Summary
We propose a new mesh refinement algorithm for computing quality guaranteed Delaunay triangulations in three dimensions. The refinement relies on new ideas for computing the goodness of the mesh, and a sampling strategy that employs numerically stable Steiner points. We show through experiments that the new algorithm results in sparse well-spaced point sets which in turn leads to tetrahedral meshes with fewer elements than the traditional refinement methods.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Bern, D. Eppstein, and J. Gilbert. Provably good mesh generation. J. Comp. System Sciences 48:384–409, 1994.
B. Chazelle, Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm, SIAM Journal on Computing 13: 488–507, 1984.
S.-W. Cheng and T. K. Dey. Quality meshing with weighted Delaunay refinement. Proc. 13th ACM-SIAM Sympos. Discrete Algorithms, 137–146, 2002.
S.-W. Cheng, T.K. Dey, E.A. Ramos, and T. Ray. Quality Meshing of Polyhedra with Small Angles. Int. J. Computational Geometry&Applications 15(4): 421–461, 2005.
S.-W. Cheng, T.K. Dey, H. Edelsbrunner,M.A. Facello, and S.-H. Teng. Sliver exudation. Proc. 15th ACM Symp. Comp. Geometry, 1–13, 1999.
S.-W. Cheng and Sheung-Hung Poon. Three-Dimensional Delaunay Mesh Generation. 419-456 Discrete&Computational Geometry, 36(3): 419–456, 2006.
L.P. Chew. Guaranteed-quality triangular meshes. TR-89-983, Cornell Univ., 1989.
D. Cohen-Steiner, E. C. de Verdiére, and M. Yvinec. Conforming Delaunay triangulations in 3d. Proc. 18th ACM Symp. Comp. Geometry, 199–208, 2002.
T.K. Dey, C. L. Bajaj, and K. Sugihara. On good triangulations in three dimensions. Int. J. Computational Geometry&Applications 2(1):75–95, 1992.
H. Edelsbrunner. Geometry and Topology for Mesh Generation. Cambridge Univ. Press, 2001.
H. Edelsbrunner and D. Guoy. Sink insertion for mesh improvement. Proc. 17th ACM Symp. Comp. Geometry, 115–123, 2001.
H. Edelsbrunner, X. Li, G.L. Miller, A. Stathopoulos, D. Talmor, S.-H. Teng, A. Üngör, and N. Walkington. Smoothing and cleaning up slivers. Proc. 32nd ACM Symp. on Theory of Computing, 273–277, 2000.
H. Erten and A. Üngör. Delaunay Refinement with Locally Optimal Steiner Points. (to appear) In Proc. EUROGRAPHICS Symposium on Geometry Processing, Barcelona, Spain July 2007.
S. Har-Peled and A. Üngör. A time-optimal Delaunay Refinement algorithm in two dimensions. Proc. ACM Symposium on Computational Geometry, 228–236, 2005.
B. Hudson, G.L. Miller, and T. Phillips. Sparse Voronoi Refinement. Proc. 15th Int. Meshing Roundtable, 339–356, 2006.
F. Labelle. Sliver Removal by Lattice Refinement. Proc. ACM Symposium on Computational Geometry, 347–356, 2006.
F. Labelle and J.R. Shewchuk. Isosurface Stuffing: Fast Tetrahedral Meshes with Good Dihedral Angles Proc. ACM SIGGRAPH 2007.
X.-Y. Li, S.-H. Teng, and A. Üngör. Biting: Advancing front meets sphere packing. Int. J. Numer. Meth. Eng. 49:61–81, 2000.
X.-Y. Li and S.-H. Teng. Generating well-shaped Delaunay meshed in 3D. Proc. ACM-SIAM Symp. on Discrete Algorithms, 28–37, 2001.
R. Löhner. Progress in grid generation via the advancing front technique. Engineering with Computers 12:186–210, 1996.
G.L. Miller. A time efficient Delaunay Refinement algorithm. Proc. ACM-SIAM Symp. on Disc. Algorithms, 400–409, 2004.
