Skip to main content
SpringerLink
Log in

Triangulation of p-Order Parametric Surfaces

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The generation of triangulations on p-order parametric surfaces is a fundamental first step to numerical solutions for multidomain problems involving complex geometries such as those encountered in biological fluid dynamics and other applications. In this study we develop a novel, computationally efficient method for generating triangulations in computational space, which, under parametric mapping, are of high geometric quality. Computational efficiency is maintained over parametric orders (p) through approximating the parametric surface by a grid of simplified vector functions. Unlike other length metric approximations, a maximum bound on the error introduced to the calculation of lengths by this approximation is defined to ensure the fidelity of the transformation. This technique is applied to three parametric functions which demonstrate its robustness in handling high mesh distortions, singularities, and high order surfaces. Further, three complex high-order biological finite element meshes are triangulated. High mesh quality and a linear relationship between triangle generation and CPU time is observed for each of these meshes.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anglada, M., Garcia, N., Crosa, P.: Directional adaptive surface triangulation. Comput. Aided Geom. Des. 16, 107–126 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Babuska, I., Aziz, A.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13(2), 214–226 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  3. Babuska, I., Rheinboldt, W.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15(4), 736–754 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baker, T.: Adaptive modification of time evolving meshes. Comput. Methods Appl. Mech. Eng. 194, 4977–5001 (2005)

    Article  MATH  Google Scholar 

  5. Bernardini, F., Mittleman, J., Rushmeier, H.: The ball-pivoting algorithm for surface reconstruction. IEEE Trans. Vis. Comput. Graph. 5(4), 349–359 (1999)

    Article  Google Scholar 

  6. Boender, E.: Reliable Delaunay-based mesh generation and mesh improvement. Commun. Numer. Meth. Eng. 10, 773–783 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boivin, C., Gooch, C.: Guaranteed-quality triangular mesh generation for domains with curved boundaries. Int. J. Numer. Methods Eng. 55, 1185–1213 (2002)

    Article  MATH  Google Scholar 

  8. Borouchaki, H., Frey, P.: Simplification of surface mesh using Hausdorff envelope. Comput. Methods Appl. Mech. Eng. 194, 4864–4884 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Borouchaki, H., George, P.: Aspects of 2d Delaunay mesh generation. Int. J. Numer. Methods Eng. 40, 1957–1975 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Borouchaki, H., George, P., Hecht, F., Laug, P., Saltel, E.: Delaunay mesh generation governed by metric specifications, Part I. Algorithms. Finite Elem. Anal. Des. 25, 61–83 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Borouchaki, H., George, P., Mohammadi, B.: Delaunay mesh generation governed by metric specifications, Part II. Applications. Finite Elem. Anal. Des. 25, 85–109 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Borouchaki, H., Laug, P., George, P.: Parametric surface meshing using a combined advancing-front generalized Delaunay approach. Int. J. Numer. Methods Eng. 49, 233–259 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bowyer, A.: Computing Dirichlet tessellations. Comput. J. 24(2), 162–166 (1981)

    Article  MathSciNet  Google Scholar 

  14. Chase, H.: Alternative convergence criteria for iterative methods of solving nonlinear equations. J. Franklin 317(2), 89–103 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cherfils, C., Hermeline, F.: Diagonal swap procedures and characterizations of 2d-Delaunay triangulations. Math. Model. Numer. Anal. 24(5), 613–625 (1990)

    MATH  MathSciNet  Google Scholar 

  16. Cuilliere, J.: An adaptive method for the automatic triangulation of 3d parametric surfaces. Comput. Aided Des. 30(2), 139–149 (1996)

    Article  Google Scholar 

  17. Dennis, J., Schnabel, R.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  18. Du, Q., Wang, D.: Anisotropic centroidal Voronoi tessellations and their applications. SIAM J. Sci. Comput. 26(3), 737–761 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Eisenstat, S., Walker, H.: Globally convergent inexact Newton methods. SIAM J. Optim. 4(2), 393–422 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ekaterinaris, J.A.: High-order accurate, low numerical diffusion methods for aerodynamics. Prog. Aerospace Sci. 41, 192–300 (2005)

    Article  Google Scholar 

  21. Farhat, C., Lesoinne, M., Maman, N.: Mixed explicit/implicit time integration of coupled aeroelastic problems: Three-field formulation, geometric conservation and distributed solution. Int. J. Numer. Methods Fluids 21, 807–835 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Filip, D., Magedson, R., Markot, R.: Surface algorithms using bounds on derivatives. Comput. Aided Geom. Des. 3, 295–311 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  23. Frey, P., Alauzet, F.: Anisotropic mesh adaption for cfd computations. Comput. Methods Appl. Mech. Eng. 194, 5068–5082 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Fuchs, A.: Automatic grid generation with almost regular Delaunay tetrahedra (1998)

  25. Gudmundsson, J., Haverkort, H., Kreveld, M.: Constrained higher order Delaunay triangulations. Comput. Geom. 30, 271–277 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Holmes, D., Snyder, D.: The Generation of Unstructured Triangular Meshes Using Delaunay Triangulations. Pineridge, Swansed (1988)

    Google Scholar 

  27. Reynolds, H., Smith, N., Hunter, P.: Construction of an anatomically accurate geometric model of the forearm and hand musculo-skeletal system. In: Annual Meeting Proceedings, 26th IEEE EMBS Annual International Conference, San Francisco, Sept. 1–5, 2004, pp. 265–269

