Robust Numerical Integration on Curved Polyhedra Based on Folded Decompositions

TL;DR

This paper introduces a flexible numerical integration method for curved polyhedra using folded decompositions, allowing for accurate and robust integration even with complex geometries and self-intersecting cells.

Contribution

The method enables robust integration over curved polyhedra by allowing self-intersecting cells, simplifying decomposition, and maintaining accuracy, which was challenging with traditional positive-Jacobian constraints.

Findings

Folded cells do not compromise integration accuracy.

The method handles complex geometries with sharp features.

Folded decompositions simplify the integration process.

Abstract

We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the given parametric surfaces. Each pyramid serves as an integration cell with a geometric mapping from a standard parent domain (e.g., a unit cube), where the tensor-product Gauss quadrature is adopted. As no constraint is imposed on the decomposition, certain resulting pyramids may intersect with themselves, and thus their geometric mappings may present negative Jacobian values. We call such cells the folded cells and refer to the corresponding decomposition as a folded decomposition. We show that folded cells do not cause any issues in practice as they are only used to numerically compute certain integrals of interest. The same idea can be applied to planar curved polygons as well. We demonstrate both theoretically and numerically that folded cells can retain the same accuracy as the cells with strictly positive Jacobians. On the other hand, folded cells allow for a much easier and much more flexible decomposition for general curved polyhedra, on which one can robustly compute integrals. In the end, we show that folded cells can flexibly and robustly accommodate real-world complex geometries by presenting several examples in the context of immersed isogeometric analysis, where involved sharp features can be well respected in generating integration cells.