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We present high-order, finite element–based Second Moment Methods (SMMs) for solving radiation transport problems in two spatial dimensions. We leverage the close connection between the Variable Eddington Factor (VEF) method and SMM to convert existing discretizations of the VEF moment system to discretizations of the SMM moment system. The moment discretizations are coupled to a high-order Discontinuous Galerkin discretization of the SN transport equations. We show that the resulting methods achieve high-order accuracy on high-order (curved) meshes, preserve the thick diffusion limit, and are effective on a challenging multimaterial problem both in outer fixed-point iterations and in inner preconditioned iterative solver iterations for the discrete moment systems. We also present parallel scaling results and provide direct comparisons to the VEF algorithms from which the SMM algorithms were derived. [ABSTRACT FROM AUTHOR]

A technique for the reduction in the CPU-time in the analysis of electromagnetic problems using the Characteristic Basis Function Method (CBFM) is presented here, allowing for analysis of electrically large cases where an iterative solution process cannot be avoided. This technique is based on the use of the Adaptive Cross Approximation (ACA) for the fast computation of the coupling matrix between CBFs belonging to adjacent blocks, as well as the Multilevel Fast Multipole Method (MLFMM) for the computation of matrix−vector products in the solution of the full system. This combination allows for a noticeable reduction in the computational resources during the analysis of electrically large and complex scenarios while maintaining a very good degree of accuracy. A number of test cases serve to validate the presented approach in terms of accuracy, memory and CPU-time compared with conventional techniques. [ABSTRACT FROM AUTHOR]

We present an efficient matrix-free point spread function (PSF) method for approx-imating operators that have locally supported nonnegative integral kernels. The PSF-based method computes impulse responses of the operator at scattered points and interpolates these impulse responses to approximate entries of the integral kernel. To compute impulse responses efficiently, we apply the operator to Dirac combs associated with batches of point sources, which are chosen by solving an ellipsoid packing problem. The ability to rapidly evaluate kernel entries allows us to construct a hierarchical matrix (H-matrix) approximation of the operator. Further matrix computations are then performed with fast H-matrix methods. This end-to-end procedure is illustrated on a blur problem. We demonstrate the PSF-based method's effectiveness by using it to build preconditioners for the Hessian operator arising in two inverse problems governed by PDEs: inversion for the basal friction coefficient in an ice sheet flow problem and for the initial condition in an advective-diffusive transport problem. While for many ill-posed inverse problems the Hessian of the data misfit term exhibits a low-rank structure, and hence a low-rank approximation is suitable, for many problems of practical interest, the numerical rank of the Hessian is still large. The Hessian impulse responses, on the other hand, typically become more local as the numerical rank increases, which benefits the PSF-based method. Numerical results reveal that the preconditioner clusters the spectrum of the preconditioned Hessian near one, yielding roughly 5x-10x reductions in the required number of PDE solves, as compared to classical regularization-based preconditioning and no preconditioning. We also present a comprehensive numerical study for the influence of various parameters (that control the shape of the impulse responses and the rank of the Hessian) on the effectiveness of the advection-diffusion Hessian approximation. The results show that the PSF-based method is able to form good approximations of high-rank Hessians using only a small number of operator applications. [ABSTRACT FROM AUTHOR]

A technique for the reduction in the CPU-time in the analysis of electromagnetic problems using the Characteristic Basis Function Method (CBFM) is presented here, allowing for analysis of electrically large cases where an iterative solution process cannot be avoided. This technique is based on the use of the Adaptive Cross Approximation (ACA) for the fast computation of the coupling matrix between CBFs belonging to adjacent blocks, as well as the Multilevel Fast Multipole Method (MLFMM) for the computation of matrix−vector products in the solution of the full system. This combination allows for a noticeable reduction in the computational resources during the analysis of electrically large and complex scenarios while maintaining a very good degree of accuracy. A number of test cases serve to validate the presented approach in terms of accuracy, memory and CPU-time compared with conventional techniques.

The generalised gamma distribution (GGD) is one of the most widely used statistical distributions used extensively in several scientific and engineering application areas due to its high adaptability with the normal and exponential, lognormal distributions, among others. However, the estimation of the unknown parameters of the model is a challenging task. Many algorithms were developed for parameter estimation, but none can find the best solution. In this study, a simulated annealing (SA) algorithm is proposed for the assessment of effectiveness in determining the parameters for the GDD using modified internal rate of return (MIRR) data extracted from the financial report of the publicly traded Malaysian property companies for long term investment periods (2010-2019). The performance of the SA is compared to the moment method (MM) based on mean absolute error (MAE) and root mean squares errors (RMSE) based on the MIRR data set. The performance of this study reveals that the SA algorithm has a better estimate with the increases in sample size (long-term investment periods) compared to MM, which reveals a better estimate with a small sample size (short-time investment periods). The results show that the SA algorithm approach provides better estimates for GGD parameters based on the MIRR data set for the long-term investment period. [ABSTRACT FROM AUTHOR]

