Authors: Jovana Jezdimirović, Alexandre Chemin, Jean François Remacle
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 28th International Meshing Roundtable)
Reviving the Search for Optimal Tetrahedralizations
For the 28th International Meshing Roundtable held this year in Buffalo NY, we are revisiting an operation to find the optimal triangulation of a cavity (considering fixed points). This operation was named “Small Polyhedron Reconnection” (SPR) by Liu et al. in their 2006 paper which introduced it. Our implementation contains various optimizations, that differs from the optimization that Liu et al. described in their 2009 paper, notably adding memoization and robust exact intersection tests. We probably have to apologize that there is no reference or comparison to the optimized method of Liu et al. in our IMR paper. The unfortunate reasons for this are described in the README.md of our source code, available on Gitlab. Hope you enjoy(ed) our presentation at the IMR
First Workshop on Frame-based hex meshing Université catholique de Louvain, Louvain-la-Neuve, July 1-2, 2019
The goal of the FRAMES workshop is to gather both theoretical (computer science & applied mathematics) and practical (engineering & industry) specialists in the field of frame-based hex-meshing, in view of numerical computations.
(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.
Marot, Célestin, Jeanne Pellerin et Jean-Francois Remacle. « One machine, one minute, three billion tetrahedra », International Journal for Numerical Methods in Engineering, noja. <https://doi.org/10.1002/nme.5987>.
We recently published a paper in which we describe a new mesh of Schneiders’ pyramid (see images below). The paper describes two new algorithms used to construct it:
A procedure to enumerate all hexahedral meshes with a specific boundary
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.
We developed an innovative way to compute cross fields in order to spawn points which are consistent with a square grid. The mathematical background is built step by step to highlight the meaningful use of Ginzburg-Landau functional. An interesting result is obtained over the sphere: the anti-cube. The computation is extended to asterisk fields for equilateral triangular grid.