Method Of Moments Impedance Matrix Research Articles

An Improved RCS Calculation Method for Power Lines Combining Characteristic Mode with SMWA

The radar cross-section (RCS) of power lines has an important significance for detection of the power lines. The method of moments (MoM) can calculate the RCS of power lines. However, the efficiency of the MoM is limited by the time-consuming computing process, as well as the expensive storage overhead. In order to enhance the efficiency and reduce the storage of the RCS calculation of power lines, we propose an RCS calculation method that combines the characteristic mode (CM) with a Sherman–Morrison–Woodbury formula-based algorithm (SMWA), which is referred to as a CM-SMWA. CMs are used as the basis functions for reducing the dimension of the MoM impedance matrix, and the SMWA is applied to directly solve the CMs-reduced matrix equation, which can reduce the computational time and storage. The numerical results demonstrate that the proposed method can obtain the RCS of power lines, with different incident angles and different polarizations, at a higher efficiency. At 35 GHz, compared with the conventional MoM, for a typical LGJ50-8 power line with a length of 0.276 m, the computation time is reduced by 62.4% and the memory occupation is reduced by 96.4%.

Exact Solution of New Magnetic Current Based Surface-Volume-Surface EFIE and Analysis of Its Spectral Properties

A novel magnetic current based Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE-M) is presented for the problem of scattering on homogeneous non-magnetic dielectric objects. The exact Galerkin Method of Moments (MoM) utilizing both the rotational and irrotational vector spherical harmonics as orthogonal basis and test functions according to the Helmholtz decomposition is implemented to solve SVS-EFIE-M analytically for the case of dielectric sphere excited by an electric dipole. The field throughout the sphere is evaluated and compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. This exact solution validates the rigorous nature of the new SVS-EFIE-M formulation. It also reveals the spectral properties of its individual operators, their products and their linear combination. The spectrum of the MoM impedance matrix is also obtained. It is shown that upon choosing basis and test functions in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}(S)$</tex-math></inline-formula> space and evaluating testing inner products in the same space, the MoM impedance matrix features bounded condition number with increasing order of discretization and/or at low frequencies. This makes the proposed SVS-EFIE-M formulation free of oversampling and low-frequency breakdowns giving it advantage both over its SVS-EFIE-J predecessor and classical double-source integral equations such as PMCHWT, Muller, and others suffering from this type of numerical instabilities inherent to their inferior spectral properties.

Analytic Solution of Surface–Volume–Surface Electric Field Integral Equation on Dielectric Sphere and Analysis of Its Spectral Properties

The exact solution of the surface–volume–surface electric field integral equation (SVS-EFIE) is presented for the problem of radiation in the vicinity of the homogeneous dielectric sphere. The solution is obtained via the Galerkin method of moments (MoM) utilizing rotational and irrotational complete sets of orthogonal vector spherical harmonics as basis and test functions according to the Helmholtz decomposition. In the case of radial or tangential electric dipole radiation, the electric field throughout the sphere evaluated via the analytic MoM solution of the SVS-EFIE is compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. The exact solution confirms and validates the rigorous nature of the SVS-EFIE formulation. It also reveals the spectral properties of its individual operators, their products, and their linear combination, as well as the spectrum of the MoM impedance matrix. It is shown that, upon choosing basis and test functions in the Sobolev space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{1/2}(S)$ </tex-math></inline-formula> and performing testing inner products evaluation in the space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{-1/2}(S)$ </tex-math></inline-formula> , with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> being the surface of the sphere, the MoM impedance matrix features bounded condition number with increasing order of discretization similar with analogous exact MoM solution of the surface EFIE (S-EFIE) on the perfectly electrically conducting (PEC) sphere.

Scattering From Quasi-Planar and Moderate Rough Surfaces: Efficient Method to Fill the EFIE-Galerkin MoM Impedance Matrix and to Solve the Linear System

First, an acceleration to compute the impedance matrix obtained from the electric field integral equation (EFIE) discretized by the Galerkin method of moments (MoM) with Rao–Wilton–Glisson basis functions is addressed. It is based on a far-field approximation and makes it possible to avoid looping on the source and observation triangles. Next, the impedance matrix is split into strong and weak interactions; the latter is compressed by expressing it from Toeplitz submatrices. Then, the linear system is efficiently solved by a bi-iterative scheme. For a given order, the least-squares QR (LSQR) algorithm is applied to solve the sparse linear system related to the strong interactions, while the matrix–vector products, related to the weak interactions, are accelerated by using FFTs. Numerical results of the field scattered by perfectly conducting paraboloid-shape object and Gaussian rough surface are shown.

