Variational delaunay approach to the generation of tetrahedral finite element meshes (original) (raw)
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A mesh generator for tetrahedral elements using Delaunay triangulation
Magnetics, IEEE Transactions on, 1993
A tetrahedral mesh generator has been developed. The generator is based on the Delaunay triangulation which is implemented b y employing the insertion polyhedron algorithm. In this paper some n e w methods to deal w i t h the problems associated w i t h the three-dimensional Delaunay triangulation and the insertion polyhedron algorithm are presented: degeneracy, the crossing situation, identification of the internal elements and internal point generation. The generator works both for convex and non-convex domains, including those w i t h high aspectratio subdomains. Some examples are given in this paper to illustrate the capability of the generator.
ACM Transactions on Graphics, 2009
We present a practical approach to isotropic tetrahedral meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of user-defined criteria. This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone. A careful treatment of boundaries and their features is presented, offering a versatile framework for designing smoothly graded tetrahedral meshes.
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.
An approach to refining three-dimensional tetrahedral meshes based on Delaunay transformations
International Journal for Numerical Methods in Engineering, 1994
A technique for refining three-dimensional tetrahedral meshes is proposed in this paper. The proposed technique is capable of treating arbitrary unstructured tetrahedral meshes, convex or non-convex with multiple regions resulting in high quality constrained Delaunay triangulations. The tetrahedra generated are of high quality (nearly equilateral). Sliver tetrahedra, which present a real problem to many algorithms are not produced with the new method. The key to the generation of high quality tetrahedra is the iterative application of a set of topological transformations based on the Voronoi-Delaunay theory and a reposition of nodes technique. The computational requirements of the proposed technique are in linear relationship with the number of nodes and tetrahedra, making it ideal for direct employment in a fully automatic finite element analysis system for 3-D adaptive mesh refinement. Application to some test problems is presented to show the effectiveness and applicability of the new method.
Tetrahedral mesh generation using Delaunay refinement with non-standard quality measures
International Journal for Numerical Methods in Engineering, 2011
This paper studies the practical performance of Delaunay refinement tetrahedral mesh generation algorithms. By using non-standard quality measures to drive refinement, we show that sliver tetrahedra can be eliminated from constrained Delaunay tetrahedralizations solely by refinement. Despite the fact that quality guarantees cannot be proven, the algorithm can consistently generate meshes with dihedral angles between 18 • and 154 •. Using a fairer quality measure targeting every type of bad tetrahedron, dihedral angles between 14 • and 154 • can be obtained. The number of vertices inserted to achieve quality meshes is comparable to that needed when driving refinement with the standard circumradius-to-shortest-edge ratio. We also study the use of mesh improvement techniques on Delaunay refined meshes and observe that the minimum dihedral angle can generally be pushed above 20 • , regardless of the quality measure used to drive refinement. The algorithm presented in this paper can accept geometric domains whose boundaries are piecewise smooth.
Boundary Refinement in Delaunay Mesh Generation Using Arbitrarily Ordered Vertex Insertion
In general, guaranteed-quality Delaunay meshing algo- rithms are dicult to parallelize because they require strictly ordered updates to the mesh boundary. We show that, by replacing the Delaunay cavity in the Bowyer-Watson algorithm with what we call the cir- cumball intersection set, updates to the mesh can occur in any order, especially at the mesh boundary. To demonstrate this new idea, we describe a 2D con- strained Delaunay meshing algorithm that does not en- force strict ordering of vertex insertions near the mesh boundary. We prove that the sequential version of this algorithm generates a mesh in which the circumradius to shortest edge ratio of every triangle is p 2 or greater, as long as every angle interior to the polygonal input do- main is at least 90o. We briefly touch upon the parallel version of this algorithm, but we relegate a more com- plete discussion (with extension to 3D) to a forthcoming paper.
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.
Computer Methods in Applied Mechanics and Engineering, 2000
A method of unstructured tetrahedral-mesh generation for general three-dimensional domains is presented. A c o n ventional boundary representation is adopted as the basis for the description of solids with evolving geometry and topology. The geometry of surfaces is represented either analytically of by piecewise polynomial interpolation. A preliminary surface mesh is generated by an advancing front method, with the nodes inserted by hard-sphere packing in physical space in accordance with a prescribed mesh density. Interior nodes are inserted in a face-centered-cubic (FCC) crystal lattice arrangement coupled to octree spatial subdivision, with the local lattice parameter determined by a prespeci ed nodal density function. Prior to triangulation of the volume, the preliminary surface mesh is preprocessed by a combination of local transformations and subdivision in order to guarantee that the surface triangulation bea subcomplex of the volume Delaunay triangulation. A joint Delaunay triangulation of the interior and boundary nodes which preserves the modi ed surface mesh is then constructed via an advancing front approach. The resulting mesh is nally improved upon by the application of local transformations. The overall time complexity of the mesher is O(N log N). The robustness and versatility of the approach, as well as the good quality o f the resulting meshes, is demonstrated with the aid of selected examples.
Symposium on Geometry Processing, 2007
We present algorithms to produce Delaunay meshes from arbitrary triangle meshes by edge flipping and geometry- preserving refinement and prove their correctness. In particular we show that edge flipping serves to reduce mesh surface area, and that a poorly sampled input mesh may yield unflippable edges necessitating refinement to ensure a Delaunay mesh output. Multiresolution Delaunay meshes can be obtained
Variational tetrahedral meshing
ACM Transactions on Graphics, 2005
In this paper, a novel Delaunay-based variational approach to isotropic tetrahedral meshing is presented. To achieve both robustness and efficiency, we minimize a simple mesh-dependent energy through global updates of both vertex positions and connectivity. As this energy is known to be the L 1 distance between an isotropic quadratic function and its linear interpolation on the mesh, our minimization procedure generates well-shaped tetrahedra. Mesh design is controlled through a gradation smoothness parameter and selection of the desired number of vertices. We provide the foundations of our approach by explaining both the underlying variational principle and its geometric interpretation. We demonstrate the quality of the resulting meshes through a series of examples.