How is trait diversity in a community apportioned between and within coevolving species? Disruptive selection may result in either a few species with large intraspecific trait variation (ITV) or many species with different mean traits but little ITV. Similar questions arise in spatially structured communities: heterogeneous environments could result in either a few species that exhibit local adaptation or many species with different mean traits but little local adaptation. To date, theory has been well-equipped to either include ITV or to dynamically determine the number of coexisting species, but not both. Here, we devise a theoretical framework that combines these facets and apply it to the above questions of how trait variation is apportioned within and between species in unstructured and structured populations, using two simple models of Lotka-Volterra competition. For unstructured communities, we find that as the breadth of the resource spectrum increases, ITV goes from being unimportant to crucial for characterizing the community. For spatially structured communities on two patches, we find no local adaptation, symmetric local adaptation, or asymmetric local adaptation, depending on how much the patches differ. Our framework provides a general approach to incorporate ITV in models of eco-evolutionary community assembly. [ABSTRACT FROM AUTHOR]

Supersaturation as the thermodynamic driving force of particulate precipitation couples the mixing of solute compounds with the solid formation consisting of nucleation and molecular growth. Because the process spans many orders of magnitude in space and time, gathering spatio-temporal information by experiments is hardly achievable. Numerical simulation of the nonlinear, non-local balance equations for mass, momentum and dispersed phase provide detailed solution information in space and time. We here introduce a novel, highly accurate and very efficient numerical formulation of the population balance equations for the dispersed phase coupled to the concentration and velocity field. We reformulate the problem in terms of a fixed-point equation solving the population balance equation along Lagrangian paths with the recently developed exact Method of Moments (eMoM, Pflug et al., 2020) coupled to the concentration field (Eulerian frame). The analytical reformulation provides the information about the full particle size distribution while the computational effort remains in the order of classical moment methods and is much less than numerical schemes approximating the full particle size distribution such as finite volume methods. Furthermore, we explore the difference between an Euler-Euler and Euler-Lagrange coupling considering moderate to large P é clet numbers for the solutes and particles. It turns out that the mean particle size as a solution of the population balance is much more sensitive to the numerical discretization of the advection term than the approximation of the number density function. Based on this finding, we eventually explore the impact of particle diffusion on the solid formation process and give guidelines, when particle diffusion should be taken into account. • Reformulation of coupled population balance equation with CFD as fixed-point problem. • Novel numerical scheme for coupling CFD and PBE. • Full particle size distribution based on evolution of solute concentration. • Comparison of Euler-Euler and Euler-Lagrange coupling for precipitation simulation. • Influence of particle diffusion on solid formation process. [ABSTRACT FROM AUTHOR]

The approach outlined in this paper takes advantage of the existence of stationary states and similarity solutions to reduce a one-dimensional population balance model with particle volume as the internal dimension from a partial integro-differential equation to a single ordinary differential equation for the evolution of the mean size. The mechanisms of accretional growth, collisional growth and breakage, either individually or in all possible combinations, are considered. Power-law rate kernels are employed and a self-similar fragment distribution is assumed for the breakage term. When expressed in terms of the evolution of the scaled number-mean particle volume, the model parameters consist of kernel orders and the characteristic times associated with each mechanism. Equations for the scaled moments of the stationary states and similarity solutions are exhibited for all possible mechanism combinations as are equations for the trajectories of scaled number-mean particle volume. Stability of the stationary states and similarity solutions is addressed in an appendix. It should be clear that when this level of simplification is possible, the subject of population balance modeling becomes considerably easier to teach while its use becomes much easier to promote in industry. The use of this simplified approach for extraction of model parameters from experiment is demonstrated in an example, followed by a brief discussion of using the results for scale up. To close out the paper, the same approach, but now in two internal dimensions (particle volume and surface area), is used to demonstrate the capability of the model to unify a set of operating conditions for carbon black manufacture across multiple grades and two different types of synthesis reactors. This is done by extracting the characteristic coalescence time at multiple temperatures and correlating the results in an Arrhenius plot. The model matches known values of primary particle size, and predicts a key indicator of the product's fractal aggregate structure, namely the number of primary particles per aggregate. The model results are then pushed to yield insight into the nature of the possible collision and coalescence mechanisms. [Display omitted] • Similarity solutions/stationary states enable reduction of 1-D models to a single ODE for mean size evolution. • Distribution shape is expressed in the form of scaled moments that are constant under these conditions. • Certain of these combine with the rate kernels to identify characteristic timescales. • These simplifications enable extraction of rate kernel orders and timescales from data. • These 1-D model simplifications are extended to a 2-D collision-coalescence example. [ABSTRACT FROM AUTHOR]

