A data modeling abstraction for describing triangular mesh algorithms

  • R. Bruce Simpson


The use of a relational data model is proposed as the basis for a technique for expressing algorithms for unstructured meshes, or geometric incidences more generally. The result is an abstraction for meshes that is conceptually intermediate between the geometric abstractions of mathematical descriptions and the list representations of source code implementations. Algorithms are then described in terms of:
  • -conventional structured pseudocode,

  • -the mesh conceived as a globally accessible database,

  • -variables typed using the data model, and

  • -commands based on a database query and modification syntax.

It is argued that the resulting paradigm supports descriptions that are precise, complete, and relatively representation independent. It thus supports the role of algorithms in providing templates for coding, while also supporting the role of algorithms as vehicles for the communication and documentation of ideas.

A simple data model for planar triangular meshes is defined. Using it, the technique is demonstrated by a suite of familiar algorithms for constructing the Delaunay triangulation, using the divide-and-conquer and incremental approaches. Data dependencies in the model of triangular meshes are noted, and their implications for relating the model to several standard mesh representations is discussed.

AMS subject classification

G.4 I.3.5 

Key words

Triangulation algorithms Delaunay triangulation relational model divide-and-conquer finite element mesh 


  1. 1.
    F. Aurenhammer,Voronoi diagrams—a survey of a fundamental geometric data structure, ACM Computing Surveys, 23 (1991) pp. 345–405.CrossRefGoogle Scholar
  2. 2.
    O. Axelsson and V. A. Barker,Finite Element Solution of Boundary Value Problems, Academic Press, New York, 1984.MATHGoogle Scholar
  3. 3.
    R. E. Bank, A. H. Sherman, and A. Weiser,Refinement algorithms and data structures for regular local mesh refinement, in Scientific Computing; IMACS Conference, R. S. Stepleman ed., North Holland, 1983, pp. 3–18.Google Scholar
  4. 4.
    E. Bruzzone and L. De Floriani,An efficient data structure for three-dimensional triangulations, in Proceedings of CG International 90, T. S. Chua and T. L. Kunii, eds., Springer-Verlag, June 1990, pp. 425–441.Google Scholar
  5. 5.
    E. Bruzzone and L. De Floriani,Extracting adjacency relationships from a modular boundary model, Computer Aided Design, 23(5) (1991), pp. 344–356.MATHCrossRefGoogle Scholar
  6. 6.
    A. Bykat,Automatic generation of triangular grid: i-subdivision of a general polygon into convex subregions. ii-triangulation of convex polygons, Internat. J. Numer. Methods Engrg., 10 (1976), pp. 1329–1342.MATHCrossRefGoogle Scholar
  7. 7.
    J. C. Cavendish, Automatic triangulation of arbitrary planar domains for the finite element method, Internat. J. Numer. Methods Engrg., 8 (1974), pp. 679–696.MATHCrossRefGoogle Scholar
  8. 8.
    A. K. Cline and R. J. Renka,A storage-efficient method for construction of a Thiessen triangulation, Rocky Mountain J. Math., 14 (1984), pp. 119–139.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    A. K. Cline and R. J. Renka,A constrained two-dimensional triangulation and the solution of closest node problems in the presence of barriers, SIAM J. Numer. Anal., 27 (1990), pp. 1305–1321.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    D. P. Dobkin and M. J. Laszlo,Primitives for the manipulation of three-dimensional subdivisions, in Proceedings of Third Annual Symposium on Computational Geometry, ACM Press, 1987, pp. 246–259.Google Scholar
  11. 11.
    S. Farestam and R. B. Simpson,On correctness and efficiency for advancing front techniques of finite element mesh generation, BIT, 35 (1995), pp. 1210–232.MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Fortune,Numerical stability of algorithms for 2d Delaunay triangulations, Proceedings of Eighth Annual Symposium on Computational Geometry, ACM Press, 1992, pp. 208–217.Google Scholar
