Surface Integral Equation Solver Research Articles

A 3-D MoM for Uniaxial Objects in a Uniaxial Background Medium

A 3-D method of moments (MoM) for solving surface integral equations (SIEs) is presented for the electromagnetic (EM) scattering of 3-D uniaxial EM objects in a uniaxial EM background with arbitrary optical axes. The Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) formulation for uniaxially anisotropic media is applied to find the induced electric and magnetic current densities on the uniaxially anisotropic objects. The conventional Rao–Wilton–Glisson (RWG) functions are used as the basis functions and test functions to convert the PMCHWT equations into weak-form equations. As the main challenge in impedance matrix assembly, the singularity problem in uniaxial EM media is addressed. The new contribution of this work is the development of an MoM solver allowing both the scatterers and the background medium to be uniaxially EM with arbitrary optical axes. To verify the accuracy and efficiency of the proposed method, three numerical examples are presented. Compared with the simulation results of mature commercial software, the proposed method has high accuracy and efficiency and has great advantages in computing time and resource consumption.

Multibranch Curvilinear Rao–Wilton–Glisson Basis Functions for Electromagnetic Scattering From Perfect Electric Conducting Targets

A multibranch curvilinear Rao-Wilton-Glisson (MB-CRWG) basis function is proposed to solve surface integral equations (SIEs) with nonconformal curvilinear meshes. The derivation of the MB-CRWG basis function is given. The structure of the MB-CRWG basis function can be divided into positive and negative parts. The positive part is one curvilinear triangular patch, and the negative part is composed of several curvilinear triangular patches. The MB-CRWG can conveniently connect two surfaces discretized with curvilinear triangular patches of different sizes. Numerical examples on the electromagnetic scattering from multiscale perfect electrically conducting (PEC) targets are given to demonstrate the effectiveness of the proposed basis function.

A Multitrace Surface Integral Equation Solver to Simulate Graphene-Based Devices

A multitrace surface integral equation (MT-SIE) solver is proposed to analyze electromagnetic field interactions on composite devices involving magnetized and nonmagnetized graphene sheets. The computation domain is decomposed into two subdomains: an exterior subdomain that represents the unbounded background medium where the device resides in an interior subdomain that represents the dielectric substrate. Resistive Robin transmission conditions (RRTCs) are formulated to describe the infinitesimally thin graphene sheet that partially covers the surface between the two subdomains. On the rest of this surface, traditional Robin transmission conditions (RTCs) are enforced. The electric and magnetic field equations are used as the governing equations in each subdomain. The governing equations of a subdomain are locally coupled to the governing equations of its neighbor using RRTCs and RTCs. The accuracy and the applicability of the proposed MT-SIE solver are demonstrated by various numerical examples.

Fully Analytical Evaluation of Singular Integrals With RWG and Rooftop Basis Functions

When the method of moments is used to solve surface integral equations, it is inevitable to encounter the problem of singularity because of Green's function in integral kernel. In this article, by expanding Green's function with Taylor series and applying appropriate coordinate transformation, closed-form expression of each term is derived. After proper linear combination, analytical solutions of singular integrals on triangular and rectangular domains with Rao–Wilton–Glisson and rooftop basis function can be achieved. Compared with numerical solutions, this approach is more efficient and can considerably decrease the computational cost, for a given precision.

Evaluation of 4-D Reaction Integrals Via Double Application of the Divergence Theorem

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.

Electron energy loss spectroscopy simulation by a frequency domain surface integral equation solver

Plasmonic nanoparticles have been mostly studied using conventional light sources. Recently, electron energy loss spectroscopy (EELS) has started to be used to analyze plasmonic nanoparticles where incident electromagnetic fields are created by swift electrons. To accurately simulate EELS experiments, several numerical methods have been adapted. In this paper, a frequency domain surface integral equation (FDSIE) solver is modified to simulate EELS for plasmonic nanoparticles of gold and silver. Accuracy and versatility of the proposed FDSIE solver are shown by several numerical examples and compared to existing numerical, analytical, and experimental results.

