Analytic preconditioners for the electric field integral equation

Since the advent of the fast multipole method, large‐scale electromagnetic scattering problems based on the electric field integral equation (EFIE) formulation are generally solved by a Krylov iterative solver. A well‐known fact is that the dense complex non‐hermitian linear system associated to the EFIE becomes ill‐conditioned especially in the high‐frequency regime. As a consequence, this slows down the convergence rate of Krylov subspace iterative solvers. In this work, a new analytic preconditioner based on the combination of a finite element method with a local absorbing boundary condition is proposed to improve the convergence of the iterative solver for an open boundary. Some numerical tests precise the behaviour of the new preconditioner. Moreover, comparisons are performed with the analytic preconditioner based on the Calderòn's relations for integral equations for several kinds of scatterers. Copyright © 2004 John Wiley & Sons, Ltd.     

Direct use of discrete complex image method for evaluating electric field expressions in a lossy half space

A modelling technique is proposed for direct use of the discrete complex image method (DCIM) to derive closed-form expressions for electric field components encountered in the electric field integral equation (EFIE) representing a lossy half space problem. The technique circumvents time consuming numerical computation of Sommerfeld integrals by approximating the kernel of the integrals with appropriate mathematical functions. This is done by appropriate use of either the least-square Prony (LS-Prony) method or the matrix pencil method (MPM) to represent electric field expressions in terms of spherical waves and their derivatives. A comparison is made between the two methods based on the computation time and accuracy and it is shown that the LS-Prony method performs two?three times faster than the MPM in approximating the integral kernels depending on the platform. The main feature of the proposed technique is its ability for direct inclusion in the kernel of computational tools based on the method of moments solution of the EFIE. This can be viewed as an advantage over the conventional DCIM approximation of spatial Green's functions for mixed potential integral equation for cases where the problem in hand can be more efficiently represented by the EFIE (e.g. the thin-wire EFIE). The accuracy of the proposed technique is validated against numerical integration of Sommerfeld integrals for an arbitrary electric dipole inside a lossy half space.     

Improving the accuracy of the magnetic field integral equation with the use of a new impedance matrix element formulation

The electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) are widely used in conjunction with method of moments for electromagnetic scattering analysis of three-dimensional conducting objects with closed surfaces. However, the MFIE suffers from an accuracy problem compared with the EFIE with the use of the Rao?Wilton?Glisson (RWG) basis function. This accuracy problem is more serious for objects with sharp edges or corners. To solve this problem, a new technique to compute the impedance matrix elements (IME) of the MFIE using an RWG basis function is presented here. Details to compute the IME and the advantage of this new formulation are displayed. In addition, the relationship between this new IME formulation and the formulation using the low-order curl-conforming basis function for the MFIE is given. Through the computation of the RCS of several relatively small sharp-edged conducting objects, it is shown that the accuracy of the MFIE can be greatly improved by the use of the new IME formulation.     

Modeling considerations for rigorous boundary element method analysis of diffractive optical elements.

Critical modeling issues relating to rigorous boundary element method (BEM) analysis of diffractive optical elements (DOEs) are identified. Electric-field integral equation (EFIE) and combined-field integral equation (CFIE) formulations of the BEM are introduced and implemented. The nonphysical interior resonance phenomenon and thin-shape breakdown are illustrated in the context of a guided-mode resonant subwavelength grating. It is shown that modeling such structures by using an open geometric configuration eliminates these problems that are associated with the EFIE BEM. Necessary precautions in defining the incident fields are also presented for the analysis of multiple-layer DOEs.     

On the spectrum of the electric field integral equation and the convergence of the moment method

Existing convergence estimates for numerical scattering methods based on boundary integral equations are asymptotic in the limit of vanishing discretization length, and break down as the electrical size of the problem grows. In order to analyse the efficiency and accuracy of numerical methods for the large scattering problems of interest in computational electromagnetics, we study the spectrum of the electric field integral equation (EFIE) for an infinite, conducting strip for both the TM (weakly singular kernel) and TE polarizations (hypersingular kernel). Due to the self‐coupling of surface wave modes, the condition number of the discretized integral equation increases as the square root of the electrical size of the strip for both polarizations. From the spectrum of the EFIE, the solution error introduced by discretization of the integral equation can also be estimated. Away from the edge singularities of the solution, the error is second order in the discretization length for low‐order bases with exact integration of matrix elements, and is first order if an approximate quadrature rule is employed. Comparison with numerical results demonstrates the validity of these condition number and solution error estimates. The spectral theory offers insights into the behaviour of numerical methods commonly observed in computational electromagnetics. Copyright © 2001 John Wiley & Sons, Ltd.     

