TY - JOUR

T1 - Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions

AU - VanderZee, Evan

AU - Hirani, Anil N.

AU - Guoy, Damrong

AU - Zharnitsky, Vadim

AU - Ramos, Edgar A.

N1 - Funding Information:
The authors thank Doug West for a helpful discussion. The work of Anil N. Hirani and Evan VanderZee was supported by an NSF Grant No. DMS-0645604 . Evan VanderZee was also partially supported by a fellowship jointly funded by the Computational Science and Engineering Program and the Applied Mathematics Program of the University of Illinois at Urbana-Champaign . Vadim Zharnitsky was partially supported by NSF grant DMS 08-07897 .

PY - 2013/8/1

Y1 - 2013/8/1

N2 - An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, together with some other observations, are used to describe restrictions on the local combinatorial structure of simplicial meshes in which every simplex is well-centered. In particular, it is shown that in a 3-well-centered (2-well-centered) tetrahedral mesh there are at least 7 (9) edges incident to each interior vertex, and these bounds are sharp. Moreover, it is shown that, in stark contrast to the 2-dimensional analog, where there are exactly two vertex links that prevent a well-centered triangle mesh in R2, there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in R3.

AB - An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These conditions, together with some other observations, are used to describe restrictions on the local combinatorial structure of simplicial meshes in which every simplex is well-centered. In particular, it is shown that in a 3-well-centered (2-well-centered) tetrahedral mesh there are at least 7 (9) edges incident to each interior vertex, and these bounds are sharp. Moreover, it is shown that, in stark contrast to the 2-dimensional analog, where there are exactly two vertex links that prevent a well-centered triangle mesh in R2, there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in R3.

KW - Acute triangulations

KW - Circumcentric dual

KW - Discrete exterior calculus

KW - Finite element method

KW - Mesh generation

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U2 - 10.1016/j.comgeo.2012.11.003

DO - 10.1016/j.comgeo.2012.11.003

M3 - Article

AN - SCOPUS:84992518984

VL - 46

SP - 700

EP - 724

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 6

ER -