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High-order numerical integration on regular embedded surfaces

Gentian Zavalani, Michael Hecht

TL;DR

Efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries, as well as efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries.

Abstract

We present a high-order surface quadrature (HOSQ) for accurately approximating regular surface integrals on closed surfaces. The initial step of our approach rests on exploiting square-squeezing--a homeomorphic bilinear square-simplex transformation, re-parametrizing any surface triangulation to a quadrilateral mesh. For each resulting quadrilateral domain we interpolate the geometry by tensor polynomials in Chebyshev--Lobatto grids. Posterior the tensor-product Clenshaw-Curtis quadrature is applied to compute the resulting integral. We demonstrate efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries.

High-order numerical integration on regular embedded surfaces

TL;DR

Efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries, as well as efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries.

Abstract

We present a high-order surface quadrature (HOSQ) for accurately approximating regular surface integrals on closed surfaces. The initial step of our approach rests on exploiting square-squeezing--a homeomorphic bilinear square-simplex transformation, re-parametrizing any surface triangulation to a quadrilateral mesh. For each resulting quadrilateral domain we interpolate the geometry by tensor polynomials in Chebyshev--Lobatto grids. Posterior the tensor-product Clenshaw-Curtis quadrature is applied to compute the resulting integral. We demonstrate efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries.
Paper Structure (5 sections, 1 theorem, 26 equations, 5 figures)

This paper contains 5 sections, 1 theorem, 26 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Square-squeezing re-parametrization
  3. Interpolation in the square
  4. Numerical experiments
  5. Conclusion

Key Result

theorem 1

Consider a $C^r$ surface $S$, where $r \geq 1$, with $r$-regular quadrilateral re-parametrization $\varphi_i : \square_2 \to S$, Let $\mathrm{p}\in P$, and $\omega_\mathrm{p}$ be the points and weights of the tensorial $(n+1)$-order Clenshaw-Curtis quadrature rule, $f : S \to \mathbb{R}$ be a function with absolutely continuous derivatives up to order $(r-1)$ and bounded variation $\|f^{(r)}\|_T

Figures (5)

  • Figure 1: Bilinear square--simplex transformations: Deformations of equidistant grids, under Duffy's transformation (b) and square-squeezing (c)
  • Figure 2: Construction of a surface parametrization over $\triangle_2$ by closest-point projection from a piecewise affine approximate mesh, and re-parametrization over the square $\Box_2$.
  • Figure 3: Relative errors and runtimes between DCG and $\text{HOSQ-CC}$ computing the surface area of the unit sphere (\ref{['fig:US3']}) and torus (\ref{['fig:US4']}) respectively.
  • Figure 4: Gauss-Bonnet validation for Dziuk's surface.
  • Figure 5: Gauss-Bonnet validation for a double torus.

Theorems & Definitions (5)

  • definition 1: Quadrilateral re-parametrization
  • definition 2: Lagrange polynomials MIPminterpy
  • definition 3: $k^{\text{th}}$-order quadrilateral re-parametrization
  • theorem 1
  • proof