## Tetrahedral Mesh Generation by Delaunay Refinement (1998)

Venue: | Proc. 14th Annu. ACM Sympos. Comput. Geom |

Citations: | 114 - 7 self |

### BibTeX

@INPROCEEDINGS{Shewchuk98tetrahedralmesh,

author = {Jonathan Richard Shewchuk},

title = {Tetrahedral Mesh Generation by Delaunay Refinement},

booktitle = {Proc. 14th Annu. ACM Sympos. Comput. Geom},

year = {1998},

pages = {86--95}

}

### Years of Citing Articles

### OpenURL

### Abstract

Given a complex of vertices, constraining segments, and planar straight-line constraining facets in E 3 , with no input angle less than 90 ffi , an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradius-to-shortest edge ratios are no greater than two. The sizes of the tetrahedra can provably grade from small to large over a relatively short distance. An implementation demonstrates that the algorithm generates excellent meshes, generally surpassing the theoretical bounds, and is effective in eliminating tetrahedra with small or large dihedral angles, although they are not all covered by the theoretical guarantee. 1 Introduction Meshes of triangles or tetrahedra have many applications, including interpolation, rendering, and numerical methods such as the finite element method. Most such applications demand more than just a triangulation of the object or domain being rendered or simulated. To ensure accurate results, the triangles or tetr...

### Citations

195 | A Delaunay Refinement Algorithm for quality 2Dimensional Mesh Generation
- Ruppert
- 1995
(Show Context)
Citation Context ...same size. Uniform meshes sometimes have many more triangles or tetrahedra than are necessary, and thus impose an excessive computational load upon the applications that make use of them. Jim Ruppert =-=[13]-=- and Paul Chew [3] have each proposed two-dimensional Delaunay refinement algorithms that produce meshes of well-shaped triangles whose sizes are graded, and Ruppert has furthermore proven that his al... |

186 |
Computing the n-dimensional Delaunay tessellation with applications to Voronoi polytopes
- Watson
- 1981
(Show Context)
Citation Context ...tex at the circumcenter of a triangle or tetrahedron of poor quality. The Delaunay property is maintained, perhaps by the Bowyer/Watson algorithm for the incremental update of Delaunay triangulations =-=[1, 16]-=-. The poor simplex cannot survive, because its circumsphere is no longer empty. For brevity, the act of inserting a vertex at a simplex 's circumcenter is called splitting a simplex. If poor simplices... |

171 |
Computing Dirichlet tessellations
- Bowyer
- 1981
(Show Context)
Citation Context ...tex at the circumcenter of a triangle or tetrahedron of poor quality. The Delaunay property is maintained, perhaps by the Bowyer/Watson algorithm for the incremental update of Delaunay triangulations =-=[1, 16]-=-. The poor simplex cannot survive, because its circumsphere is no longer empty. For brevity, the act of inserting a vertex at a simplex 's circumcenter is called splitting a simplex. If poor simplices... |

103 |
Guaranteed-quality triangular meshes
- Chew
- 1989
(Show Context)
Citation Context ...Foundation under Grant CMS-9318163, and in part by a grant from the Intel Corporation. geometry community in the early 1990s. The first provably good Delaunay refinement algorithm is due to Paul Chew =-=[2]-=-, and takes as its input a set of vertices and segments that define the region to be meshed. By inserting additional vertices, Chew's algorithm generates a two-dimensional constrained Delaunay triangu... |

82 | Delaunay Refinement Mesh Generation
- Shewchuk
- 1997
(Show Context)
Citation Context ...d of 2, 1:63, or 1:15. The projection condition can be somewhat weakened; for example, a 60 ffi separation between incident segments suffices. All these improvements are described in detail elsewhere =-=[14]-=-. The improvements in circumradius-to-shortest edge ratio, in particular, are significant and further propel the approach described here beyond the pioneering tetrahedral mesh generation algorithms of... |

