Method Of Moments Matrix Research Articles

Scalable and Fast Characteristic Mode Analysis using GPUs

Characteristic mode analysis (CMA) is used in the design and analysis of a wide range of electromagnetic devices such as antennas and nanostructures. The implementation of CMA involves the evaluation of a large method of moments (MoM) complex impedance matrix at every frequency. In this work, we use different open-source software for the GPU acceleration of the CMA. This open-source software comprises a wide range of computer science numerical and machine learning libraries not typically used for electromagnetic applications. Specifically, this paper shows how these different Python-based libraries can optimize the computational time of the matrix operations that compose the CMA algorithm. Based on our computational experiments and optimizations, we propose an approach using a GPU platform that is able to achieve up to 16×× and 26×× speedup for the CMA processing of a single 15k ×× 15k MoM matrix of a perfect electric conductor scatterer and a single 30k ×× 30k MoM matrix of a dielectric scatterer, respectively. In addition to improving the processing speed of CMA, our approach provided the same accuracy as independent CMA simulations. The speedup, efficiency, and accuracy of our CMA implementation will enable the analysis of electromagnetic systems much larger than what was previously possible at a fraction of the computational time.

A Butterfly-Accelerated Volume Integral Equation Solver for Large-Scale Electromagnetic Analysis

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.

Fast Direct Solution of 2D Scalar Volume Integral Equation via Tensor Train Decomposition for Scatterers of Arbitrary Shape

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.

Fast Computation by MLFMM-FFT with NURBS in Large Volumetric Dielectric Structures

A refinement for the computation of the rigorous part of the multi-level fast multipole method (MLFMM) of analyzing volumetric objects is presented. A scheme based on the fast Fourier technique (FFT) is proposed with the objective of reducing the computational resources required to accurately analyze large homogeneous and non-homogeneous dielectric volumes. In order to reduce the memory requirements, the storage of the near-field terms of the method of moments (MoM) matrix is performed only for the positions corresponding to a parallelepiped with the size of the level 1 block of the MLFMM, computed with the vacuum permittivity, taking advantage of the Toeplitz symmetry present in regular hexahedral meshes. The FFT avoids applying the near-field MoM matrix in the iterative solution process. The application of this approach results in huge improvements in terms of memory usage, but also a speeds up the iterative solution process because the use of three-dimensional (3D) FFTs is very efficient for computing convolutions when the number of unknowns of the problems becomes very large as happens in volumetric problems. We also propose a new approach for the numerical treatment of the transition of the dielectric permittivity between different dielectrics or between a dielectric and a free space. To validate the computation technique, the radar cross section (RCS) of several dielectric bodies is computed using the classical MLFMM approach and it is compared with the presented FFT-based-MLFMM solution. The results demonstrate that the efficient memory and computation time usage of the proposed approach.

Direct Characterization of Grounding System Wide-Band Input Impedance

In this article, direct characterization of the wide-band input impedance of grounding system buried in multilayer soil is presented. The problem is formulated by mixed-potential integral equations solved by the method of moments (MoM). The method accommodates both delta-gap voltage and impressed-current excitations. The input impedance is then directly obtained from the MoM matrix. The proposed method does not rely on the integration of the electric field on the ground surface. Hence, the calculated input impedance is unique and path-independent. The accuracy of the proposed method is evaluated for the case of vertical ground electrodes buried in homogeneous soil. The cases of vertical grounding electrode buried in multilayer soil as well as square and circle-plates buried in homogeneous soil are studied. It is shown that for plate-shape grounding systems, the total surface area determines the grounding system input impedance and the effect of plate geometry is negligible.

Comparison between Specialized Quadrature Rules for Method of Moments with NURBS Modelling Applied to Periodic Multilayer Structures

A comparison between Ma-Rokhlin-Wandzura (MRW) and double exponential (DE) quadrature rules for numerical integration of method of moments (MoM) matrix entries with singular behavior is presented for multilayer periodic structures. Non Uniform Rational B-Splines (NURBS) modelling of the layout surfaces is implemented to provide high-order description of the geometry. The comparison is carried out in order to show that quadrature rule is more suitable for MoM matrix computation in terms of sampling, accuracy of computation of MoM matrix, and CPU time consumption. The comparison of CPU time consumption shows that the numerical integration with MRW samples is roughly 15 times faster than that numerical integration using DE samples for results with similar accuracies. These promising results encourage to carry out a comparison with results obtained in previous works where a specialized approach for the specific analysis of split rings geometries was carried out. This previous approach uses spectral MoM version with specific entire domain basis function with edge singularities defined on split ring geometry. Thus, the previous approach provides accurate results with low CPU time consumption to be compared. The comparison shows that CPU time consumption obtained by MRW samples is similar to the CPU time consumption required by the previous work of specific analysis of split rings geometries. The fact that similar CPU time consumptions are obtained by MRW quadrature rules for modelling of general planar geometries and by the specialized approach for split ring geometry provides an assessment for the usage of the MRW quadrature rules and NURBS modelling. This fact provides an efficient tool for analysis of reflectarray elements with general planar layout geometries, which is suitable for reflectarray designs under local periodicity assumption where a huge number of periodic multilayer structures have to be analyzed.