This article addresses the problem of efficiently and accurately evaluating the singular or near-singular double volume (6-D) reaction integrals fundamental to the solution of volume integral equations (VIEs) using the method of moments (MoM). VIE techniques based on the MoM are used to obtain accurate solutions for electromagnetic (EM) problems of all types, and the accurate computation of the singular and near-singular reaction integrals that occur in the resulting system matrices is crucial for accurate EM modeling. Here we propose a scheme to treat, as a whole, the 6-D reaction integrals appearing in the MoM. The divergence theorem is directly applied twice in the physical space domain, not a transformed domain, thus eliminating restrictions to well-shaped elements. With appropriate integration reordering, the 6-D volume integrals are expressed as two radial integrals followed by two surface integrals over source and observation domain boundaries. We further smooth the integral by removing the static form of the kernel and evaluating it separately using a semi-analytical approach. The method is numerically validated for static and dynamic kernels arising in the electric field VIEs (i.e., for kernels having $1/R$ singularities) and for linear basis functions. [ABSTRACT FROM AUTHOR]

The acoustic scattering by highly inhomogeneous objects is analyzed by a method-of-moment solver for the volume integral equation. To enable the treatment of acoustically large scatterers of various topologies, the iterative numerical solution of the resulting system is accelerated via a kernel independent algebraic compression scheme: blocks of the hierarchically partitioned moment stiffness matrix are expressed in butterfly form that, for volume problems, scales favorably compared to the popular low-rank approximation. A detailed description of the algorithm, as implemented in this work, is provided. Validations of the numerical formulation, parameter tuning, and performance study of the fast method for acoustically large objects are presented, in various settings and for a range of examples, representative of biomedical and oceanographic applications. [ABSTRACT FROM AUTHOR]

In this paper the Finite Integration Technique (FIT) hybridized with the Method-of-Moments (MoM) is used to find resonance frequencies and quality factors of open cylindrical dielectric resonators (CDRs). The technique is based on the previously developed formulation for scattering problems, with the application of root searching algorithm to find zeros of the final matrix determinant in the complex frequency plane. The method is validated by comparison of obtained results with the results of simulations done using other methods, and with measurement data found in the literature.

Static potential integrals for constant and linear sources on triangles are derived in a straightforward way. The new representations, as presented, are robust with respect to machine evaluation in important limiting cases. The potential integrals comprise up to six functions, each dependent on the relative position and orientation (with respect to an observation point) of a vertex and edge, respectively, of the source triangle. Gradients of the potentials are derived by differentiation, thus preserving relations between the representations. Each such vertex function reveals any anomalous functional behavior near its associated vertex or edge, which is useful information for devising test integral schemes. Potential plots in the source plane of an equilateral triangle illustrate such behavior, as do similar plots for each vertex function and gradient components near their associated edge and vertex.

In this paper, we explore the excitation of magnetic current surface waves in truncated periodic arrays of slots in a conducting screen. A specialized Method of Moments (MoM) implementation is presented, which makes it possible to efficiently solve the scattering problem involving truncated arrays of several thousands of slots. By making use of the dispersion diagrams of surface waves propagating along infinite periodic arrays of slots, we are able to explain the absence of magnetic current surface waves in the arrays at frequencies in the neighborhood of the transmission peak associated to the slots natural resonances (length roughly equal to half the wavelength), while they are present when the arrays are excited under extraordinary transmission (EOT) conditions. In order to experimentally check this different behavior, an aluminium plate periodically perforated with slots has been fabricated and fed by means of a pyramidal horn, and the electric field behind the plate has been measured with a planar near-field system at a few centimeters from the plate. Our experimental results and MoM simulations agree, demonstrating the presence of a standing wave pattern of magnetic current surface waves at the EOT frequency, and the absence of surface waves at the slots natural resonant frequency.

This communication presents a formulation of the spectral-domain Method of Moments (spectral MoM) for the analysis of periodic multilayered structures with a number of metallized interfaces alternating patches and apertures. Edge singularity entire-domain basis functions are used to model the electric currents on the patches and the tangential electric fields in the apertures. The formulation is especially suitable to characterize high-order bandpass frequency selective surfaces (FSSs) made up of nonresonant elements. Also in this communication, the nonuniform fast Fourier transform (NUFFT) is used in the spectral MoM to cover the case of unit cells in skewed grids, which enables the analysis of nonrectangular periodic lattices with tightly packed elements. Numerical examples of the design of third-order and fifth-order bandpass FSSs are presented. The FSSs are made up of hexagonal patches and apertures arranged in an equilateral triangular lattice. Results are cross-checked against commercial software CST Microwave Studio (CST MWS), and excellent agreement is found, our in-house software being more than one order of magnitude faster than CST MWS. This CPU time saving makes the proposed formulation very convenient for full-wave optimization and design. [ABSTRACT FROM AUTHOR]

