A Complete Helmholtz Decomposition on Multiply Connected Subdivision Surfaces and Its Application to Integral Equations

The analysis of electromagnetic (EM) scattering in the isogeometric analysis (IGA) framework based on the Loop subdivision has long been restricted to simply connected geometries. The inability to analyze multiply connected objects is a glaring omission. In this article, we address this challenge. IGA provides seamless integration between the geometry and analysis using the same basis set to represent both. In particular, IGA methods using subdivision basis sets exploit the fact that the basis functions used for surface description are smooth (with continuous second derivatives) almost everywhere. On simply connected surfaces, this permits the definition of basis sets that are divergence-free and curl-free. What is missing from this suite is a basis set that is both divergence-free and curl-free, a necessary ingredient for a complete Helmholtz decomposition of currents on multiply connected structures. In this article, we achieve this missing ingredient numerically using random polynomial vector fields. We show that this basis set is analytically divergence-free and curl-free. Furthermore, we show that these bases recover curl-free, divergence-free, and both curl-free and divergence-free fields. Finally, we use this basis set to discretize a well-conditioned integral equation for analyzing perfectly conducting objects and demonstrate excellent agreement with other methods.