Constrained CVT meshes and a comparison of triangular mesh generators (original) (raw)
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Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations
International Journal for Numerical Methods in Engineering, 2003
The centroidal Voronoi tessellation based Delaunay triangulation (CVDT) provides an optimal distribution of generating points with respect to a given density function and accordingly generates a high-quality mesh. In this paper, we discuss algorithms for the construction of the constrained CVDT from an initial Delaunay tetrahedral mesh of a three-dimensional domain. By establishing an appropriate relationship between the density function and the speciÿed sizing ÿeld and applying the Lloyd's iteration, the constrained CVDT mesh is obtained as a natural global optimization of the initial mesh. Simple local operations such as edges=faces ippings are also used to further improve the CVDT mesh. Several complex meshing examples and their element quality statistics are presented to demonstrate the e ectiveness and e ciency of the proposed mesh generation and optimization method. Copyright
Triangulation of arbitrary polyhedra to support automatic mesh generators
International Journal for Numerical Methods in Engineering, 2000
An algorithm is presented for the triangulation of arbitrary non-convex polyhedral regions starting with a prescribed boundary triangulation matching existing mesh entities in the remainder of the domain. The algorithm is designed to circumvent the termination problems of volume meshing algorithms which manifest themselves in the inability to successfully create tetrahedra within small subdomains to be referred to herein as cavities. To address this need, a robust Delaunay algorithm with an e cient and termination guaranteed face recovery method is the most appropriate approach. The algorithm begins with Delaunay vertex insertion followed by a face recovery method that conserves the boundary of the cavity by utilizing local mesh modi cation operations such as edge split, collapse and swap and a new set of tools which we call complex splits. The local mesh modi cations are performed in such a manner that each original surface triangulation is represented either as was, or as a concatenation of triangles. When done in this manner, it is always possible to split the matching mesh entities, ensuring that a compatible mesh is created. The algorithm is robust and has been tested against complex manifold and non-manifold cavities resulting in a valid mesh of the entire domain.
Constrained Centroidal Voronoi Tessellations for Surfaces
SIAM Journal on Scientific Computing, 2003
Centroidal Voronoi tessellations are useful for subdividing a region in Euclidean space into Voronoi regions whose generators are also the centers of mass, with respect to a prescribed density function, of the regions. Their extensions to general spaces and sets are also available; for example, tessellations of surfaces in a Euclidean space may be considered. In this paper, a precise definition of such constrained centroidal Voronoi tessellations (CCVTs) is given and a number of their properties are derived, including their characterization as minimizers of an "energy." Deterministic and probabilistic algorithms for the construction of CCVTs are presented and some analytical results for one of the algorithms are given. Computational examples are provided which serve to illustrate the high quality of CCVT point sets. Finally, CCVT point sets are applied to polynomial interpolation and numerical integration on the sphere.
Geoscientific Model Development, 2013
A new algorithm, featuring overlapping domain decompositions, for the parallel construction of Delaunay and Voronoi tessellations is developed. Overlapping allows for the seamless stitching of the partial pieces of the global Delaunay tessellations constructed by individual processors. The algorithm is then modified, by the addition of stereographic projections, to handle the parallel construction of spherical Delaunay and Voronoi tessellations. The algorithms are then embedded into algorithms for the parallel construction of planar and spherical centroidal Voronoi tessellations that require multiple constructions of Delaunay tessellations. This combination of overlapping domain decompositions with stereographic projections provides a unique algorithm for the construction of spherical meshes that can be used in climate simulations. Computational tests are used to demonstrate the efficiency and scalability of the algorithms for spherical Delaunay and centroidal Voronoi tessellations....
Anisotropic Centroidal Voronoi Tessellations and Their Applications
SIAM Journal on Scientific Computing, 2005
In this paper, we introduce a novel definition of the anisotropic centroidal Voronoi tessellation (ACVT) corresponding to a given Riemann metric tensor. A directional distance function is used in the definition to simplify the computation. We provide algorithms to approximate the ACVT using the Lloyd iteration and the construction of anisotropic Delaunay triangulation under the given Riemannian metric. The ACVT is applied to the optimization of two-dimensional anisotropic Delaunay triangulation, to the generation of surface CVT, and high-quality triangular mesh on general surfaces. Various numerical examples demonstrating the effectiveness of the proposed method are presented.