Modeling of Scattering in Arbitrary-Shape Waveguide Using Broadband Green's Function With Higher Order Low Wavenumber Extractions

The broadband Green's function with low wavenumber extractions (BBGFL) consists of low wavenumber method of moments (MoM) solutions combined with modal expansions. BBGFL is suitable for broadband simulations because the modal functions are independent of frequencies. The modal expansion in this paper has sixth-order convergence. Using wavenumber derivatives of the surface current, it is shown that the higher order extraction requires only a single MoM impedance matrix on the left-hand side. For the same number of modes, the sixth-order convergence has significant improvement in accuracy over the ${\text{second}}$ - and ${\text{fourth}}$ -order convergence. Scattering in an arbitrary-shape waveguide is modeled by using a surface integral equation formulated with the sixth-order BBGFL. Using BBGFL, the unknowns are only on the surface of the scatterer. Results of simulations are illustrated and the results are in good agreement with those of direct MoM.

Gaussian Ring Basis Functions for the Analysis of Modulated Metasurface Antennas

This paper presents a novel type of basis functions, whose spectral- and space-domain properties can be exploited for the efficient method of moments (MoM) analysis of planar metasurface (MTS) antennas. The effect of the homogenized MTS is introduced in the integral equation as an impedance boundary condition (IBC). The proposed basis functions are shaped as Gaussian-type rings with small width and linear azimuthal phase. The analytical form of the spectrum of the Gaussian ring basis allows for a closed-form evaluation of the MoM impedance matrix’s entries. Moreover, these basis functions account for the global evolution of the surface current density in an effective manner, reducing the size of the MoM system of equations with respect to the case of subdomain basis functions. These features allow one to carry out a direct solution for problems with a diameter of up to 15 wavelengths in less than 1 min using a conventional laptop. The applicability on practical antennas has been tested through the full-wave analysis of MTS antennas implemented with small printed elements.

MLFMA-MoM for Solving the Scattering of Densely Packed Plasmonic Nanoparticle Assemblies

In this paper, we present a judicious combination of two renowned surface integral equation (SIE)-based techniques, namely, the multilevel fast multipole algorithm (MLFMA) and the method of moments (MoM), which synergize into a hybrid method that allows to address the analysis of large densely packed particle assemblies in an efficient and accurate way. This hybridization takes advantage of the repetition pattern inherent to these kinds of structures. Basically, the repeated self-coupling problems are squarely solved throughout the factorization of their MoM impedance matrix, whereas the cross-couplings through the surrounding medium are expedited via the MLFMA in the framework of a global iterative scheme. Some results are presented here to demonstrate the suitability of the proposed hybrid method to address large-scale nanoparticle arrays in the framework of nanoplasmonic biosensing applications.

On the Relationship Between Multiple-Scattering Macro Basis Functions and Krylov Subspace Iterative Methods

A mathematical link is provided between the Krylov subspace iterative approach based on the full orthogonalization method (FOM) and the macro basis functions (MBF) approach based on a multiple-scattering methodology. The link refers to the subspaces created by those methods as well as to the orthogonality conditions which they satisfy. Both approaches are applied to the same method-of-moments (MoM) system of equations that is preconditioned based on a closest-interaction rule, and where blocks of the MoM impedance matrix are compressed using a rank-revealing method. MBF and FOM approaches are compared numerically, with a special attention given to accuracy, for perfectly conducting objects, comprising an array of tapered-slot antennas, spheres and an aircraft. The respective advantages of both methods are briefly discussed and further prospects are given.

Accelerated Macro Basis Functions Analysis of Finite Printed Antenna Arrays Through 2D and 3D Multipole Expansions

An efficient technique is presented for the analysis of finite printed antenna arrays made of identical elements. It is based on a closed-form expression for the spatial-domain Green's function (GF) given as a finite sum of cylindrical waves (obtained through rational function fitting) plus one spherical wave. From there, a multipole expansion can be obtained for planar layered medium GFs. The macro basis function (MBF) technique is applied to the method of moments (MoM) solution of a mixed-potential integral equation, this reduces the size of the MoM impedance matrix and allows for a direct solution. However, the evaluation of the entries of this reduced matrix becomes the dominant contribution to the total computation time. The aforementioned multipole expansion is exploited to provide a fast construction of the reduced MoM matrix, whose elements are the reaction integrals between the MBFs considered to characterize the currents on the array element. The complexity of evaluating the interactions between MBFs is found to be dominated by the calculations related to the spherical wave term. Thus, taking into account the layered medium does not increase the order of the complexity with respect to a multipole-accelerated computation of reaction integrals in a homogeneous medium.

