Third-order accurate initialization of VOF volume fractions on unstructured meshes with arbitrary polyhedral cells

TL;DR

This paper presents a third-order accurate method for initializing volume fractions in VOF on unstructured polyhedral meshes, leveraging a novel face-based integral transformation for high precision in 3D simulations.

Contribution

It introduces a new face-based analytical approach for volume fraction computation on unstructured meshes with arbitrary polyhedral cells, achieving high accuracy and convergence.

Findings

Achieves third- to fourth-order convergence in volume fraction calculations.

Applicable to convex and non-convex hypersurfaces in 3D meshes.

Simplifies numerical procedures for complex unstructured meshes.

Abstract

This paper introduces a novel method for the efficient and accurate computation of volume fractions on unstructured polyhedral meshes, where the phase boundary is an orientable hypersurface, implicitly given as the iso-contour of a sufficiently smooth level-set function. Locally, i.e. in each mesh cell, we compute a principal coordinate system in which the hypersurface can be approximated as the graph of an osculating paraboloid. A recursive application of the \textsc{Gaussian} divergence theorem then allows to analytically transform the volume integrals to curve integrals associated to the polyhedron faces, which can be easily approximated numerically by means of standard \textsc{Gauss-Legendre} quadrature. This face-based formulation enables the applicability to unstructured meshes and considerably simplifies the numerical procedure for applications in three spatial dimensions. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes, showing both high accuracy and third- to fourth-order convergence with spatial resolution.