# Identifying combinations of tetrahedra into hexahedra: a vertex based strategy

## TL;DR

This paper presents a vertex-based algorithm for efficiently identifying all possible hexahedra, prisms, and pyramids within a tetrahedral mesh, significantly improving detection speed and coverage for engineering analysis.

## Contribution

The authors introduce a novel vertex-based method that rapidly detects all valid combinations of tetrahedra forming complex polyhedral cells, surpassing pattern-based approaches.

## Key findings

- Identifies over 3 million potential hexahedra in 10 seconds on a laptop.
- Builds a larger set of potential hexes than predefined subdivision patterns.
- Early quality checks improve efficiency by discarding poor-quality cells.

## Abstract

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements.