Delaunay Meshing of Isosurfaces

2010

In the Euclidean plane, a Delaunay triangulation can be characterized by the requirement that the circumcircle of each triangle be empty of vertices of all other triangles. For triangulating a surface S in R^3, the Delaunay paradigm has typically been employed in the form of the restricted Delaunay triangulation, where the empty circumcircle property is defined by using the Euclidean metric in R^3 to measure distances on the surface. More recently, the intrinsic (geodesic) metric of S has also been employed to define the Delaunay condition. In either case the resulting mesh M is known to approximate S with increasing accuracy as the density of the sample points increases. However, the use of the reference surface S to define the Delaunay criterion is a serious limitation. In particular, in the absence of the original reference surface, there is no way of verifying if a given mesh meets the criterion. We define a self-Delaunay mesh as a triangle mesh that is a Delaunay triangulation ...

Proceedings of the 15th International Meshing Roundtable, 2006

The contribution of the current paper is threefold. First, we generalize the existing sequential point placement strategies for guaranteed quality Delaunay refinement: instead of a specific position for a new point, we derive a selection disk inside the circumdisk of a poor quality triangle. We prove that any point placement algorithm that inserts a point inside the selection disk of a poor quality triangle will terminate and produce a size-optimal mesh. Second, we extend our theoretical foundation for the parallel Delaunay refinement. Our new parallel algorithm can be used in conjunction with any sequential point placement strategy that chooses a point within the selection disk. Third, we implemented our algorithm in C++ for shared memory architectures and present the experimental results. Our data show that even on workstations with a few cores, which are now in common use, our implementation is significantly faster the best sequential counterpart.

Computational Mathematics and Mathematical Physics, 2012

A method is proposed for the generation of three dimensional tetrahedral meshes from incomplete, weakly structured, and inconsistent data describing a geometric model. The method is based on the construction of a piecewise smooth scalar function defining the body so that its boundary is the zero isosurface of the function. Such implicit description of three dimensional domains can be defined analytically or can be constructed from a cloud of points, a set of cross sections, or a "soup" of individual vertices, edges, and faces. By applying Boolean operations over domains, simple primi tives can be combined with reconstruction results to produce complex geometric models without resorting to specialized software. Sharp edges and conical vertices on the domain boundary are repro duced automatically without using special algorithms. Refs. 42. Figs. 25.

Proceedings of the 16th International Meshing Roundtable, 2008

Many modern research areas face the challenge of meshing level sets of sampled scalar functions. While many algorithms focus on ensuring geometric qualities of the output mesh, recent attention has been paid to building topologically accurate Delaunay conforming meshes of any level set from such volumetric data.

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.

Computer Graphics Forum, 2011

Delaunay refinement, recognized as a versatile tool for meshing a variety of geometries, has the deficiency that it does not scale well with increasing mesh size. The bottleneck can be traced down to the memory usage of 3D Delaunay triangulations. Recently an approach has been suggested to tackle this problem for the specific case of smooth surfaces by subdividing the sample set in an octree and then refining each subset individually while ensuring termination and consistency. We extend this to localized refinement of volumes, which brings about some new challenges. We show how these challenges can be met with simple steps while retaining provable guarantees, and that our algorithm scales many folds better than a state-of-the-art meshing tool provided by CGAL.

1997

Abstract A new surface-based approach to triangulation of an implicit surface calledMarching Triangles'(MT) is introduced in this paper. MT enables reconstruction of an e cient triangular mesh representation of an open manifold implicit surface of arbitrary topology. The surface-based approach polygonises the implicit surface by growing a triangulated mesh according to the local geometry and topology.

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.

Proceedings of the 14th International Meshing …, 2005

Polygonal meshes areu sed to model smooth surfaces in manya pplications. Often these meshes need to be remeshed for improving the quality, density or gradedness. We applyt he Delaunayr efinementp aradigm to design ap rovable algorithm for isotropicr emeshing of ap olygonal mesh that approximates as mooth surface. The proofs provide new insights and our experimental results corroborate the theory.

Computer Graphics Forum, 2000

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.

IEEE Antennas and Propagation Magazine, 1997

In this paper, we present an intrinsic algorithm for isotropic mesh simplification. Starting with a set of unevenly distributed samples on the surface, our method computes the geodesic Delaunay triangulation with regard to the sample set and iteratively evolves the Delaunay triangulation such that the Delaunay edges become almost equal in length. Finally, our method outputs the simplified mesh by replacing each curved Delaunay edge with a line segment. We conduct experiments on numerous real-world models of complicated geometry and topology. The promising experimental results demonstrate that the proposed method is intrinsic and insensitive to initial mesh triangulation.

Applied Sciences, 2022

Triangular meshes play critical roles in many applications, such as numerical simulation and additive manufacturing. However, the triangular meshes transformed from computer-aided design models using common algorithms may have many undesirable narrow triangles, which tends to affect the downstream applications. In this paper, we proposed two algorithms for Delaunay mesh construction and simplification to improve the quality of the triangular meshes. Two improved mesh operations of inserting vertices and collapsing vertices based on the principle of minimum volume destruction were designed. The improved vertex inserting operation is able to modify the local mesh so that it will conform to the local Delaunay property. The improved vertex collapsing operation can realize the simplification of the original mesh while maintaining the local Delaunay property. The results of visualized rendering and thermal diffusion simulations verified the improvement of the proposed algorithms in the as...

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.