Exact computation of the color function for triangular element interfaces

TL;DR

This work addresses exact computation of the color function inside a cube cut by curved interfaces represented as triangles. It introduces Front2VOF (F2V), a two-step geometric method: first clip the triangulated interface against the cube faces to obtain polygons inside the cube, then evaluate the volume via a Gauss/Green-based surface integral that combines polygon contributions and cube-face terms. The approach robustly handles configurations with multiple interfaces and corner cases, validated against analytical solutions for single and multiple elements and showing second-order convergence for spherical interfaces. The method provides an efficient, geometry-based initialization of volume fractions for VOF/front-tracking simulations and extends naturally to unstructured tetrahedral meshes, with broad applications in CFD and computer graphics.

Abstract

The calculation of the volume enclosed by curved surfaces discretized into triangular elements, and a cube is of great importance in different domains, such as computer graphics and multiphase flow simulations. We propose a robust algorithm, the Front2VOF (F2V) algorithm, to address this problem. The F2V algorithm consists of two main steps. First, it identifies the polygons within the cube by segmenting the triangular elements on the surface, retaining only the portions inside the cube boundaries. Second, it computes the volume enclosed by these polygons in combination with the cube faces. To validate the algorithm's accuracy and robustness, we tested it using a range of synthetic configurations with known analytical solutions.

Figures (6)

• Figure 1: Schematic of the Front2VOF algorithm: (a) triangular elements intersecting with a cube; (b) polygons inside the cube obtained using the clipping algorithm; (c) intersection regions between the reference phase and cube face $\partial \Omega{c, x = 1}$; (d) line segments contributing to area computation.
• Figure 2: Schematic of the clipping algorithm, showing a polygon at different stages of the process: (a) initial triangle, denoted by red solid line; (b) after the clipping by two faces $\partial \Omega_{c, x=0, 1}$, denoted by blue solid line; (c) after the clipping by two faces $\partial \Omega_{c, y=0, 1}$, denoted by green solid line (d) after the clipping by two faces $\partial \Omega_{c, z=0, 1}$, denoted by orange solid line.
• Figure 3: A Cube cut by a single element with different normal vectors: $\hat{{\bf n}} = \pm {\bf m}$, (a) ${\bf m} = (1, 0, 0)$; (b) ${\bf m} = (1/2, 1/2, 0)$; (c) ${\bf m} = (1/3, 1/3, 1/3)$.
• Figure 4: A Cube cut by multiple elements: $\hat{{\bf n}_1} = -\hat{{\bf n}_2}= \pm {\bf m}_1$, ${\bf m}_1 = {\bf m}_2 = (1/3, 1/3, 1/3)$.
• Figure 5: A cube cut by spherical interfaces discretized by triangular elements with difference size: (a) $\bar{l} = 0.05$; (b) $\bar{l} = 0.025$; (c) $\bar{l}=0.0125$.