High-Accuracy Mesh-Free Quadrature for Trimmed Parametric Surfaces and Volumes. (December 2021) - Gunderman, David; Weiss, Kenneth; Evans, John A.
Abstract: This work presents a high-accuracy, mesh-free, generalized Stokes theorem-based numerical quadrature scheme for integrating functions over trimmed parametric surfaces and volumes. The algorithm relies on two fundamental steps: (1) We iteratively reduce the dimensionality of integration using the generalized Stokes theorem to line integrals over trimming curves, and (2) we employ numerical antidifferentiation in the generalized Stokes theorem using high-order quadrature rules. The scheme achieves exponential convergence up to trimming curve approximation error and has applications to computation of geometric moments, immersogeometric analysis, conservative field transfer between high-order curvilinear meshes, and initialization of multi-material simulations. We compare the quadrature scheme to commonly-used quadrature schemes in the literature and show that our scheme is much more efficient in terms of number of quadrature points used. We provide an open-source implementation of the scheme in MATLAB as part of QuaHOG, a software package for Quadrature of High-Order Geometries. Highlights: We introduce new quadrature rules for trimmed NURBS surfaces and volumes. The rules converge spectrally up to geometric error due to trimming curve approximation. The rules achieve high-order convergence under refinement of the trim approximations. The rules do not require mesh generation of the region of interest. The rules converge with less quadrature points than state-of-the-art methods.