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.

Efficient Iterative Method of Moments—Physical Optics Hybrid Technique for Electrically Large Objects

The conventional hybrid method of moments (MoM)-physical optics (PO) technique provides a possible way to handle electrically large objects with affordable computer memory. However, its efficiency is not very good because the evaluation of the PO contribution to the MoM impedance matrix is very time-consuming. An efficient implementation of the iterative MoM-PO hybrid technique is presented in this paper to avoid the calculation of the PO contribution in matrix form. For electrically large objects, the proposed efficient iterative MoM-PO (EI-MoM-PO) method can greatly reduce the computational time and maintain the same or better accuracy with the same number of unknowns compared with the conventional MoM-PO method. Several examples of large-scale structures are analyzed by the EI-MoM-PO method, the conventional MoM-PO method, and the multilevel fast multipole algorithm. The excellent efficiency and accuracy are achieved by the proposed EI-MoM-PO technique.

Measurements and Analysis of a Probe-Fed Circularly Polarized Loop Antenna Printed on a Layered Dielectric Sphere

Radiation characteristics of a probe-fed open circular loop antenna are presented when the antenna is printed on a layered dielectric sphere. A smaller parasitic circular loop has been incorporated in the configuration to enhance the circular polarization (CP) bandwidth. The structure has been analyzed rigorously using the method of moments (MoM), where the sphere has been represented using the spherical dyadic Green's function (DGF). The computation efficiency has been enhanced using asymptotic extraction as well as closed form integrations to compute the MoM impedance matrix elements. The computed results are in close agreement with those obtained from measurements.

Nondestructive electromagnetic material characterization using a dual waveguide probe: A full wave solution

A nondestructive technique for determining the complex permittivity and permeability of a perfect electric conductor backed magnetic shielding material using a dual waveguide probe is presented. The dual waveguide probe allows for the simultaneous collection of reflection and transmission coefficients which distinguishes it from single probe methods common in the literature. Theoretical development of these coefficients, which is accomplished through a coupled magnetic field integral equations formulation using Love's equivalence principle and solved via the method of moments (MOM), is discussed. Evaluation of the resulting MOM impedance matrix elements is performed using complex plane integration leading to enhanced computational efficiency and physical insight. Comparison of the theoretical and measured reflection and transmission coefficients using a root finding algorithm leads to the desired permittivity and permeability. Measurement results of a magnetic shielding material are presented and compared to traditional methods for the purpose of validating the new technique. The probe's sensitivity to aperture alignment, sample thickness, and flange thickness is also investigated.

LOW RCS DIPOLE ARRAY SYNTHESIS BASED ON MOM-PSO HYBRID ALGORITHM

In this paper, a hybrid algorithm combined method of moments (MoM) with particle swarm optimization (PSO) is used to realize low radar cross section (RCS) array synthesis. Both the scattering factor and the radiation factor are involved in the proposed objective function to achieve the promising dipole array with a reduced RCS and satisfied radiation performance. To improve the optimization efficiency, radiation constraint conditions are adopted to avoid unnecessary scattering calculation. The symmetric matrix and block treatment are also used to fill the MoM impedance matrix. The optimization results show that the proposed algorithm is able to achieve RCS reduction of 5.5 dB for dipole array.

Fast Direct Solution of Method of Moments Linear System

A novel algorithm, the compressed block decomposition (CBD), is presented for highly accelerated direct (noniterative) method of moments (MoM) solution of electromagnetic scattering and radiation problems. The algorithm is based on a block-wise subdivision of the MoM impedance matrix. Impedance 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 preserving the compression. Examples are presented of typical problems in the range of 5000 to 70000 unknowns. The total execution time for the largest problem is about 1 h and 20 min for a single excitation vector. The main strength of the method is for problems with multiple excitation vectors (monostatic RCS computations) due to the negligible extra cost for each new excitation. For radiation and scattering problems in free space, the numerical complexity of the algorithm is shown to be N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and the storage requirements scale with N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> .

