integral equations. The key result in this approach is an asymptotic expansion of the solution of the integral equations in singularity functions reflecting the singular behaviour of the solutions near corners of Γ. With such expansions we show that the solutions of the integral equations belong to countably normed spaces. Therefore these solutions can be approximated exponentially fast in the energy norm via the hp- Galerkin solutions of those integral equations. This result is not restricted to integral equations which stem from boundary value problems for the Laplacian but applies to Helmholtz problems as well. Further applications are 2D crack problems in linear elasticity. For numerical experiments with hp-version (BEM) see [165, 340].

Accurate numerical solutions of scattering phenomena can be achieved by method of moments (MoM) solutions of integral equations. For the evaluation of the monostatic radar cross section (RCS) of an object, the same system of equations has to be solved many times for different excitation fields incident from many directions. Especially for electrically large bodies involving many millions of unknowns, this becomes computationally demanding. In this work, we study and compare the performance of block and projection iterative methods for MoM solutions of surface integral equations related to perfect electrically conducting objects. Both methods are capable of reducing the overall number of matrix-vector products in the considered iterative solution process.

An efficient method of moments (MoM) solution of the volume-surface integral equation (VSIE) is presented for the analysis of electromagnetic (EM) scattering and radiation of composite objects comprising arbitrarily contacted dielectrics and conductors. In contrast to the conventional Schaubert-Wilton-Glisson (SWG) and Rao-Wilton-Glisson (RWG) basis functions, which were widely used in the MoM solutions of the VSIE, a new hybrid pulse-rooftop basis function is proposed to explicitly enforce the boundary condition of vanishing tangential electric field at the dielectric-conductor interface. As a consequence, the displacement currents in the dielectrics can be efficiently modeled by a reduced number of volume bases on the same set of meshes, particularly when thin dielectric slabs are involved in the composite structures. The numerical results of EM scattering and radiation of several composite objects are shown to illustrate the accuracy and efficiency of the proposed scheme. Since the new hybrid basis function can be easily implemented into the existing fast algorithms, the VSIE solution presented in this paper is promising for EM problems of composite targets with large size.

An ${\cal H}^{2}$ -matrix based linear complexity direct matrix solution is developed for the volume integral equation (VIE) based broadband full-wave extraction of general 3-D circuits. Such circuits are in general electrically small or moderate, but contain arbitrarily shaped lossy conductors immersed in inhomogeneous dielectrics with ports located anywhere in the physical layout of the circuit. In the proposed direct solver, we first develop a well-conditioned VIE formulation without sacrificing the rigor and the advantages of the prevailing formulations. This formulation facilitates a robust direct solution of good accuracy even with a rank-1 representation. We then overcome the numerical challenge of solving the resultant highly unstructured system matrix mixed with both square and rectangular dense and sparse matrices by developing a fast linear complexity direct solution. This direct solution is capable of inverting dense matrices involving over 2 million unknowns in less than 1 h on a single CPU core running at 3 GHz. Numerical simulations of large-scale 3-D circuits and comparisons with state-of-the-art linear complexity iterative VIE solvers have demonstrated the accuracy, efficiency, and linear complexity of the proposed direct VIE solver.

A new integration algorithm based on use of Quasi Monte Carlo Integration (QMCI) technique is proposed for Method of Moments (MoM) solution of Integral equations. As an example, solution of the integral equation for unknown charge distribution on metallic structures is obtained. It is observed that the proposed method is not only capable of dealing with the problem of singularity efficiently but also provides accurate computation of the static charge distribution on thin metallic rods.

Modeling of electromagnetic wave propagation and scattering in random media play an important role in geoscience and remote sensing research. Considerable efforts have been made to elucidate and understand the wave interaction processes involved in such problems, and various models have been developed for microwave active and passive remote sensing applications. With the rapid advances in computer technology and fast computational electromagnetic algorithms, numerical simulations of scattering by random media allow us to solve Maxwell's equations exactly without the limitations of approximate analytical models. The numerical models can provide a valuable means for evaluating the validity regimes of analytical scattering theories; in addition, they can potentially aid in the future development of extended analytical models. In this article, updated developments in the numerical scattering models for discrete random scatterers are presented, with emphasis on the applications of microwave remote sensing in snowcover, seafoam, and vegetation canopy.The frequency dependence of scattering by dense media at microwave frequencies is the first numerical model described, which is an important topic because multifrequency measurements are usually used in remote sensing applications. The approach used is based on the Monte Carlo simulations where the three‐dimensional solutions of Maxwell's equations are solved. The properties of absorption, scattering, and extinction are calculated for dense media consisting of sticky and nonsticky particles. Numerical solutions of Maxwell's equations indicate that the frequency dependence of densely packed sticky small particles is much weaker than that of independent scattering. Numerical results are illustrated using parameters of snow in microwave remote sensing. Comparisons are made with extinction measurements as a function of frequency.Polarimetric microwave emissions form foam‐covered ocean surfaces are studied by modeling foam as densely packed air bubbles coated with thin seawater. The absorption, scattering, and extinction coefficients are computed from the Monte Carlo solutions of Maxwell's equations for a collection of coated particles. These quantities are then applied in the dense media radiative transfer theory to calculate the polarimetric microwave emissivities of ocean surfaces with foam cover. The theoretical results of Stokes brightness temperatures with typical parameters of foam in passive remote sensing at 10.8 and 36.5 GHz are illustrated and compared with experimental measurements.It follows by describing an efficient computational model based on the sparse matrix iterative approach (SMIA) for tree scattering at VHF/UHF frequencies. The method of moments is applied to solve the volume integral equation for the tree scattering signatures. The SMIA decomposes the impedance matrix into a sparse matrix for the near interactions, and a complementary matrix for the far interactions of the tree structure. Solutions obtained from the SMIA method agree very well with the solutions obtained using extract matrix inversion and the conjugate gradient method (CGM). The key feature of the SMIA approach is that very little iteration is required to obtain convergent solutions; compared to the CGM, the SMIA approach may reduce the number of iterations by a factor of >100.The UV multilevel partitioning (UV‐MLP) method is also presented for solving the general volume scattering problems. The method consists of setting up a rank table of transmitting and receiving block sizes and their separations. The table can be set up speedily using coarse–coarse sampling. For a specific scattering problem with given geometry, the scattering structure is partitioned into multilevel blocks. By looking up the rank in the pre‐determined table, the impedance matrix for a given transmitting and receiving block is expressed by a product ofUandVmatrices. We demonstrate the method for two‐dimensional volume scattering by discrete scatterers. Multiple scattering is cast into the Foldy–Lax equations of partial waves. We show that the UV decomposition can be applied directly to the impedance matrix of partial waves of higher order than the usual lowest‐order Green function. Numerical results are illustrated for randomly distributed cylinders with a diameter of 1 wavelength. For scattering by 1024 cylinders on a single PC processor with 2.6 GHz CPU and 2 GB memory, only 14 CPU minutes is needed to obtain the numerical solution and for 4096 cylinders, only 7.34 s is needed for one matrix–vector multiplication.

Numerial Methods for Volterra Integral Equations with Singular Kernels

Method of moments (MOM) is applied extensively in electromagnetic, but the impedance matrix from MOM is so dense that it will consume prohibitive time if using the direct solver. In this paper, we will sparse the MOM-matrix using discrete wavelets transform method (DWT), and take the Daubechies wavelets of order N=3 for example, introducing the theory of DWT and the construction of the transformation matrix. At last, we will provide a new example of a 2D conducting plate with a groove.