Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions
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 des...
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| Published in | Computational geometry : theory and applications Vol. 46; no. 6; pp. 700 - 724 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
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Elsevier B.V
01.08.2013
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| Abstract | 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. |
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| AbstractList | 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 R 2 , there are infinitely many vertex links that prohibit a well-centered tetrahedral mesh in R 3 . 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. |
| Author | Ramos, Edgar A. Guoy, Damrong Zharnitsky, Vadim VanderZee, Evan Hirani, Anil N. |
| Author_xml | – sequence: 1 givenname: Evan surname: VanderZee fullname: VanderZee, Evan organization: Argonne National Laboratory, Lemont, IL, USA – sequence: 2 givenname: Anil N. surname: Hirani fullname: Hirani, Anil N. email: hirani@illinois.edu organization: Department of Computer Science, University of Illinois at Urbana-Champaign, 201 N. Goodwin Avenue, Urbana, IL 61801, USA – sequence: 3 givenname: Damrong surname: Guoy fullname: Guoy, Damrong organization: Synopsys Inc., Mountain View, CA, USA – sequence: 4 givenname: Vadim surname: Zharnitsky fullname: Zharnitsky, Vadim organization: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, USA – sequence: 5 givenname: Edgar A. surname: Ramos fullname: Ramos, Edgar A. organization: Escuela de Matemáticas, Universidad Nacional de Colombia, Colombia |
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| Keywords | Finite element method Acute triangulations Circumcentric dual Discrete exterior calculus Mesh generation |
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| References_xml | – year: 1984 ident: br0140 article-title: Elements of Algebraic Topology contributor: fullname: Munkres – volume: 29 start-page: 362 year: July 1973 end-page: 371 ident: br0040 article-title: A complete catalogue of polyhedra with eight or fewer vertices publication-title: Acta Crystallographica Section A contributor: fullname: Dunitz – volume: 51 start-page: 317 year: 2009 end-page: 335 ident: br0030 article-title: On nonobtuse simplicial partitions publication-title: SIAM Review contributor: fullname: Šolc – volume: 3 start-page: 59 year: 1995 end-page: 69 ident: br0120 article-title: On diagonal dominance of stiffness matrices in 3D publication-title: East–West Journal of Numerical Mathematics contributor: fullname: Qun – volume: vol. 10 year: 2006 ident: br0010 article-title: Algorithms in Real Algebraic Geometry publication-title: Algorithms and Computation in Mathematics contributor: fullname: Roy – start-page: 1 year: 2012 end-page: 26 ident: br0100 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10.1016/j.comgeo.2012.11.003_br0180 article-title: Pitching tents in space–time: Mesh generation for discontinuous Galerkin method publication-title: International Journal of Foundations of Computer Science doi: 10.1142/S0129054102001059 contributor: fullname: Üngör – ident: 10.1016/j.comgeo.2012.11.003_br0020 doi: 10.1145/2382585.2382588 – volume: 29 start-page: 362 issue: 4 year: 1973 ident: 10.1016/j.comgeo.2012.11.003_br0040 article-title: A complete catalogue of polyhedra with eight or fewer vertices publication-title: Acta Crystallographica Section A doi: 10.1107/S0567739473000963 contributor: fullname: Britton – volume: 3 start-page: 59 issue: 1 year: 1995 ident: 10.1016/j.comgeo.2012.11.003_br0120 article-title: On diagonal dominance of stiffness matrices in 3D publication-title: East–West Journal of Numerical Mathematics contributor: fullname: Křížek – volume: 58 start-page: 323 issue: 2 year: 2007 ident: 10.1016/j.comgeo.2012.11.003_br0130 article-title: Fast generation of 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| SubjectTerms | Acute triangulations Algebra Circumcentric dual Combinatorial analysis Computational geometry Constrictions Discrete exterior calculus Finite element method Links Mesh generation Triangles Triangulation |
| Title | Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions |
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