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Quad layouts with high valence singularities for flexible quad meshing

Jovana Jezdimirović, Alexandre Chemin, Maxence Reberol, François Henrotte, Jean François Remacle

TL;DR

This work addresses robust quad meshing by controlling cross-field singularities to generate block-structured quad layouts. It introduces a two-step cross-field construction: first a nonlinear energy-based cross-field to reveal a singularity pattern, then a second cross-field computed with an imposed pattern on a refined mesh, ensuring accurate separatrices and adherence to topological constraints. A correction scheme fixes limit cycles and non-quadrilateral patches, and a per-partition parameterization enables high-quality bilinear remeshing into conformal quadrilaterals. The approach is interactive, compatible with Gmsh, and suitable for producing quad meshes with high-valence singularities that provide strong size gradients and topological control, with potential extensions to manifolds and varying size maps.

Abstract

A novel algorithm that produces a quad layout based on imposed set of singularities is proposed. In this paper, we either use singularities that appear naturally, e.g., by minimizing Ginzburg-Landau energy, or use as an input user-defined singularity pattern, possibly with high valence singularities that do not appear naturally in cross-field computations. The first contribution of the paper is the development of a formulation that allows computing a cross-field from a given set of singularities through the resolution of two linear PDEs. A specific mesh refinement is applied at the vicinity of singularities to accommodate the large gradients of cross directions that appear in the vicinity of singularities of high valence. The second contribution of the paper is a correction scheme that repairs limit cycles and/or non-quadrilateral patches. Finally, a high quality block-structured quad mesh is generated from the quad layout and per-partition parameterization.

Quad layouts with high valence singularities for flexible quad meshing

TL;DR

This work addresses robust quad meshing by controlling cross-field singularities to generate block-structured quad layouts. It introduces a two-step cross-field construction: first a nonlinear energy-based cross-field to reveal a singularity pattern, then a second cross-field computed with an imposed pattern on a refined mesh, ensuring accurate separatrices and adherence to topological constraints. A correction scheme fixes limit cycles and non-quadrilateral patches, and a per-partition parameterization enables high-quality bilinear remeshing into conformal quadrilaterals. The approach is interactive, compatible with Gmsh, and suitable for producing quad meshes with high-valence singularities that provide strong size gradients and topological control, with potential extensions to manifolds and varying size maps.

Abstract

A novel algorithm that produces a quad layout based on imposed set of singularities is proposed. In this paper, we either use singularities that appear naturally, e.g., by minimizing Ginzburg-Landau energy, or use as an input user-defined singularity pattern, possibly with high valence singularities that do not appear naturally in cross-field computations. The first contribution of the paper is the development of a formulation that allows computing a cross-field from a given set of singularities through the resolution of two linear PDEs. A specific mesh refinement is applied at the vicinity of singularities to accommodate the large gradients of cross directions that appear in the vicinity of singularities of high valence. The second contribution of the paper is a correction scheme that repairs limit cycles and/or non-quadrilateral patches. Finally, a high quality block-structured quad mesh is generated from the quad layout and per-partition parameterization.

Paper Structure

This paper contains 21 sections, 26 equations, 19 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Cross-field generation for meshing purposes
  4. Singularities with high indices
  5. Quad layout partition guided by cross-field
  6. First cross-field computation
  7. Singularity pattern
  8. Cross-field computation with imposed singularity pattern
  9. Computation of the $H$ field
  10. Computation of the $\theta$ field
  11. Generate the branch cut
  12. Solve the Cauchy-Riemann equations
  13. Non quad-meshable cross-fields
  14. Computing the per-partition parameterization
  15. Quad layout: generating and correcting the partitions
  16. ...and 6 more sections

Figures (19)

  • Figure 1: Three quad layouts of a simple domain defined by a square minus a disk. Colored points are the irregular corners of the layouts.
  • Figure 2: Displacement of singular points due to a cross-field mesh dependency. Cross-field computation follows the work from Beaufort:2017.
  • Figure 3: Illustration of 4-steps pipeline
  • Figure 4: Illustration of the reduced 3-steps pipeline
  • Figure 5: 2D cross definition as a set of four orthogonal vectors
  • ...and 14 more figures