2020/05/15 by David Gunderman, Gunderman, David, Kenneth M. Weiss +3 · 1 citation
Engineering · Physics and Astronomy · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #Computational Geometry (cs.CG) #Electromagnetic Scattering and Analysis #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2005.07780
openalex publication_date 2020/05/15 · openalex created_date 2022/07/24 · openalex updated_date 2026/07/28
This work presents spectral, mesh-free, Green's theorem-based numerical\nquadrature schemes for integrating functions over planar regions bounded by\nrational parametric curves. Our algorithm proceeds in two steps: (1) We first\nfind intermediate quadrature rules for line integrals along the region's\nboundary curves corresponding to Green's theorem. (2) We then use a high-order\nquadrature rule to compute the numerical antiderivative of the integrand along\na coordinate axis, which is used to evaluate the Green's theorem line integral.\nWe present two methods to compute the intermediate quadrature rule. The first\nis spectrally accurate (it converges faster than any algebraic order with\nrespect to number of quadrature points) and is relatively easy to implement,\nbut has no guarantee of polynomial exactness. The second guarantees exactness\nfor polynomial integrands up to a pre-specified degree k with an a priori-known\nnumber of quadrature points and retains the convergence properties of the\nfirst, but is slightly more complicated. The quadrature schemes have\napplications to computation of geometric moments, immersogeometric analysis,\nconservative field transfer between high-order meshes, and initialization of\nsimulations with rational geometry. We compare the quadrature schemes produced\nusing our method to other methods in the literature and show that they are much\nmore efficient both in terms of number of quadrature points and computational\ntime.\n