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Interleaving Delaunay Refinement and Optimization for 2D Triangle Mesh Generation
"... We address the problem of generating 2D quality triangle meshes from a set of constraints provided as a planar straight line graph. The algorithm first computes a constrained Delaunay triangulation of the input set of constraints, then interleaves Delaunay refinement and optimization. The refinemen ..."
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We address the problem of generating 2D quality triangle meshes from a set of constraints provided as a planar straight line graph. The algorithm first computes a constrained Delaunay triangulation of the input set of constraints, then interleaves Delaunay refinement and optimization. The refinement stage inserts a subset of the Voronoi vertices and midpoints of constrained edges as Steiner points. The optimization stage optimizes the shape of the triangles through the Lloyd iteration applied to Steiner points both in 1D along constrained edges and in 2D after computing the bounded Voronoi diagram. Our experiments show that the proposed algorithm inserts fewer Steiner points than Delaunay refinement alone, and improves over the mesh quality.
Optimizing voronoi diagrams for polygonal finite element computations
 In Proceedings of the 19th International Meshing Roundtable
, 2010
"... 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 effectiv ..."
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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. 1
NONOBTUSE TRIANGULATIONS OF PSLGS
, 2010
"... We show that any planar PSLG with n vertices has a conforming triangulation by O(n2.5) nonobtuse triangles; they may be chosen to be all acute or all right. This result also improves a previous O(n3) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the ..."
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We show that any planar PSLG with n vertices has a conforming triangulation by O(n2.5) nonobtuse triangles; they may be chosen to be all acute or all right. This result also improves a previous O(n3) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only O(n2) elements are needed, improving an O(n4) bound of Bern and Eppstein. We also show that for any ǫ> 0, every PSLG has a conforming triangulation with O(n2 /ǫ2) elements and with all angles bounded above by 90 ◦ +ǫ. This improves a result of S. Mitchell when ǫ = 3 8π = 67.5 ◦ and Tan when ǫ = 7 30π = 42 ◦. Finally, we prove that any PSLG has a conforming quadrilateral mesh with O(n2) elements and all new angles between 60 ◦ and 120 ◦ (the complexity and angle bounds are both sharp). Moreover, all but O(n) of the angles may be taken in a smaller interval, say [89◦,91 ◦].
Contents lists available at ScienceDirect ComputerAided Design
"... journal homepage: www.elsevier.com/locate/cad ..."
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A Joint Deployment and Routing Strategy for Directional Wireless Mesh Networks
, 2008
"... Abstract—For the emerging wireless mesh networks with multiple radios and directional antennas, this paper first proposes a positionbased deployment and routing strategy, and then gives a concrete approach under this strategy. The main idea of this strategy is to deploy the network in certain kind ..."
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Abstract—For the emerging wireless mesh networks with multiple radios and directional antennas, this paper first proposes a positionbased deployment and routing strategy, and then gives a concrete approach under this strategy. The main idea of this strategy is to deploy the network in certain kind of geometric graph and then design a positionbased routing protocol accordingly, so as to achieve efficiency and scalability for the mesh networks. The proposed approach comprises two parts: (1) a topology generation algorithm based on Delaunay triangulations and (2) a routing protocol based on the greedy forwarding algorithm. Both parts have low complexity of computation, as well as possess appealing properties for deployment or routing. Formal proofs to these claims are provided when applicable. The extensive simulation results show that our proposed approach is indeed efficient and scalable.
A survey of point insertion techniques in bidimensional Delaunay
"... Abstract. Triangulations are geometric discretizations essential in many scientific applications, such as engineering simulations, visualizations, and geographic information systems. The preferred shape of a triangle depends on the applications. Theoretical and experimental analysis of numerical met ..."
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Abstract. Triangulations are geometric discretizations essential in many scientific applications, such as engineering simulations, visualizations, and geographic information systems. The preferred shape of a triangle depends on the applications. Theoretical and experimental analysis of numerical methods that are used in conjunction with triangulations suggest that triangles with no large angles and/or small angles serve well in most applications. This paper is a brief review of a point insertion in 2D Delaunay Triangulations. Important works on the insertion of vertices in Delaunay Triangulations are described as a start point for one who needs to build a quality mesh using adaptive triangularmesh refinement.