Computation of Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo Method

Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization, the number of MC samples has to be large. In this paper, to address this challenge, the continuation multilevel Monte Carlo (CMLMC) method is used together with a surface integral equation solver. The CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational cost. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.

Skeletonization Accelerated MLFMA Solution of Volume Integral Equation for Plasmonic Structures

Skeletonization is employed to accelerate the multilevel fast multipole algorithm (MLFMA) solution of the method of moments matrix system arising from volume integral equations (VIEs). The acceleration is conducted in two aspects: 1) speedup of the calculation of near-field interactions of the MLFMA and 2) improvement of the iterative convergence. Numerical experiments indicate that the proposed algorithm can significantly reduce the runtime and memory usage without degrading the accuracy, in comparison with the VIE solver purely accelerated by the MLFMA. Additionally, for piecewise homogeneous complex plasmonic structures, the proposed algorithm can be more efficient and accurate than the commonly used surface integral equation solver.

Compression of Translation Operator Tensors in FMM-FFT-Accelerated SIE Solvers via Tucker Decomposition

Tucker decompositions (also known as higher order singular value decompositions) are used to lessen the memory requirements of translation operator tensors in fast multipole method-fast Fourier transform accelerated surface integral equation solvers. For many practical examples, the proposed drop-in code enhancement results in over 90% reduction in these tensors' storage requirements while imposing negligible computational overhead, thus significantly enhancing the solvers' application range on fixed computational resources.

Transient analysis of electromagnetic wave interactions on plasmonic nanostructures using a surface integral equation solver.

Transient electromagnetic interactions on plasmonic nanostructures are analyzed by solving the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) surface integral equation (SIE). Equivalent (unknown) electric and magnetic current densities, which are introduced on the surfaces of the nanostructures, are expanded using Rao-Wilton-Glisson and polynomial basis functions in space and time, respectively. Inserting this expansion into the PMCHWT-SIE and Galerkin testing the resulting equation at discrete times yield a system of equations that is solved for the current expansion coefficients by a marching on-in-time (MOT) scheme. The resulting MOT-PMCHWT-SIE solver calls for computation of additional convolutions between the temporal basis function and the plasmonic medium's permittivity and Green function. This computation is carried out with almost no additional cost and without changing the computational complexity of the solver. Time-domain samples of the permittivity and the Green function required by these convolutions are obtained from their frequency-domain samples using a fast relaxed vector fitting algorithm. Numerical results demonstrate the accuracy and applicability of the proposed MOT-PMCHWT solver.

Efficient and Accurate Computational Electromagnetics Approach to Precipitation Particle Scattering Analysis Based on Higher-Order Method of Moments Integral Equation Modeling

AbstractA new full-wave computational electromagnetics (CEM) approach to precipitation particle scattering analysis based primarily on a higher-order method of moments (MoM) for solving surface integral equations (SIEs) is proposed, as an alternative and addition to the conventionally used tools in this area. This is a well-established CEM approach that has not been applied, evaluated, discussed, or compared with other approaches in the scattering analysis of precipitation particles so far. Several characteristic examples of scattering from precipitation particles of various shapes demonstrate the capabilities and potential of the presented numerical methodology, and discuss its advantages over both discrete dipole approximation (DDA) and -matrix methods in cases considered. In particular, it is shown that the higher-order MoM-SIE approach is much faster, more accurate, and more robust than the DDA method, and much more general and versatile than the -matrix method. In addition, the paper illustrates problems with the convergence of the DDA method in some cases with high-contrast dielectric materials and large electrical sizes of particles and with the convergence of the -matrix method in some cases with electrically large or geometrically complex (viz., with a large aspect ratio) particles. For simulations of continuously inhomogeneous scatterers (e.g., melting ice particles), a higher-order MoM volume integral equation (VIE) technique is used, as the study’s secondary methodology. The results also indicate the necessity for numerically rigorous and computationally efficient realistic precipitation particle modeling in weather scattering applications, which is becoming even more important as the sensor frequencies of radar/radiometric systems are increasing.

