There are 174 Subdivisions of the Hexahedron into Tetrahedra

SessionMeshing

Event Type

Technical Papers

TimeThursday, 6 December 20185:33pm - 5:59pm

LocationHall B5(1) (5F, B Block)

DescriptionThis 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.

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.