Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings
We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewi...
| Main Authors: | , , , |
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| Format: | Article |
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Elsevier BV
2018
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| Online Access: | http://hdl.handle.net/1721.1/115254 https://orcid.org/0000-0001-5652-9913 |
| _version_ | 1826214289309958144 |
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| author | Caplan, Philip Claude Haimes, Robert Darmofal, David L Galbraith, Marshall C. |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Caplan, Philip Claude Haimes, Robert Darmofal, David L Galbraith, Marshall C. |
| author_sort | Caplan, Philip Claude |
| collection | MIT |
| description | We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. Motivated by the work of Lévy and Dassi [10-12,20], we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and use data structures based on convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries. The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both 3d and 4d problems. Keywords: Anisotropic mesh generation; metric; Nash embedding theorem; isometric; geometry-conforming; restricted Voronoi diagram; constrained centroidal Voronoi tessellation; Steiner vertices; dimension-independent |
| first_indexed | 2024-09-23T12:56:33Z |
| format | Article |
| id | mit-1721.1/115254 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T12:56:33Z |
| publishDate | 2018 |
| publisher | Elsevier BV |
| record_format | dspace |
| spelling | mit-1721.1/1152542022-09-28T11:03:24Z Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings Caplan, Philip Claude Haimes, Robert Darmofal, David L Galbraith, Marshall C. Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Caplan, Philip Claude Haimes, Robert Darmofal, David L Galbraith, Marshall C. We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. Motivated by the work of Lévy and Dassi [10-12,20], we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and use data structures based on convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries. The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both 3d and 4d problems. Keywords: Anisotropic mesh generation; metric; Nash embedding theorem; isometric; geometry-conforming; restricted Voronoi diagram; constrained centroidal Voronoi tessellation; Steiner vertices; dimension-independent 2018-05-07T19:56:13Z 2018-05-07T19:56:13Z 2017-10 2018-03-19T14:34:09Z Article http://purl.org/eprint/type/JournalArticle 1877-7058 http://hdl.handle.net/1721.1/115254 Caplan, Philip Claude, et al. “Anisotropic Geometry-Conforming d-Simplicial Meshing via Isometric Embeddings.” Procedia Engineering, vol. 203, 2017, pp. 141–53. © 2017 The Authors https://orcid.org/0000-0001-5652-9913 http://dx.doi.org/10.1016/J.PROENG.2017.09.798 Procedia Engineering Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier BV Elsevier |
| spellingShingle | Caplan, Philip Claude Haimes, Robert Darmofal, David L Galbraith, Marshall C. Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings |
| title | Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings |
| title_full | Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings |
| title_fullStr | Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings |
| title_full_unstemmed | Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings |
| title_short | Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings |
| title_sort | anisotropic geometry conforming d simplicial meshing via isometric embeddings |
| url | http://hdl.handle.net/1721.1/115254 https://orcid.org/0000-0001-5652-9913 |
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