  13. 13.
    P. L. George,Automatic Mesh Generation, John Wiley, Paris, 1991.MATHGoogle Scholar
  14. 14.
    L. Guibas and J. Stolfi,Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams, ACM Trans. on Graphics, 4 (1985), pp. 74–123.MATHCrossRefGoogle Scholar
  15. 15.
    C. M. Hoffman,The problems of accuracy and robustness in geometric computation, Computer, Volume 22:3 (1989) pp. 31–42.CrossRefGoogle Scholar
  16. 16.
    C. M. Hoffmann,Geometric and Solid Modeling: An Introduction, Morgan Kaufmann, San Mateo, CA, 1989.Google Scholar
  17. 17.
    B. Joe,Delaunay triangular meshes in convex polygons, SIAM J. Sci. Stat. Comp, 7 (1986), pp. 514–539.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    B. Joe, Three-dimensional triangulations from local transformations, SIAM J. Sci. Stat. Comp, 10 (1989), pp. 718–741.MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    B. Joe and R. B. Simpson, Corrections to Lee's visibility polygon algorithm, BIT, 27 (1987), pp. 458–473.MATHCrossRefGoogle Scholar
  20. 20.
    M. Korth and A. Silberschatz,Database System Concepts, 2nd ed. McGraw-Hill, New York, 1991.MATHGoogle Scholar
  21. 21.
    C. L. Lawson,Software for c1 surface interpolation, in Mathematical Software III, J. R. Rice, ed., Academic Press, 1977.Google Scholar
  22. 22.
    D. T. Lee and B. J. Schachter,Two algorithms for constructing a Delaunay triangulation, Internat. J. Comp. Information Science, 9 (1980), pp. 219–242.MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    B. A. Lewis and J. S. Robinson,Triangulation of planar regions with applications, Comput. J., 21 (1978), pp. 324–329.MATHCrossRefGoogle Scholar
  24. 24.
    J. O'Rourke,Computational Geometry in C, Cambridge University Press, 1994.Google Scholar
  25. 25.
    O. Palacios-Velez and B. C. Renaud,A dynamic heirarchical subdivision algorithm for computing Delaunay triangulations and other closest-point problems, ACM Trans. Math. Software, 16 (1990), pp. 275–292.MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    F. P. Preparata and M. I. Shamos,Computational Geometry: an Introduction, Springer-Verlag, New York, 1985.Google Scholar
  27. 27.
    R. B. SimpsonA two-dimensional mesh verification algorithm, SIAM J. Sci. Stat. Comp., 2 (1981), pp. 455–473.MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    R. B. Simpson,A database abstraction for unstructured triangular mesh algorithms, CERFACS, Tech. Report TR/PA/92/66, January 1992, Toulouse, France.Google Scholar
  29. 29.
    D. C. Tsichritzis and F. H. Lochovsky,Data Models, Prentice-Hall, Englewood Cliffes, NJ, 1982.Google Scholar
  30. 30.
    G. E. Weddell,Rdm reference manual, Tech. Report CS-89-41, Computer Science Department, University of Waterloo, Canada, 1989.Google Scholar
  31. 31.
    N. Wirth,Algorithms+Data Structures=Programs, Prentice-Hall, Englewood Cliffs, NJ, 1976.MATHGoogle Scholar
  32. 32.
    T. C. Woo,A combinatorial analysis of boundary data structure schemata, IEEE Computer Graphics and Applications, 5(3) (1985), pp. 19–27.MathSciNetCrossRefGoogle Scholar
  33. 33.
    T. C. Woo and J. D. Wolter,A constant expected time, linear storage data structure for representing three-dimensional objects, IEEE Transactions on Systems, Man and Cybernetics, SMC-14(3 (1984), pp. 510–515.Google Scholar
  34. 34.
    M. A. Yerry and M. S. Shephard,Automatic three-dimensional mesh generation by the modified-octree technique, Internat. J. Numer. Methods Engrg., 20 (1984), pp. 1965–1990.MATHCrossRefGoogle Scholar
  35. 35.
    P. Zave and W. C. Rheinboldt,Design of an adaptive, parallel finite-element system, ACM Trans. math. Software, 5(1) (1979), pp. 1–17.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© BIT Foundation 1997

Authors and Affiliations

  • R. Bruce Simpson
    • 1
  1. 1.Department of Computer ScienceUniversity of WaterlooWaterlooCanada

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