Abstract
A d-dimensional simplicial mesh is a Delaunay triangulation if the circumsphere of each of its simplices does not contain any vertices inside. A mesh is well-shaped if the maximum aspect ratio of all its simplices is bounded from above by a constant. It is a long-term open problem to generate well-shaped d-dimensional Delaunay meshes for a given polyhedral domain. In this paper, we present a re?nement-based method that generates well-shaped d-dimensional Delaunay meshes for any PLC domain with no small input angles. Furthermore, we show that the generated well-shaped mesh has O(n) d-simplices, where n is the smallest number of d-simplices of any almost-good meshes for the same domain. A mesh is almost-good if each of its simplices has a bounded circumradius to the shortest edge length ratio.
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References
Bern, M., Chew, P., Eppstein, D., AND Ruppert, J. Dihedral bounds for mesh generation in high dimensions. In Proc. 6th ACM-SIAM Symposium on Discrete Algorithms (1995), ACM, pp. 189–196.
Cheng, S.W., Dey, T.K., Edelsbrunner, H., Facello, M.A., AND Teng, S.H. Sliver exudation. In Proc. 15th ACM Symposium on Computational Geometry (1999), pp. 1–13.
Chew, L.P. Constrained Delaunay triangulations. Algorithmica 4 (1989), 97–108.
Chew, L.P. Guaranteed-quality delaunay meshing in 3d (short version). In 13th ACM Sym. on Comp. Geometry (1997), pp. 391–393.
Edelsbrunner, H., Li, X.Y., Miller, G., Stathopoulos, A., Talmor, D., Teng, S.H., Üngör, A., AND Walkington, N. Smoothing and cleaning up slivers. In ACM Symposium on Theory of Computing (STOC00) (2000).
Li, X.Y. Sliver-free Three Dimensional Delaunay Mesh Generation. PhD thesis, University of Illinois at Urbana-Champaign, 2000.
L, X.Y., AND Teng, S.H. Sliver-free three dimensional delaunay mesh generation. In Symposium on Discrete Algorithms (SODA) (2001).
Miller, G.L., Talmor, D., Teng, S.H., AND Walkington, N.A delaunay based numerical method for three dimensions: generation, formulation, and partition. In Proc. 27thAnnu. ACM Sympos. Theory Comput. (1995), pp. 683–692.
Ruppert, J. A new and simple algorithm for quality 2-dimensional mesh generation. In Third Annual ACM-SIAM Symposium on Discrete Algorithms (1992), pp. 83–92.
Shewchuk, J.R. Delaunay Refinement Mesh Generation. PhD thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania, 1997.
Shewchuk, J.R. A condition guaranteeing the existence of higher-dimensional constrained delaunay triangulations. In Proc. 14th Annual Symposium on Computational Geometry (1998), ACM, pp. 76–85.
Ta;mor, D. Well-Spaced Points for Numerical Methods. PhD thesis, Carnegie Mellon University, 1997.
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© 2001 Springer-Verlag Berlin Heidelberg
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Li, XY. (2001). Generating Well-Shaped d-dimensional Delaunay Meshes. In: Wang, J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44679-6_11
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DOI: https://doi.org/10.1007/3-540-44679-6_11
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