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A linear MARS method for three-dimensional interface tracking

Yunhao Qiu, Qinghai Zhang

TL;DR

This paper introduces a 3D linear MARS method for explicit interface tracking that preserves topological invariants and geometric features by representing 3D continua as Yin sets, decomposed into a poset of glued surfaces triangulated with meshes. It enforces a robust mesh regularity, $(r_{tiny},h_L,\theta)$, through a cascade of mesh adjustments: fast elementary operations (EMA), energy-based vertex relocation (VREM), and local triangulation regeneration (LTR). The method demonstrates second- and third-order accuracy depending on the interface-tobulk scale relation $h_L=O(h^{\alpha})$ and outperforms several VOF/MOF approaches in topology preservation and geometry under complex deformations (sphere, armadillo). Numerical tests confirm stability, convergence, and efficiency, while preserving topology across nontrivial 3D flows, indicating strong potential for coupling with high-order flow solvers in moving-boundary problems.

Abstract

For explicit interface tracking in three dimensions, we propose a linear MARS method that (a) represents the interface by a partially ordered set of glued surfaces and approximates each glued surface with a triangular mesh, (b) maintains an $(r,h,θ)$-regularity on each triangular mesh so that the distance between any pair of adjacent markers is within the range $[rh,h]$ and no angle in any triangle is less than $θ$, (c) applies to three-dimensional continua with arbitrarily complex topology and geometry, (d) preserves topological structures and geometric features of moving phases under diffeomorphic and isometric flow maps, and (e) achieves second-order and third-order accuracy in terms of the Lagrangian and Eulerian length scales, respectively. Results of classic benchmark tests verify the effectiveness of the novel mesh adjustment algorithms in enforcing the $(r,h,θ)$-regularity and demonstrate the high accuracy and efficiency of the proposed linear MARS method.

A linear MARS method for three-dimensional interface tracking

TL;DR

This paper introduces a 3D linear MARS method for explicit interface tracking that preserves topological invariants and geometric features by representing 3D continua as Yin sets, decomposed into a poset of glued surfaces triangulated with meshes. It enforces a robust mesh regularity, , through a cascade of mesh adjustments: fast elementary operations (EMA), energy-based vertex relocation (VREM), and local triangulation regeneration (LTR). The method demonstrates second- and third-order accuracy depending on the interface-tobulk scale relation and outperforms several VOF/MOF approaches in topology preservation and geometry under complex deformations (sphere, armadillo). Numerical tests confirm stability, convergence, and efficiency, while preserving topology across nontrivial 3D flows, indicating strong potential for coupling with high-order flow solvers in moving-boundary problems.

Abstract

For explicit interface tracking in three dimensions, we propose a linear MARS method that (a) represents the interface by a partially ordered set of glued surfaces and approximates each glued surface with a triangular mesh, (b) maintains an -regularity on each triangular mesh so that the distance between any pair of adjacent markers is within the range and no angle in any triangle is less than , (c) applies to three-dimensional continua with arbitrarily complex topology and geometry, (d) preserves topological structures and geometric features of moving phases under diffeomorphic and isometric flow maps, and (e) achieves second-order and third-order accuracy in terms of the Lagrangian and Eulerian length scales, respectively. Results of classic benchmark tests verify the effectiveness of the novel mesh adjustment algorithms in enforcing the -regularity and demonstrate the high accuracy and efficiency of the proposed linear MARS method.

Paper Structure

This paper contains 21 sections, 2 theorems, 32 equations, 8 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Yin sets for modeling 3D continua
  3. Uniquely representing Yin sets via glued surfaces
  4. Approximating glued surfaces with triangular meshes
  5. Theoretical and algorithmic significance of Yin sets for IT
  6. The IT problem
  7. The MARS framework
  8. Algorithms
  9. The 3D linear MARS method
  10. Elementary mesh adjustments (EMA)
  11. Vertex relocation via energy minimization (VREM)
  12. Minimizing the total potential energy of a spring system
  13. A local projection to $|{\@fontswitch{}{\mathcal{}} T}|$ at $p$
  14. The VREM algorithm and its convergence
  15. Local triangulation regeneration (LTR)
  16. ...and 6 more sections

Key Result

Theorem 2.5

\newlabelthm:connectedYinsetrep0 The boundary of any connected Yin set $\mathcal{Y} \neq \emptyset, \mathbb{R}^3$ can be uniquely oriented and partitioned into a finite poset $\mathcal{G}_{\partial \mathcal{Y}}=\{G_j\subset \partial {\@fontswitch{}{\mathcal{}} Y}\}$ of pairwise almost disjoint ori where $\mathrm{int}(G_j)$ is the internal complement of $G_j$ in def:internalComplementOfGluedSurfac

Figures (8)

  • Figure 1: Three elementary mesh adjustments. In subplot (a), edges $p_{0}p_{2}$, $p_1p_2$, $p_2p_3$ and $p_1p_3$ are longer than $h_L$ and are split at their midpoints; new edges are then added locally to obtain a new triangular mesh. In subplot (b), the edge $p_1p_{2}$ shorter than $r_{\mathrm{tiny}}h_{L}$ is collapsed into its endpoint $p_2$. In subplot (c), $\angle p_{3}p_1p_{2}$ and $\angle p_3p_2p_{1}$ are smaller than $\theta$ and the diagonal $p_1p_2$ is flipped to $p_3p_{4}$ to remove the two small angles. In subplot (d), the edge flipping fails to fulfill the $\theta$-regularity.
  • Figure 1: Results of the linear MARS method for solving the vortical shear test in \ref{['tab:vorticalshearset']} with $h = \frac{1}{64}$ and $h_L=0.5h$. The distances between adjacent markers at the initial time $t=0$ are set to the uniform constant $0.25h$.
  • Figure 2: The net force of a vertex $p_0$ moves $p_0$ so that the lengths of edges at $p_0$ become closer to the resting length. On the left, only $p_0p_1$ exerts a repulsive force on $p_0$ while all other edges exert attractive forces on $p_0$.
  • Figure 2: Results of the 3D linear MARS method in solving the deformation test in \ref{['tab:deformationTestSetup']} with $h = \frac{1}{64}$ and $h_L=0.5h$. The initial distances between adjacent markers are set to the uniform $0.25h$.
  • Figure 3: The VREM algorithm updates positions of the interior vertices of a triangular mesh by first moving each $p_i\in V_I$ along the direction of its net force to $p'$ and then projecting $p'$ onto the star of $p$ to obtain $p"$ as the updated position of $p$. As defined in \ref{['sec:triangular-mesh-as']} and shown in subplot (a), the star of $p$ is the union of the six triangles with $p$ as their common vertex while the link of $p$ is the simple closed curve formed by the line segments " $p_{1}\rightarrow p_{2} \rightarrow p_{3} \rightarrow p_{4} \rightarrow p_{5} \rightarrow p_{6} \rightarrow p_{1}$."
  • ...and 3 more figures

Theorems & Definitions (14)

  • Definition 2.1: Yin space ZhFo16ZhLi20
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Unique boundary representation of connected 3D Yin sets ZhQi
  • Definition 2.6
  • Definition 3.1
  • Definition 4.1
  • Definition 4.2: $(r, h, \theta)$-regularity
  • Definition 5.1
  • ...and 4 more