# Finding Hexahedrizations for Small Quadrangulations of the Sphere

## TL;DR

This paper presents a practical algorithm for constructing small combinatorial hexahedral meshes from quadrangulations of the sphere, significantly improving previous solutions by automating the process and reducing the number of hexahedra needed.

## Contribution

The paper introduces the first practical algorithm for generating small hexahedral meshes from sphere quadrangulations, exploiting quad flips and symmetry reduction techniques.

## Key findings

- Successfully meshed all 54,943 quadrangulations with up to 20 quadrangles using no more than 72 hexahedra.
- Proved that any ball-shaped domain with n quadrangles can be meshed with at most 78n hexahedra, greatly improving previous bounds.
- Developed a symmetry-aware backtracking search to efficiently find small hexahedral meshes.

## Abstract

This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n.