Iso-Geometric Integral Equation Solvers and Their Compression via Manifold Harmonics

Abstract

The state of art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low frequency and dense discretization breakdown, preconditioning, and so on. Likewise, the community has seen extensive investment in development of methods for higher order analysis, in both geometry and physics. Unfortunately, these standard geometric descriptors are continuous, but their normals are discontinuous at the boundary between triangular tessellations of control nodes, or patches, with a few exceptions; as a result, one needs to define additional mathematical infrastructure to define physical basis sets for vector problems. In stark contrast, the geometric representation used for design are second order differentiable almost everywhere on the surfaces. Using these description for analysis opens the door to several possibilities, and is the area we explore in this paper. Our focus is on Loop subdivision based isogeometric methods. In this paper, our goals are two fold: (i) development of computational infrastructure for isogeometric analysis of electrically large simply connected objects, and (ii) to introduce the notion of manifold harmonics transforms and its utility in computational electromagnetics. Several results highlighting the efficacy of these two methods are presented.