Abstract: A novel algorithm that produces a quad layout based on an 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. 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 paper’s second contribution 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 parametrization.

Paper in proceedings of the 29^th International Meshing Roundtable

Abstract: An in-depth method to generate multi-block decomposition of the arbitrary 2D domain using 2D cross fields solution of Ginzburg-Landau partial differential equation (PDE) is presented. It is relied on parameterization of multi-block decomposition of the domain, obtained by using particular PDE for the purpose of generating direction fields, appropriate number and localization of singular points and their separatrices. We have proved that solutions of particular PDE imply locally integrable vector fields and have adequate distribution of singularities, advocating its usage. Multi-block graph was generated by the separatrices and extraordinary vertices of the domain (singularities, corners and separatrices intersections) and obtained blocks were parameterized/remeshed. As a result, a mechanism to obtain multi-block structured all-quad mesh in automatic manner is developed.

• Paper (in proceedings of the 28^th International Meshing Roundtable)
• The code will soon be available in Gmsh

(The meshes above can be rotated by clicking on them and dragging your cursor.)

Suppose you drew a set of n quadrangles on a sphere, covering its entirety. How many cubes do you need to fill the interior of that sphere? This question is surprisingly difficult to answer, even with only a few quadrangles (the case n = 8 is notoriously hard, and of the 3 cases where n = 10, only one is easy to solve).

Our contribution to SIGGRAPH 2019 shows that with n quadrangles on the boundary, 78n combinatorial cubes (or hexahedra) are always enough. This is a significant improvement over the previous upper bound: 5396n (Carbonera and Shepherd, 2010). Our result is based on a proof by Erickson (2014): we computed hexahedral meshes of the two base cases that it uses (shown above).

In most cases, there are significantly smaller solution. The paper gives an algorithm to search for such solutions. This was fast enough to compute hexahedral meshes for all 54,943 quadrangulations of up to 20 quadrangles for which a solution exists. The worst case required only 72 hexahedra.

• Paper (arXiv)
• Slides
• Code
• Results

*on an AMD EPYC 7551 (2×32 core)

• Submitted version on arXiv.org
• The sequential Delaunay code
• The parallel Delaunay code (being intregated into Gmsh)

The paper will be presented at SIGGRAPH Asia 2018.

1. A procedure to enumerate all hexahedral meshes with a specific boundary
2. A procedure which locally modifies a hexahedral mesh to reduce the number of hexahedra without changing the boundary.

Our implementation of the algorithms described in the paper and our results can be downloaded below.

Abstract: This paper shows that constraint programming techniques can successfully be used to solve challenging hex-meshing problems. Schneiders’ pyramid is a square-based pyramid whose facets are subdivided into three or four quadrangles by adding vertices at edge midpoints and facet centroids. In this paper, we prove that Schneiders’ pyramid has no hexahedral meshes with fewer than 18 interior vertices and 17 hexahedra, and introduce a valid mesh with 44 hexahedra. We also construct the smallest known mesh of the octagonal spindle, with 40 hexahedra and 42 interior vertices. These results were obtained through a general purpose algorithm that computes the hexahedral meshes conformal to a given quadrilateral surface boundary. The lower bound for Schneiders’pyramid is obtained by exhaustively listing the hexahedral meshes with up to 17 interior vertices and which have the same boundary as the pyramid. Our 44-element mesh is obtained by modifying a prior solution with 88 hexahedra. The number of elements was reduced using an algorithm which locally simplifies groups of hexahedra. Given the boundary of such a group, our algorithm is used to find a mesh of its interior that has fewer elements than the initial subdivision. The resulting mesh is untangled to obtain a valid hexahedral mesh.

• Paper (from the proceedings of the 27th International Meshing Roundtable)
• Slides from the presentation at the 27th International Meshing Roundtable
• 44-element mesh of the pyramid
• 40-element mesh of the octagonal spindle
• Source code (GPL-licensed)

• The paper has been pubPaper on Wiley Online Librarylished in Elsevier – Procedia Engineering (Volume 203, 201, Pages 219-231).
• The C code will soon be available in Gmsh.
• This study has been funded by the ARC WAVES 15/19-03.

• The paper is available on arXiv: arxiv.org/abs/1706.01613.
• The C++ code will soon be available in Gmsh.
• Input tetrahedral meshes are available here.

• The paper is available on arXiv:   arxiv.org/abs/1705.02451.
• The C++ code will soon be available in Gmsh.
• Input tetrahedral meshes are available here.