Applying Loop-Flower Basis Functions to Analyze Electromagnetic Scattering Problems of PEC Scatterers

This paper discusses the application of loop-flower basis functions for solving surface integral equations involved in electromagnetic scattering problems on perfectly electrically conducting surfaces. Flower-shaped basis functions are proposed to replace the conventional star basis functions. The flower basis functions are defined based on mesh nodes instead of surface triangles. It is shown that the loop-flower basis functions not only can be used to handle the electromagnetic scattering problems at very low frequencies, but also can be directly used to implement Calderon preconditioners for EFIEs.

Homogenization of three-dimensional metamaterial objects and validation by a fast surface-integral equation solver

A homogenization model is applied to describe the wave interaction with finite three-dimensional metamaterial objects composed of periodic arrays of magnetodielectric spheres and is validated with full-wave numerical simulations. The homogenization is based on a dipolar model of the inclusions, which is shown to hold even in the case of densely packed arrays once weak forms of spatial dispersion and the full dynamic array coupling are taken into account. The numerical simulations are based on a fast surface-integral equation solver that enables the analysis of scattering from complex piecewise homogeneous objects. We validate the homogenization model by considering electrically large disk- and cube-shaped arrays and quantify the accuracy of the transition from an array of spheres to a homogeneous object as a function of the array size. Simulation results show that the fields scattered from large arrays with up to one thousand spheres and equivalent homogeneous objects agree well, not only far away from the arrays but also near them.

Efficient Evaluation of Weakly Singular Integrals Arising From Electromagnetic Surface Integral Equations

Solving electromagnetic (EM) problems by integral equation methods relies on the accurate evaluation of singular integrals related to the Green's function. In the method of moments (MoM) with the Rao-Wilton-Glisson (RWG) basis function for solving surface integral equations (SIEs), the gradient operator on the scalar Green's function can be moved onto the basic function and testing function, resulting in a 1/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> weak singularity in the integral kernel, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> is the distance between an observation point and a source point. The weakly singular integral can be evaluated with the well-known Duffy's method, but it requires a two-fold numerical integration. In this work, we develop a novel approach to evaluate the singular integral by using a local polar coordinate system. The approach can automatically cancel the singularity and reduce the integral to a one-fold numerical integration by deriving a closed-form expression for the integral over the polar coordinate. Numerical examples are presented to demonstrate the effectiveness of the approach.

Selected theoretical and numerical aspects of fast volume and surface integral equation solvers for simulation of elasto-acoustic waves in complex inhomogeneous media

Comparative analysis is considered of two fast FFT-matrix-compression based elasto-acoustic integral equation solvers, employing volumetric and surface formulations, and designed to analyze sound propagation inside a human head; in particular to examine mechanisms of energy transfer to the inner ear through airborne as well as non-airborn path-ways, and to assess effectiveness of noise-protection devices. Verification tests involving the fast surface and volume integral equation solvers are carried out comparing their predictions with those following from an analytical solution of field distribution in an elasto-acoustic layered sphere.Resultsiare presented of representative numerical simulations of acoustic energy transfer to the cochlea for a human head model containing a detailed geometry representation of the outer, middle, and inner ear. The geometry model consists of: (1) the outer surface of the skin surrounding the skull and containing (2) the outer ear represented by its exterior surface, the surface of the auditory canal, and the tympanic membrane modeled as a finite-thickness surface, (3) the middle ear consisting of the system of ossicles and supporting structures, (4) the skull described by its external surfaces and including (5) a set of surfaces representing the inner ear (boundaries of the cochlea, the vestibule, and the semi-circular canals). *This work is supported by a grant from US Air Force Office of Scientific Research.