On the use of piecewise linear wavelets for fast construction of sparsified moment matrices in solving the thin-wire EFIE

Multiresolution wavelet expansion technique has been successfully used in the method of moments (MoM), and sparse matrix equations have been attained. Solving boundary integral equations arising in electromagnetic (EM) problems by the wavelet-based moment method (WMM) involves a time-consuming double numerical integration for each entry of the resultant matrix which in turn can outweigh the advantages of achieving a sparse matrix. The paper presents an alternative computational model to speed up the WMM by excluding double numerical integrations in the evaluation of matrix elements. In this regard, pieces of linear wavelet bases are replaced by proper sinusoidal functions for which closed-form analytical expressions are available. In addition, by introducing approximate closed-form expressions for radiating EM fields of wavelet current elements, the thresholding procedure is modified so that one can compute only the matrix elements of interest. To demonstrate the effectiveness of the proposed method, the thin-wire electric field integral equation (EFIE) is numerically solved by non-orthogonal linear spline wavelet bases.     

Non-orthogonal spline wavelets for boundary element analysis

Non-orthogonal spline wavelets are developed for Galerkin boundary element method. The proposed wavelets have compact supports and closed-form expressions. Besides, one can choose arbitrarily the order of vanishing moments of the wavelets independently of order of B-splines. Sparse coefficient matrices are obtained by truncating the small elements a priori. The memory requirement and computational time can be controled by changing the order of vanishing moments of the wavelets. As an iterative technique for solving the boundary element equations, GMRES(m) method is employed. Diagonal scaling and incomplete LU factorization (ILU(0)) are considered for the preconditioning. The ILU(0) becomes an effective preconditioner for higher order vanishing moments. Through numerical examples, availability of the proposed wavelets is investigated.     

Numerical solution of the EFIE by the meshfree collocation method

This paper contains guidelines for numerical solution of the EFIE meshfree. Degrees of freedom including mathematical statement, the meshfree method, shape functions and their parameters are considered and proper choices are selected by logical deduction, experience or previous reports. The method is based on decomposing the differential and integral parts of the EFIE, which is an integro-differential equation. These two independent parts could be processed in parallel. The differential and the integral parts are expanded over interpolants and approximant meshless shape functions, respectively. The final arrangement is applied to various scattering problems. Even though we applied the method mostly to linear and rectangular structures, the approach is applicable to all Electromagnetic integral equations of arbitrary geometries. For simple geometries with equidistance node arrangements and considering only the central node for support of the approximants, suggestions are made for bypassing numerical integration. In this case, although the differential part is still meshless, the integral part cannot be regarded as a meshless method in a strict sense and it may be considered a high-order collocation method. The results are compared with the low-order method of moments, previous reports and the FEKO software.     

A fast wavelet-based moment method for solving thin-wire EFIE

Solving the thin-wire electric field integral equation (EFIE) by the multiresolution wavelet expansion method involves a time-consuming double numerical integration for each nonzero element of the moment matrix which in turn can outweigh the advantages of achieving a sparse matrix. To speed up the matrix fill process in wavelet-based moment method codes, first, the triangular scaling functions of a nonorthogonal piecewise liner wavelet at the finest spatial resolution are appropriately replaced by sinusoidal dipoles for which mutual impedances are available in closed-form analytical expressions. The fast wavelet bases transform is then exploited to effectively transfer the resultant matrix equation to multiresolution wavelet domain. Numerical results obtained by the compactly supported semi-orthogonal linear B-spline wavelet demonstrate dramatic reduction of the overall solution time without any degradation in the accuracy of the final solution.     

A Block Diagonal Preconditioner for Generalised Saddle Point Problems

A lopsided alternating direction iteration (LADI) method and an induced block diagonal preconditioner for solving block two-by-two generalised saddle point problems are presented. The convergence of the LADI method is analysed, and the block diagonal preconditioner can accelerate the convergence rates of Krylov subspace iteration methods such as GMRES. Our new preconditioned method only requires a solver for two linear equation sub-systems with symmetric and positive definite coefficient matrices. Numerical experiments show that the GMRES with the new preconditioner is quite effective.     

