The moment method is the predominant approach for the solution of electromagnetic boundary integral equations. Traditional moment method discretizations rely on the projection of solution currents onto basis sets that must satisfy strict continuity properties to model physical currents. The choice of basis sets is further restricted by the tight coupling of traditional functional descriptions to the underlying geometrical approximation of the scattering or radiating body. As a result, the choice of approximation function spaces and geometry discretizations for a given boundary integral equation is significantly limited. A quasi-meshless partition of unity based method called the Generalized Method of Moments (GMM) was recently introduced to overcome some of these limitations. The GMM partition of unity scheme affords automatic continuity of solution currents, and therefore permits the use of a much wider range of basis functions than traditional moment methods. However, prior to the work in this thesis, GMM was limited in practical applicability because it was only formulated for a few geometry types, could not be accurately applied to arbitrary scatterers, e.g. those with mixtures of geometrical features, and was not amenable to traditional acceleration methodologies that would permit its application to electrically large problems. The primary contribution of this thesis is to introduce several new GMM formulations that significantly expand the capabilities of the method to make it a practical, broadly applicable approach for solving boundary integral equations and overcoming the limitations inherent in traditional moment method discretizations. Additionally, several of the topics covered address continuing open problems in electromagnetic boundary integral equations with applicability beyond GMM. The work comprises five broad contributions. The first is a new GMM formulation capable of mixing both GMM-type basis sets and traditional basis sets in the same discretization. The scheme handles geometries with mixtures of smooth and sharp features using an expanded range of basis functions classes and geometry representations, thereby permitting the application of GMM to arbitrary scatterers with high fidelity. Second, an acceleration method based on the Multilevel Fast Multipole Algorithm is developed that is equally applicable to GMM and traditional higher order moment methods. The third contribution is an interior penalty integral equation formulation that regularizes GMM surface currents and affords significant savings in computational cost. Fourth, a technique for creating arbitrarily smooth GMM current representations based on conformal mapping is advanced. Finally, a GMM formulation for a new surface description for boundary integral equations, the subdivision surface, is proposed. The resulting subdivision-based method holds promise for optimization and uncertainty quantification of scattering and radiating structures through rapid geometry morphing and subwavelength deformation; furthermore it is readily extensible to traditional moment methods. Taken together, these contributions address several open problems and yield a GMM boundary integral equation solver that is highly accurate, flexible, fast, and applicable to a wide range of electromagnetics engineering problems.
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