On polytope intersection by half-spaces and hyperplanes for unsplit geometric volume of fluid methods on arbitrary grids

Abstract

Several tools with improved accuracy and computational efficiency, along with added capabilities, are presented for performing analytic and geometric operations involving intersections between half-spaces or hyperplanes and 2- or 3-polytopes (polygons or polyhedra, either convex or non-convex) that typically arise in advanced volume of fluid (VOF) methods on arbitrary grids. In particular, tools for the intersection between a polyhedron and a half-space, and for the computation of the interface position to cut off a given area/volume fraction from an arbitrary polygonal/polyhedral grid cell in PLIC (piecewise linear interface calculation) reconstruction have been improved in terms of accuracy and computational efficiency. In addition, specific tools are also proposed for (1) the intersection in 3D between a planar polygon and a half-space, and the computation of the resulting intersected area, which can be useful for unsplit methods based on wetted-area time-integration advection (WATIA) schemes, and for (2) the efficient computation of the geometric center of the set of clipped vertices and volume resulting from the intersection between an arbitrary polyhedron and a half-space, which can be useful for, respectively, several PLIC reconstruction methods and unsplit methods based on fluid-volume intersection advection (FVIA) schemes. Finally, a convenient strategy to perform the recursive intersections required during the advection step in advanced geometric VOF methods based on FVIA schemes is also proposed. To assess the performance and accuracy of the proposed tools, different tests were carried out for several convex and non-convex polytopes. Speedups of ×5 and improvements in accuracy of up to 5 orders of magnitude over previous tools have been achieved for certain geometries and operations. The proposed tools have been included in two advection methods (an FVIA method and a new WATIA method proposed in this work) and in a PLIC reconstruction method to demonstrate their improved performance and capabilities on structured and unstructured grids.