Efficient electromagnetic analysis for curved open thin-wire structures using intervallic wavelets in the method of moments

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

The integral equations are often discretized into matrix equation by the method of moment (MoM) in EM analysis. However, the MoM evaluates each matrix entry at the cost of double integrations. Furthermore, to construct the divergence-conforming basis functions, the grids have to meet strict conforming conditions, which is a burden on mesh generation and preprocessing procedure. In the contrast, the Nyström method has advantages over the MoM in the aforementioned aspects. To accelerate its rate of convergence and accuracy, a higher-order Nyström discretization scheme is proposed to solve surface integral equations for electromagnetic (EM) scattering by penetrable objects in this paper. A numerical example is presented to demonstrate the scheme and good results have been observed.

Hybrid finite element methods (HFEM) as used in the present paper refer to a coupling of the finite element method (FEM) applied to Maxwell's equations in differential form and the method of moments (MOM) applied to Maxwell's equations in integral form. This coupling allows the strengths of each method to supplant the other. The finite element method models penetrable materials and complicated geometry, while the method of moments provides the exact radiation boundary condition where the finite element mesh is terminated. In the paper a three-dimensional HFEM implementation is presented. As in Boyse and Seidl (1991), the finite element formulation is coupled to a highly efficient BOR MOM code which provides the exact near field radiation condition. The scalar and vector potential formulation completely eliminates spurious modes when using nodal based elements, allowing iterative methods to rapidly converge and the computation of very accurate internal fields. However, calculations involving models with sharp corners initially produced very inaccurate results. A new element (Boyse et al., 1992 and Bardi et al., 1993) was used along corners and edges which combines both nodal and edge basis functions. The addition of these new elements produces very accurate results while retaining conventional nodal elements throughout most of the model.

A probe-fed rectangular dielectric resonator antenna (RDRA) placed on a finite ground plane is numerically investigated using method of moments (MoM). The whole structure of the antenna is exactly modeled in our simulation. The feed probe, coaxial cable and ground plane are modeled as surface electric currents, while the dielectric resonator (DR) and the internal dielectric of coaxial cable is modeled as volume polarization currents. Each of the objects is treated as a set of combined field integral equations. The associated couplings are then formulated with sets of integral equations. The coupled integral equations are solved using MoM in spatial domain. The effects of ground plane size, air gap between dielectric resonator and ground plane, probe length, and position on the radiation performance of the antenna including resonant frequency, input impedance, radiation patterns, and bandwidth are investigated. The results obtained for the antenna parameters based on the MoM investigation shows that there is a close agreement with those obtained by measurement. Moreover it is shown that the MoM results are more accurate than other simulation results using software package such as High Frequency Structure Simulator (HFSS).

Computational electromagnetics has become a real cornerstone in the design of microwave and millimeter wave components and systems. One of the major computational techniques used in these designs is to solve integral equations with the Method of Moments (MoM). Traditionally, MoM has been used extensively for antennas, transmission lines, filters, couplers, etc., mainly in the MHz and GHz range. However, MoM can also be applied at much higher frequencies. It will be shown that at these frequencies, this computational technique is just emerging. One of the reasons is that there are serious differences with the classical implementations, which prohibits the direct use of the well-known commercially available MoM solvers. For example, one of the differences is the fact that the concept of a surface current, confined to a very thin layer at the surface of a conductor, is not valid any more. In this presentation, the different computational techniques will be reviewed and compared, not only in the traditional microwave and millimeter wave range, but especially at THz, IR, and optical frequencies. In concreto, the MAGMAS MoM solver will be compared to a numerical FDTD solver well-known in the plasmonics research community: Lumerical. It will be shown that also at these very high frequencies, MoM has its proper place in the pallet of computational algorithms.

Very simple electrostatic problems were being formulated as integral equations and solved using digital computers beginning in the early ‘60’s. By the next decade, researchers were solving wire, cylinder, and body-of-revolution problems in both frequency- and time-domain using many of the now-familiar integral equation formulations. But it was Harrington's 1968 monograph on the Method of Moments (MoM) that generalized, systematized, introduced the terminology, and popularized the projection-based approach. Though the monograph convincingly established MoM's variational character and its applicability to linear operator equations of all types, the combination of its emphasis on the projection method and its immediate application to a variety of scattering and antenna problems led to its becoming almost exclusively associated with integral equation applications. Meanwhile, the finite element method (FEM) developed almost independently, driven largely by the microwave circuits and guided wave communities, and became firmly associated with differential equation formulations based on variational principles. Though the overlapping applicability of MoM, FEM, and related methods are now widely recognized, many of the original associations persist today.

