Skip to main content
SpringerLink
Log in

A Laguerre Voronoi based scheme for meshing particle systems

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

We present Laguerre Voronoi based subdivision algorithms for the quadrilateral and hexahedral meshing of particle systems within a bounded region in two and three dimensions, respectively. Particles are smooth functions over circular or spherical domains. The algorithm first breaks the bounded region containing the particles into Voronoi cells that are then subsequently decomposed into an initial quadrilateral or an initial hexahedral scaffold conforming to individual particles. The scaffolds are subsequently refined via applications of recursive subdivision (splitting and averaging rules). Our choice of averaging rules yield a particle conforming quadrilateral/ hexahedral mesh, of good quality, along with being smooth and differentiable in the limit. Extensions of the basic scheme to dynamic re-meshing in the case of addition, deletion, and moving particles are also discussed. Motivating applications of the use of these static and dynamic meshes for particle systems include the mechanics of epoxy/glass composite materials, bio-molecular force field calculations, and gas hydrodynamics simulations in cosmology.

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. R. Gingold and J. Monaghan, Smoothed particle hydrodynamics: theory and application to non-spherical stars. Royal Astronomical Society,181 (1977), 375–389.

    MATH  Google Scholar 

  2. K. Vemaganti and T. Oden, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials,Part II: A computational environment for adaptive modeling of heterogeneous elastic solids. Computer Methods in Applied Mechanics and Engineering,190, No. 46-47 (2001), 6089–6124.

    Article  MATH  MathSciNet  Google Scholar 

  3. R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles. McGraw-Hill, 1981.

  4. Y. Song, Y. Zhang, T. Shen, C.L. Bajaj, J.A. McCammon and N.A. Baker, Finite element solution of the steady-state Smoluchowski equation for rate constant calculations. Biophysical Journal,86, No. 4 (2004), 1–13.

    Article  Google Scholar 

  5. H. Imai, M. Iri and K. Murota, Voronoi diagram in the Laguerre geometry and its applications. SIAM Journal on Computing,14, No. 1 (1985), 93–105.

    Article  MATH  MathSciNet  Google Scholar 

  6. M.D. Berg, M.V. Kreveld, M. Overmars, O. Schwarzkopf and M.V. Kreveld, Computational Geometry. Springer Verlag, 2000.

  7. A. Okabe, B. Boots and K. Sugihara, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley and Sons, 1992.

  8. T.J. Tautges, T.D. Blacker and S.A. Mitchell, The Whisker weaving algorithm: A connectivity based method for constructing all hexahedral finite element meshes. International Journal for Numerical Methods in Engineering,39 (1996), 3327–3349.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Sheffer, M. Etalon, A. Rapports and M. Beckoner, Hexahedral mesh generation using the embedded Voronoi graph. Proceedings 7th International Meshing Roundtable, Sandia National Labs, 1998, 347–364.

  10. T.S. Li, R. McKeag and C.G. Armstrong, Hexahedral meshing using midpoint subdivision and integer programming. Computer Methods in Applied Mechanics and Engineering,124 (1995), 171–193.

    Article  Google Scholar 

  11. M. Price and C. Armstrong, Hexahedral mesh generation by medial surface subdivision: Part I. International Journal of Numerical Methods in Engineering,38, No. 19 (1995), 3335–3359.

    Article  MATH  Google Scholar 

  12. S. Owen, A survey of unstructured mesh generation technology. 7th International Meshing Roundtable, 1998, 26–28.

  13. S.H. Teng and C.W. Wong, Unstructured mesh generation: Theory, practice, and perspectives. International Journal of Computational Geometry and Applications,10, No. 3 (2000), 227–266.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. Eppstein, Linear complexity hexahedral mesh generation. Symposium on Computational Geometry, 1996, 58–67.

  15. J.F. Pascal and P.L. George, Mesh Generation. Hermes Science Publishing, 2000.

  16. R. Schneiders, R. Schindler and F. Weiler, Octree-based generation of hexahedral element meshes. 5th International Meshing Roundtable, Sandia National Labs, 1996, 205–216.

  17. Y. Wada, Effective adaptation technique for hexahedral mesh. Concurrency and Computation: Practice and Experience,14 (2002), 451–463.

    Article  MATH  Google Scholar 

  18. Y. Zhang, C. Bajaj and B. Sohn, Adaptive and quality 3D meshing from imaging data. 8th ACM Symposium on Solid Modeling and Applications, 2003, 286–291.