A closed‐form solution to the asymptotic part of the MOM impedance matrix and the MOM excitation vector for printed structures on planar grounded dielectric slabs

AbstractIn the spectral domain method of moments (MoM) solution of printed structures on planar grounded dielectric slabs, the infinite double integrals which appear in the asymptotic parts of the MoM impedance matrix and the MoM excitation vector elements, have been previously transformed to one‐dimensional finite integrals, which have been numerically computed using the highly specialized “International Mathematics and Statistics Library” subroutines. In this paper, these one‐dimensional integrals are evaluated in closed‐form, resulting in an improved efficiency and accuracy for the rigorous investigation of printed antennas and complex millimeter and microwave integrated circuits. Numerical results in the form of mutual impedance between two expansion functions and input impedance of various microstrip antennas are presented to assess the accuracy of these closed‐form expressions. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 882–886, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22274

The general 2-D moments via integral transform method for acoustic radiation and scattering

The moments via integral transform method (MITM) is a technique to analytically reduce the 2-D method of moments (MoM) impedance double integrals into single integrals. By using a special integral representation of the Green’s function, the impedance integral can be analytically simplified to a single integral in terms of transformed shape and weight functions. The reduced expression requires fewer computations and reduces the fill times of the MoM impedance matrix. Furthermore, the resulting integral is analytic for nearly arbitrary shape and weight function sets. The MITM technique is developed for mixed boundary conditions and predictions with basic shape and weight function sets are presented. Comparisons of accuracy and speed between MITM and brute force are presented. [Work sponsored by ONR and NSWCCD ILIR Board.]

An Efficient MoM Formulation for Finite-by-Infinite Arrays of Two-Dimensional Antennas Arranged in a Three-Dimensional Structure

In strongly coupled antenna arrays, the behavior of the elements near the edge can exhibit very large deviations with respect to the infinite periodic array solution. Insight into these truncation effects can be obtained by simulating finite-by-infinite arrays. This paper describes an efficient method-of-moments (MoM) scheme for simulating such arrays. This scheme is capable of handling arrays of two-dimensional metallic antennas placed perpendicularly to the array plane, in lossless media. This formulation relies on the free-space Green's function related to arrays infinite in one direction only, with linear phase excitation. After extraction of its singular part, this function is tabulated. Then, the elements of the MoM impedance matrix are computed in the space domain, with the help of a limited number of integration points. The computation time needed for establishing the MoM system of equations and for solving it is comparable to the time needed in the linear array case. An extension of this formulation is also developed to study infinite-by-infinite arrays and semi-infinite arrays. The latter solutions also provide standard current distributions, which are used to obtain a fast approximate solution of the MoM system of equations. Simulation results are shown for broadband arrays, made of tapered slot antennas consisting of metallic plates.

Some convergence considerations in space-domain moment-method analysis of a class of wide-band microstrip antennas

The method of moments (MoM) analysis of probe-fed rectangular microstrip patches requires the inclusion of a probe-to-patch attachment mode-expansion function when the substrate thickness d/spl ges/0.02/spl lambda/, where /spl lambda/ is the free-space wavelength. The results for the input impedance showed increased divergence with measurements when the attachment mode was omitted from the full-wave analysis. The attachment mode can be expressed as an infinite eigenfunction series that increases the fill time of the impedance matrix in an MoM analysis. In an earlier investigation, the infinite eigenfunction series was reduced to a residue series that required one or two terms compared to about 55 terms for the eigenfunction series. In this paper, the convergence properties of the eigenfunction and residue series are investigated in view of rigorous MoM analysis. The relative errors resulting from replacing the eigenfunction by the residue series for the attachment mode, are compared by numerically evaluating a class of two-dimensional (2-D) spatial integrals shown to be closely related to the elements of an MoM impedance matrix. Additionally, the computation times for the evaluation of these integrals for the two forms of the attachment mode-expansion function are also included. Based on the superior convergence properties of the residue series for the attachment mode-expansion function, it is mathematically justified that this form can readily be used for analytic reduction of the spatial, reaction integrals from four to 2-D forms. This feature allows further reduction of the fill time of the MoM impedance matrix, suggesting the possibility of developing an efficient space-domain MoM technique for modeling of wide-band microstrip antennas.

Novel closed-form expressions for MoM impedance matrix elements for numerical modeling of shielded passive components

Recent advances in multi-layer insulating substrates have necessitated the development of efficient numerical modeling tools capable of handling complex three dimensional structures. In this paper, a closed-form solution is presented for the method of moments (MoM) impedance matrix elements obtained in the integral equation modeling of shielded passive circuits. This closed-form solution is written in terms of rapidly-computable special functions and has proven to be two orders of magnitude faster than direct numerical integration.