This article presents a technique for the computation of the monostatic radar cross section of complex objects based on a combination of macro-basis functions (MBFs) and the multilevel fast multipole algorithm. An initial pool of excitation-independent MBFs is first obtained, generating the corresponding reduced coupling matrix as well as the multipole data. For each excitation, ray-tracing processing is performed, extracting a number of critical points that are used to obtain a mask that allows to dynamically select the basis functions to be considered in the analysis. This strategy allows a noticeable reduction in the size of the problems with minimal CPU-time preprocessing overhead. [ABSTRACT FROM AUTHOR]

The development of time-domain integral equation solvers with robust late-time stability properties has been a long-standing quest. Among the many methods that have been investigated, those leveraging quasi-exact integration techniques appear to be most successful. This article presents stable and accurate marching-on-in-time (MOT) solvers for time-domain electric, magnetic, and combined field integral equations (EFIE, MFIE, and CFIE) based on quasi-exact integration techniques. The novel MOT solvers exhibit excellent stability while yielding highly accurate results, as demonstrated by various numerical examples. In addition, the solvers’ excellent stability and accuracy properties are used to examine spurious modes encountered when time-domain integral equations are applied to closed surfaces. It is demonstrated that MOT solutions to the time domain EFIE and MFIE often are polluted by spurious modes at the cavity’s resonant frequencies whereas those of the CFIE solver are devoid of such contamination. [ABSTRACT FROM AUTHOR]

The butterfly (BF)-compressibility of moment matrix blocks is studied quantitatively for various geometrical configurations and compression algorithm parameters. To enable investigation of electrically large geometries, the methodology employs simplified expressions for the compressed memory and fast low-rank (LR)-approximation techniques. The study indicates that a block’s optimal BF-compression is obtained for a nontrivial choice of the BF’s depth, determined by the underlying geometrical configuration and compression threshold. It is also shown that reduced-dimensionality interactions are more BF-compressible. In particular, the optimal compressibility for the recently introduced generalized equivalence and generalized source integral equations is shown to be dependent on their effective dimensionality, which is dictated by the compression threshold. At high thresholds, these formulations are more BF-compressible than their conventional counterparts. The implications for the design of BF-compression-based solvers, in a geometrically adaptive manner, are discussed. [ABSTRACT FROM AUTHOR]

In this paper, wavelets transformation (WT) and wavelet packet transformation (WPT) are used in solving, by the method of moments, a semicircular array of parallel wires electric field integral equation. First, the integral equation is solved by applying the direct method of moments via point-matching procedure, results in a linear system with a dense matrix. Therefore, wavelet transformation and wavelet packet transformation are used to sparsify the impedance matrix, using two categories of wavelets functions, Biorthogonal (bior2.2) and Orthogonal (db4) wavelets. The far-field scattering patterns and the comparison between wavelets transformation and wavelet packet transformation in term number of zeros in impedance matrix and CPU Time reduction are presented. Numerical results are presented to identify which technique is best suited to solve such scattering electromagnetic problems and compared with published results.

The Weibull distribution function with shape (k) and scale (c) parameters predicts wind speed distribution for different locations. The k and c parameters have been estimated by using several analytical methods, but it provides different results in the same wind speed distribution data. In this work, the correlation technique was applied to evaluate the performance of analytical methods using several empirical methods. The analytical methods presented including Method of Moments (MoM), Energy Pattern Density (EPD), Graphic Method (GM), and Maximum Likelihood Estimator (MLE). Five empirical methods were evaluated graphically by using the GM method to estimate the Weibull two-parameter. The Weibull two-parameter of correlation coefficient (r) of greater than 0.9503 indicated from four empirical methods. Again, five empirical methods for k parameter and four empirical methods for c parameter were used in the special case of MoM and EPD methods. The results indicated four empirical methods for estimating k parameter and two empirical methods for estimating c parameter having the r of greater than 0.7. When empirical methods of k and c parameter correlated, six matching patterns were identified from 20 possible pairs. Comparison of GM, MLE, and selected pairs of k and c values predicted from empirical methods resulted from MoM and EPD indicated the average coefficient of best-fit of 0.8851 and root-mean-square-error of 0.2083. Therefore, the use of more than one empirical method in one analytical method of MoM, EPD, and GM would improve the prediction of Weibull distribution. [ABSTRACT FROM AUTHOR]

In this paper the Finite Integration Technique (FIT) hybridized with the Method-of-Moments (MoM) is used to find resonance frequencies and quality factors of open cylindrical dielectric resonators (CDRs). The technique is based on the previously developed formulation for scattering problems, with the application of root searching algorithm to find zeros of the final matrix determinant in the complex frequency plane. The method is validated by comparison of obtained results with the results of simulations done using other methods, and with measurement data found in the literature. [ABSTRACT FROM AUTHOR]

