An Interface Tracking Method with Triangle Edge Cuts

TL;DR

This paper introduces a volume-conserving interface tracking algorithm on unstructured triangle meshes by discretizing the interface via triangle edge cuts which represent the intersections between the interface and the triangle mesh edges using a compact 6 numbers per triangle.

Abstract

This paper introduces a volume-conserving interface tracking algorithm on unstructured triangle meshes. We propose to discretize the interface via triangle edge cuts which represent the intersections between the interface and the triangle mesh edges using a compact 6 numbers per triangle. This enables an efficient implicit representation of the sub-triangle polygonal material regions without explicitly storing connectivity information. Moreover, we propose an efficient advection algorithm for this interface representation that is based on geometric queries and does not require an optimization process. This advection algorithm is extended via an area correction step that enforces volume-conservation of the materials. We demonstrate the efficacy of our method on a variety of advection problems on a triangle mesh and compare its performance to existing interface tracking methods including VOF and MOF.

Figures (23)

• Figure 1: The pre-image of a point $\bm{p}^n$ at the previous time step $n-1$, and its image at the subsequent time step $n+1$.
• Figure 2: Different advection methods of direction-splitting PLIC-VOF, MOF, PAM, and EBIT. (a) The cell is translated along the $x$-axis, and its intersection with the interface (red area) is calculated as the flux. (b) In MOF advection, the pre-image $\overleftarrow{C_{i,j}}$ of cell $(i,j)$ is first calculated, and then the zeroth and first moments are calculated by intersecting it with the liquid region at the last time step. (c) PAM advection is similar to MOF, but it accommodates more interface segments inside a cell. (d) EBIT advection along the $x-$axis. Red and grey points represent new marker points located on grid lines.
• Figure 3: A simple example of a triangle edge cut. The small blue triangle denotes the liquid region.
• Figure 4: An example of deriving the interface reconstruction of a triangle edge cut $\mathcal{E}^2$ from a basic case $\mathcal{E}^1$. First, we exchange the roles of the liquid and the air, changing the value of $c$ from $0$ to $1$; accordingly, the air polygon is swapped with the liquid polygon. Second, we apply a cyclic permutation of vertex indices $1\ 2\ 3\to 3\ 1\ 2$ to obtain $\mathcal{E}^2$.
• Figure 5: Six basic cases of triangle edge cuts. Black dots represent liquid, white dots represent air, and red dots represent valid cuts. Blue polygons represent the liquid regions inside triangles. In cases $1,2,3,4$, all vertices are air, while in cases $5,6$, there is one liquid vertex and two air vertices.