Loop subdivision has become a standard tool in computer graphics forgenerating (second order) smooth surfaces of arbitrary topology. A few of its most attractive features include that it is inherently multiresolution, scalable, allows for local adaptive refinement, compression, and can readily handle smooth deformations of the surface. These features make Loop subdivision an attractive candidate for use in an isogeometric analysis (IGA) framework, in which geometry and physical quantities (e.g., surface currents, pressures, charges, etc.) are expressed in the same basis. The goal of this thesis is to develop a collection of efficient and accurate methods that inherit these features for both analysis and design in acoustics and electromagnetics problems.This thesis addresses some of the fundamental problems that arise in IGA of acoustic and electromagnetic phenomena. First, we develop representation of scalar and vector quantities on both simply connected and multiply connected Loop subdivision surfaces. This leverages the high order nature of Loop subdivision to construct a complete Helmholtz decomposition that avoids issues that plague standard methods. We use this complete Helmholtz decomposition to discretize well-conditioned integral equations for scattering analysis. Next, we address the challenge of computational cost associated with evaluating the many additional inner products that arise in these integral equation formulations; this bottleneck is overcome by using a wideband multilevel fast multipole algorithm (MLFMA) and mixed-potential field representations to accelerate computations in the iterative solver. Then, we introduce the notion of manifold harmonics as a means to (i) compress geometry and integral equation matrices, (ii) enrich physical quantities hierarchically for acoustic and electromagnetic scattering, and (iii) utilize local enrichment to better handle representation of physical quantities on structures with challenging features. Finally, we utilize the manifold harmonics basis on a Loop subdivision surface for the shape reconstruction problem; specifically, they allow for multi-resolution representation (and editing) of complex surfaces and functions defined thereupon. Taken together, these contributions address several open problems and yield an integrated design-analysis approach that is highly accurate, flexible, and fast with applicability to a range of engineering problems.
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