In this paper, we present an electromagnetic analysis for three‐dimensional arbitrarily shaped large structures illuminated by a short pulse. The time domain electric field integral equation (EFIE) formulation is used to obtain the early time data with a central differencing scheme to evaluate the time derivatives of the potential functions. The solution method is based on the method of moments (MoM) and marching‐on‐in‐time (MOT) technique. Low frequency data is generated using a frequency domain integral equation method. The hybrid method is based on the singular value decomposition (SVD) of the frequency and the time domain data for a direct extrapolation of a causal signal. The detailed mathematical procedures are included along with representative numerical results from an aircraft model. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 898–904, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22253

Thin-wire structures in the presence or absence of a ground plane are analyzed numerically in the time domain (TD) with the aid of the Cagniard–DeHoop method of moments (CdH-MoM). It is demonstrated that the TD solution of such problems can be cast into the form of discrete-time-convolution equations. Under the assumption of piecewise linear space–time axial current distribution, the elements of the TD impedance array are derived analytically in terms of elementary functions. Their approximations applying to multiconductor transmission lines are discussed. Illustrative numerical examples validating the TD solution are presented.

In realistic landmine fields, the anti-personnel plastic mine is often buried nearby a clutter-object under the ground. The presence of a second object buried near the mine under a two-dimensional (2-D) rough ground can easily obscure the target and/or cause a false alarm. The separation distance between the AP mine and clutter-object plays a significant role on the probability of true or false alarm in this situation. A rigorous electromagnetic model has been developed to analyze the scattering mechanism between the target and the clutter-object, between the target and the rough ground, between clutter-object and the rough ground and the multiple scattering between different spots on the rough ground itself. The new rigorous model is based on the classical electromagnetic equivalence theorem leading to producing six new integral equations. Using the Method of Moment (MoM), the new integral equations are transformed into a linear system of equations to be solved for the unknown electric and magnetic currents on the surface of three scatterers; 1) rough ground, 2)target and 3)clutter-object. The MoM impedance matrix completely represents every interaction between these three scatterers. The superior Steepest Descent Fast Multipole Method (SDFMM) is used to tremendously speed up the computations of the unknown MoM surface currents.

This paper presents a computationally efficient method for solving automotive low-frequency electromagnetic compatibility (EMC) problems by using integral equations. We consider the interaction of magnetic fields with thin, finite, conducting 3-D metallic structures, obtaining the fields radiated by these structures by using single- and double-layer equivalent currents. Our proposed numerical solution is unique in its representation of equivalent currents as the sum of solenoidal and nonsolenoidal components found using the method of moments (MoM) in two steps: first, the solenoidal currents are found using loop basis functions, after which the nonsolenoidal currents are found. Decomposing the equivalent currents into solenoidal and nonsolenoidal components provides a total solution that is computationally efficient for problems dominated by magnetic fields. We validated this numerical electromagnetic solution against semianalytical solutions and measured data and illustrated its applicability by analyzing three practical automotive problems. We then analyzed the magnetic fields generated by a power cable inside a car, suggested a methodology for optimizing the locations of antennas for smart-entry systems, and studied the EMC implications of an inductive-charging system in an electric vehicle.

A three-dimensional pre-corrected fast Fourier transform (PFFT) algorithm for the rapid solution of the full-dyadic Michalski-Zheng's mixed potential integral equation is presented. The integral equation is discretized with the Rao-Wilton-Glisson (RWG) method of moments. Handling the method of moments interactions with the dyadic kernel is simplified via representation of the RWG functions in terms of barycentric shape functions. The proposed three-dimensional precorrected FFT method distributes two-dimensional FFT grids nonuniformly along the direction of stratification according to conductor locations within the layers. For <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> two-dimensional FFT grids each with an average of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Np</i> associated triangular elements the method exhibits <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Np</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Np</i> ) computational complexity and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P Np</i> ) memory usage. The low-frequency breakdown of the integral equation is eliminated via loop-tree decomposition. A unique combination of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) computational complexity, fully three-dimensional boundary-element modeling in layered substrates, and full-wave modeling from dc to multi-gigahertz frequencies makes the algorithm particularly useful for characterizing large interconnect networks embedded in multilayered substrates. The method is implemented as the electromagnetic solver in Cadence's Virtuoso RF Designer software.