Polydisperse multiphase flows can be simulated by coupling a population balance model to the multi‐fluid model. For the first time, this simulation is carried out using the dual‐quadrature method of generalized moments (DuQMoGeM) and its direct version to solve the population balance model. The main disadvantage of these methods is the high computational cost of the embedded cubature. Herein, this challenge was addressed by parallelization on graphics processing units, which resulted in a significant acceleration of the simulations, with speedups that can be larger than 1000. Numerical simulations considering simultaneous particles breakage and aggregation were conducted with direct DuQMoGeM‐FC and DQMoM‐FC, whose results were different due to the existence of quadrature error in the latter. [ABSTRACT FROM AUTHOR]

This article presents a mapping-based optimization method for pattern synthesis via array element rotation. The optimization is carried out through optimizing the orientation of each array element to minimize the sidelobe and cross-polar level while maximizing the co-polar level of the main beam in a desired direction. The basic idea of the proposed method is to convert a variable definition domain problem into a fixed definition domain problem through coordinate transformation. Based on this, the optimization is well integrated with the method of moments (MoM), and it is convenient to calculate the radiation pattern properties and their sensitivities with respect to the design variables. Thus, the design optimization problem can be efficiently solved by use of any gradient-based algorithm. Additionally, benefiting from the combination of the mapping transformation and the full-wave MoM simulation, the mutual coupling between array elements is rigorously taken into account and the complex meshing of the surrounding radiation region is avoided. Furthermore, the change in the array performance is derived only associating with the transformation variables, thus remeshing for iteratively changed arrangements is avoided and the computing cost is saved. Typical numerical examples are provided to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A previous paper (Fikioris and Wu, 2001) analyzed in detail the difficulties associated with the application of numerical methods to Hallén's integral equation with the approximate kernel for the case of a lossless surrounding medium. This letter extends to the case where the medium is conducting and points out similarities and differences between the two cases. Our main tool is an analytical/asymptotic study of the antenna of infinite length. [ABSTRACT FROM AUTHOR]

In this communication an efficient method of moments (MoM) code is used for the analysis of the extraordinary transmission (EOT) through a periodic array of rectangular slots in a conducting screen, in the case where the number of slots is finite in one direction and infinite in the orthogonal direction. The slots can be arbitrarily rotated within the periodic unit cell. Once the magnetic current density on the slots is obtained by means of MoM, both the transmission coefficient and the far-field radiated by the array of slots are computed. The onset of EOT turns out to be strongly dependent on the orientation of the slots with respect to the direction in which the array is infinite. If the slots are perpendicular to this direction, EOT appears for a single infinite chain of slots. However, tens of parallel chains of slots are required to reproduce the EOT response when the slots are aligned along the parallel chains direction. The obtained radiation patterns show the excitation of grating lobes as the number of slots grow in the direction where the arrays are finite. [ABSTRACT FROM AUTHOR]

This paper presents a technique for the full-wave analysis of large problems using the method of moments, in combination with the multilevel fast multipole algorithm, based on a decomposition of the scenario into two domains using a ray-tracing analysis. These domains are computed considering the proximity to the critical points. The currents over the main scenario are obtained first and used to modify the excitation of the secondary domain, which in turn can be analyzed using a relaxed iteration error that very noticeably reduces the iteration time without a significant impact on the accuracy. The procedure used for the separation of both domains minimizes the amount of information exchanged between them. [ABSTRACT FROM AUTHOR]

The accurate and efficient evaluation of surface source integrals is a key step in obtaining reliable solutions to electromagnetic problems using integral equation formulations. In this paper, we propose to combine two of the most extensively used schemes to efficiently evaluate strongly singular and near-singular source integrals, such as those arising in the ${\mathcal {K}}$ operator. In the proposed approach, singularity subtraction is first applied to remove the most dominant singular part; singularity cancellation, using the radial-angular variable transformation, is then employed to treat the remaining integrand. The method is compared to other common numerical schemes to demonstrate its effectiveness. [ABSTRACT FROM AUTHOR]

We present a new entropy-based moment method for the velocity discretization of kinetic equations. This method is based on a regularization of the optimization problem defining the original entropy-based moment method, and this gives the new method the advantage that the moment vectors of the solution do not have to take on realizable values. We show that this equation still retains many of the properties of the original equations, including hyperbolicity, an entropy-dissipation law, and rotational invariance. The cost of the regularization is mismatch between the moment vector of the solution and that of the ansatz returned by the regularized optimization problem. However, we show how to control this error using the parameter defining the regularization. This suggests that with proper choice of the regularization parameter, the new method can be used to generate accurate solutions of the original entropy-based moment method, and we confirm this with numerical simulations. [ABSTRACT FROM AUTHOR]

The paper is concerned with the numerical solution of the linear steady-state regularized 13-moment equations in two space dimensions. To facilitate the understanding of the specific challenges, the equations are first divided into two subsystems before the full system is approached. The arising problems mainly stem from the complicated saddle-point structure as well as the non-standard nature of the boundary conditions. A continuous interior penalty method is presented and the pronounced advantages of utilizing high order basis functions in this setting are illustrated. To render the presented approach more efficient, a hybridization technique is presented that originates in the discontinuous Galerkin method. [ABSTRACT FROM AUTHOR]

