Skip to main content
SpringerLink
Log in
Book cover

Proceedings of the 15th International Meshing Roundtable pp 509–528Cite as

On Refinement of Constrained Delaunay Tetrahedralizations

  • Conference paper

Abstract

This paper discusses the problem of refining constrained Delaunay tetrahedralizations (CDTs) into good quality meshes suitable for adaptive numerical simulations. A practical algorithm which extends the basic Delaunay refinement scheme is proposed. It generates an isotropic mesh corresponding to a sizing function which can be either user-specified or automatically derived from the geometric data. Analysis shows that the algorithm is able to produce provable-good meshes, i.e., most output tetrahedra have their circumradius-to-shortest edge ratios bounded, except those in the neighborhood of small input angles. Good mesh conformity can be obtained for smoothly changing sizing information. The algorithm has been implemented. Various examples are provided to illustrate the theoretical aspects and practical performance of the algorithm.

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. Schönhardt E (1928) Über die Zerlegung von Dreieckspolyedern in Tetraeder. Mathematische Annalen 98:309–312

    Article  Google Scholar 

  2. 2. Shepard D (1968) A Two-Dimensional Interpolation Function for Irregularly Space Data. In: Proc. 23th Nat. Conf. ACM, 517–524.

    Google Scholar 

  3. 3. Babuška I, Aziz, A K (1976) On the Angle Condition in the Finite Element Method. SIAM Journal on Numerical Analysis 13(2): 214–226.

    Article  Google Scholar 

  4. 4. Chew P L (1989) Guaranteed-Quality Triangular Meshes. Technical Report TR-89–983, Department of Computer Science, Cornell University.

    Google Scholar 

  5. 5. Chew P L (1997) Guaranteed-Quality Delaunay Meshing in 3D. In: Proc. 19th Annu. Sympos. Comput. Geom., 274–280.

    Google Scholar 

  6. 6. George P L (1991) Automatic Mesh Generation. Application to Finite Element Methods. Wiley.

    Google Scholar 

  7. 7. Weatherill N P, Hassan O (1994) E.cient Three-Dimensional Delaunay Triangulation with Automatic Point Creation and Imposed Boundary Constraints. Int. J. Numer. Meth. Engng 37: 2005–2039.

    Article  MATH  Google Scholar 

  8. 8. Ruppert J (1995) A Delaunay Re.nement Algorithm for Quality 2-Dimensional Mesh Generation. J Algorithms 18(3): 548–585

    Article  MATH  Google Scholar 

  9. 9. Frey P J, Borouchaki H, George P L (1996) Delaunay Tetrahedralization Using an Advancing Front Approach. In: Proc. of 5th Intl. Meshing Roundtable.

    Google Scholar 

  10. 10. Frey P J, George P L (2000) Mesh Generation. Application to Finite Elements. Hermes, Paris.

    Google Scholar 

  11. 11. Miller G L, Talmor D, Teng S H, Walkington N, Wang H (1996) Control Volume Meshes using Sphere Packing: Generation, Re.nement and Coarsening. In: Proc. of 5th Intl. Meshing Roundtable.

    Google Scholar 

  12. 12. Freitag L, Ollivier-Gooch C (1996) A Comparsion of Tetrahedral Mesh Improvement Techniques. In: Proc. 5th Intl. Meshing Roundtable.

    Google Scholar 

  13. 13. Shewchuk J R (1998) Tetrahedral Mesh Generation by Delaunay Re.nement. In: Proc. 14th Annu. Sympos. Comput. Geom.

    Google Scholar 

  14. 14. Shewchuk J R (1998) Tetrahedral Mesh Generation by Delaunay Re.nement. In: Proc. 14th Annu. Sympos. Comput. Geom.

    Google Scholar 

  15. 15. Shewchuk J R (2000) Mesh Generation for Domains with Small Angles. In: Proc 16th Annu. ACM Sympos. Comput. Geom.

    Google Scholar 

  16. 16. Shewchuk J R (2002) Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery. In: Proc. 11th Intl. Meshing Roundtable.

