D. Dault

The Generalized Method of Moments for Electromagnetic Boundary Integral Equations

The generalized method of moments (GMM) is a partition of unity based technique for solving electromagnetic and acoustic boundary integral equations. Past work on GMM for electromagnetics was confined to geometries modeled by piecewise flat tessellations and suffered from spurious internal line charges. In the present article, we redesign the GMM scheme and demonstrate its ability to model scattering from PEC scatterers composed of mixtures of smooth and non-smooth geometrical features. Furthermore, we demonstrate that because the partition of unity provides both functional and effective geometrical continuity between patches, GMM permits mixtures of local geometry descriptions and approximation function spaces with significantly more freedom than traditional moment methods.

Accelerated Cartesian Expansions for the Rapid Solution of Periodic Multiscale Problems

We present an algorithm for the fast and efficient solution of integral equations that arise in the analysis of scattering from periodic arrays of PEC objects, such as multiband frequency selective surfaces (FSS) or metamaterial structures. Our approach relies upon the method of Accelerated Cartesian Expansions (ACE) to rapidly evaluate the requisite potential integrals. ACE is analogous to FMM in that it can be used to accelerate the matrix vector product used in the solution of systems discretized using MoM. Here, ACE provides linear scaling in both CPU time and memory. Details regarding the implementation of this method within the context of periodic systems are provided, as well as results that establish error convergence and scalability. We also demonstrate the applicability of this algorithm by studying several exemplary electrically dense systems.

Subdivision based isogeometric analysis technique for electric field integral equations for simply connected structures

The analysis of electromagnetic scattering has long been performed on a discrete representation of the geometry. This representation is typically continuous but not differentiable. The need to define physical quantities on this geometric representation has led to development of sets of basis functions that need to satisfy constraints at the boundaries of the elements/tessellations (viz., continuity of normal or tangential components across element boundaries). For electromagnetics, these result in either curl/div-conforming basis sets. The geometric representation used for analysis is in stark contrast with that used for design, wherein the surface representation is higher order differentiable. Using this representation for both geometry and physics on geometry has several advantages, and is elucidated in Hughes et al. (2005) 7. Until now, a bulk of the literature on isogeometric methods have been limited to solid mechanics, with some effort to create NURBS based basis functions for electromagnetic analysis. In this paper, we present the first complete isogeometry solution methodology for the electric field integral equation as applied to simply connected structures. This paper systematically proceeds through surface representation using subdivision, definition of vector basis functions on this surface, to fidelity in the solution of integral equations. We also present techniques to stabilize the solution at low frequencies, and impose a Calderon preconditioner. Several results presented serve to validate the proposed approach as well as demonstrate some of its capabilities.

A quasianalytical time domain solution for scattering from a homogeneous sphere.

A transient spherical multipole expansion-like solution for acoustic scattering from a spherical object is derived within a mesh-free and singularity-free time domain integral equation (TDIE) framework for the sound-soft, sound-rigid and penetrable cases. The method is based on an expansion of the time domain Greens function that allows independent evaluation of spatial and temporal convolutions. The TDIE system is solved by descretizing the integral equations in space and time, forming a matrix system via the method of moments, and solving the system with the marching on in time algorithm. Spatial discretization using tesseral harmonics leads to closed form expressions for spatial integrals, and use of a strictly band limited temporal interpolant permits efficient, accurate computation of temporal convolutions via numerical quadrature. The accuracy of these integrations ensures late time stability and accuracy of the deconvolution data. Results presented demonstrate the accuracy and convergence of the approach for broadband simulations compared with Fourier transformed analytical data.

A Mixed Potential MLFMA for Higher Order Moment Methods With Application to the Generalized Method of Moments

The application of the multilevel fast multipole algorithm (MLFMA) to higher order moment method discretizations is a continuing open problem. Herein, we present a point-based mixed potential variant of the MLFMA algorithm that exhibits an MLFMA tree of arbitrary height with no restriction related to basis function support size, efficient nearfield precomputation, and that maintains favorable scaling for any mixture of low- and high-order bases. The flexibility of the algorithm is also leveraged to accelerate the precomputation of algebraic preconditioners. We demonstrate the method through application to the generalized method of moments (GMM), a recently introduced moment method discretization capable of combining both low- and high-order bases and geometries in the same simulation; however, the method may be used to accelerate other higher order moment methods as well.

A flexible framework for the solution to surface scattering problems using integral equations

Most method of moment solutions to integral equations in electromagnetics use the Rao Wilton Glisson (RWG) basis functions. These functions, which are constructed on a triangulation of the geometry are limited by their inherent need to satisfy continuity conditions. Recent developments by the authors have resulted in a new basis function scheme for integral equations called the Generalized Method of Moments (GMM). This scheme permits a wide variety of basis functions and their arbitrary mixtures. These functions are described on a locally smooth representation of the underlying surface. However, it is not possible to provide such a locally smooth representation in the presence of geometric singularities such as edges and corners. In this work, we address this problem by hybridizing the GMM patch construction and basis function definitions with RWG basis functions and piecewise flat triangulations. We will describe a scheme to automatically choose and mix GMM and RWG patch definitions, and basis functions. This hybridization can also be extended to other basis function types (and corresponding geometry descriptions). Preliminary results are presented that validate the method.

Smooth Surface Blending for the 2-D Generalized Method of Moments

Accurate computation of scattered electromagnetic fields is challenging due to approximation inherent: 1) in the geometrical description of the object, and 2) functions that are defined on this geometric representation. In this letter, we present a partition of a unity-based scheme capable of recreating scattering geometries to very low error with an arbitrary degree of smoothness and global continuity of normals in two dimensions. This method is then coupled with the Generalized Method of Moments, a recently developed meshless boundary integral equation method, to compute scattered fields in two dimensions. Several examples illustrating accuracy and convergence of this approach are discussed.

Subdivision surfaces for electromagnetic integral equations

Subdivision surfaces are a powerful geometrical modeling tool that has been used extensively in computer graphics. In this paper we develop a general subdivision-based basis scheme that can be applied to a wide range of electromagnetic integral equations, and demonstrate several features and advantages.

Isogeometric analysis of integral equations using subdivision

Isogeometric analysis (IGA) has recently become popular in computational science during the past decade or so. IGA tries to to unify both geometric and field representation; in other words, both the geometry and the fields are represented using the same underlying basis set. However, while the concept of IGA for differential equations is more common, extension to an integral equation framework is significantly more challenging. In this work, we present for the first time, the IGA as applied to integral equations encountered in electromagnetics. The presented approach relies on the subdivision scheme for both geometry and function representation. Results presented attest to the viability of the method.

Method of Moments: As Applied to the Solution of Electromagnetic Integral Equations

Integral equation-based techniques are one of the primary solution approaches for analysis of electromagnetic systems. Most electromagnetic integral equations cannot be solved analytically, and numerical methods are therefore required. The most common numerical solution technique for these types of equations is the moment method. Over the past few decades, several bottlenecks in terms of accuracy and cost of the moment method have been solved, so much so that analysis of problems in excess of millions of degrees of freedom is now almost routine. Given the ubiquity of these numerical methods, and because the moment method and the formulation of integral equations are inextricably linked, this article presents a gentle introduction to some common classes of electromagnetic integral equations and accompanying moment method formulations to numerically solve them. Recent progress and unsolved and open problems in moment methods are also presented. Keywords: electromagnetics; integral equations; method of moments