Point estimation methods for the three-parameter lognormal distribution are investigated and compared. The lognormal distribution is required in many topical areas, but so far there have been no comparative studies of the various estimation methods. We show that despite the large number of traditional estimation methods, the lognormal distribution requires special methods. We accordingly consider specializations of the main parameter estimation approaches, including the actively developing distance minimization methods. Their accuracy and speed are compared on simulated data. We show that specialized parameterestimation methods may outperform the highly popular maximum likelihood method. [ABSTRACT FROM AUTHOR]

Objective: This paper proposes a new methodology to solve one-dimensional cases of integral equations with difference kernels using Fourier analysis. Methodology: In this study, it was proven that any Fredholm equation of the first kind can be expressed as an extended convolutional problem; consequently, a new approach to solve that problem, using the nonideal instantaneous sampling theory and Fourier analysis, can be developed. Results and Discussion: The proposal was extensively evaluated and compared with the method of moments by considering two benchmarks. The first was a narrowband problem related to a second-order differential equation with specific boundaries. The second was a standard wideband problem related to wire antenna radiation in electrodynamics, known as the Pocklington equation. In both cases, we derived new interpretations and different approaches to solve the problems efficiently. Conclusions: The new proposal generalized the method of moments via new interpretations, strategies and design rules. We found that the techniques based on the method of moments are point-matching procedures independent of the weighting functions; the basis functions can be designed as generalized interpolation functions with more information provided by the original domain; the weighting functions literally represent a sampled linear filter; the unknown continuous function can be approximated without using the classical variational approach; and several new strategies based on the Fourier transform can be used to reduce the computational cost. [ABSTRACT FROM AUTHOR]

The method of moments (MoM) in the spectral domain is used for the analysis of the scattering of a plane wave by a multilayered periodic structure containing conducting concentric split rings in the unit cell. Basis functions accounting for edge singularities are used in the approximation of the current density on the split rings, which makes it possible a fast convergence of MoM with respect to the number of basis functions. Since the 2-D Fourier transforms of the basis functions cannot be obtained in closed-form, judicious tricks (controlled truncation of infinite summations, interpolations, etc.) are used for the efficient numerical determination of these Fourier transforms. The implemented spectral domain MoM software has been used in the design of a circularly polarized reflectarray antenna based on split rings under the local periodicity condition. The antenna has been analyzed with our spectral domain MoM software, with CST and with HFSS, and good agreement has been found among all sets of results. Our software has proven to be around 27 times faster than CST and HFSS. [ABSTRACT FROM AUTHOR]

The use of the method of moments to solve surface integral equations is one of the most popular numerical techniques in electromagnetic modeling and analysis. This method requires the accurate and efficient numerical evaluation of iterated surface integrals over both source and testing domains. In this paper, we propose a scheme for evaluating these 4-D interaction integrals between pairs of arbitrarily positioned and oriented elements. The approach is based on applying the surface divergence theorem twice, once on the source and once on the test domain. When the integrations are reordered as two outer contour integrals plus two inner radial integrals, the initial radial integrations provide significant smoothing of the underlying singular integrands. The method is numerically validated for static and dynamic kernels arising in the electric field integral equation, i.e., for kernels with $1/R$ singularities, and linear basis functions. The proposed formula to evaluate 4-D reaction integrals can be extended to different kernels and to different elements, e.g., to curved or volumetric elements, and to basis functions of higher order. [ABSTRACT FROM AUTHOR]

A method of moments-based numerical technique for the calculation of the electric-type Green’s function of inhomogeneous media is presented for a 2-D configuration. The problem is reduced to an integral equation by the use of a special representation through the decomposition of Green’s function into two parts, which allows the singularity is being expressed conveniently. However, this representation yields multiple singular terms to appear on the right-hand side of the integral equation system, which are treated by an approximation of the related integral through an equivalent cell approach. The method is validated by comparisons with classical solutions and some numerical examples are also given in order to show the applicability and the effectiveness. [ABSTRACT FROM AUTHOR]

This paper describes a method of determining the first two moments of the response for basic components of electrical circuits, i.e. resistors, inductors and capacitors. The paper goal was to obtain closed form formulae for the moments describing voltage or current stochastic processes. It has been assumed that the element parameters R (resistance), L (inductance) and C (capacitance) could be random variables, deterministic functions or stochastic processes and excitations are second order stochastic processes. Moreover, two cases of dependence between the random parameters and the excitation stochastic processes have been considered. The obtained results enable determination of exact solutions for the first two moments without application of numerical algorithms.