Singularity Cancellation for Accurate MoM Analysis of Periodic Planar Structures in Layered Media

One of the singularity cancellation schemes proposed in the literature is applied to calculate singular integrals arising in the method of moments (MoM) analysis of 2-D periodic planar structures in multilayered media. Discrete complex image method is utilized for the accurate approximation of Green's function which also makes possible the application of the Ewald transformation for the efficient computation of the series associated with the periodic structure. This approximation and transformation modifies the kernel of the integral appearing in the computation of the MoM matrix entries. Formulation of the problem for this new kernel is presented together with numerical results demonstrating the achieved accuracy.

Efficient Solution of Electromagnetic Scattering Problems Using Multilevel Adaptive Cross Approximation and LU Factorization

The multilevel adaptive cross approximation (MLACA), previously described in the literature, extends the single-level ACA with a recursive multilevel algorithm that significantly improves compression of off-diagonal matrix blocks resulting from electromagnetic integral equations (IE) discretized via the method of moments (MoM). In this article, the MLACA approach is extended and applied to a direct solution of the MoM matrix system via LU factorization. It will be shown through numerical experiments that the off-diagonal LU blocks are also compressible using MLACA, yielding a compression rate superior to the single-level ACA and a memory complexity of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4/3</sup> log N). In addition, the MLACA LU block updates are performed in rank-reduced form, yielding a very efficient software implementation via a Level 3 BLAS optimized for the CPU or GPU.

Sparsity-Aware Precorrected Tensor Train Algorithm for Fast Solution of 2-D Scattering Problems and Current Flow Modeling on Unstructured Meshes

Acceleration of the method of moments (MoM) solution of the volume integral equation (VIE) on unstructured meshes is performed using a precorrected tensor train (P-TT) algorithm. The elements of the MoM’s unstructured mesh are projected onto a regular Cartesian grid. This enables representation of the MoM matrix as the Toeplitz matrix of point-to-point interactions pre- and post-multiplied by sparse matrices projecting MoM’s basis and testing functions on the Cartesian grid. The Toeplitz matrix is subsequently cast into the form of a multidimensional tensor. The latter is decomposed into the product of smaller dimensional matrices also known as tensor train (TT). TT allows to store Toeplitz matrix in $O(\log N)$ memory for VIE with the Laplace kernel and in $O(N \log N)$ memory for VIE with the Helmholtz kernel. Unlike the FFT-based fast algorithms, the P-TT method enables further memory reduction due to the sparsity of sources on the Cartesian grid. This sparsity pattern can be accounted for in construction of TT cores. It further reduces memory use as well as CPU time required for the multiplication of the Toeplitz matrix with a vector. It is shown that upon sufficient sparsity in representation of the sources on the grid, the P-TT algorithm can outperform both in CPU time and memory its FFT-based counterpart known as the precorrected FFT algorithm.

A novel CEM technique for modeling electromagnetic scattering from metasurfaces

AbstractThis paper presents a novel computational electromagnetics (CEM) technique, which hybridizes the periodic finite element method (FEM) with the method of moments (MoM), for efficient numerical modeling of electromagnetic scattering from metasurfaces consisting of truncated periodic or locally varying quasi‐periodic array of structures. Based on the quasi‐periodic nature of metasurfaces, the periodic FEM is employed to generate high‐level macro basis functions (MBFs). Following that, a reduced MoM matrix is formed by using the MBFs with unknown coefficients belonging to each cell of the periodic array. The proposed hybrid method reduces the computational load significantly when compared with conventional numerical methods, especially for electrically large truncated metasurfaces involving arbitrarily inhomogeneous unit cells. Various numerical results are presented, and the accuracy and performance of the proposed method are tested against commercial codes, as well as an in‐house FEM code.

