Rao-Wilton-Glisson Functions Research Articles

Fast Convergent Quadrature Method for Evaluating the RWG- and SWG-Related Convolutional Integrals

The Rao–Wilton–Glisson (RWG) basis function (BF)-related convolutional surface integrals are intrinsically 2-D. The Schaubert–Wilton–Glisson (SWG) BF-related convolutional volume integrals are 3-D in nature. It has been shown that such integrals can be reduced to a summation of several 1-D integrals by repeatedly applying the divergence theorem. To the best of our knowledge, there are no known analytic formulas for the 1-D integrals, and consequently, one must choose a quadrature rule to get the final results. The 1-D integrals are prone to numerical errors when the observation point is close to the source. We propose that sinh-related transformations can be used to improve the accuracy, which has been shown to have exponential convergence with respect to the number of Gauss–Legendre quadrature points in the numerical examples. We can reach ten or more significant digits in the convolutional integrals pertinent to RWG or SWG functions with a small number of quadrature points.

On the Use of Entire-Domain Basis Functions and Fast Factorizations for the Design of Modulated Metasurface

Entire-domain, spectral basis functions (BFs) have witnessed recent interest in the integral-equation analysis of large metasurface (MS) antennas modeled via homogenized impedance boundary conditions. We present a formulation employing classical Galerkin test via Rao–Wilton–Glisson functions yet assembled to represent entire-domain div-conforming BFs for the shape of interest (e.g., circular/coaxial waveguide modes). On the one hand, the rationale is that entire-domain, spectral BFs afford a significant economy in the number of necessary unknowns; on the other hand, being expressed as combination of Rao–Wilton–Glisson functions, reaction integrals are computed with optimum cost via fast methods. This is applied to reduce the cost of the optimization process used to design MS antennas based on spatially modulated reactance profiles. The authors support the method proposed, presenting criteria to define the entire-domain functions, considering the overall numerical complexity in an optimization framework, and providing a convergence analysis and the numerical results for holographic leaky-wave antennas, relevant in the MS context.

RWG Basis Functions for Accurate Modeling of Substrate Integrated Waveguide Slot-Based Antennas

In this article, we propose the implementation of a triangular mesh along with Rao-Wilton-Glisson (RWG) basis functions to improve the performance and flexibility of a mixed mode-matching (MM)/method-of-moments (MoM) algorithm dedicated to substrate integrated waveguide (SIW) antennas analysis and optimization. These basis functions represent equivalent magnetic currents on either coupling or radiating apertures. In the initial implementation, the entire domain sine basis function was presented, providing fast and accurate simulations for electrically thin rectangular slots. In order to treat wider slots and other geometries, we add here RWG functions, present the theoretical framework of this approach, and highlight its potential through concrete examples.

Efficient Numerical Analysis of Dielectric Resonator Antenna Arrays

Synthetic Functions eXpansion (SFX) is adopted to analyze dielectric resonator antenna (DRA) arrays. First, a two-independent current exciting method is introduced for radiating problems that can be perfectly combined with Rao–Wilton–Glisson functions and T-junctions. Then, a multilevel decomposition scheme is presented to organize the impedance matrix and to generate synthetic functions. This scheme not only expands the application of SFX to a multidielectric system but also benefits the efficiency of SFX. Finally, a four-element DRA array is analyzed using the proposed approach, and results well validate the accuracy and great advantages on efficiency and memory cost of the approach.

Solutions for General-Purpose Electromagnetic Problems Using the Random Auxiliary Sources Method

A 3-D general-purpose implementation of the random auxiliary sources (RASs) method is proposed, benefiting from the Rao–Wilton–Glisson (RWG) testing functions. The testing procedure is presented in terms of the RWG function parameters. The performance of the proposed RAS with the RWG testing function is compared with point matching. Also, a simple treatment for mixed boundary problems is introduced to enable solving the complex electromagnetic problems. Furthermore, a waveport formulation is proposed that is better suited for the iterative approach of the RAS method to facilitate the modular analysis of components. A gradient-based testing formulation for the transmitting port is exploited to enforce the matching to the forward wave only on a waveport. Nevertheless, several test cases are presented to evaluate the accuracy and performance of the proposed method in comparison to the commercial full-wave solvers showing a reasonable agreement.

An Efficient Fast Algorithm for Accelerating the Time-Domain Integral Equation Discontinuous Galerkin Method

An efficient fast algorithm for accelerating the time-domain integral equation discontinuous Galerkin (TDIEDG) method for analyzing the transient scattering from electrically large complex objects is proposed. The TDIEDG is discretized using the monopolar Rao–Wilton–Glisson (RWG) function and shifted Lagrange polynomial function in spatial and time domain, respectively. The final system of equations can be solved iteratively using the classical marching-on-in-time scheme. Taking advantage of the approximate evaluation of the vector and scalar potential terms combined with the Taylor series expansion of the transient far field, the computational burden of the TDIEDG solver has been significantly reduced. In addition, it is flexible and convenient to analyze object with nonconformal discretization due to the employment of the monopolar RWG functions.