In the grounding systems with a two layers soil, the calculation of the surface potential using the image method is sometimes impossible due to singularities, avoiding researchers to use the method for electrodes in the bottom layer. In the literature this problem solution is refereed as unreliable or solved with other more complex methods. This paper presents a new approach to calculate the surface potentials in a two. layer soil, for electrodes in the bottom layer, when images are at surface. The singularities in computing surface voltage, when the first image upwards lies at surface, are analysed and it's shown that a small change in top layer thickness allows an approximate solution. Surface potentials due to grid conductor are also considered and the values of resistance are compared with those from other methodologies. Singularities for a ground rod that crosses the two layers are also treated. The obtained values of resistance are not satisfactory, due to lower segments images that overlap the upper segments. This paper also proposes shifting the surface of the upper part of the ground rod, in the upper layer, or taking the modulus of the mutual resistance, to overcome this difficulty.

Quadrature is an approximation of the definite integral of a function by a weighted sum of function values at specified points, or nodes, within the domain of integration. Gaussian quadratures are constructed to yield exact results for any polynomials of degree 2r−1 or less by a suitable choice of r nodes and weights. Cubature is a generalization of quadrature in higher dimension. In this article, we elaborate algorithms to compute all minimal cubatures for a given domain and a given degree. We propose first algorithm in symbolic computation to characterize all cubatures of a given degree with a fixed number of nodes. The determination of the nodes and weights is then left to the computation of the eigenvectors of the matrix identified at the characterization stage and can be performed numerically. The characterization of cubatures on which our algorithms are based stems from moment theory. We formulate the results there in a basis‐independent way: Rather than considering the moment matrix, the central object in moment problems, we introduce the underlying linear map from the polynomial ring to its dual, the Hankel operator. This makes natural the use of bases of polynomials other than the monomial basis, and proves to be computationally relevant, either for numerical properties or to exploit symmetry. [ABSTRACT FROM AUTHOR]

A physics-based rank-revealing multilevel algorithm to more efficiently compute low-rank (LR) approximations of the method of moments matrix blocks ${\mathbf{Z}}^{\textrm {os}}$ is presented. Using surface subsets of volumetric nonuniform spherical grids (proxy grids), an LR approximation ${\mathbf{Z}}^{\textrm {os}}\approx {\mathbf{AB}}^{\dagger }$ is obtained in two stages: 1) an upward pass and a downward pass on a multilevel tree, via truncated singular value decomposition-based analyses of interactions between source subclusters and their attendant proxy grids, rapidly estimates the rank and yields a domain of ${\mathbf{Z}}^{\textrm {os}}$ , in the form of a matrix ${\mathbf{B}}$ with orthonormal columns and 2) a fast matrix–matrix multiplication step yields ${\mathbf{A}}$. The algorithm reduces the ${\mathcal{ O}}(N^{3})$ computational costs of revealing the rank to ${\mathcal{ O}}(N^{2})$ or ${\mathcal{ O}}(N^{3/2})$ operations and that of finding an LR approximation to ${\mathcal{ O}}(N^{2})$ or ${\mathcal{ O}}(N^{3/2}\log N)$ operations by adopting enclosing or belt-like proxy grids for electromagnetically large basis/testing function distributions that are either densely packed in a volume or quasi-planar, respectively. [ABSTRACT FROM AUTHOR]

An approach for the generation of macro basis functions considering the directivity of the sources and the ray-tracing characteristics of the scenario is described in this letter. The number of basis functions generated is highly optimized by dynamically generating the threshold to be applied for the retrieval of the macro basis functions after the application of the singular value decomposition of the currents induced by the plane wave spectrum surrounding each block. Significant performance gains are shown due to the reduction of the number of unknowns compared to the conventional generation of macro basis functions. [ABSTRACT FROM AUTHOR]

This paper proposes an efficient method to analyze scattering from an inhomogeneous body of revolution (BOR) with anisotropic electromagnetic properties. Both the permittivity and the permeability are assumed to be generalized tensors while there are no restrictions on the geometry of the BOR. The proposed method involves three stages. First, the volume equivalence principle is used to replace the BOR with volume electric and magnetic current densities. Requisite volume integral equations (VIEs) are, then, derived with the electric and magnetic flux densities inside the BOR being the unknown quantities. Finally, the VIEs are solved by Galerkin’s method of moments. This is done by expanding the unknown flux densities in terms of appropriate basis functions compatible with the problem symmetry, and hence, reducing the integral equations to a matrix equation. The formation of the resultant matrix equation is elucidated in detail, whereby it is shown that the computation burden is remarkably reduced as no double volumetric-type integrals are to be computed. The efficiency of the proposed method is demonstrated by comparing the simulation results of several case studies with those available in the literature, or computed by commercial numerical codes. [ABSTRACT FROM AUTHOR]

This communication presents a technique that allows an efficient selection of the number of macro basis functions for each block by assigning dynamic thresholds applied after the singular value decomposition of the currents induced by the plane wave spectrum. It makes use of a ray-tracing preprocessing of the scenario, where a number of critical points are identified. The threshold assigned for the selection of the high-level basis functions on each block is based on the distance of the center of the block to the nearest critical point. Representative examples are included in order to illustrate the efficiency and accuracy of the proposed approach. [ABSTRACT FROM AUTHOR]

