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Meshing Piecewise Linear Complexes by Constrained Delaunay Tetrahedralizations

  • Conference paper
Proceedings of the 14th International Meshing Roundtable

Summary

We present a method to decompose an arbitrary 3D piecewise linear complex (PLC) into a constrained Delaunay tetrahedralization (CDT). It successfully resolves the problem of non-existence of a CDT by updating the input PLC into another PLC which is topologically and geometrically equivalent to the original one and does have a CDT. Based on a strong CDT existence condition, the redefinition is done by a segment splitting and vertex perturbation. Once the CDT exists, a practically fast cavity retetrahedralization algorithm recovers the missing facets. This method has been implemented and tested through various examples. In practice, it behaves rather robust and efficient for relatively complicated 3D domains.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Bagemihl F (1948) On Indecomposable Polyhedra. American Mathematical Monthly 55: 411–413

    Article  MATH  MathSciNet  Google Scholar 

  3. Chazelle B. (1984) Convex Partition of Polyhedra: A lower Bound and Worst-case Optimal Algorithm. SIAM Journal on Computing 13(3): 488–507

    Article  MATH  MathSciNet  Google Scholar 

  4. Lee D T, Lin A K (1986) Generalized Delaunay Triangulations for Planar Graphs. Discrete Comput Geom 1:201–217

    Article  MathSciNet  MATH  Google Scholar 

  5. Chew P L(1989) Constrained Delaunay Triangulation. Algorithmica 4(1):97–108

    Article  MATH  MathSciNet  Google Scholar 

  6. Edelsbrunner H, Mücke M P (1990) Simulation of Simplicity: A Technique to Cope with Degenerate Cases in Geometric Algorithm. ACM Transactions on Graphics 9(1): 66–104

    Article  MATH  Google Scholar 

  7. Edelsbrunner H, Shah N R (1996) Incremental Topological Flipping Works for Regular Triangulations. Algorithmica 15: 223–241

    Article  MathSciNet  MATH  Google Scholar 

  8. Ruppert J (1992) On the Difficulty of Triangulating Three-dimensional Non-convex Polyhedra. Discrete Comput Geom 7: 227–253

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Google Scholar 

  12. Shewchuk J R (1998) A Condition Guaranteeing the Existence of Higher-Dimensional Constrained Delaunay Triangulations. In: Proc. of the 14th Annu. Sympos. Comput. Geom., 76–85, Minneapolis, Minnesota

    Google Scholar 

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

    Google Scholar 

  14. Shewchuk J R (2002) Sweep Algorithms for Constructing Higher-Dimensional Constrained Delaunay Triangulations. In: Proc. of the 16th Annu. Sympos. Comput. Geom.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  17. Pébay P (1998) A Priori Delaunay-Conformity. In: Proc. of 7th Intl Meshing Roundtable, SANDIA

    Google Scholar 

  18. Murphy M, Mount D M, Gable C W (2000) A Point-Placement Strategy for Conforming Delaunay Tetrahedralization. In: Proc. of the 11th Annu. Sympos. on Discrete Algorithms

    Google Scholar 

  19. Cohen-Steiner D, Colinde Verdière E, Yvinec M (2002) Conforming Delaunay Triangulations in 3D. In: Proc. of 18th Annu. Sympos. Comput. Geom. Barcelona

    Google Scholar 

  20. George P L, Borouchaki H, Saltel E (2003) ‘Ultimate’ Robustness in Meshing an Arbitrary Polyhedron. Int. J. Numer. Meth. Engng 58: 1061–1089

    Article  MathSciNet  MATH  Google Scholar 

  21. Rambau J. (2003) On a Generalization of Schönhardt’s Polyhedron. MSRI Preprint 2003-13

    Google Scholar 

  22. Du Q, Wang D (2004) Constrained Boundary Recovery for Three Dimensional Delaunay Triangulations. Int. J. Numer. Meth. Engng 61: 1471–1500

    Article  MathSciNet  MATH  Google Scholar 

  23. 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 

  24. Pav S, Walkington N (2004) A Robust 3D Delaunay Refinement Algorithm. In: Proc. Intl. Meshing Roundtable.

    Google Scholar 

  25. Si H, Gärtner K (2004) An Algorithm for Three-Dimensional Constrained Delaunay Triangulations. In: Proc of 4th Intl. Conf. on Engineering Computational Technology, Lisbon

    Google Scholar 

  26. Si H (2004) TetGen, A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator, v1.3 User’s Manual. WIAS Technical Report No. 9

    Google Scholar 

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Author information

Authors and Affiliations

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

    Hang Si & Klaus Gärtner

Authors
  1. Hang Si
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  2. Klaus Gärtner
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Editor information

Editors and Affiliations

  1. Computer Modeling and Sciences Dept., Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM, 87185, USA

    Byron W. Hanks

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© 2005 Springer-Verlag Berlin Heidelberg

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Si, H., Gärtner, K. (2005). Meshing Piecewise Linear Complexes by Constrained Delaunay Tetrahedralizations. In: Hanks, B.W. (eds) Proceedings of the 14th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-29090-7_9

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  • DOI: https://doi.org/10.1007/3-540-29090-7_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25137-8

  • Online ISBN: 978-3-540-29090-2

  • eBook Packages: EngineeringEngineering (R0)

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