Most indirect hexahedral meshing methods rely on 10 patterns of subdivision of the hexahedron into tetrahedra . A recent observation at least one more pattern exists raise the question of the actual number of subdivisions  of the hexahedron into tetrahedra. In this articleanswers we enumerate these subdivisions by exhausting all possible ways to combine tetrahedra into hexahedra. We introduce a combinatorial algorithm that (1) generates all the combinations of tetrahedra that can be built from eight vertices and (2) tests if they subdivide a hexahedron or not. We use topological arguments and an efficient pruning strategy to drastically reduce this number. Our main result is that there are 6,966 combinatorialtriangulations of the hexahedron which can be classified into 174 equivalence classes. Our results are consistent with theoretical results available on the subdivision of the cube .
This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that a cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving 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 main result. Each of the 174 triangulations has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. This result is purely combinatorial. We also demonstrate that a minimum of 114 triangulations do correspond to hexahedron geometrical triangulations. Each hexahedron is valid for finite element computations (strictly positive Jacobian determinant). We exhibit the tetrahedral meshes for these geometries and show in particular triangulations of the hexahedron with 14 tetrahedra .
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Most indirect hexahedral meshing methods rely on 10 patterns of subdivision of
This article answers an important theoretical question: How many different subdivisions of
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. A recent observation at least one more pattern exists raise the question of the actual
are there? It is well known that a cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving vertex positions increases the
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. 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
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tetrahedra. In this articleanswers we enumerate these subdivisions by exhausting all possible ways to combine tetrahedra into hexahedra. We introduce a combinatorial algorithm that (1) generates all the combinations of tetrahedra that can be built from eight vertices and (2) tests if they subdivide a hexahedron or not. We use topological arguments and an efficient pruning strategy to drastically reduce this number. Our main result is that there are 6,966 combinatorialtriangulations of the hexahedron which can be classified into
tetrahedra is our main result. Each of the
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equivalence classes. Our results are consistent with theoretical results available on the subdivision of the cube
triangulations has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. This result is purely combinatorial. We also demonstrate that a minimum of 114 triangulations do correspond to hexahedron geometrical triangulations. Each hexahedron is valid for finite element computations (strictly positive Jacobian determinant). We exhibit the tetrahedral meshes for these geometries and show in particular triangulations of the hexahedron with 14 tetrahedra