An efficient approach for the generation of sparse preconditioners, tailored for the application of macrobasis function methods, is described in this letter. The preconditioner is highly scalable and generated by the computing groups of rows that share a common sparsity pattern, calculated by using a distance threshold between geometrical blocks that controls the amount of data to be considered. This approach is suitable for the analysis of very large and realistic problems, as shown in some test cases included. Due to the clustering nature of this technique, the preconditioners can be generated in a substantially lower amount of time than that required by conventional approaches. [ABSTRACT FROM AUTHOR]

Derivation of the dynamic correction to Grad's moment system from kinetic equations (regularized Grad's 13 moment system, or R13) is revisited. The R13 distribution function is found as a superposition of eight modes. Three primary modes, known from the previous derivation (Karlin et al. 1998 Phys. Rev. E 57, 1668-1672. (doi:10.1103/PhysRevE.57.1668)), are extended into the nonlinear parameter domain. Three essentially nonlinear modes are identified and two ghostmodes which do not contribute to the R13 fluxes are revealed. The eight-mode structure of the R13 distribution function implies partition of R13 fluxes into two types of contributions: dissipative fluxes (both linear and nonlinear) and nonlinear streamline convective fluxes. Physical interpretation of the latter non-dissipative and non-local in time effect is discussed. A non-perturbative R13-type solution is demonstrated for a simple Lorentz scattering kinetic model. The results of this study clarify the intrinsic structure of the R13 system. This article is part of the theme issue 'Hilbert's sixth problem'. [ABSTRACT FROM AUTHOR]

We present a novel deterministic method capable of calculating statistical moments of transverse electric polarized fields scattered by perfect electric conductor gratings with small surface random perturbations. Based on a first-order shape Taylor expansion, the resulting electric field integral equations are solved via the method of moments with constant hierarchical basis or Haar wavelets. This allows for a sparse tensor approximation, significantly reducing the number of required unknowns and yielding a higher rate of convergence than a dense approximation. Moreover, the proposed approach converges faster than Monte Carlo simulations with significantly less computational effort. Validation of the proposed approach is performed for several cases, and realistic simulations applied to the calculation and prediction of grating efficiency reveal the applicability of the method. [ABSTRACT FROM PUBLISHER]

In this paper we consider a viscoelastic plate (linear viscoelasticity of the Maxwell- Boltzmann type) and we compare its controllability properties with the (known) controllability of a purely elastic plate (the control acts on the boundary displacement or bending). By combining operator and moment methods, we prove that the viscoelastic plate inherits the controllability properties of the purely elastic plate. [ABSTRACT FROM AUTHOR]

In this study, we extend the Taylor-series expansion method of moment to two-component aggregation problem undergoing Brownian coagulation with kernels that are independent of composition. A set of closed particle population balance equation for lower-order moments is then derived. Numerical results and its asymptotic solutions are validated by comparing with Monte Carlo simulation method both in free molecular regime and continuum regime. It is shown that three dimensionless particle moments M C 1 , M C 2 , M C 3 almost approach to a same value over large evolution time. The normalized variance of excess component A decreases as 1 / v ¯ and it tends to zero over large evolution time. [ABSTRACT FROM AUTHOR]

Currently the number of reduction methods used in practice in climate applications is vast and tends to be difficult to access for researchers searching for an overview of the area. In this work, we review a range of reduction methods that have been, or may be useful for connecting models of the Earth's climate system of differing complexity. We particularly focus on methods where rigorous reduction is possible. We aim to highlight the main mathematical ideas of each reduction method and also provide several benchmark examples for the reduction from climate modelling. In particular, our goal is to provide a broad overview of available methods. • We present a brief but broad overview of reduction methods applicable to climate dynamics. • Each method is illustrated using a model example and/or a discussion of the ideas behind the method. • Mathematical underpinnings of the methods are explained and compared. [ABSTRACT FROM AUTHOR]

In numerical weather prediction models, parameterisations are used as an alternative to spectral modelling. One type of parameterisations are the so-called methods of moments. In the present study, two different methods of moments, a presumed-number-density-function method with finite upper integration limit and a quadrature method, are applied to a one-dimensional test case (‘rainshaft’) for drop sedimentation. The results are compared with those of a reference spectral model. An error norm is introduced, which is based on several characteristic properties of the drop ensemble relevant to the cloud microphysics context. This error norm makes it possible to carry out a quantitative comparison between the two methods. It turns out that the two moment methods presented constitute an improvement regarding two-moment presumed-number-density-function methods from literature for a variety of initial conditions. However, they are excelled by a traditional three-moment presumed-number-density-function method which requires less computational effort. Comparisons of error scores and moment profiles reveal that error scores alone should not be taken for a comparison of parameterisations, since moment profile characteristics can be lost in the integral value of the error norm.

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. [ABSTRACT FROM AUTHOR]