On Delaunay refinement for curved geometries

Computer Graphics Forum, 2010

The technique of Delaunay refinement has been recognized as a versatile tool to generate Delaunay meshes of a variety of geometries. Despite its usefulness, it suffers from one lacuna that limits its application. It does not scale well with the mesh size. As the sample point set grows, the Delaunay triangulation starts stressing the available memory space which ultimately stalls any effective progress. A natural solution to the problem is to maintain the point set in clusters and run the refinement on each individual cluster. However, this needs a careful point insertion strategy and a balanced coordination among the neighboring clusters to ensure consistency across individual meshes. We design an octtree based localized Delaunay refinement method for meshing surfaces in three dimensions which meets these goals. We prove that the algorithm terminates and provide guarantees about structural properties of the output mesh. Experimental results show that the method can avoid memory thrashing while computing large meshes and thus scales much better than the standard Delaunay refinement method.

Lecture Notes in Computer Science, 2006

The use of edge based refinement in general, and Delaunay terminal edge refinement in particular are well established for planar meshing, but largely on a heuristic basis. In this paper, we present a series of theoretical results on the geometric mesh improvement properties of these methods. The discussion is based on refining a mesh to meet a specified angle tolerance.

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.

IEEE Antennas and Propagation Magazine, 1997

Lecture Notes in Computer Science, 2009

This paper surveys Delaunay-based meshing techniques for curved objects, and their application in medical imaging and in computer vision to the extraction of geometric models from segmented images. We show that the so-called Delaunay refinement technique allows to mesh surfaces and volumes bounded by surfaces, with theoretical guarantees on the quality of the approximation, from a geometrical and a topological point of view. Moreover, it offers extensive control over the size and shape of mesh elements, for instance through a (possibly non-uniform) sizing field. We show how this general paradigm can be adapted to produce anisotropic meshes, i.e. meshes elongated along prescribed directions. Lastly, we discuss extensions to higher dimensions, and especially to space-time for producing time-varying 3D models. This is also of interest when input images are transformed into data points in some higher dimensional space as is common practice in machine learning.

2004

The Delaunay Refinement Algorithm for quality meshing is extended to three dimensions. The algorithm accepts input with arbitrarily small angles, and outputs a Conforming Delaunay Tetrahedralization where most tetrahedra have radius-to-shortest-edge ratio smaller than some user chosen µ > 2. Those tets with poor quality are in well defined locations: their circumcenters are describably near input segments. Moreover, the output mesh is well graded to the input: short mesh edges only appear around close features of the input. The algorithm has the added advantage of not requiring a priori knowledge of the "local feature size," and only requires searching locally in the mesh.

Proceedings of the 16th International …, 2008

Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes . This class includes polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all non-manifolds. In contrast to previous approaches, the algorithm does not impose any restriction on the input angles. Although this algorithm has a provable guarantee about topology, certain steps are too expensive to make it practical.

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.

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.

International Journal for Numerical Methods in Engineering, 1988

One approach to fully automatic mesh generation in two and three dimensions is to generate and triangulate a set of points within and on the boundary of a geometry using the properties of the Delaunay triangulation. Because the point data generate mesh topology of greater dimension, it is necessary to insure topological compatibility and perform classification of the resulting mesh with respect to the original geometry.

Proceedings of the thirteenth annual ACM- …, 2002

Delaunay meshes with bounded circumradius to shortest edge length ratio have been proposed in the past for quality meshing. The only poor quality tetrahedra called slivers that can occur in such a mesh can be eliminated by the sliver exudation method. This method has been shown to work for periodic point sets, but not with boundaries. Recently a randomized point-placement strategy has been proposed to remove slivers while conforming to a given boundary. In this paper we present a deterministic algorithm for generating a weighted Delaunay mesh which respects the input boundary and has no poor quality tetrahedron including slivers. This success is achieved by combining the weight pumping method for sliver exudation and the Delaunay refinement method for boundary conformation. We show that an incremental weight pumping can be mixed seamlessly with vertex insertions in our weighted Delaunay refinement paradigm.

Proceedings of the 14th International Meshing Roundtable

An algorithm for quality Delaunay meshing of 2D domains with curved boundaries is presented. The algorithm uses Ruppert's "corner lopping" heuristic [1]. In addition to admitting a simple termination proof, the algorithm can accept curved input without any bound on the tangent angle between adjoining curves. In the limit case, where all curves are straight line segments, the algorithm returns a mesh with a minimum angle of arcsin`1/2 √ 2´, except "near" input corners. Some loss of output quality is experienced with the use of curved input, but this loss is diminuished for smaller input curvature.

1997

This paper proposes a Delaunay-type mesh generation algorithm governed by a metric map. The classical method is briefly established and then the different steps it involves are extended. It will be shown that the proposed method applies in three dimensions. The work is divided in two parts. Part I, i.e. the present paper, is devoted to the algorithmical aspects while Part II will present numerous application examples in the context of finite element computations.

International Journal for Numerical Methods in Engineering, 1997

This paper aims to outline the di erent phases necessary to implement a Delaunay-type automatic mesh generator. First, it summarizes this method and then describes a variant which is numerically robust by mentioning at the same time the problems to solve and the di erent solutions possible. The Delaunay insertion process by itself, the boundary integrity problem, the way to create the ÿeld points as well as the optimization procedures are discussed. The two-dimensional situation is described fully and possible extensions to the three-dimensional case are brie y indicated. ? 1997 by John Wiley & Sons, Ltd.

International Journal for Numerical Methods in Engineering, 2002

Guaranteed-quality unstructured meshing algorithms facilitate the development of automatic meshing tools. However, these algorithms require domains discretized using a set of linear segments, leading to numerical errors in domains with curved boundaries. We introduce an extension of Ruppert's Delaunay refinement algorithm to twodimensional domains with curved boundaries and prove that the same quality bounds apply with curved boundaries as with straight boundaries. We provide implementation details for two-dimensional boundary patches such as lines, circular arcs, cubic parametric curves, and interpolated splines. We present guaranteed-quality triangular meshes generated with curved boundaries, and propose solutions to some problems associated with the use of curved boundaries.