Summary form only given. Fast direct integral equations methods for analyzing time-harmonic scattering phenomena constitute an active area of research in the CEM community. These solvers present an interesting alternative to FMM-based iterative schemes for problems that are inherently ill-conditioned and/or involve many excitations. Present fast direct methods leverage the low-rank (LR) nature of blocks of the LU factors or inverse of the method of moment (MoM) matrix (if and when this property manifests itself). In essence, they leverage the limited number of degrees of freedom of the fields radiated by sources in the presence of the scatterer (or portions thereof), observed across a finite region of the scatterer (whenever the concept makes sense). LR compression strategies do not lead to low-complexity solvers for electrically large concave structures. This fact was demonstrated in [H. Guo et al., Antennas and Wireless Propagation Letters, IEEE, vol. 11, 2012], where blocks of LU factors that proved incompressible by LR methods were recompressed by MLMDA/butterfly methods. Somewhat unexpectedly, this study showed that MLMDA/butterfly scheme produced compression above and beyond that provided by LR schemes, requiring only O(N log2 N) final storage irrespective of the nature or electrical size of the scatterer. The practical applicability of the scheme however was severely hindered by the fact that the MLMDA/butterfly scheme acted as an add-on to a LR solver. Here, we report on a new direct integral equation solver by that utilizes MLMDA/butterfly schemes for compressing the LU factors of a MoM matrix. When compared to its predecessor, the new solver has three important features of note. (i) The solver entirely bypasses the LR compression step of its predecessor and operates directly on butterfly-compressed blocks. The latter is achieved using new schemes for rapidly adding and multiplying butterfly-compressed operators. (ii) The solver executes in parallel, using a hybrid OpenMP-MPI strategy to accelerate the hierarchical inverse/decomposition of the MoM matrix. Despite the inherently sequential nature of this operation, the solver exhibits excellent scaling properties up to several hundred processors. (iii) The solver applies to large-scale 3D (as opposed to 2D) analysis and was implemented to invert a combined field (as opposed to an electric field) integral operator. The solver is capable of inverting MoM matrices that discretize combined field integral equations modeling scattering from electrically large structures involving millions of unknowns on a small cluster using just a few hours of CPU time.

The use of the method of moments (MM) based on the electric-field integral equation is considered for conducting bodies with closed regions. For such a calculation, the scattered field is theoretically well defined, but numerical problems have been found near resonant frequencies. To study these problems, the use of the singular value decomposition (SVD) in MM calculations is developed, and the reason the SVD is the appropriate tool for the numerical analysis of any MM calculation (even when the matrix equation is to be solved by an iterative technique) is shown. In particular, the SVD casts a MM calculation into a diagonal form, which allows a careful numerical analysis. The problems near resonance are found to be due to the creation of an approximate impedance matrix Z, which has the wrong resonant frequency (i.e., it is not consistent with that of the radiation problem). One numerical method that avoids these difficulties by using the SVD is discussed, and other more efficient ways of avoiding the usual numerical difficulties are suggested.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A rigorous treatment on the computation of the various integrals that arise in the method of moments (MoM) formulation of the straight-wire electric field integral equation is provided. For triangle basis functions along with delta function, pulse or triangle weights, particular attention is given to integrals whose integrands are weakly singular. A singularity extraction technique is employed that splits the integral under question into two parts: one that is numerically integrable and one that is analytically integrable. Closed-form approximations based on Taylor series techniques are also provided for the former. These approximations are very robust resulting in errors less than 0.1% when 2α/δ <; 1 ; here δ is the length of a single MoM segment and α is the wire radius. Results are compared with data from the literature to demonstrate the robustness of the presented approach for fat wires.

In this paper we analyze the numerical aspects of the various methods that have been utilized to analyze thin wire antennas. First, we derive the properties of the operators for Pocklington's and Hallen's integral equation. On the basis of these properties, we discuss the various iterative methods used to find current distribution on thin wire structures. An attempt has been made to resolve the question of numerical stability associated with various entire domain and subdomain expansion functions in Galerkin's method. It has been shown that the sequence of solutions generated by the iterative methods monotonically approaches the exact solution provided the excitations chosen for these problems are in the range of the operator. Such a statement may not hold for Galerkin's methods if the inverse operator is unbounded. Moreover, if the excitation function is not in the range of the operator, then the sequence of solutions forms an asymptotic series. Examples have been presented to illustrate this point.