High-accuracy mesh-free quadrature for trimmed parametric surfaces and volumes

TL;DR

This paper introduces a high-accuracy, mesh-free quadrature method for integrating over trimmed parametric surfaces and volumes, leveraging the generalized Stokes theorem for exponential convergence and efficiency.

Contribution

It presents a novel, generalized Stokes theorem-based quadrature scheme that reduces dimensionality and employs high-order numerical antidifferentiation, outperforming existing methods in efficiency.

Findings

Achieves exponential convergence up to trimming curve error

More efficient than traditional quadrature schemes in point usage

Applicable to geometric moments, analysis, and multi-material simulations

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.