An electromagnetic model of lightning return stroke channel using electric field integral equation in time domain

We use an electromagnetic approach based on antenna theory (AT) to evaluate the lightning return stroke current as a function of time and height. The lightning channel is modeled as a lossy, straight, and vertical monopole antenna above a perfectly conducting ground, which is excited by a source voltage at the base of the channel. This voltage source is a function of the current assumed at the ground level and the input impedance of the monopole antenna. An electric field integral equation (EFIE) is employed to describe the electromagnetic behavior of the antenna. The numerical solution of EFIE by the method of moments in time domain provides the time-space distribution of the current along the lightning channel. This AT model with specified current at the channel base requires only two adjustable parameters, namely the return-stroke propagation speed and the channel resistance per unit length. To demonstrate the accuracy of the proposed model, we compare it to the most commonly used models in terms of the temporal and spatial distributions of channel current and predicted electromagnetic fields. We also present results to show the effectiveness of the model in the analysis of lightning-related problems dealing with complex structures. In this regard, the lightning induced overvoltages on the neighboring overhead lines and the lightning strikes to tall structures are investigated.     

On the implementation of 3D Galerkin boundary integral equations

In this article, a reverse contribution technique is proposed to accelerate the construction of the dense influence matrices associated with a Galerkin approximation of hypersingular boundary integral equations of mixed-type in potential theory. In addition, a general-purpose sparse preconditioner for boundary element methods has also been developed to successfully deal with ill-conditioned linear systems arising from the discretization of mixed boundary-value problems on non-smooth surfaces. The proposed preconditioner, which originates from the precorrected-FFT method, is sparse, easy to generate and apply in a Krylov subspace iterative solution of discretized boundary integral equations. Moreover, an approximate inverse of the preconditioner is implicitly built by employing an incomplete LU factorization. Numerical experiments involving mixed boundary-value problems for the Laplace equation are included to illustrate the performance and validity of the proposed techniques.     

Using MBPE technique to accelerate solving the thin-wire EFIE used in numerical simulation of lightning

The antenna theory (AT) model is widely used to numerically simulate the propagation of current wave along lightning return-stroke channels and compute the radiated electromagnetic fields. In this model, the return stroke channel is considered as a vertical monopole antenna above perfectly conducting ground for which the numerical solution of the governing electric field integral equation (EFIE) in the frequency domain by the conventional method of moment (MoM) is prohibitively slow. In this paper, a model-based parameter estimation (MBPE) technique is proposed to reduce the number of frequency-domain calculation points required for the evaluation of space-time current distribution along a lightning return stroke channel. In applying this technique to a rational function model for the channel current distribution, a uniform-like sampling strategy is investigated. In order to accelerate the building of the moment impedance matrix, the reciprocal closed-form mutual impedance of sinusoidal electric dipoles and the symmetry of the model are used. The proposed technique is validated against the conventional inverse fast Fourier transform algorithm which uses a MoM solution for all frequencies within the channel base current spectrum. It is shown that considerable computation efficiency is achieved in terms of CPU time without losing accuracy.     

Application of new fast multipole boundary integral equation method to crack problems in 3D

Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large scale problems. This paper discusses an application of the new version of FMM to three-dimensional boundary integral equation method (BIEM) for crack problems for the Laplace equation. The boundary integral equation is discretised with collocation method. The resulting algebraic equation is solved with generalised minimum residual method (GMRES). The numerical results show that the new version of FMM is more efficient than the original FMM.     

An efficient preconditioner for the fast simulation of a 2D stokes flow in porous media

We consider an efficient preconditioner for a boundary integral equation (BIE) formulation of the two‐dimensional Stokes equations in porous media. While BIEs are well‐suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as generalized minimial residual method (GMRES). In this paper, we apply a fast inexact direct solver, the inverse fast multipole method, as an efficient preconditioner for GMRES. This solver is based on the framework of ‐matrices and uses low‐rank compressions to approximate certain matrix blocks. It has a tunable accuracy ε and a computational cost that scales as . We discuss various numerical benchmarks that validate the accuracy and confirm the efficiency of the proposed method. We demonstrate with several types of boundary conditions that the preconditioner is capable of significantly accelerating the convergence of GMRES when compared to a simple block‐diagonal preconditioner, especially for pipe flow problems involving many pores.     