Method of Moments Simulation of Modulated Metasurface Antennas With a Set of Orthogonal Entire-Domain Basis Functions

A family of orthogonal and entire-domain basis functions (named Fourier-Bessel) is proposed for the analysis of circular modulated metasurface (MTS) antennas. In the structures at hand, the MTS is accounted for in the electric field integral equation (EFIE) as a sheet transition impedance boundary condition on the top of a grounded dielectric slab. The closed-form Hankel transform of the Fourier-Bessel basis functions (FBBFs) allows one to use a spectral domain formulation in the method-of-moments (MoM) solution of the EFIE. Moreover, these basis functions are fully orthogonal, which implies that they are able to represent the global evolution of the current distribution in a compact form. FBBFs also present a better filtering capability of their spectrum compared to other well-known orthogonal families such as the Zernike functions. The obtained MoM matrix is sparse and compact, and it is thus very well-conditioned and can be efficiently computed and inverted. The numerical results based on the proposed decomposition are presented and compared with those based on the use of the Gaussian ring basis functions and with the full-wave analysis of MTS antennas implemented with small printed elements. A very good agreement is observed.

Entire Domain Basis Function With Accurate Edge Condition for Rectangular Microstrip Antennas

This letter presents the study of a segmented entire domain basis function suitable for fast analysis of rectangular microstrip antennas with the method of moments (MoM). In contrast to other basis functions, whereby the edge condition is valid only for narrow lines, the proposed basis function can be applied to microstrip lines and antennas with arbitrary width. For validation, a probe-fed rectangular microstrip antenna was modeled and the results were validated with measurements. The deviation of the calculated resonance frequency from the measured value could be reduced to less than 0.1%. This has been obtained with the use of only two expansion modes along each direction for the construction of the MoM matrices.

Implementation of loop co‐tree basis functions for EFIE formulation at low frequency

In this study, a method of moments (MoM) of electric field integral equation (EFIE) formulation using the loop co-tree basis functions has been implemented, analysed, and applied to the electromagnetic solution of perfect electric conducting objects at low frequency. The EFIE with Rao-Wilton-Glisson (RWG) basis functions suffer from low-frequency instability. Even for an appropriate frequency, the EFIE with RWG basis functions is ill-posed. However, the matrix form of the EFIE with loop co-tree basis functions is well-conditioned when combined with an iterative solver such as a type of Krylov subspace methods. The approach is shown to be very effective for resolving both the low-frequency instability and the ill-condition of the MoM matrix with a generalised minimal residual method.

Computation of EFIE Matrix Entries With Singular Basis Functions

The use of singular basis functions enhances the convergence of method-of-moments (MoM) solutions for structures containing edges. While standard algorithms for computing the MoM matrix entries treat Green’s function singularities, these are not well-suited for integrating the singular basis functions: conventional quadrature routines exhibit slow convergence and may produce inaccurate results. In this paper, new algorithms are proposed for handling the combination of edge singularities and Green’s function singularities on quadrilateral cells.

Harmonic Impedance of Grounding Electrodes Buried in a Horizontally Stratified Multilayer Ground: A Full-Wave Approach

The effect of multilayer stratified ground on the harmonic impedance of grounding systems is evaluated. To this aim, the mixed-potential integral equation is used. The method is general as it allows taking into account 1) stratified soil structure with any number of layers and 2) grounding electrodes with arbitrary shapes. Moreover, the method has no limitation in terms of the maximum working frequency. The impressed-current model is used as the excitation and the harmonic impedance is directly obtained from the method of moments matrix. This is in contrast with the conventional methods where the harmonic impedance is calculated from the integration of the electric field at ground surface. The cases of horizontal and vertical electrodes buried in soils with different layers are analyzed. The accuracy of the proposed approach is validated by comparing the results with those obtained by independent numerical methods.

Notice of Retraction: Direct Characterization of Grounding System Wide-Band Input Impedance

In this article, direct characterization of the wide-band input impedance of grounding system buried in multilayer soil is presented. The problem is formulated by mixed-potential integral equations solved by the method of moments (MoM). The method accommodates both delta-gap voltage and impressed-current excitations. The input impedance is then directly obtained from the MoM matrix. The proposed method does not rely on the integration of the electric field on the ground surface. Hence, the calculated input impedance is unique and path-independent. The accuracy of the proposed method is evaluated for the case of vertical ground electrodes buried in homogeneous soil. The cases of vertical grounding electrode buried in multilayer soil as well as square and circle-plates buried in homogeneous soil are studied. It is shown that for plate-shape grounding systems, the total surface area determines the grounding system input impedance and the effect of plate geometry is negligible.