A Novel Meshless Scheme for Solving Surface Integral Equations With Flat Integral Domains

Numerical solutions for electromagnetic (EM) integral equations rely on the discretization of integral domains and the use of meshes for geometric description. Meshing geometries is very tedious, especially for complicated structures with many details (tiny parts) and geometric discontinuities (corners or edges), and remeshing could be required in many scenarios. To reduce the costs of generating quality meshes, meshless or mesh-free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in EM community. The meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally. In this work, we first address the traditional meshless scheme for solving EM integral equations based on the moving least square (MLS) approximation for unknown currents and the use of background meshes for integrations, and then develop a novel meshless scheme by applying the Green's lemma to the EM surface integral equations with flat domains. The novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain. Since only the domain boundaries are discretized and no background meshes are needed, the scheme is truly meshless. Numerical examples for EM scattering by flat-surface objects are presented to demonstrate the effectiveness of the novel scheme.

Efficient Electromagnetic Analysis for Interconnect and Packaging Structures Using Dual Basis Function in the Method of Moments

Efficient electromagnetic analysis for interconnect and packaging structures is developed by solving surface integral equations (SIEs) with the method of moments. Since the SIEs include both electric and magnetic currents as unknowns on the interfaces of dielectric materials, two basis functions are needed to represent them. The robust Rao-Wilton-Glisson (RWG) basis function is a natural choice to represent the electric current, but how one represents the magnetic current is less obvious. One may employ n̂ × RWG basis function, where n̂ is a unit normal vector on the material interfaces or the RWG basis function again to represent the magnetic current, but both choices are not ideal. In this paper, we use the dual basis function proposed by Chen and Wilton in 1990 to represent the magnetic current, and find that it is robust though complicated in implementation. The interconnect and packaging analysis requires extra cares on numerical procedure because of the multiscale features and low-frequency effects of the structures. With the use of RWG and dual basis functions together, both electric and magnetic currents are well represented and the system matrix is well conditioned. Therefore, the low-frequency effects are alleviated and the poor mesh quality can be sustained without special treatment. Numerical examples for typical structures are presented to illustrate the merits of the scheme.

Parameters Extraction of Millimeter Wave Circuits by Hybrid Vector Finite Element and Moment Method Combined with SOC Technique

This paper presents a hybrid method, which couples the vector finite element method (FEM) and method of moment (MOM) for analyzing the field and current distribution of the millimeter wave circuits. The FEM is applied to handle the interior region of dielectric bodies and MOM is used to solve surface integral equations. Then, These integral expressions are coupled into the FEM equations through the continuity of the tangential fields across the connection boundaries. Simultaneously, the short-open calibration (SOC) technique is used for predicting accurately the scattering parameters of the circuits. Numerical results are well compared with those published in the previous literatures.

Eddy current analysis by integral equation method utilizing loop electric and surface magnetic currents as unknowns

The surface integral equations whose unknowns are the surface electric and magnetic currents are widely used in eddy current analysis. However, when the skin depth is thick, computational error is increased especially in obtaining electromagnetic fields near the edge of the conductor. In order to obtain the electromagnetic field accurately, we propose an approach to solve surface integral equations utilizing loop electric and surface magnetic currents as unknowns.

Automatic mesh generation using single- and double-node segmentation techniques

The desired properties of quadrilateral mesh-generation techniques, well-suited for solving surface integral equations by the method of moments, are discussed. Based on this investigation, two iterative techniques for segmentation of electrically large surfaces are developed. Each iteration consists of two steps: (1) long edges are subdivided into the minimal number of short edges; and (2) large plates are subdivided by using either one interior node per long edge (the single-node technique), or two interior nodes per long edge (the double-node technique). In each of these cases, the subdivision of a large plate is performed by using three specific basic schemes, not affected by the subdivision of neighboring large plates. In addition, a constraint for segmentation of long interior edges is proposed, enabling generation of more uniform meshes. The proposed techniques are applied to the methods based on both subdomain and entire-domain approximation. It is found that, in most cases, the double-node technique is superior to the single node technique.