EFIE-Tuned Testing Functions for MFIE and CFIE

A recently developed numerical technique for improving the accuracy of the magnetic-field integral equation and the combined-field integral equation with low-order discretizations using the Rao–Wilton–Glisson functions is demonstrated on iterative solutions of large-scale complex problems, in order to prove the effectiveness of the proposed strategy as an alternative way for accurate and efficient analysis of multifrequency applications.

Simulation of multiscale structures using equivalence principle algorithm with grid-robust higher order vector basis

In this paper, equivalence principle algorithm (EPA) with grid-robust higher order vector basis (GRHOVB) is proposed to solve the multiscale problems. The EPA is a kind of domain decomposition method which transforms the interaction of objects into the interaction between virtual equivalence surfaces. Compared with traditional Curvilinear Rao–Wilton–Glisson function, GRHOVB can be used on the equivalence surface to reduce the number of unknowns and also to improve the accuracy. The tap basis is utilized to deal with current continuity when the object is intercepted by an equivalence surface. The equations are simplified further to improve the tap basis scheme. Moreover, to accelerate the solution, multilevel fast multipole algorithm is utilized in EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

Analytical Shape Derivatives of the MFIE System Matrix Discretized With RWG Functions

An analytical formula for the shape derivative of the magnetic field integral equation (MFIE) method of moments (MoM) system matrix (or impedance matrix) is derived and validated against finite difference formulas. The motivation for computing the shape derivatives stems from adjoint variable methods (AVM). The Galerkin system matrix is constructed by means of Rao-Wilton-Glisson (RWG) basis and testing functions. The shape derivative formula yields an integral representation which is of same singularity order as the integrals appearing in the traditional MFIE system matrix.

An efficient method for electromagnetic scattering analysis

We present a novel method to solve the magnetic field integral equation (MFIE) using the method of moments (MoM) efficiently. This method employs a linear combination of the divergence-conforming Rao–Wilton–Glisson (RWG) function and the curl-conforming n×RWG function to test the MFIE in MoM. The discretization process and the relationship of this new testing function with the previously employed RWG and n×RWG testing functions are presented. Numerical results of radar cross section (RCS) data for objects with sharp edges and corners show that accuracy of the MFIE can be improved significantly through the use of the new testing functions. At the same time, only the commonly used RWG basis functions are needed for this method.

Solutions of large-scale electromagnetics problems involving dielectric objects with the parallel multilevel fast multipole algorithm

Fast and accurate solutions of large-scale electromagnetics problems involving homogeneous dielectric objects are considered. Problems are formulated with the electric and magnetic current combined-field integral equation and discretized with the Rao-Wilton-Glisson functions. Solutions are performed iteratively by using the multilevel fast multipole algorithm (MLFMA). For the solution of large-scale problems discretized with millions of unknowns, MLFMA is parallelized on distributed-memory architectures using a rigorous technique, namely, the hierarchical partitioning strategy. Efficiency and accuracy of the developed implementation are demonstrated on very large problems involving as many as 100 million unknowns.

Generalized method of moments: a framework for analyzing scattering from homogeneous dielectric bodies

In this paper, we present a novel framework for discretizing integral equations--specifically, those used for analyzing scattering from dielectric bodies. The candidate integral equations chosen for the analysis are the well-known Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) and the Müller equations. Discrete solutions to these equations are typically obtained by representing the spatial variation of the currents using the Rao-Wilton-Glisson (RWG) basis functions or their higher order equivalents. In this paper, we propose a framework for defining basis functions that departs significantly from those of RWG functions in that approximation functions can be chosen independent of continuity constraints. We will show that using this framework together with a quasi-Helmholtz type representation has a number of benefits. Namely, (i) it shows excellent convergence, (ii) it permits inclusion of different orders of polynomials or different functions as basis functions without imposition of additional constraints, (iii) the method can easily handle nonconformal meshes, and (iv) the method is well conditioned at all frequencies. These features will be demonstrated via a number of numerical experiments.

A Well-Conditioned Integral-Equation Formulation for Efficient Transient Analysis of Electrically Small Microelectronic Devices

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A hierarchically regularized coupled set of time-domain surface and volume electric field integral-equations (TD-S-EFIE and TD-V-EFIE) for analyzing electromagnetic wave interactions with electrically small and geometrically intricate composite structures comprising perfect electrically conducting surfaces and finite dielectric volumes is presented. A classically formulated coupled set of TD-S- and V-EFIEs is shown to be ill-conditioned at low frequencies owing to the hypersingular nature of the TD-S-EFIE. To eliminate low-frequency breakdown in marching-on-in-time solvers for these coupled equations, a hierarchical regularizer leveraging generalized Rao–Wilton–Glisson functions is applied to the TD-S-EFIE; no regularization is applied to the TD-V-EFIE as it is protected from low-frequency breakdown by an identity term. The resulting hierarchically regularized hybrid TD-S- and V-EFIE solver is applicable to the analysis of wave interactions with electrically small and densely meshed structures of arbitrary topology. The accuracy, efficiency, and applicability of the proposed solver are demonstrated by analyzing crosstalk in a six-port transmission line, radiation from a miniature radio-frequency identification antenna, and, plane-wave coupling onto a partially-shielded and fully loaded two-layer computer board. </para>

A Three-Dimensional Precorrected FFT Algorithm for Fast Method of Moments Solutions of the Mixed-Potential Integral Equation in Layered Media

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 P two-dimensional FFT grids each with an average of Np associated triangular elements the method exhibits O(P 2 Np logNp) computational complexity and O(P Np) memory usage. The low-frequency breakdown of the integral equation is eliminated via loop-tree decomposition. A unique combination of O(N logN) 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.