Use of Computational Electromagnetics to Enhance the Accuracy and Efficiency of Antenna Pattern Measurements

The objective of this paper is to illustrate that computational electromagnetics can be used very effectively to improve the accuracy and efficiency of antenna pattern measurements. This is accomplished by moving a single probe over the measurement plane to generate the enhanced accuracy in planar near-field (NF) to far-field (FF) transformation than over the classical Fourier-based modal expansion methods. It is also illustrated that this method provides reliable results for cases when the conventional method fails, particularly, for the case, when the actual source plane and the measurement plane are approximately equal in size. Also, in this approach, there is no need to incorporate probe correction unlike in the existing approaches. In addition, a methodology can be designed where one can use an array of probes instead of moving a single probe over the measurement plane, thus improving the accuracy and efficiency of the measurements. In the use of the probe array, there is also no need to perform the probe correction. This proposed novel methodology is accomplished by solving for the equivalent magnetic current over a plane near the original source antenna under test, and then, employing the method-of-moments (MoM) approach to solve for the equivalent magnetic currents on this fictitious surface. For this proposed methodology, even though, there is no need to satisfy the Nyquist sampling criteria in the measurement plane, a super resolution can be achieved in the solution of the equivalent magnetic current so that one can predict the status of the operation of each antenna elements in an array. Also, the presence of evanescent fields in the measurements does not make this methodology unstable unlike in the conventional Fourier-based techniques. The two components of the equivalent magnetic currents can be solved independently from the two measured components of the electric fields by solving the resultant MoM matrix equation very efficiently and accurately by using the iterative conjugate gradient method enhanced through the incorporation of the fast Fourier transform techniques. Sample numerical results are presented to illustrate the potential of a novel planar NF to FF transformation applied to the planar NF measurement technique.

Optimized Periodic MoM for the Analysis and Design of Dual Polarization Multilayered Reflectarray Antennas Made of Dipoles

A hybrid version of the Method of Moments (MoM) is applied to the analysis of the scattering of plane waves by periodic multilayered structures containing dipoles at two metallization levels. The MoM matrix entries involving basis functions (BFs) at different metallization levels are computed in the spectral domain as double infinite summations with fast exponential convergence. The MoM matrix entries involving BFs at the same metallization level are computed in the spatial domain as double integrals, which require low-order quadrature rules. The integrands are cross correlations between BFs times multilayered periodic Green’s functions (MPGFs). The cross correlations between BFs are obtained in terms of elliptic integrals of first and second kind. Also, the MPGFs are accurately interpolated in 4-D in terms of both the spatial variables and the angles of incidence. The hybrid MoM proposed is used in the design of dual polarization reflectarray antennas under the local periodicity assumption. Thanks to the 4-D interpolation of the MPGFs, which minimizes the total number of MPGFs that have to be computed per reflectarray element, the proposed hybrid MoM is shown to be around 15 times faster than the standard spectral domain MoM in the design of the antennas.

Load-Balanced Fortran-Based Out-of-GPU Memory Implementation of the Method of Moments

In this letter, an efficient Fortran-based out-ofgraphics processing unit (GPU) memory implementation of the method of moments (MoM) on heterogeneous low-cost single CPU/GPU computing platform is presented. The MoM solution process is based on the frequency-domain integral equation formulation and employs a wire-grid approach to evaluate the radiation pattern of the glide path antenna system. To handle processor heterogeneity and balance the workload between the host and the device, a novel task scheduling strategy, both for the matrix assembly phase and the solution of the MoM matrix equation, is proposed. Using the CUDA-capable GTX 680 GPU from NVIDIA, a speedup ratio of about 1.44× is reported when compared to the classical hybrid CPU/GPU approach, and about 2.21× when compared to the multicore CPU implementation.

The CBFM-Enhanced Jacobi Method for Efficient Finite Antenna Array Analysis

An enhancement to the iterative Jacobi technique with the characteristic basis function method is presented. The resulting method is intended for efficient method of moments (MoM)-based analysis of large, disjoint finite antenna arrays. The enhancement improves the convergence rate of the Jacobi method by better accounting for mutual coupling between array elements. This involves solving a small, global problem at each iterative step, using macrobasis functions that are iteratively generated. Results are presented, and it is found that the proposed method can recover fast convergence in cases where the Jacobi method diverges or converges slowly. The proposed method also converges significantly faster than a conventional, preconditioned, iterative solution of the MoM matrix equation.