Spectral mesh-free quadrature for planar regions bounded by rational parametric curves

TL;DR

This paper introduces spectral, mesh-free quadrature schemes based on Green's theorem for efficiently integrating over planar regions bounded by rational parametric curves, with high accuracy and applications in geometric computations.

Contribution

It develops two novel spectral quadrature methods for rational regions, improving efficiency and accuracy over existing techniques, with guarantees on polynomial exactness.

Findings

Spectrally accurate quadrature converges faster than algebraic rates.

Methods outperform existing quadrature schemes in efficiency and computational time.

Two quadrature schemes offer a trade-off between ease of implementation and polynomial exactness.

Abstract

This work presents spectral, mesh-free, Green's theorem-based numerical quadrature schemes for integrating functions over planar regions bounded by rational parametric curves. Our algorithm proceeds in two steps: (1) We first find intermediate quadrature rules for line integrals along the region's boundary curves corresponding to Green's theorem. (2) We then use a high-order quadrature rule to compute the numerical antiderivative of the integrand along a coordinate axis, which is used to evaluate the Green's theorem line integral. We present two methods to compute the intermediate quadrature rule. The first is spectrally accurate (it converges faster than any algebraic order with respect to number of quadrature points) and is relatively easy to implement, but has no guarantee of polynomial exactness. The second guarantees exactness for polynomial integrands up to a pre-specified degree k with an a priori-known number of quadrature points and retains the convergence properties of the first, but is slightly more complicated. The quadrature schemes have applications to computation of geometric moments, immersogeometric analysis, conservative field transfer between high-order meshes, and initialization of simulations with rational geometry. We compare the quadrature schemes produced using our method to other methods in the literature and show that they are much more efficient both in terms of number of quadrature points and computational time.