Skip to main content
SpringerLink
Log in
Book cover

Proceedings of the 15th International Meshing Roundtable pp 543–562Cite as

A Study on Delaunay Terminal Edge Method

  • Conference paper

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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1. M. Bern, D. Eppstein and J. Gilbert, Provably good mesh generation. Journal Computer System Science, 48, 1994, 384–409.

    Article  MATH  MathSciNet  Google Scholar 

  2. 2. L.P.Chew, Guaranteed-quality triangular meshes. Technical report TR-98-983, Computer Science Department, Cornell University, Ithaca, NY, 1989.

    Google Scholar 

  3. 3. P L George and H Borouchaki, Delaunay Triangulation and Meshing. Hermes, 1998.

    Google Scholar 

  4. 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.

    Article  MATH  MathSciNet  Google Scholar 

  5. 5. R.E.Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Users' Guide 8.0. SIAM, 1998.

    Google Scholar 

  6. 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.

    Article  MATH  MathSciNet  Google Scholar 

  7. 7. M. C. Rivara. Selective refinement/derefinement algorithms for sequences of nested triangulations. International Journal for Numerical Methods in Engineering, 28, 1989, 2889–2906.

    Article  MATH  MathSciNet  Google Scholar 

  8. 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.

    Article  MATH  Google Scholar 

  9. 9. P Morin, R H Nochetto, and K G Siebert, Convergence of Adaptive Finite Element Methods, SIAM Review. 44 631–658.

    Google Scholar 

  10. 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.

    Article  MATH  Google Scholar 

  11. 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.

    Article  MATH  Google Scholar 

  12. 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.

    Article  MATH  MathSciNet  Google Scholar 

  13. 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.

    Article  MATH  MathSciNet  Google Scholar 

  14. 14. J Ruppert. A Delaunay refinement algorithm for quality 2-dimensional mesh generation. J. of Algorithms, 18, 1995, 548–585.

    Article  MATH  MathSciNet  Google Scholar 

  15. 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.

    Article  MATH  Google Scholar 

  16. 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.

    Google Scholar 

  17. 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.

    Article  Google Scholar 

  18. 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.

    Google Scholar 

  19. 19. T.J. Baker, Automatic mesh generation for complex three dimensional regions using a constrained Delaunay triangulation. Engineering with Computers, 5, 1989, 161–175.

    Article  Google Scholar 

  20. 20. T. J. Baker, Triangulations, Mesh Generation and Point Placement Strategies. Computing the Future, ed. D Caughey,John Wiley, 61–75.

    Google Scholar 

  21. 21. J R Shewchuk, Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. First Workshop on Applied Computational Geometry, ACM, 1996, 124–133.

    Google Scholar 

  22. 22. R.B. Simpson, N. Hitschfeld and M.C. Rivara, Approximate quality mesh generation, Engineering with computers, 17, 2001, 287–298.

    Article  MATH  Google Scholar 

  23. 23. M. Bern, Triangulations, In Handbook of Discrete and Computational Geometry J. E. Goodman and J O'Rourke (eds.), CRC Press Boca Raton, 1997.

    Google Scholar 

  24. 24. D. T. Lee and A. Lin Generalized Delaunay triangulation for planar graphs. Disc and Comp Geom, bf 1, 1986, 201–217.

    Article  MATH  MathSciNet  Google Scholar 

  25. 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.

    Article  MATH  Google Scholar 

  26. 26. D.L. Marcum and N.P. Weatherill, Aerospace applications of solution adaptive finite element analysis. CAGEOD, bf 12, 1995, 709–731

    MATH  MathSciNet  Google Scholar 

  27. 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.

    Article  MATH  MathSciNet  Google Scholar 

  28. 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.

    Google Scholar 

  29. 29. B. Simpson and M.C. Rivara, Geometrical mesh improvement properties of Delaunay terminal edge refinement, Geometric Modeling and Processing 2006, Pittsburgh, 2006.

    Google Scholar 

  30. 30. M.de Berg, M Van Kreveld, M. Overmars, O. Schwarzkopf, Computational Geometry, algorithms and applications, second edition, Springer, 2000.

    Google Scholar 

  31. 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.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Universidad de Chile, Chile

    Maria-Cecilia Rivara

Authors
  1. Maria-Cecilia Rivara
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Sandia National Laboratories, 969, MS 9051, Livermore, CA, U.S.A

    Philippe P. Pébay

Rights and permissions

Reprints 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)

Publish with us

Policies and ethics

Search

Navigation