Conventional Method Of Moments Research Articles

A Single-Level Dual Rank IE-QR Algorithm to Model Large Microstrip Antenna Arrays

A single-level dual rank IE-QR algorithm is introduced so that the resulting dense method of moments (MOM) matrix is efficiently compressed. For a system of N equations, an amount of work of the order O(N/sup 2/) has traditionally been required for both matrix assembly and matrix-vector multiplication in an iterative matrix solver. The algorithm of the present paper reduces the memory requirement and CPU time for both matrix assembly and matrix-vector multiplication to O(N/sup 3/2/) making it practical to solve for large antenna arrays with full wave approach. In conjunction with a "geometric-neighboring" preconditioner for matrix solution using GMRES, the current approach solves problems involving large antenna arrays using only a fraction of what are needed by conventional MOM both in term of memory and total CPU time.

Conditional simulations of flow in randomly heterogeneous porous media using a KL-based moment-equation approach

In this study, we extend the KLME approach, a moment-equation approach based on the Karhunen–Loève decomposition (KL), developed by Zhang and Lu [An efficient, higher-order perturbation approach for flow in randomly heterogeneous porous media via Karhunen–Loève decomposition. J Comput Phys 2004;194(2):773–94] to efficiently incorporate existing direct measurements of the log hydraulic conductivity. We first decompose the conditional log hydraulic conductivity Y = ln K s as an infinite series on the basis of a set of orthogonal Gaussian standard random variables { ξ i }. The coefficients of this series are related to eigenvalues and eigenfunctions of the conditional covariance function of the log hydraulic conductivity. We then write head as an infinite series whose terms h ( n) represent the head contribution at the nth order in terms of σ Y , the standard deviation of Y, and derive a set of recursive equations for h ( n) . We assume that h ( n) can be expressed as infinite series in terms of the products of n Gaussian random variables. The coefficients in these series are determined by substituting decompositions of Y and h ( m) , m < n, into those recursive equations. We solve the conditional mean head up to fourth-order in σ Y and the conditional head covariances up to third-order in σ Y 2 . The higher-order corrections for the conditional mean flux and flux covariance can be determined directly from the higher-order moments of the head, using Darcy’s law. We compare the results from the KLME approach against those from Monte Carlo (MC) simulations and the conventional first-order moment method. It is evident that the KLME approach with higher-order corrections is superior to the conventional first-order approximations and is computationally more efficient than both the Monte Carlo simulations and the conventional first-order moment method.

Neural Networks-Method of Moments (NN-MoM) for the Efficient Filling of the Coupling Matrix

In this paper, novel radial basis function-neural network (RBF-NN) models are presented for the efficient filling of the coupling matrix of the method of moments (MoM). Two RBF-NNs are trained to calculate the majority of elements in the coupling matrix. The rest of elements are calculated using the conventional MoM, hence the technique is referred to as neural network-method of moments (NN-MoM). The proposed NN-MoM is applied to the analysis of a number of microstrip patch antenna arrays. The results show that NN-MoM is both accurate and fast. The proposed technique is general and it is convenient to integrate with MoM planar solvers.

Magnetic field integral equation at very low frequencies

It is known that there is a low-frequency breakdown problem when the method of moments (MOM) with Rao-Wilton-Glisson (RWG) basis is used in the electric field integral equation (EFIE); it can be solved through the loop and tree basis decomposition. The behavior of the magnetic field integral equation (MFIE) at very low frequencies is investigated using MOM, where two approaches are presented based on the RWG basis and loop and tree bases. The study shows that MFIE can be solved by the conventional MOM with the RWG basis at arbitrarily low frequencies, but there exists an accuracy problem in the real part of the electric current. Although the error in the current distribution is small, it results in a large error in the far-field computation. This is because a big cancellation occurs during the far field computation. The source of error in the current distribution is easily detected through the MOM analysis using the loop and tree basis decomposition. To eliminate the error, a perturbation method is proposed, from which a very accurate real part of the tree current has been obtained. Using the perturbation method, the error in the far-field computation is also removed. Numerical examples show that both the current distribution and the far field can be accurately computed at extremely low frequencies by the proposed method.

Rapidly converging spectral-domain analysis of rectangularly shielded layered microstrip lines

A moment-method-oriented direct integral-equation technique is presented for the exact analysis of rectangularly shielded layered microstrip lines. This technique retains the simplicity of conventional moment methods while optimizing them by recasting all matrix elements into rapidly converging series. Filling up the matrix requires no numerical integration. The proposed algorithms yield highly accurate results both for the modal currents and propagation constants.

