A Weighted Delaunay Triangulation Framework for Merging Triangulations in a Connectivity Oblivious Fashion

Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework fo...

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Vydané v:	Computer graphics forum Ročník 33; číslo 6; s. 18 - 30
Hlavní autori:	Silva, Luis F., Scheidegger, Luiz F., Etiene, Tiago, Comba, João L. D., Nonato, Luis G., Silva, Cláudio T.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Oxford Blackwell Publishing Ltd 01.09.2014
Predmet:	Analysis Computer graphics Computer Graphics I.3.5 Computational Geometry and Object Modelling Geometric algorithms languages and systems Connectivity merging algorithms Numerical analysis regular triangulations Simulation Studies triangulations weighted Delaunay triangulations
ISSN:	0167-7055, 1467-8659
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Abstract	Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework for updating simplicial meshes that undergo geometric and topological changes. Instead of explicitly maintaining connectivity information, we keep a collection of weights associated with mesh vertices, using a Weighted Delaunay Triangulation (WDT). These weights implicitly define mesh connectivity and allow direct merging of triangulations. We propose two formulations for computing the weights, and two techniques for merging triangulations, and finally illustrate our results with examples in two and three dimensions. Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework for updating simplicial meshes that undergo geometric and topological changes. Instead of explicitly maintaining connectivity information, we keep a collection of weights associated with mesh vertices, using a Weighted Delaunay Triangulation (WDT). These weights implicitly define mesh connectivity and allow direct merging of triangulations
AbstractList	Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework for updating simplicial meshes that undergo geometric and topological changes. Instead of explicitly maintaining connectivity information, we keep a collection of weights associated with mesh vertices, using a Weighted Delaunay Triangulation (WDT). These weights implicitly define mesh connectivity and allow direct merging of triangulations. We propose two formulations for computing the weights, and two techniques for merging triangulations, and finally illustrate our results with examples in two and three dimensions. [PUBLICATION ABSTRACT] Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework for updating simplicial meshes that undergo geometric and topological changes. Instead of explicitly maintaining connectivity information, we keep a collection of weights associated with mesh vertices, using a Weighted Delaunay Triangulation (WDT). These weights implicitly define mesh connectivity and allow direct merging of triangulations. We propose two formulations for computing the weights, and two techniques for merging triangulations, and finally illustrate our results with examples in two and three dimensions. Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework for updating simplicial meshes that undergo geometric and topological changes. Instead of explicitly maintaining connectivity information, we keep a collection of weights associated with mesh vertices, using a Weighted Delaunay Triangulation (WDT). These weights implicitly define mesh connectivity and allow direct merging of triangulations Simplicial meshes are useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, making the simulations unstable. In this paper we propose a framework for updating simplicial meshes that undergo geometric and topological changes. Instead of explicitly maintaining connectivity information, we keep a collection of weights associated with mesh vertices, using a Weighted Delaunay Triangulation (WDT). These weights implicitly define mesh connectivity and allow direct merging of triangulations. We propose two formulations for computing the weights, and two techniques for merging triangulations, and finally illustrate our results with examples in two and three dimensions.
Author	Silva, Luis F. Etiene, Tiago Comba, João L. D. Silva, Cláudio T. Nonato, Luis G. Scheidegger, Luiz F.
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References_xml	– reference: [DLPT98] Devillers O., Liotta G., Preparata F., Tamassia R.: Checking the convexity of polytopes and the planarity of subdivisions. Computational Geometry 11 (1998), 187-208. – reference: [BGH*97] Borouchaki H., George P. L., Hecht F., Laug P., Saltel E.: Delaunay mesh generation governed by metric specifications. Part I. Algorithms. Finite Elements in Analysis and Design 25 (1997), 61-83. – reference: [ES96] Edelsbrunner H., Shah S.: Incremental topological flipping works for regular triangulations. Algorithmica 15 (1996), 223-241. – reference: [Ede01] Edelsbrunner H.: Geometry and Topology for Mesh Generation. Cambridge University Press, Cambridge, UK, 2001. – reference: [DLRS10] De Loera J., Rambau J., Santos F.: Triangulations: Structures for Algorithms and Applications. Springer, Berlin, Heidelberg, 2010. – reference: [DT97] Dantzig G., Thapa M.: Linear Programming: 1: Introduction. Springer Series in Operations Research and Financial Engineering, Springer, NY, USA. 1997. – reference: [GP07] Gross M., Pfister H. (Eds.): Point-Based Graphics. Morgan Kaufmann, Burlington, MA, USA, 2007. – reference: [She02] Shewchuk J.: Delaunay refinement algorithms for triangular mesh generation. Computational Geometry: Theory and Applications 22, 2-3 (2002), 21-74. – reference: [dCTAD09] de Castro P. M. M., Tournois J., Alliez P., Devillers O.: Filtering relocations on a Delaunay triangulation. Computer Graphics Forum 28, 5 (2009), 1465-1474. – reference: [ES86] Edelsbrunner H., Seidel R.: Voronoi diagrams and arrangements. Discrete & Computational Geometry 1 (1986), 25-44. – reference: [BO05] Boissonnat J.-D., Oudot S.: Provably good sampling and meshing of surfaces. Graphical Models 67 (2005), 405-451. – reference: [ACK01] Amenta N., Choi S., Kolluri R.: The power crust. Computatinal Geometry 19 (2001), 127-153. – reference: [Aur87] Aurenhammer F.: Power diagrams: Properties, algorithms and applications. SIAM Journal on Computing 16, 1 (1987), 78-96. – reference: [TWAD09] Tournois J., Wormser C., Alliez P., Desbrun M.: Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Transactions on Graphics 28, 3 (2009), 1-9. – reference: [VPC02] Vigo M., Pla N., Cotrina J.: Regular triangulations of dynamic sets of points. Computer Aided Geometric Design 19, 2 (2002), 127-149. – reference: [NCVMO05] Nonato L. G., Cuadros-Vargas A., Minghim R., Oliveira M.: Beta-connection: Generating a family of models from planar cross sections. 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SubjectTerms	Analysis Computer graphics Computer Graphics I.3.5 Computational Geometry and Object Modelling Geometric algorithms languages and systems Connectivity merging algorithms Numerical analysis regular triangulations Simulation Studies triangulations weighted Delaunay triangulations
Title	A Weighted Delaunay Triangulation Framework for Merging Triangulations in a Connectivity Oblivious Fashion
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