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An efficient particle locating method on unstructured meshes in two and three dimensions based on patch searching

Shuang Chen, Fanyi Yang

TL;DR

The paper tackles efficient particle locating on unstructured 2D/3D meshes by introducing a patch-searching method. It first maps a query point to a vertex-based patch using a background Cartesian grid, then determines the host element within that patch, with 2D relying on a fast angle-based binary search and 3D employing a moving step to reduce to an edge-based planar search. The approach eliminates point-in-element tests in the initial step, uses a prescribed grid spacing, and guarantees linear initialization cost in the number of elements. Numerical experiments in 2D and 3D demonstrate significant efficiency gains and robustness compared to traditional neighbourhood-search and auxiliary-grid methods, including extensions to polygonal meshes in preliminary form.

Abstract

We present a particle locating method for unstructured meshes in two and three dimensions. Our algorithm is based on a patch searching process, and includes two steps. We first locate the given point to a patch near a vertex, and then the host element is determined within the patch domain. Here, the patch near a vertex is the domain of elements around this vertex. We prove that in the first step the patch can be rapidly identified by constructing an auxiliary Cartesian grid with a prescribed resolution. Then, the second step can be converted into a searching problem, which can be easily solved by searching algorithms. Only coordinates to particles are required in our method. We conduct a series of numerical tests in two and three dimensions to illustrate the robustness and efficiency of our method.

An efficient particle locating method on unstructured meshes in two and three dimensions based on patch searching

TL;DR

The paper tackles efficient particle locating on unstructured 2D/3D meshes by introducing a patch-searching method. It first maps a query point to a vertex-based patch using a background Cartesian grid, then determines the host element within that patch, with 2D relying on a fast angle-based binary search and 3D employing a moving step to reduce to an edge-based planar search. The approach eliminates point-in-element tests in the initial step, uses a prescribed grid spacing, and guarantees linear initialization cost in the number of elements. Numerical experiments in 2D and 3D demonstrate significant efficiency gains and robustness compared to traditional neighbourhood-search and auxiliary-grid methods, including extensions to polygonal meshes in preliminary form.

Abstract

We present a particle locating method for unstructured meshes in two and three dimensions. Our algorithm is based on a patch searching process, and includes two steps. We first locate the given point to a patch near a vertex, and then the host element is determined within the patch domain. Here, the patch near a vertex is the domain of elements around this vertex. We prove that in the first step the patch can be rapidly identified by constructing an auxiliary Cartesian grid with a prescribed resolution. Then, the second step can be converted into a searching problem, which can be easily solved by searching algorithms. Only coordinates to particles are required in our method. We conduct a series of numerical tests in two and three dimensions to illustrate the robustness and efficiency of our method.
Paper Structure (10 sections, 4 theorems, 26 equations, 10 figures, 4 tables, 4 algorithms)

This paper contains 10 sections, 4 theorems, 26 equations, 10 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Particle Locating based on Patch Searching
  4. Locating in two dimensions
  5. Locating in three dimensions
  6. Computer Implementation
  7. Numerical Results
  8. Conclusions
  9. Geometrical Relations
  10. list of Notation

Key Result

Lemma 1

Under the condition $s \leq \frac{w^* \sin \alpha}{ \sqrt{2}(1 + \sin \alpha)}$, for any $T \in \mathcal{C}_s^{\circ}$, there exists a node $\bm{\nu} \in \mathcal{N}_h$ such that $(T \cap \Omega) \subset \mathcal{D}(\mathcal{T}_{\bm{\nu}})$.

Figures (10)

  • Figure 1: red: nodes in $\mathcal{N}_h^b$, blue: nodes in $\mathcal{N}_h^i$ (left) / the patch domain $\mathcal{D}(\mathcal{T}_{\bm{\nu}})$ (right)
  • Figure 2: The Cartesian cells in $\mathcal{C}_{s}^{\circ, \bm{\mathrm{c}}}$ and $\mathcal{C}_{s}^{\circ, \bm{\mathrm{b}}}$.
  • Figure 3: localizing the point in the patch.
  • Figure 4: The tetrahedron $K$ and the points $\bm{c}_1, \bm{c}_2, \bm{\chi}, \bm{\omega}$.
  • Figure 5: The set $\mathcal{T}_{e}$ and the plane $P_e$.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 2
  • Lemma 4
  • proof