Identifying combinations of tetrahedra into hexahedra: a vertex based strategy

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.Comment: Preprint submitted to CAD (26th IMR special issue

There are 174 Subdivisions of the Hexahedron into Tetrahedra

This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian

The moduli space of hex spheres

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $\frac{2\pi}{3}$ but less than $2\pi$. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result gives us as a corollary that each unit-area hex sphere $M$ satisfies the following properties: (1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of $M$; (2) it embeds isometrically in the 3-dimensional Euclidean space as the boundary of a tetrahedron; and (3) there is a simple closed geodesic $\gamma$ in $M$ such that a fractional Dehn twist along $\gamma$ converts $M$ to the double of a parallelogram

Flipping Cubical Meshes

We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th International Meshing Roundtable. This version removes some unwanted paragraph breaks from the previous version; the text is unchange