An advanced numerical model in solving thin-wire integral equations by using semi-orthogonal compactly supported spline wavelets

In this paper, the semi-orthogonal compactly supported spline wavelets are used as basis functions for the efficient solution of the thin-wire electric field integral equation (EFIE) in frequency domain. The method of moments (MoM) is used via the Galerkin procedure. Conventional MoM directly applied to the EFIE, leads to dense matrix which often becomes computationally intractable when large-scale problems are approached. To overcome these difficulties, wavelets can be used as a basis set so obtaining the generation of a sparse matrix; this is due to the local supports and the vanishing moments properties of the wavelets. In the paper, this technique is applied to analyze electromagnetic transients in a lightning protection systems schematized as a thin-wire structure. The study is carried out in frequency domain; a discrete fast Fourier transform algorithm can be used to compute time profiles of the electromagnetic interesting quantities. The unknown longitudinal currents are expressed by using multiscale wavelet expansions. Thus, the thin-wire EFIE is converted into a matrix equation by the Galerkin method. Results for linear spline wavelets along with their comparison with conventional MoM that uses triangular basis functions and the point matching procedure are presented, for two case studies. Good agreement has been reached with a strong reduction of the computational complexity.

Dynamical mean-field theory of spiking neuron ensembles: response to a single spike with independent noises.

A semianalytical dynamical mean-field theory has been developed for a study of dynamics of an ensemble of N-unit FitzHugh-Nagumo neurons subject to white noises. Assuming weak noises and the Gaussian distribution of state variables, we have driven equations of motions for moments of local and global variables. Dynamical mean-field approximation (DMA) has replaced original, 2N-dimensional stochastic differential equations (DEs) by eight-dimensional deterministic DEs, whereas the conventional moment method yields N(2N+3)-dimensional deterministic DEs for local variables. We have discussed the dependence of the spike firing precision and the synchronization on the noise intensity, synaptic coupling, and the size of the neuron ensemble. The spike timing precision is shown to be improved by increasing the size of the neuron ensemble, even when there are no couplings among neurons. When the coupling is introduced, neurons in ensembles respond to an input spike with a partial synchronization. Results calculated by our DMA theory are in good agreement with those obtained by direct simulations. DMA theory is extended to a large cluster which can be divided into multiple subclusters according to their functions. A model calculation has demonstrated that when the noise intensity is moderate, the spike propagation with a fairly precise timing is possible among noisy subclusters with feed-forward couplings, as in the synfire chain. We have compared DMA theory with the conventional moment method, showing that the former may be alternatively derived from the latter by a reduction in the number of moments with the mean-field approximation.

Efficient analysis of arbitrarily shaped microstrip structures

AbstractA new, efficient method of moments (MoM) is developed for accurate, full‐wave analysis of microstrip components and circuits. The new method is applied for the analysis of a two‐gap filter. The new method is found to be as accurate as conventional moment methods, and also significantly faster. It has a good potential for use in the microstrip design of CAD tools. The key to the high efficiency of the new MoM is the use of a set of newly developed closed‐form Green's functions, which allow the computation of the MoM matrix elements without any numerical integration when using the RWG function as the basis functions. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 37: 246–248, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10883

A Shielded Two-Arc-Shaped-Conductor Transmission Line Solved by a Fast Converging Moment Method

A balanced transmission line consisting of two arc-shaped thick conductors and shielded by a circular pipe is solved for the Transverse Electromagnetic (TEM) mode by an efficient moment method in which the basis functions contain the exact behavior of the electric field at the right-angle wedges. It is shown that in most cases two expansion functions are sufficient for an excellent convergence of the solution. The dependance of the characteristic impedance upon the geometric parameters is calculated and presented graphically. We also present the convergence plots of the solution, compared to a conventional moment method, in which the basis functions do not contain the edge singularity of the electric field. It is shown that using the effective moment method can significantly improve the convergence rate of the solution.

A novel algorithm based on the domain-decomposition method for the full-wave analysis of 3-D electromagnetic problems

A novel technique based on the domain-decomposition method and frequency-domain finite-difference method is presented for the full-wave analysis of three-dimensional electromagnetic problems. In this method, the original domain is decomposed into sub-domains, Maxwell's equations are then solved in each sub-domain independently, and the global solution is achieved finally by an iterative procedure. It greatly reduces the computational complexity and the memory requirement compared with the conventional finite-difference method and method of moments, etc. To reduce CPU time, some techniques, such as the relaxation iterative algorithm, overlapped domain decomposition, and multimesh resolution are also investigated and adopted to accelerate the algorithm. The validity of this algorithm is verified by numerical examples, including the analysis of a multilayered aperture-coupled patch antenna, the scattering characteristic analysis of conducting pillars, and the S-parameters extraction of the air-bridge discontinuity.

Adaptive multiscale moment method (AMMM) for analysis of scattering from three-dimensional perfectly conducting structures

The adaptive multiscale moment method (AMMM) is presented for the analysis of scattering from three-dimensional (3D) perfectly conducting bodies. This algorithm employs the conventional moment method (MM) using the subsectional triangular patch basis functions and a special matrix transformation, which is derived from solving the Fredholm equation of the first kind by the multiscale technique. This methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time. The widely used triangular patch vector basis functions developed by Rao et al., (1982), are used for expansion and testing functions in the conventional MM. The objective here is to compress the unknowns in existing MM codes, which solves for the currents crossing the edges of the triangular patch basis functions. By use of a matrix transformation, the currents, source terms, and impedance matrix can be arranged in the form of different scales. From one scale to another scale, the initial guess for the solution can be predicted according to the properties of the multiscale technique. AMMM can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional MM. The basic difference between this methodology and the other wavelet-based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown. Two numerical results are presented, which demonstrate that the AMMM is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3D perfectly conducting bodies.

Efficient hybrid discrete Fourier transform-moment method for fast analysis of large rectangular arrays

A novel approach combining the moment method (MoM) and the discrete Fourier transform (DFT) is developed for the fast analysis of electromagnetic (EM) radiation/scattering from electrically large, finite, planar rectangular arrays. In particular, the unknown array distribution to be solved is represented in terms of the DFT within the MoM for a given array excitation. The proposed DFT-MoM approach for large arrays has the advantage that it can overcome the inefficiency of the conventional MoM approach by drastically reducing the number of unknowns. The latter is possible because only a relatively few DFT terms are significant in this DFT-MoM. A useful criterion to select significant DFT terms is described. Numerical results are presented to indicate the efficiency and accuracy of the DFT-MoM analysis for determining the array distribution and the radiation pattern of large rectangular arrays with uniform excitation. It is found that the efficiency and accuracy of the DFT-MoM increases dramatically with an increase in array size. Furthermore, the DFT representation employed within the MoM can provide an asymptotic closed form solution for both the near and far fields of the array, which can be described in the ray format of the uniform geometrical theory of diffraction (UTD).

Neural Computation of the MoM Matrix Elements for Planar Configurations

Summary In this paper a new artificial neural network (ANN) based model for the calculation of the method of moments (MoM) matrix elements is presented. Training sets that characterize the matrix elements are first constructed. These sets are then utilized to effectively train two radial basis function (RBF) neural networks to accurately estimate all the elements of the MoM matrix for any mesh used. The potential of the proposed approach is demonstrated in the case of a narrow microstrip line. The current distribution on the microstrip line produced by the trained RBF networks agrees very well with the exact distribution. In addition, the proposed ANN model is much faster than the conventional MoM procedure.

Wavelet matrix transform approach for electromagnetic scattering by a dielectric cylinder

The wavelet matrix transform approach, in combination with the conventional method of moments, is used to solve the problem of electromagnetic scattering by a dielectric cylinder. First, the problem is formulated in terms of the integral equation for the field of a harmonic source in the presence of the cylinder. Then the equation is discretized to obtain a dense system of linear equations by dividing the cylinder into square cells which are small enough so that the field intensity is nearly uniform in each cell. Finally, the wavelet matrix transform is applied to produce a sparse system of linear equations which is treated effectively by a sparse linear system solver. © 2001 Scripta Technica, Electr Eng Jpn, 137(3): 1–9, 2001

Wavelet matrix transform approach for electromagnetic scattering by a dielectric cylinder

AbstractThe wavelet matrix transform approach, in combination with the conventional method of moments, is used to solve the problem of electromagnetic scattering by a dielectric cylinder. First, the problem is formulated in terms of the integral equation for the field of a harmonic source in the presence of the cylinder. Then the equation is discretized to obtain a dense system of linear equations by dividing the cylinder into square cells which are small enough so that the field intensity is nearly uniform in each cell. Finally, the wavelet matrix transform is applied to produce a sparse system of linear equations which is treated effectively by a sparse linear system solver. © 2001 Scripta Technica, Electr Eng Jpn, 137(3): 1–9, 2001

Three‐dimensional weak‐form conjugate‐ and biconjugate‐gradient FFT methods for volume integral equations

A large-scale three-dimensional volume integral equation solution for electromagnetic radiation and scattering problems remains a great challenge in spite of many ongoing research efforts. The conventional method of moments, although accurate and flexible, is limited to small-scale problems because of its large requirement of computer memory and computation time. In this paper, we develop two fast methods, the weak-form conjugate- and biconjugate-gradient FFT methods, to solve the Fredholm integral equation of the second kind arising from Maxwell's equations in three dimensions. The weak form is a modified version of the Zwamborn–van den Berg formulation, where the singularity is circumvented by employing the weak-form discretization by rooftop vectorial basis and testing functions. Both weak- form CG–FFT and BCG–FFT methods require O(N log2 N) CPU time, and O(N) computer memory, but the latter converges three–six times faster than the CG–FFT method. We validate the numerical results by comparing them with analytical solutions to multilayer spherical media, and with other published results. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 29: 350–356, 2001.

