Efficient three-material PLIC interface positioning on unstructured polyhedral meshes

This paper introduces an efficient algorithm for the sequential positioning (or nested dissection) of two planar interfaces in an arbitrary polyhedron, such that, after each truncation, the respectively remaining polyhedron admits a prescribed volume. This task, among others, is frequently encountered in the numerical simulation of three-phase flows when resorting to the geometric Volume-of-Fluid method. For two-phase flows, the recent work of Kromer and Bothe (arXiv:2101.03861) addresses the positioning of a single plane by combining an implicit bracketing of the sought position with up to third-order derivatives of the volume fraction. An analogous application of their highly efficient root-finding scheme to three-material configurations requires computing the volume of a twice truncated arbitrary polyhedron. The present manuscript achieves this by recursive application of the Gaussian divergence theorem in appropriate form, which allows to compute the volume as a sum of quantities associated to the faces of the original polyhedron. With a suitable choice of the coordinate origin, accounting for the sequential character of the truncation, the volume parametrization becomes co-moving with respect to the planes. This eliminates the necessity to establish topological connectivity and tetrahedron decomposition after each truncation. After a detailed mathematical description of the concept, we conduct a series of carefully designed numerical experiments to assess the performance in terms of polyhedron truncations. The high efficiency of the two-phase positioning persists for sequential application, thereby being robust with respect to input data and possible intersection topologies. In comparison to an existing decomposition-based approach, the number of truncations was reduced by up to an order of magnitude, further highlighting the superiority of the divergence-based volume computation.