Application of the combined RWG basis functions to the MFIE for the scattering analysis of small sharp-edged objects

We present a novel implementation of the magnetic field integral equation (MFIE) with the use of the combined Rao–Wilton–Glisson (RWG) basis and testing functions. Implementation details are outlined in the context of the method of moments. For a wide variety of small sharp-edged objects, we show that accuracy of the MFIE can be greatly improved with a combination parameter of about 0.2 for the combined RWG functions. This is achieved with relatively coarse discretization for the scatterers and with only minor modifications in the existing codes based on the RWG and n ×RWG functions.

Discretization error due to the identity operator in surface integral equations

We consider the accuracy of surface integral equations for the solution of scattering and radiation problems in electromagnetics. In numerical solutions, second-kind integral equations involving well-tested identity operators are preferable for efficiency, because they produce diagonally-dominant matrix equations that can be solved easily with iterative methods. However, the existence of the well-tested identity operators leads to inaccurate results, especially when the equations are discretized with low-order basis functions, such as the Rao–Wilton–Glisson functions. By performing a computational experiment based on the nonradiating property of the tangential incident fields on arbitrary surfaces, we show that the discretization error of the identity operator is a major error source that contaminates the accuracy of the second-kind integral equations significantly.

Linear-Linear Basis Functions for MLFMA Solutions of Magnetic-Field and Combined-Field Integral Equations

We present the linear-linear (LL) basis functions to improve the accuracy of the magnetic-field integral equation (MFIE) and the combined-field integral equation (CFIE) for three-dimensional electromagnetic scattering problems involving closed conductors. We consider the solutions of relatively large scattering problems by employing the multilevel fast multipole algorithm. Accuracy problems of MFIE and CFIE arising from their implementations with the conventional Rao-Wilton-Glisson (RWG) basis functions can be mitigated by using the LL functions for discretization. This is achieved without increasing the computational requirements and with only minor modifications in the existing codes based on the RWG functions

Műller Formulation for Analysis of Scattering from 3-D Dielectric Objects with Triangular Patching Model

In this paper, we present a set of numerical schemes to solve the Muller integral equation for the analysis of electromagnetic scattering from arbitrarily shaped three-dimensional (3-D) dielectric bodies by applying the method of moments (MoM). The piecewise homogeneous dielectric structure is approximated by planar triangular patches. A set of the RWG (Rao, Wilton, Glisson) functions is used for expansion of the equivalent electric and magnetic current densities and a combination of the RWG function and its orthogonal component is used for testing. The objective of this paper is to illustrate that only some testing procedures for the Muller integral equation yield a valid solution even at a frequency corresponding to an internal resonance of the structure. Numerical results for a dielectric sphere are presented and compared with solutions obtained using other formulations.

A Multiresolution System of Rao–Wilton–Glisson Functions

A multiresolution (MR) basis is described for the method of moments (MoM) analysis of a generic 3-D conductor discretized with triangular cells. The MR basis functions are constructed as linear combination of Rao-Wilton-Glisson (RWG) basis functions, thus allowing direct applicability on existing MoM codes. With respect to previous work by the authors, the generation of the MR functions is significantly simplified, yet keeping similar applicability and performance, in terms of conditioning of the MoM matrix, convergence of the iterative solvers, and possibility of sparsifying the MoM matrix by clipping. Moreover, with the proposed basis the basis-change matrix from RWG to MR is highly sparse at all levels; this allows a more efficient generation of the MR-MoM matrix

Improving the accuracy of the magnetic field integral equation with the linear‐linear basis functions

Basis functions with linear variations are investigated in terms of the accuracy of the magnetic field integral equation (MFIE) and the combined‐field integral equation (CFIE), on the basis of recent reports indicating the inaccuracy of the MFIE. Electromagnetic scattering problems involving conducting targets with arbitrary geometries, closed surfaces, and planar triangulations are considered. Specifically, two functions with linear variations along the triangulation edges in both tangential and normal directions (linear normal and linear tangential (LN‐LT) type) are defined. They are compared to the previously employed divergence‐conforming Rao‐Wilton‐Glisson (RWG) and curl‐conforming × RWG functions. Examples are presented to demonstrate the significant improvement in the accuracy of the MFIE and the CFIE gained by replacing the commonly used RWG functions with the LN‐LT type functions.