77 | Quality mesh generation in three dimensions
- Mitchell, Vavasis
- 1992
(Show Context)
Citation Context ...good circumradius-to-shortest edge ratios. Two other tetrahedral mesh generation algorithms (not based on Delaunay refinement) have provable bounds. The octree-based algorithm of Mitchell and Vavasis =-=[11, 12]-=- is a theoretical tour de force, obtaining provable bounds on dihedral angles and grading, but its bounds are too weak to offer any practical reassurance (and have not been explicitly stated). An algo... |

74 | A Delaunay based numerical method for three dimensions: generation, formulation, and partition
- Miller, almor, et al.
- 1995
(Show Context)
Citation Context ...result, the present algorithm typically uses fewer vertices by several orders of magnitude; details are provided elsewhere [14]. 2 A Quality Measure for Simplices Miller, Talmor, Teng, and Walkington =-=[9]-=- have pointed out that the most natural and elegant measure for analyzing Delaunay refinement algorithms is the circumradius-to-shortest edge ratio of a triangle or tetrahedron. The circumsphere of a ... |

56 |
Software for C Surface Interpolation
- Lawson
- 1977
(Show Context)
Citation Context ... is a local operation, and is inexpensive except in unusual cases. In two dimensions, Delaunay triangulations maximize the minimum angle, compared with all other triangulations of the same vertex set =-=[8]. The grea-=-test advantage of Delaunay triangulations is less obvious. The central question of any Delaunay refinement algorithm is "where should the next vertex be inserted?" As Section 3 will demonstr... |

34 |
On Good Triangulations in Three Dimensions
- Dey, Bajaj, et al.
- 1992
(Show Context)
Citation Context ...eshed. By inserting additional vertices, Chew's algorithm generates a two-dimensional constrained Delaunay triangulation whose angles are bounded between 30 ffi and 120 ffi . Dey, Bajaj, and Sugihara =-=[5]-=- and Chew [4] generalize Chew's algorithm to three dimensions, but only for unconstrained point set inputs. All three algorithms produce uniform meshes, whose triangles or tetrahedra are of roughly th... |

29 | Ollivier-Gooch C. A comparison of tetrahedral mesh improvement techniques
- Freitag
- 1996
(Show Context)
Citation Context ...radius-toshortest edge ratios of their tetrahedra are an excellent starting point for mesh smoothing and optimization methods that remove slivers and otherwise improve the quality of an existing mesh =-=[6]-=-. Even if slivers are not removed, the Voronoi dual of a tetrahedralization with bounded circumradius-to-shortest edge ratios has nicely rounded cells, and is sometimes ideal for use in the control vo... |

19 |
Selective Refinement: A New Strategy for Automatic Node Placement in Graded Triangular Meshes
- Frey
- 1987
(Show Context)
Citation Context ...lt to tell where the suggestion arose to use the triangulation to guide vertex creation. These ideas have been intensively studied in the engineering community since the mid-1980s; for instance, Frey =-=[7]-=- eliminates poorly shaped triangles from a triangulation by inserting new vertices at their circumcenters (defined in Section 2), whereas Weatherill [17] inserts new vertices at their centroids. These... |

14 |
Delaunay Triangulation in Computational Fluid Dynamics
- Weatherill
- 1992
(Show Context)
Citation Context ...mmunity since the mid-1980s; for instance, Frey [7] eliminates poorly shaped triangles from a triangulation by inserting new vertices at their circumcenters (defined in Section 2), whereas Weatherill =-=[17]-=- inserts new vertices at their centroids. These ideas bore vital theoretical fruit when the problem of mesh generation began to attract interest from the computational Supported in part by the Advance... |

6 |
Control Volume Meshes using Sphere
- Miller, Talmor, et al.
- 1996
(Show Context)
Citation Context ...bounds on dihedral angles and grading, but its bounds are too weak to offer any practical reassurance (and have not been explicitly stated). An algorithm by Miller, Talmor, Teng, Walkington, and Wang =-=[10]-=- generates its final vertex set before triangulating it. Their algorithm has provable bounds similar to those of the algorithm described herein, but the algorithm herein has the opportunity to stop ea... |