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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...
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...
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.
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.
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...
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 general framework based on the Generalized Method of Moments that enables inclusion of multiple local geometry descriptions and orders and multiple basis function types and orders in the solution of Moment Method systems. The resulting method allows arbitrary mixing of these parameters in different regions of the scatterer so that both geometric and surface current approximations may...
Hierarchical and multiresolution basis function schemes have been successfully employed as preconditioners for solution of the Combined Field Integral Equation (CFIE) in the context of Method of Moments (MoM) solvers. A hierarchical approach is attractive because it reduces susceptibility of the descretized CFIE-MoM system to low-frequency breakdown, and also acts as an effective preconditioner for...
The study of light-matter interactions is of great importance to many applications ranging from optical device characterization to material design to analysis of photonic band gap (PBG) structures etc. Many nanostructures of interest possess complex geometries and topologies that are difficult to model geometrically, and even more difficult to characterize in terms of electromagnetic response; for...
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...
Analysis of periodic structures in time domain is an increasingly important tool in the design of a wide range of novel structures. Time Domain Integral Equation based methods provide an accurate means of solving transient periodic problems, but require costly convolutions that scale as O(Nt2Ns2), where Nt and Ns are the temporal and spatial degrees of freedom. This work proposes a fast method for...
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...
In this work we derive a time domain Green's function for a periodic source distribution above a layered medium and a corresponding acceleration scheme. A representation using sums of complex exponentials allows closed form evaluation of the requisite integrals, leading to a fully closed form expression for the Green's function. A separable expansion based on this expression is accelerated in space...
Transient analysis of periodic structures finds application in a number of areas in applied electromagnetics. The cost of these analyses scales as O(Ns2Nt2), where Ns and Nt are the number of spatial and temporal degrees of freedom. In this paper, we (1) present a method that reduces the cost of this evaluation to O(NsNt log2 Nt) for quasiplanar structures and (2) integrate this into a late time stable...
In this work, the authors present a method for analyzing field enhancement in dense, disordered, semiconductor nanowire forests coated in noble metal nanoparticles using the volume integral equation. A hybrid fast solver employing the fast multipole method (FMM) and accelerated Cartesian expansions (ACE) is utilized to rapidly solve for fields in geometries which would otherwise be computationally...
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