G.L. Miller, S. Pav, and N. Walkington. When and why Ruppert’s algorithm works. Proc. 12th Int. Meshing Roundtable, 91–102, 2003.
G.L. Miller, S. Pav, and N. Walkington. Fully Incremental 3D Delaunay Refinement Mesh Generation. Proc. 11th Int. Meshing Roundtable, 75–86, 2002.
G.L. Miller, D. Talmor, S.-H. Teng, N. Walkington, and H. Wang. Control volume meshes using sphere packing: Generation, Refinement, and coarsening. Proc. 5th Int. Meshing Roundtable, 47–61, 1996.
S. Mitchell and S. Vavasis. Quality mesh generation in three dimensions. Proc. 8th ACM Symp. Comp. Geometry, 212–221, 1992.
M. Murphy, D. M. Mount, and C. W. Gable. A point-placement strategy for conforming Delaunay tetrahedralization. Proc. 11th ACM-SIAM Symp. on Discrete Algorithms, 67–74, 2000.
M.S. Paterson and F.F. Yao, Binary partitions with applications to hiddensurface removal and solid modeling, Proc. 5th ACM Symp. Computational Geometry, 23–32, 1989.
J. Ruppert. A new and simple algorithm for quality 2-dimensional mesh generation. Proc. 4th ACM-SIAM Symp. on Disc. Algorithms, 83–92, 1993.
E. Schönhardt. Über die Zerlegung von Dreieckspolyedern in Tetraeder, Math. Annalen, 98: 309–312, 1928.
J.R. Shewchuk. Delaunay Refinement Mesh Generation. Ph.D. thesis, Carnegie Mellon University, 1997.
J.R. Shewchuk. Tetrahedral mesh generation by Delaunay Refinement. Proc. 14th Annual ACM Symposium on Computational Geometry, 86–95, 1998.
J.R. Shewchuk. Mesh generation for domains with small angles. Proc. 16th ACM Symposium on Computational Geometry, 1–10, 2000.
J.R. Shewchuk. Sweep algorithms for constructing higher-dimensional constrained Delaunay triangulations. Proc. 16th ACM Symposium on Computational Geometry, 350–359, 2000.
J.R. Shewchuk. Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery. Proc. 11th Int. Meshing Roundtable, 193–204, 2002.
J.R. Shewchuk. Delaunay Refinement algorithms for triangular mesh generation. Computational Geometry: Theory and Applications 22(1–3):21–74, 2002.
K. Shimada. Physically-based Mesh Generation: Automated Triangulation of Surfaces and Volumes via Bubble Packing. Ph.D. thesis, MIT, 1993.
D.A. Spielman, S.-H. Teng, and A. Üngör. Parallel Delaunay Refinement: Algorithms and analyses. Proc. 11th Int. Meshing Roundtable, 205–217, 2002.
D.A. Spielman, S.-H. Teng, and A. Üngör. Parallel Delaunay Refinement: Algorithms and analyses. Int. J. Comput. Geometry Appl. 17(1):1-30 (2007)
H. Si. On Refinement of Constrained Delaunay Tetrahedralizations. Proc. 11th Int. Meshing Roundtable, 2006.
D.A. Spielman, S.-H. Teng, and A. Üngör. Parallel Delaunay Refinement with off-centers. Proc. EUROPAR, LNCS 3149, 812–819, 2004.
D. Talmor. Well-Spaced Points for Numerical Methods. Ph.D. thesis, Carnegie Mellon University, 1997.
A. Üngör. Off-centers: A new type of Steiner points for computing size-optimal quality-guaranteed Delaunay triangulations. Proc. LATIN, LNCS 2976, 152–161, 2004.
A. Üngör. Quality meshes made smaller. Proc. 20th European Workshop on Computational Geometry, 5–8, 2005.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jampani, R., Üngör, A. (2008). Construction of Sparse Well-spaced Point Sets for Quality Tetrahedralizations. In: Brewer, M.L., Marcum, D. (eds) Proceedings of the 16th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75103-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-75103-8_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75102-1
Online ISBN: 978-3-540-75103-8
eBook Packages: EngineeringEngineering (R0)