Vector near-field calculation of scanning near-field optical microscopy probes using Borgnis potentials as auxiliary functions

A new boundary integral equation method for solving the near field in three-dimensional vector form in scanning near-field optical microscopy (SNOM) using Borgnis potentials as auxiliary functions is presented. A boundary integral equation of the electromagnetic fields, expressed by Borgnis potentials, is derived based on Green's theorem. The harmonic expansion in rotationally symmetric SNOM probe--sample systems is studied, and the three-dimensional electromagnetic problem is partly simplified into a two-dimensional one. The boundary conditions of Borgnis potentials both on dielectric boundaries and on perfectly conducting boundaries are derived. Relevant algorithms were studied, and a computer program was written. As an example, a SNOM probe-sample system composed of a round metal-covered probe and a sample with a flat surface has been numerically studied, and the computational results are given. This new method can be used efficiently for other electromagnetic field problems with round subwavelength structures.     

Nonlinear three-dimensional magnetic field computations usingLagrange finite-element functions and algebraic multigrid

The paper proposes an efficient solution strategy for nonlinear three-dimensional (3-D) magnetic field problems. The spatial discretization of Maxwell's equations uses Lagrange finite-element functions. The paper shows that this discretization is appropriate for the problem class. The nonlinear equation is linearized by the standard fixed-point scheme. The arising sequence of symmetric positive definite matrices is solved by a preconditioned conjugate gradient method, preconditioned by an algebraic multigrid technique. Because of the relatively high setup time of algebraic multigrid, the preconditioner is kept constant as long as possible in order to minimize the overall CPU time. A practical control mechanism keeps the condition number of the overall preconditioned system as small as possible and reduces the total computational costs in terms of CPU time. Numerical studies involving the TEAM 20 and the TEAM 27 problem demonstrate the efficiency of the proposed technique. For comparison, the standard incomplete Cholesky preconditioner is used     

Performance of a Petrov–Galerkin algebraic multilevel preconditioner for finite element modeling of the semiconductor device drift‐diffusion equations

This study compares the performance of a relatively new Petrov–Galerkin smoothed aggregation (PGSA) multilevel preconditioner with a nonsmoothed aggregation (NSA) multilevel preconditioner to accelerate the convergence of Krylov solvers on systems arising from a drift‐diffusion model for semiconductor devices. PGSA is designed for nonsymmetric linear systems, Ax=b, and has two main differences with smoothed aggregation. Damping parameters for smoothing interpolation basis functions are now calculated locally and restriction is no longer the transpose of interpolation but instead corresponds to applying the interpolation algorithm to A^T and then transposing the result. The drift‐diffusion system consists of a Poisson equation for the electrostatic potential and two convection–diffusion‐reaction‐type equations for the electron and hole concentration. This system is discretized in space with a stabilized finite element method and the discrete solution is obtained by using a fully coupled preconditioned Newton–Krylov solver. The results demonstrate that the PGSA preconditioner scales significantly better than the NSA preconditioner, and can reduce the solution time by more than a factor of two for a problem with 110 million unknowns on 4000 processors. The solution of a 1B unknown problem on 24 000 processor cores of a Cray XT3/4 machine was obtained using the PGSA preconditioner. Copyright © 2010 John Wiley & Sons, Ltd.     

Incompressible flow past oscillatory wings of low aspect ratio by the integral equations method

In the framework of the linearized perturbation theory, the pressure jump over the oscillating wing is the solution of a two‐dimensional integral equation. Performing an asymptotic expansion with respect to the aspect ratio and keeping the leading terms, we reduce the integral equation to a one‐dimensional one, obtaining a simplified method of solving the lifting surface integral equation for a class of thin wings of low aspect ratio with a straight trailing edge. The one‐dimensional integral equation is solved for the delta flat plate and the pressure coefficient field and the lift and moment coefficients are calculated. The range of validity of the new method is discussed in the final part of the paper. Copyright © 1999 John Wiley & Sons, Ltd.     

Mutually constrained partial differential and integral equations for an exterior field problem

A new implementation of the mutually constrained partial differential and integral equation method for the exterior 2-dimensional field problem is described. It is shown, that the method is applicable to exterior problems in an inhomogeneous medium. The inhomogeneity is considered in the finite element procedure and in boundary element method, where an adequate Green's function is applied. The temperature distribution around a three-cable system is then computed as an illustration. The eddy-current losses in the cable sheaths are calculated using the Fredholm integral equation of the second kind.