  19. Y. Zhang, C. Bajaj and B.S. Sohn, 3D finite element meshing from imaging data. To appear in the special issue of Computer Methods in Applied Mechanics and Engineering (CMAME) on Unstructured Mesh Generation, www.ices.utexas.edu/∼jessica/meshing, 2004.

  20. P. Schroder, D. Zorin, T. DeRose, L. Kobbelt, J. Stam and J. Warren, Subdivision for Modeling and Animation. Special course notes, Siggraph, 1999.

  21. W.A. Cook and W.R. Oakes, Mapping methods for generating three-dimensional meshes. Computers in Mechanical Engineering, 1982, 67–72.

  22. CUBIT Mesh Generation Toolkit. Web site: http://sassl693.sandia.gov/cubit.

  23. D. Field, Laplacian smoothing and Delaunay triangulations. Communications in Applied Numerical Methods,4 (1988), 709–712.

    Article  MATH  Google Scholar 

  24. S. Canann, J. Tristano and M. Staten, An approach to combined Laplacian and optimization-based smoothing for triangular, quadrilateral and quad-dominant meshes. 7th International Meshing Roundtable, 1998, 479–494.

  25. L. Freitag, On combining Laplacian and optimization-based mesh smoothing techniques. AMD-Vol. 220 Trends in Unstructured Mesh Generation, 1997, 37–43.

  26. L. Freitag and C. Ollivier-Gooch, Tetrahedral mesh improvement using swapping and smoothing. International Journal for Numerical Methods in Engineering,40 (1997), 3979–4002.

    Article  MATH  MathSciNet  Google Scholar 

  27. C. Bajaj, S. Schaeffer, J. Warren and G. Xu, A subdivision scheme for hexahedral meshes. The Visual Computer,18, No. 5-6 (2002), 343–356.

    Article  Google Scholar 

  28. CL. Bajaj and W.J. Bouma, Dynamic Voronoi diagrams and Delaunay triangulations. Proc. 2nd Canadian Conference on Computational Geometry, 1990, 273–277.

  29. H. Imai, M. Iri and K. Murota, Voronoi diagram in the Laguerre geometry and its applications. SIAM Journal of Computing,14, No. 1 (1985), 93–105.

    Article  MATH  MathSciNet  Google Scholar 

  30. P. Chew, Near-quadratic bounds for the L1 Voronoi diagram of moving points. Computational Geometry Theory and Applications,7, No. 1-2 (1997), 73–80.

    MATH  MathSciNet  Google Scholar 

  31. J.J. Fu and R.C.T. Lee, Voronoi diagrams of moving points in the plane. International Journal of Computational Geometry and Applications,1, No. 1 (1991), 23–32.

    Article  MATH  MathSciNet  Google Scholar 

  32. L. Guibas, J.S.B. Mitchell and T. Roos, Voronoi diagrams of moving points in the plane. Proc. 17th Internat. Workshop Graph-Theoret. Concepts Comput. Sci., Lecture Notes in Computer Science570, Springer-Verlag, 1991, 113–125.

  33. G. Albers and T. Roos, Voronoi diagrams of moving points in higher dimensional spaces. Proc. 3rd Scand. Workshop Algorithm Theory, Lecture Notes in Computer Science621, Springer-Verlag, 1992, 399–409.

  34. T. Roos, Voronoi diagrams over dynamic scenes. Discrete Applied Mathematics. Combinatorial Algorithms, Optimization and Computer Science,43, No. 3 (1993), 243–259.

    MATH  MathSciNet  Google Scholar 

  35. T. Roos, New upper bounds on Voronoi diagrams of moving points. Nordic Journal of Computing,4, No. 2 (1997), 167–171.

    MATH  MathSciNet  Google Scholar 

  36. K. Imai and H. Imai, On weighted dynamic Voronoi diagrams. Snapshots of Computational and Discrete Geometry (eds. D. Avis and P. Bose), Vol. 3, 1994, 268–277.

  37. F. Anton and D. Mioc, Dynamic Laguerre diagrams made easy. European Workshop on Computational Geometry, 2000, 10–13.

Download references

Author information

Authors and Affiliations

  1. Center for Computational Visualization, Department of Computer Sciences & Institute for Computational Engineering and Sciences, University of Texas at Austin, 78712, Austin, TX, USA

    Chandrajit Bajaj

Authors
  1. Chandrajit Bajaj
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Bajaj, C. A Laguerre Voronoi based scheme for meshing particle systems. Japan J. Indust. Appl. Math. 22, 167–177 (2005). https://doi.org/10.1007/BF03167436

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03167436

Key words

Search

Navigation