  28. Hunter, P., Pullan, A., Smaill, B.: Modelling total heart function. Ann. Rev. Biomed. Eng. 5, 147–177 (2003)

    Article  Google Scholar 

  29. Joe, B.: Delaunay triangular meshes in convex polygons. SIAM J. Sci. Comput. 7(2), 514–539 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  30. Joe, B.: Delaunay versus max-min solid angle triangulations for three-dimensional mesh generation. Int. J. Numer. Methods Eng. 31, 987–997 (1991)

    Article  MATH  Google Scholar 

  31. Joe, B.: Construction of three-dimensional improved-quality triangulations using local transformations. SIAM J. Sci. Comput. 16(6), 1292–1307 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kuo, C., Yau, H.: A Delaunay-based region growing approach to surface reconstruction from unorganized points. Comput. Aided Des. 37, 825–835 (2005)

    Article  Google Scholar 

  33. Labelle, F., Shewchuk, J.: Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. SoGC, pp. 191–200 (2003)

  34. Lee, C.: Automatic metric advancing front triangulation over curved surfaces. Eng. Comput. 17(1), 48–74 (2000)

    Article  MATH  Google Scholar 

  35. Lee, C.: On curvature element-size control in metric surface mesh generation. Int. J. Numer. Methods Eng. 50, 787–807 (2001)

    Article  MATH  Google Scholar 

  36. LeGrice, I., Hunter, P., Young, A., Smaill, B.: The architecture of the heart: A data-based model. Phil. Trans. R. Soc. Lond. 359, 1217–1232 (2001)

    Article  MATH  Google Scholar 

  37. Liu, A., Joe, B.: Relationship between tetrahedron shape measures. BIT 34, 268–287 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  38. Lo, S.: Automatic mesh generation and adaption by using contours. Int. J. Numer. Methods Eng. 31, 689–707 (1991)

    Article  MATH  Google Scholar 

  39. Lohner, R.: Regridding surface triangulations. J. Comput. Phys. 126, 1–10 (1996)

    Article  Google Scholar 

  40. Lohner, R.: Automatic unstructured grid generators. Finite Elem. Anal. Des. 25, 111–134 (1997)

    Article  MathSciNet  Google Scholar 

  41. Mavriplis, D.: An advancing front Delaunay triangulation algorithm designed for robustness. J. Comput. Phys. 117, 90–101 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  42. Nash, M., Hunter, P.: Computational mechanics of the heart. J. Elast. 61, 113–141 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  43. Nickerson, D., Niederer, S., Stevens, C., Nash, M., Hunter, P.: A computational model of cardiac electromechanics. In: 28th Annual International Conference of the IEEE Engineering in Medicine and Biology Society. New York, September 2006

  44. Okabe, A., Boots, B., Sugihara, K., Chiu, S.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2 edn. Wiley, New York (2000)

    MATH  Google Scholar 

  45. Pang, M., Pan, Z., Zhang, M., Zhang, F.: An adaptive and efficient algorithm for polygonization of implicit surfaces, pp. 245–255 (2005)

  46. Parthasarathy, V., Graichen, C., Hathaway, A.: A comparison of tetrahedron quality measures. Finite Elem. Anal. Des. 15, 255–261 (1993)

    Article  Google Scholar 

  47. Rebay, S.: Efficient unstructured mesh generation by means of Delaunay triangulation and Bowyer-Watson algorithm. J. Comput. Phys. 106, 125–138 (1991)

    Article  Google Scholar 

  48. Stevens, C., Remme, E., LeGrice, I., Hunter, P.: Ventricular mechanics in diastole: Material parameter sensitivity. J. Biomech. 36, 737–748 (2003)

    Article  Google Scholar 

  49. Tremel, U., Deister, F., Hassan, O., Weatherill, N.: Automatic unstructured surface mesh generation for complex configurations. Int. J. Numer. Methods Fluids 45, 341–364 (2004)

    Article  MATH  Google Scholar 

  50. Tremel, U., Deister, F., Hassan, O., Weatherill, N.: Parallel generation of unstructured surface grids. Eng. Comput. 21, 36–46 (2005)

    Article  Google Scholar 

  51. van Essen, N., Anderson, I., Hunter, P., Carman, J., Clarke, R., Pullan, A.: Anatomically based model of the human skull and jaw. Cells Tissues Organs 180, 44–53 (2005)

    Article  Google Scholar 

  52. Wang, Z.: Spectral (finite) volume method for conservation laws on unstructured grids. J. Comput. Phys. 178, 210–251 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  53. Wang, Z., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids. J. Comput. Phys. 179, 665–697 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  54. Watson, D.: Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Comput. J. 24(2), 167–172 (1981)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Oxford Computing Laboratory, University of Oxford, Parks RD, OX1 3JP, Oxford, UK

    D. Nordsletten & N. P. Smith

  2. Bioengineering Institute, University of Auckland, Auckland, New Zealand

    D. Nordsletten & N. P. Smith

Authors
  1. D. Nordsletten
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. P. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Nordsletten.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nordsletten, D., Smith, N.P. Triangulation of p-Order Parametric Surfaces. J Sci Comput 34, 308–335 (2008). https://doi.org/10.1007/s10915-007-9167-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-007-9167-3

Keywords

Search

Navigation