Tiling space and slabs with acute tetrahedra

Eppstein, David; Sullivan, John M.; Üngör, Alper

doi:10.1016/j.comgeo.2003.11.003

Cited by 51 publications

(82 citation statements)

References 36 publications

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“…These examples are missing because they do not exist; every tetrahedron that is dihedral acute is also 2-well-centered. Eppstein, Sullivan, andÜngör provide a proof of this in Lemma 2 of [6], which states, among other things, that "an acute tetrahedron has acute facets. "…”

Section: Well-centeredness and Dihedral Acuteness For A Single Tetrahmentioning

confidence: 99%

“…They also briefly discuss how high-quality tilings of space have been used to design meshing algorithms. The acute triangulations of space given in [6] all make use of copies of at least two different tetrahedra, and the authors suggest it is unlikely that there is a tiling of space with copies of a single acute tetrahedron. Their acute triangulation of the slab appears to use copies of seven distinct tetrahedra In contrast to the complexity of tiling space with acute tetrahedra, there are fairly simple completely well-centered triangulations of space.…”

Section: Tiling Space Slabs and Infinite Rectangular Prisms With Comentioning

confidence: 99%

“…8. The tiling is one of four spatial tilings discovered by Sommerville [6,12]. The other three tilings Sommerville found are neither 3-well-centered nor 2-well-centered, though one of them has a maximum face angle of 90°and is dihedral nonobtuse.…”

Section: Tiling Space Slabs and Infinite Rectangular Prisms With Comentioning

confidence: 99%

“…In [6], Eppstein, Sullivan, andÜngör show that one can tile space, R 3 , and infinite slabs, R 2 × [0, a], with dihedral acute tetrahedra. They also briefly discuss how high-quality tilings of space have been used to design meshing algorithms.…”

Section: Tiling Space Slabs and Infinite Rectangular Prisms With Comentioning

confidence: 99%

“…. when he used the second Sommerville construction," S [6,7] which is the completely well-centered tetrahedron shown in Fig. 8.…”

Section: Tiling Space Slabs and Infinite Rectangular Prisms With Comentioning

confidence: 99%

See 4 more Smart Citations

Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra

VanderZee¹,

Hirani²,

Guoy

Proceedings of the 17th International Meshing Roundtable

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Abstract. A completely well-centered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and other fields. We show how to triangulate simple domains using completely well-centered tetrahedra. The domains we consider here are space, infinite slab, infinite rectangular prism, cube, and regular tetrahedron. We also demonstrate single tetrahedra with various combinations of the properties of dihedral acuteness, 2-well-centeredness, and 3-well-centeredness.

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Section: Well-centeredness and Dihedral Acuteness For A Single Tetrahmentioning

confidence: 99%

Section: Tiling Space Slabs and Infinite Rectangular Prisms With Comentioning

confidence: 99%

Section: Tiling Space Slabs and Infinite Rectangular Prisms With Comentioning

confidence: 99%

Section: Tiling Space Slabs and Infinite Rectangular Prisms With Comentioning

confidence: 99%

“…. when he used the second Sommerville construction," S [6,7] which is the completely well-centered tetrahedron shown in Fig. 8.…”

Section: Tiling Space Slabs and Infinite Rectangular Prisms With Comentioning

confidence: 99%

See 3 more Smart Citations

Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra

VanderZee¹,

Hirani²,

Guoy

Proceedings of the 17th International Meshing Roundtable

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Tailoring unstructured meshes for use with a 3D time domain co‐volume algorithm for computational electromagnetics

Xie

Hassan

Morgan

2011

Numerical Meth Engineering

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SUMMARYThe Yee algorithm is a highly efficient time domain co-volume method for computational electromagnetics using a pair of staggered orthogonal cartesian meshes. An equivalent unstructured mesh process is readily implemented on a primal Delaunay tetrahedral mesh and its orthogonal Voronoï dual. The difficulty, for practical applications, is ensuring that both the Delaunay and the Voronoï meshes possess the necessary properties to enable an accurate and efficient solution to be obtained by this approach. This is a mesh generation problem and it is addressed here by using a combination of a point distribution provided by an ideal tetrahedral mesh, a constrained centroidal Voronoï tessellation method and a mesh quality optimization procedure. For the selected application area of electromagnetic wave scattering problems, the computational efficiency is enhanced by using a hybrid primal mesh, consisting of tetrahedral elements in the vicinity of the scatterer and hexahedral elements elsewhere. Three examples are included to demonstrate the viability of the proposed approach and to indicate the numerical performance that can be achieved.

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Universal meshes for smooth surfaces with no boundary in three dimensions

Kabaria

Lew

2016

Numerical Meth Engineering

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Summary We introduce a method to mesh the boundary Γ of a smooth, open domain in double-struckR3 immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. Copyright © 2016 John Wiley & Sons, Ltd.

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Tiling space and slabs with acute tetrahedra

Cited by 51 publications

References 36 publications

Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra

Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra

Tailoring unstructured meshes for use with a 3D time domain co‐volume algorithm for computational electromagnetics

Universal meshes for smooth surfaces with no boundary in three dimensions

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