CGALmesh: a Generic Framework for Delaunay Mesh Generation
CGALmesh is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes-surface triangular meshes or volume tetrahedral meshes-from input surfaces, 3D domains as well as 3D multi-domains, with or without sharp features. The underlying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces, and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality as well as on the size and shape of the mesh elements. It provides four optional mesh optimization algorithms to further improve the mesh quality. A distinctive property of CGALmesh is its high flexibility with respect to the input domain representation. Such a flexibility is achieved through a careful software design, gathering into a single abstract concept, denoted by the oracle, all required interface features between the meshing engine and the input domain. We already provide oracles for domains defined by polyhedral and implicit surfaces.
Triangulations and tetrahedrizations are important geometrical discretization procedures applied to several areas, such as the reconstruction of surfaces and data visualization. Delaunay and Voronoi tessellations are discretization structures of domains with desirable geometrical properties. In this work, a systematic review of algorithms with linear-time behaviour to generate 2D/3D Delaunay and/or Voronoi tessellations is presented.
Efficient computation of clipped Voronoi diagram for mesh generation
Computer-Aided Design, 2013
The Voronoi diagram is a fundamental geometric structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact domain (i.e. a bounded and closed 2D region or a 3D volume), some Voronoi cells of their Voronoi diagram are infinite or partially outside of the domain, but in practice only the parts of the cells inside the domain are needed, as when computing the centroidal Voronoi tessellation. Such a Voronoi diagram confined to a compact domain is called a clipped Voronoi diagram. We present an efficient algorithm to compute the clipped Voronoi diagram for a set of sites with respect to a compact 2D region or a 3D volume. We also apply the proposed method to optimal mesh generation based on the centroidal Voronoi tessellation. 5 biology and so on. 6 Suppose that a set of sites in a compact domain in 7 R d is given. Each site is associated with a Voronoi 8 cell containing all the points in R d closer to the site 9 than to any other sites; these cells constitute the 10 Voronoi diagram of the set of sites. Voronoi cells of 11 those sites on the convex hull are infinite, and some 12 of Voronoi cells may be partially outside of the spec-13 ified domain. However, in many applications one 14 usually needs only the parts of Voronoi cells inside 15 the specific domain. That is, the Voronoi diagram 16 restricted to the given domain, which is defined as 17 the intersection of the Voronoi diagram and the do-18 main, and is therefore called the clipped Voronoi 19 diagram [1]. The corresponding Voronoi cells are 20 called the clipped Voronoi cells (see Figure 1). 21 Computing the clipped Voronoi diagram in a con-22 42 remeshing framework to 2D/3D mesh generation. 43 To minimize the CVT energy function, one needs to 44 compute the clipped Voronoi diagram in the input 45 domain for function evaluation and gradient com-46 putation (see Section 2). 47 In this paper, we shall present practical algo-48 rithms for computing clipped Voronoi diagrams 49 based on several simple operations. The main idea 50 intersection of Voronoi diagram and the domain di-52 rectly, we first detect the Voronoi cells that have in-53 tersections with domain boundary and then apply 54 computation for those cells only. We use a simple 55 and efficient algorithm based on connectivity propa-56 gation for detecting the cells that intersect with the 57 domain boundary (i.e., polygons in 2D and mesh 58 surfaces in 3D, respectively). We also utilize the 59 presented techniques for mesh generation as appli-60 cations. The contributions of this paper include : 61 • introduce new methods for computing the 62 clipped Voronoi diagram in 2D regions (Sec-63 tion 3) and 3D volumes (Section 4); 64 • present practical algorithms for 2D/3D mesh 65 generation based on the presented clipped 66 Voronoi diagram computation techniques (Sec-67 tion 5).