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Tusqh: Topological Control of Volume-Fraction Meshes Near Small Features and Dirty Geometry

Brian Shawcroft, Kendrick M. Shepherd, Scott Mitchell

TL;DR

Tusqh addresses the challenge of generating meshes with prescribed topology from dirty geometry by coupling a background grid with volume-fraction thresholds and persistent homology. The framework introduces subgrid sampling and templates to perform topological anti-aliasing, enabling consistent control over pinches and archipelagos and allowing topology-guided mesh selection for downstream analyses. A cubical-to-simplicial filtration transfer ensures persistent-homology computations reflect the anti-aliased topology in both 2D and 3D, with theoretical bounds on rasterization and observed convergence/non-convergence behavior under grid refinement. The approach is demonstrated on 2D Chesapeake Bay and 3D mechanical/graphics models (Bearings, Bronco Buster) using a Rhinoceros/Grasshopper workflow, highlighting practical applicability and remaining limitations where topology may fail to converge for some inputs.

Abstract

This work develops a framework to create meshes with user-specified homology from potentially dirty geometry by coupling background grids, persistent homology, and a generalization of volume fractions. For a mesh with fixed grid size, the topology of the output mesh changes predictably and monotonically as its volume-fraction threshold decreases. Topological anti-aliasing methods are introduced to resolve pinch points and disconnected regions that are artifacts of user choice of grid size and orientation, making the output meshes suitable for downstream processes including analysis. The methodology is demonstrated on geographical, mechanical, and graphics models in 2D and 3D using a custom-made software called Tusqh. The work demonstrates that the proposed framework is viable for generating meshes on topologically invalid geometries and for automatic defeaturing of small geometric artifacts. Finally, the work shows that although subdividing the background grid frequently improves the topological and geometrical fidelity of the output mesh, there are simple 2D examples for which the topology does not converge under refinement for volume-fraction codes.

Tusqh: Topological Control of Volume-Fraction Meshes Near Small Features and Dirty Geometry

TL;DR

Tusqh addresses the challenge of generating meshes with prescribed topology from dirty geometry by coupling a background grid with volume-fraction thresholds and persistent homology. The framework introduces subgrid sampling and templates to perform topological anti-aliasing, enabling consistent control over pinches and archipelagos and allowing topology-guided mesh selection for downstream analyses. A cubical-to-simplicial filtration transfer ensures persistent-homology computations reflect the anti-aliased topology in both 2D and 3D, with theoretical bounds on rasterization and observed convergence/non-convergence behavior under grid refinement. The approach is demonstrated on 2D Chesapeake Bay and 3D mechanical/graphics models (Bearings, Bronco Buster) using a Rhinoceros/Grasshopper workflow, highlighting practical applicability and remaining limitations where topology may fail to converge for some inputs.

Abstract

This work develops a framework to create meshes with user-specified homology from potentially dirty geometry by coupling background grids, persistent homology, and a generalization of volume fractions. For a mesh with fixed grid size, the topology of the output mesh changes predictably and monotonically as its volume-fraction threshold decreases. Topological anti-aliasing methods are introduced to resolve pinch points and disconnected regions that are artifacts of user choice of grid size and orientation, making the output meshes suitable for downstream processes including analysis. The methodology is demonstrated on geographical, mechanical, and graphics models in 2D and 3D using a custom-made software called Tusqh. The work demonstrates that the proposed framework is viable for generating meshes on topologically invalid geometries and for automatic defeaturing of small geometric artifacts. Finally, the work shows that although subdividing the background grid frequently improves the topological and geometrical fidelity of the output mesh, there are simple 2D examples for which the topology does not converge under refinement for volume-fraction codes.

Paper Structure

This paper contains 21 sections, 2 theorems, 6 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Background Material
  3. Rasterization.
  4. Topological Anti-aliasing.
  5. Homology.
  6. Winding Numbers.
  7. Methodology
  8. Volume Fractions.
  9. Volume-Fraction Persistence Parameter.
  10. Sub-cell Volume Fractions.
  11. Anti-aliasing.
  12. Transferring Persistent Parameters to Simplices.
  13. Results
  14. Computational Results.
  15. Anti-aliasing.
  16. ...and 6 more sections

Key Result

Theorem 4.1

Given a rectangular lattice in $\mathbb{R}^2$ with characteristic length $\ell$ overlaying two parallel half-spaces separated by a length of $L$, topological rasterization may occur when $\ell(\sqrt{2}-1)<L\leq\ell$. For the subgrid sampling scheme proposed in this text, topological rasterization ma

Figures (25)

  • Figure 1: Rasterization of triangles into pixels for computer graphics. Note the pinches from the two left cyan triangles, the archipelago from the lower right pink triangle, and the multitude of additional topological errors in the lower right. Image courtesy Wikipedia https://en.wikipedia.org/wiki/Rasterisation
  • Figure 2: Small grid-aligned gaps are closed, large gaps are open, and intermediate gaps depend on their offset.
  • Figure 3: This unaligned gap is resolved inconsistently.
  • Figure 4: Rotational aliasing may cause stair-step patterns, and archipelagos of isolated islands near where two lines meet at a sharp angle. Bold-outlined cells are filled, thin are open.
  • Figure 5: Winding number point and field values, courtesy Jacobson et al. Jacobson:2013 Figures 4 and 6. Used with permission of the Association for Computing Machinery, conveyed through Copyright Clearance Center, Inc.
  • ...and 20 more figures

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 1.1