Abstract
The Delaunay terminal edge algorithm for triangulation improvement proceeds by iterative Lepp selection of a point M which is midpoint of a Delaunay terminal edge in the mesh. The longest edge bisection of the associated terminal triangles (sharing the terminal edge) can be seen as a first step in the Delaunay insertion of M. The method was introduced as a generalization of non-Delaunay longest edge algorithms but formal termination proof had not been stated until now. In this paper termination is proved and several geometric aspects of the algorithm behavior are studied.
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
1. M. Bern, D. Eppstein and J. Gilbert, Provably good mesh generation. Journal Computer System Science, 48, 1994, 384–409.
2. L.P.Chew, Guaranteed-quality triangular meshes. Technical report TR-98-983, Computer Science Department, Cornell University, Ithaca, NY, 1989.
3. P L George and H Borouchaki, Delaunay Triangulation and Meshing. Hermes, 1998.
4. H Borouchaki and P L George, Aspects of 2-D Delaunay Mesh Generation. International Journal for Numerical Methods in Engineering, 40, 1997, 1957–1975.
5. R.E.Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Users' Guide 8.0. SIAM, 1998.
6. M. C. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, International Journal for Numerical Methods in Engineering, 20, 1984, 745–756.
7. M. C. Rivara. Selective refinement/derefinement algorithms for sequences of nested triangulations. International Journal for Numerical Methods in Engineering, 28, 1989, 2889–2906.
8. M. C. Rivara and C. Levin. A 3d Refinement Algorithm for adaptive and multigrid Techniques. Communications in Applied Numerical Methods, 8, 1992, 281–290.
9. P Morin, R H Nochetto, and K G Siebert, Convergence of Adaptive Finite Element Methods, SIAM Review. 44 631–658.
10. S. N. Muthukrishnan, P. S. Shiakolos R. V. Nambiar, and K. L. Lawrence. Simple algorithm for adaptative refinement of three-dimensional finite element tetrahedral meshes. AIAA Journal, 33, 1995, 928–932.
11. N. Nambiar, R. Valera, K. L. Lawrence, R. B. Morgan, and D. Amil. An algorithm for adaptive refinement of triangular finite element meshes. International Journal for Numerical Methods in Engineering, 36, 1993, 499–509.
12. M. C. Rivara and G. Iribarren, The 4-triangles longest-edge partition of triangles and linear refinement algorithms, Mathematics of Computation, 65, 1996, 1485–1502.
13. M. C. Rivara. New longest-edge algorithms for the refinement and/or improvement of unstructured triangulations. International Journal for Numerical Methods in Engineering, 40, 1997, 3313–3324.
14. J Ruppert. A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J. of Algorithms, 18, 1995, 548–585.
15. N. Hitschfeld and M.C. Rivara. Automatic construction of non-obtuse boundary and/or interface Delaunay triangulations for control volume methods. International Journal for Numerical Methods in Engineering, 55, 2002, 803–816.
16. N. Hitschfeld, L. Villablanca, J. Krause, and M.C. Rivara. Improving the quality of meshes for the simulation of semiconductor devices using Lepp-based algorithms. to appear. International Journal for Numerical Methods in Engineering, 2003.
17. M. C. Rivara, N. Hitschfeld, and R. B. Simpson. Terminal edges Delaunay (small angle based) algorithm for the quality triangulation problem. Computer-Aided Design, 33, 2001, 263–277.
18. M. C. Rivara and M. Palma. New LEPP Algorithms for Quality Polygon and Volume Triangulation: Implementation Issues and Practical Behavior. In Trends unstructured mesh generation, Eds: S. A. Cannan. Saigal, AMD, 220, 1997, 1–8.
19. T.J. Baker, Automatic mesh generation for complex three dimensional regions using a constrained Delaunay triangulation. Engineering with Computers, 5, 1989, 161–175.
20. T. J. Baker, Triangulations, Mesh Generation and Point Placement Strategies. Computing the Future, ed. D Caughey,John Wiley, 61–75.
21. J R Shewchuk, Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. First Workshop on Applied Computational Geometry, ACM, 1996, 124–133.
22. R.B. Simpson, N. Hitschfeld and M.C. Rivara, Approximate quality mesh generation, Engineering with computers, 17, 2001, 287–298.
23. M. Bern, Triangulations, In Handbook of Discrete and Computational Geometry J. E. Goodman and J O'Rourke (eds.), CRC Press Boca Raton, 1997.
24. D. T. Lee and A. Lin Generalized Delaunay triangulation for planar graphs. Disc and Comp Geom, bf 1, 1986, 201–217.
25. N.P. Weatherill and O. Hassan, Eficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints. IJMNE, bf 37, 1994, 2005–2039.
26. D.L. Marcum and N.P. Weatherill, Aerospace applications of solution adaptive finite element analysis. CAGEOD, bf 12, 1995, 709–731
27. I.G. Rosenberg and F. Stenger, A lower bound on the angles of triangles constructed by bisecting the longest side, Mathematics of Computation, 29, 1975, 390–395.
28. A. Üngor, Off-centers: a new type of Steiner points for computing size-optimal quality-guaranteed Delaunay triangulations, Latin 2004, LNCS 2076, 2004, 152–161.
29. B. Simpson and M.C. Rivara, Geometrical mesh improvement properties of Delaunay terminal edge refinement, Geometric Modeling and Processing 2006, Pittsburgh, 2006.
30. M.de Berg, M Van Kreveld, M. Overmars, O. Schwarzkopf, Computational Geometry, algorithms and applications, second edition, Springer, 2000.
31. M. C. Rivara, New mathematical tools and techniques for the refinement and/or improvement of unstructured triangulations, Proceedings 5th International Meshing Roundtable, Pittsburgh, 77–86, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer
About this paper
Cite this paper
Rivara, MC. (2006). A Study on Delaunay Terminal Edge Method. In: Pébay, P.P. (eds) Proceedings of the 15th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34958-7_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-34958-7_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34957-0
Online ISBN: 978-3-540-34958-7
eBook Packages: EngineeringEngineering (R0)