e link between tetrahedral meshes (a) and hexahedral meshes (b) depends on hexahedron triangulations. (c) One of the 174 triangulations of hexahedron {12345678} has 8 tetrahedra {1258}, {5286}, {1826}, {1246}, {1486}, {2467}, {4867}, {2347} . 

Contexts in source publication

Context 1

... results are consistent with the previous results on the tri- angulations the 3-cube presented in [De Loera et al. 2010]. One consequence of our work is that eorem 4 in Sokolov et al. [2016], that states that the triangulations of the hexahedron have 5, 6, or 7 14_D 14_E 14_F Figure 11: e three combinatorial triangulations of the hex- ahedron in 14 tetrahedra for which we found a geometrical realization. Lee: the decomposition graphs. Right: a shrunk view of the 14 tetrahedra. e mesh les are available in sup- plemental ...

Context 2

... prove that at least 114 over 174 abstract triangulations do have a geometrical realization in R 3 by exhibiting the correspond- ing tetrahedral meshes (Figure 9). All meshes are available in the supplemental material. e names of these triangulations are high- lighted in Table 2. All triangulations that have up to 10 tetrahedra are realizable. e smallest triangulations for which we have not found any geometrical realization have 11 tetrahedra (Figure 10). e largest triangulations for which we found a geometrical re- alization have 14 tetrahedra (Figure 11). e existence of these conngurations contradict the belief that there may only be subdi- visions of the hexahedron with up to 13 tetrahedra. Since these triangulated hexahedra are valid for nite element computations, i.e. their Jacobian determinant are strictly positive at any point, their vertices are in convex ...

Context 3

... prove that at least 114 over 174 abstract triangulations do have a geometrical realization in R 3 by exhibiting the correspond- ing tetrahedral meshes (Figure 9). All meshes are available in the supplemental material. e names of these triangulations are high- lighted in Table 2. All triangulations that have up to 10 tetrahedra are realizable. e smallest triangulations for which we have not found any geometrical realization have 11 tetrahedra (Figure 10). e largest triangulations for which we found a geometrical re- alization have 14 tetrahedra (Figure 11). e existence of these conngurations contradict the belief that there may only be subdi- visions of the hexahedron with up to 13 tetrahedra. Since these triangulated hexahedra are valid for nite element computations, i.e. their Jacobian determinant are strictly positive at any point, their vertices are in convex ...

Context 4

... rst contribution is to solve the purely combinatorial prob- lem of the hexahedron triangulations. is is an important theo- retical result that links hexahedral meshes to tetrahedral meshes ( Figure ...

Context 5

... second contribution is to demonstrate that at least 114 hex- ahedron triangulations have a geometrical realization in R 3 , i.e. a valid tetrahedral mesh of a valid hexahedral nite element cell. We prove their existence by exhibiting the corresponding meshes ( Figure 1). In particular, we exhibit the meshes of three hexahedra with 14 tetrahedra contradicting the belief that there might only be subdivisions with up to 13 tetrahedra [Meshkat and Talmor 2000;Sokolov et al. 2016]. We aaached our C++ implementation and all the result les of combinatorial triangulations and tetrahedral meshes in the supplementary materials. • Corner tetrahedra are obtained from a vertex together with its three ...

Context 6

... rst contribution is to solve the purely combinatorial prob- lem of the hexahedron triangulations. is is an important theo- retical result that links hexahedral meshes to tetrahedral meshes ( Figure 1). ...

Context 7

... second contribution is to demonstrate that at least 114 hex- ahedron triangulations have a geometrical realization in R 3 , i.e. a valid tetrahedral mesh of a valid hexahedral nite element cell. We prove their existence by exhibiting the corresponding meshes ( Figure 1). In particular, we exhibit the meshes of three hexahedra with 14 tetrahedra contradicting the belief that there might only be subdivisions with up to 13 tetrahedra [Meshkat and Talmor 2000;Sokolov et al. 2016]. ...

Context 8

... triangulations that have up to 10 tetrahedra are realizable. e smallest triangulations for which we have not found any geometrical realization have 11 tetrahedra (Figure 10). e largest triangulations for which we found a geometrical re- alization have 14 tetrahedra (Figure 11). ...

Context 9

... smallest triangulations for which we have not found any geometrical realization have 11 tetrahedra (Figure 10). e largest triangulations for which we found a geometrical re- alization have 14 tetrahedra (Figure 11). e existence of these conngurations contradict the belief that there may only be subdi- visions of the hexahedron with up to 13 tetrahedra. ...

Context 10

... results are consistent with the previous results on the tri- angulations the 3-cube presented in [De Loera et al. 2010]. One consequence of our work is that eorem 4 in Sokolov et al. [2016], that states that the triangulations of the hexahedron have 5, 6, or 7 14_D 14_E 14_F Figure 11: e three combinatorial triangulations of the hex- ahedron in 14 tetrahedra for which we found a geometrical realization. Lee: the decomposition graphs. ...