Three‐dimensional weak‐form conjugate‐ and biconjugate‐gradient FFT methods for volume integral equations

AbstractA large‐scale three‐dimensional volume integral equation solution for electromagnetic radiation and scattering problems remains a great challenge in spite of many ongoing research efforts. The conventional method of moments, although accurate and flexible, is limited to small‐scale problems because of its large requirement of computer memory and computation time. In this paper, we develop two fast methods, the weak‐form conjugate‐ and biconjugate‐gradient FFT methods, to solve the Fredholm integral equation of the second kind arising from Maxwell's equations in three dimensions. The weak form is a modified version of the Zwamborn–van den Berg formulation, where the singularity is circumvented by employing the weak‐form discretization by rooftop vectorial basis and testing functions. Both weak‐ form CG–FFT and BCG–FFT methods require O(N log2 N) CPU time, and O(N) computer memory, but the latter converges three–six times faster than the CG–FFT method. We validate the numerical results by comparing them with analytical solutions to multilayer spherical media, and with other published results. © 2001 John Wiley &amp; Sons, Inc. Microwave Opt Technol Lett 29: 350–356, 2001.

Method of Moments Solutions for a Laminar Flow Aerosol Reactor Model

This study compares the capabilities of the conventional and quadrature methods of moments to describe condensational growth in a well-known laminar flow aerosol reactor model. Governing equations for energy, lower order radial moments of the particle size distribution, and vapor transport are written for a two-dimensional model using both approaches. Conventional and quadrature techniques to obtain closure of the moment equations are applied and compared. The conventional method requires certain approximations to the growth law. These include assumptions that the Kelvin effect is negligible and that the particle growth rate function is linear in particle size. In contrast, restrictive constraints of the conventional method of moments to obtain closure are avoided by applying the quadrature method. Numerical results for the laminar flow aerosol reactor model are obtained for a six moment formulation of the seed particle distribution to illustrate application of the quadrature method to continuous polydisperse distributions. We present the first calculations with the new quadrature method of moments to a two-dimensional aerosol transport and growth model.

Efficient method of moments formulation for large PEC scattering problems using asymptotic phasefront extraction (APE)

A method is introduced for reducing the exorbitant dependence on computer storage and solution time in the method of moments (MoM) for electrically large electromagnetic (EM) scattering problems. The unknown surface currents on large, smooth parts of a perfect electrical conductor (PEC) scatterer are expressed by an efficient set of linearly phased surface current basis functions. The phasefront characteristics of the surface currents are numerically extracted from known current samples obtained from a lower-frequency solution of the same configuration. The use of such basis functions for efficiently representing the surface currents that are constructed in terms of linearly phased currents at higher frequencies is justified by considering the form of the surface currents predicted by high-frequency asymptotic ray methods. The procedure for extracting the current phasefronts is purely numerical, obviating computationally expensive and nonrobust operations such as ray-tracing, and thus, is amenable to general purpose scattering codes. The new MoM with linearly phased basis functions is shown to greatly relieve the storage and solution time of the conventional MoM while accurately reproducing the induced surface currents and scattered fields of some chosen targets.

Fast spectral-domain method for acoustic scattering problems.

This paper presents the application of the conjugate-gradient (CG) fast Fourier transform (FFT) (CG-FFT) method and the CG nonuniform FFT (CG-NUFFT) method for the integral equation arising from acoustic scattering problems. In the conventional method of moments (MoM) for integral equations, the CPU and memory requirements are O(N3) and O(N2), respectively, where N is the number of unknowns in the problem. The CG-FFT method, which combines the iterative conjugate-gradient method with FFT, reduces these requirements to O(KN log2 N) and O(N), respectively, where K is the number of CG iterations. The CG-NUFFT method differs from the CG-FFT method in that it makes use of nonuniform FFT algorithms instead of FFT to allow a nonuniform discretization. Therefore, the CG-NUFFT method can solve the integral equation with both uniform and nonuniform grid while retaining the efficiency of the CG-FFT method. These two methods are applied to solve for two-dimensional constant density acoustic scattering problems. Numerical results demonstrate that they can solve much larger problems than the MoM.