Interleaving Delaunay Refinement and Optimization for 2D Triangle Mesh Generation

Tournois, Jane; Alliez, Pierre; Devillers, Olivier

doi:10.1007/978-3-540-75103-8_5

Cited by 16 publications

(20 citation statements)

References 15 publications

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“…For triangular and tetrahedral meshes quality criteria for element shapes are well understood [30], and successful mesh generation and optimization techniques based on Delaunay refinement [28,29,7] or variational optimization [38,39,41] have been proposed.…”

Section: Introductionmentioning

confidence: 99%

“…To this end, a number of very successful mesh improvement approaches have been proposed, being based on either Delaunay refinement [28,29,9] or some variational optimization [1,38,22,39,10,41]. So-called pliant methods, which combine local topological changes (e.g., Delaunay refinement) and vertex relocation (e.g., Laplacian smoothing or Lloyd relaxation) have been found to be superior over methods involving one of the techniques only [5,22,38].…”

Section: Introductionmentioning

confidence: 99%

“…As a consequence, Voronoi tessellation of the simulation domain are frequently used, since they guarantee convex elements. Well-shaped, isotropic Voronoi cells can be achieved by Lloyd clustering [38] or energy minimization [23], leading to centroidal Voronoi tessellations (CVTs, see Figure 1). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

Sieger

Alliez

Botsch

2010

Proceedings of the 19th International Meshing Roundtable

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Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectively removes the major reason for numerical instabilities-short edges in the Voronoi diagram. We evaluate our method on a 2D Poisson problem and demonstrate that our simple but effective optimization achieves a significant improvement of the stiffness matrix condition number.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

Sieger

Alliez

Botsch

2010

Proceedings of the 19th International Meshing Roundtable

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“…Alliez et al [1] describe a variational technique for mesh generation that couples Delaunay refinement with a relaxation process for vertex locations. This algorithm and later variants [31,32,34,37] base their energy minimization on a sizing field for particle density coupled with an energy minimization grounded in the notion of a centroidal Voronoi diagram [14] and its dual, the optimal Delaunay triangulation [5].…”

Section: Related Work and Backgroundmentioning

confidence: 99%

Particle Systems for Adaptive, Isotropic Meshing of CAD Models

Bronson¹,

Levine²,

Whitaker³

2010

Proceedings of the 19th International Meshing Roundtable

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We present a particle-based approach for generating adaptive triangular surface and tetrahedral volume meshes from computer-aided design models. Input shapes are treated as a collection of smooth, parametric surface patches that can meet non-smoothly on boundaries. Our approach uses a hierarchical sampling scheme that places particles on features in order of increasing dimensionality. These particles reach a good distribution by minimizing an energy computed in 3D world space, with movements occurring in the parametric space of each surface patch. Rather than using a pre-computed measure of feature size, our system automatically adapts to both curvature as well as a notion of topological separation. It also enforces a measure of smoothness on these constraints to construct a sizing field that acts as a proxy to piecewise-smooth feature size. We evaluate our technique with comparisons against other popular triangular meshing techniques for this domain.

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“…Vertices are placed heuristically to adaptively sample local features and maintain a good shape of the resulting elements, although the discussion about the optimal shape is not settled even for triangles [1]. The tessellation can be done in ad-hoc constructions [2], iteratively [3], or based on guidance fields [4,5] to name a few. In the rare occasion of a predetermined set of vertices, the choice of methods is generally limited to the constrained Delaunay triangulation or other, data-dependent, triangulations [6].…”

Section: Introductionmentioning

confidence: 99%

Geometric Properties of the Adaptive Delaunay Tessellation

Bobach

Constantiniu

Steinmann

et al. 2010

Mathematical Methods for Curves and Surfaces

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Abstract. Recently, the Adaptive Delaunay Tessellation (Adt) was introduced in the context of computational mechanics as a tool to support Voronoi-based nodal integration schemes in the finite element method. While focusing on applications in mechanical engineering, the former presentation lacked rigorous proofs for the claimed geometric properties of the Adt necessary for the computation of the nodal integration scheme. This paper gives pending proofs for the three main claims which are uniqueness of the Adt, connectedness of the Adt, and coverage of the Voronoi tiles by adjacent Adt tiles. Furthermore, this paper provides a critical assessment of the Adt for arbitrary point sets.

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Interleaving Delaunay Refinement and Optimization for 2D Triangle Mesh Generation

Cited by 16 publications

References 15 publications

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

Particle Systems for Adaptive, Isotropic Meshing of CAD Models

Geometric Properties of the Adaptive Delaunay Tessellation

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