    Google Scholar 

  17. 17. Shewchuk J R (2002) What Is a Good Linear Element? Interpolation, Conditioning, and Quality Measures. In: Proc. 11th Intl. Meshing Roundtable.

    Google Scholar 

  18. 18. Shewchuk J R (2003) Updating and Constructing Constrained Delaunay and Constrained Regular Triangulations by Flips. In: Proc. 19th Annu. Sympos. Comput. Geom.

    Google Scholar 

  19. 19. Shewchuk J R (2005) General-Dimensional Constrained Delaunay and Constrained Regular Triangulations. To appear in Discrete & Computational Geometry.

    Google Scholar 

  20. 20. Cheng S W, Dey, T K, Edelsbrunner H, Facello M A, Teng S H (1999) Sliver Exudation. J. ACM, 47, 883–904.

    Article  Google Scholar 

  21. 21. Cheng S W, Dey T K, Ramos E A, Ray T (2004) Quality Meshing for Ployhedra with Small Angels. In: Proc. 20th Annu. ACM Sympos. Comput. Geom.

    Google Scholar 

  22. 22. Cheng S W, Dey T K, Ray T (2005) Weighted Delaunay Re.nement for Polyhedra with Small Angles. In: Proc. 14th Intl. Meshing Roundtable.

    Google Scholar 

  23. 23. Edelsbrunner H, Li X Y, Miller G, Stathopoulos A, Talmor D, Teng S H, Üngör A, Walkington N (2000) Smoothing Cleans up Slivers. In: Proc. 32th Annu. ACM Sympos. Theory of Computing, 273–277.

    Google Scholar 

  24. 24. Edelsbrunner H, Guoy D. (2002) An Experimental Study of Sliver Exudation. Engineering with Computers, 18, 299–240.

    Article  Google Scholar 

  25. 25. Fuhrmann J, Langmach H (2001) Stability and Existence of Solutions of Time—Implicit Finite Volume Schemes for Viscous Nonlinear Conservation Laws. Appl. Numer. Math., 37(1–2): 201–230.

    Article  MATH  Google Scholar 

  26. 26. Du Q, Wang D. (2003) Tetrahedral Mesh Generation and Optimization Based on Centroidal Voronoi Tessellations. Int. J. Nummer. Meth. Engng, 56, 1355–1373.

    Article  MATH  Google Scholar 

  27. 27. Üngör A (2004) Off-centers: A New Type of Steiner Points for Computing Size-Optimal Guaranteed-Quality Delaunay Triangulations. In: Proc. LATIN, 152–161.

    Google Scholar 

  28. 28. Pav S, Walkington N (2004) A Robust 3D Delaunay Re.nement Algorithm. In: Proc. 13th Intl. Meshing Roundtable.

    Google Scholar 

  29. 29. Alliez P, Cohen-Steiner D, Yvinec M, Desbrun M (2005) Variational Tetrahedral Meshing. ACM Transactions on Graphics, 24(3): 617–625.

    Article  Google Scholar 

  30. 30. Oudot S, Rineau L, Yvinec M (2005) Meshing Volumes Bounded by Smooth Surfaces. In: Proc. 14th Intl. Meshing Roundtable.

    Google Scholar 

  31. 31. Si H, Gärtner K (2005) Meshing Piecewise Linear Complexes by Constrained Delaunay Tetrahedralizations. In: Proc. 14th Intl. Meshing Roundtable.

    Google Scholar 

  32. 32. TetGen, a Quality Tetrahedral Mesh Generator. http://tetgen.berlios.de.

    Google Scholar 

  33. 33. pdelib, Tool Box for Solving Partial Di.erential Equations. http://www.wiasberlin. de/software/pdelib

    Google Scholar 

  34. 34. AcuSolve, a Finite Element Incompressible Flow Solver. http://www.acusim.com.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Hang Si

Authors
  1. Hang Si
    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

Si, H. (2006). On Refinement of Constrained Delaunay Tetrahedralizations. 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_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-34958-7_29

  • 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