The most popular method for self-consistent simulation of fields interacting with charged species is using finite difference time domain (FDTD) methods together with Newton's laws of motion to evolve locations and velocities of particles. Despite their popularity, the limitations of FDTD particle-in-cell (EM-FDTDPIC) methods are well-known. To address these, there has been significant interest over the past decade in exploring alternatives. In the past few years, the advances in electromagnetic finite element methods for particle-in-cell (EM-FEMPIC) have advanced by leaps and bounds. The mathematics necessary for implicit FEM methods that are unconditionally stable and charge-conserving are now well understood. Some of these advances are more recent. The next bottleneck necessary to make EM-FEMPIC competitive with the FDTD-based scheme is overcoming computational cost. Our approach to resolving this challenge is to develop two different finite element tearing and integration approaches, and use these to create domain decomposition schemes for EM-FEMPIC. Details of the proposed methodology are presented as well as a number of electrostatic results that demonstrate charge conservation as well as amelioration of costs for a number of problems.

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the ill-conditioning behavior of CFIE at high frequencies. This paper addresses this deficiency through new Calderón-type identities derived from the Vector Potential Integral Equation (VPIE). We construct novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets.

Asymptotic solutions based on the uniform geometrical theory of diffraction (UTD) are available for creeping waves traveling along geodesic paths between source and field points on convex surfaces. The application of these UTD solutions to discretized convex surfaces requires extraction of geometry based parameters such as path length, surface curvature, Fock parameter, surface torsion and divergence factor. As unique contributions of this study, accurate and efficient closed-form expressions and algorithms are proposed to extract these parameters from a triangulated conducting convex surface, and UTD surface fields are predicted using the extracted parameters. A quadratic surface mapping is used in barycentric coordinates to define a curved surface on each triangular facet, first. Then, closed-form expressions are obtained for path length, surface curvature and Fock parameter. Algorithms are developed for surface torsion and divergence factor as well. The extracted and actual values for these parameters are compared for primary and secondary paths between source and field points on a circular cylinder and a sphere. The results obtained by applying a UTD solution to triangulated conducting convex surfaces are also validated with the analytical UTD results for surface fields on a circular cylinder, a sphere and a prolate spheroid.

An algorithm is proposed to identify geodesic rays between two antennas on a faceted electrically large convex conducting platform. The algorithm is based on launching numerous geodesic ray strips from a transmitting antenna in all directions and tracing them until a receiving antenna is reached or a predefined number of facets are traced. It was used to identify two geodesic rays between antennas on a faceted circular cylinder and a faceted ogive. The algorithm is required to predict co-site interference between antennas on an electrically large platform.

Enforcement of current continuity at material junctions in surface integral equation (SIE) formulations for composite objects has generally relied on somewhat intrusive constraint procedures that complicate system assembly and preconditioning, especially in the context of low rank solvers. The method proposed alleviates these problems by enforcing constraints in a manner equivalent to earlier approaches, but with a less invasive implementation. In addition to solving these challenges, the method is sufficiently general to handle other types of constraints, e.g., unknowns in symmetry planes.

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 Calderón preconditioner. Several results presented serve to validate the proposed approach as well as demonstrate some of its capabilities.

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.

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.

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.

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.

The generalized method of moments (GMM) is a technique to discretize integral equations that permits integration of different types of basis functions as well as different geometric descriptions using a partition of unity framework. While accuracy and efficacy of the method have been demonstrated, the integration quadratures required to compute the inner products are often high, as they have to respect spatial variation of the integrand. To overcome this problem, we introduce an interior penalty function method within the GMM framework. The penalty formulation yields solutions that are smooth and accurate both in surface currents and fields with significantly lower numerical quadrature orders than would be required for the uncompensated operator. To demonstrate and analyze the method, we conduct an analytical and numerical investigation of the properties of the penalty method applied to the 2-D TE Z electric field integral equation (EFIE) operator.

Past implementations of the Generalized Method of Moments utilized nonsmooth functions in current approximation bases that were difficult to integrate accurately. Herein we devise a method for defining arbitrarily smooth functions on polygonal GMM subdomains using Schwarz-Christoffel conformal mapping. The resulting functions are much smoother, and can be integrated more accurately using numerical quadrature.

We develop a Multilevel Fast Multipole Algorithm (MLFMA) for higher order Moment Methods that, unlike extant higher order MLFMA implementations, maintains optimal scaling independent of the patch size. We employ the new scheme to both accelerate preconditioner computation and apply it to the Generalized Method of Moments to analyze scattering from large PEC objects.

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 Green's 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.

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.

When the Electric Field Integral Equation is discretized via the Generalized Method of Moments, small current irregularities sometimes appear. We propose a cause for these current deviations and advance a solution based on a Nitsche-type constraint based stabilization method. Two dimensional results demonstrate the method, although it is directly extensible to the three dimensional case.

We present a highly flexible framework that permits easy hybridization of multiple basis function spaces, within the same simulation domain, for use in solution of integral equations. The method is constructed using the Generalized Method of Moments (GMM), that uses overlapping domains and a partition of unity functions defined on these domains to ensure continuity of currents. We leverage this feature to construct a method that combines arbitrary classes of basis functions on neighboring regions of a PEC scattering surface. Because the continuity of surface currents is inherent in the GMM description, basis sets may be chosen according to local physical and geometrical requirements, permitting engineering of function spaces for optimal representation. In this paper, we present example simulations that achieve hybridization with RWG basis functions. Examples of hybridization with other functions will be presented at the conference.