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A Novel Approach to the Weighted Laplacian Formulation Applied to 2D Delaunay Triangulations

  • Sanderson L. Gonzaga de OliveiraEmail author
  • Frederico Santos de Oliveira
  • Guilherme Oliveira Chagas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9155)

Abstract

In this work, a novel smoothing method based on weighted Laplacian formulation is applied to resolve the heat conduction equation by finite-volume discretizations with Voronoi diagram. When a minimum number of vertices is obtained, the mesh is smoothed by means of a new approach to the weighted Laplacian formulation. The combination of techniques allows to solve the resulting linear system by the Conjugate Gradient Method. The new approach to the weighted Laplacian formulation within the set of techniques is compared to other 4 approaches to the weighted Laplacian formulation. Comparative analysis of the results shows that the proposed approach allows to maintain the approximation and presents smaller number of vertices than any of the other 4 approaches. Thus, the computational cost of the resolution is lower when using the proposed approach than when applying any of the other approaches and it is also lower than using only Delaunay refinements.

Keywords

Weighted laplacian formulation Delaunay triangulation Heat conduction equation Voronoi diagram Laplacian smoothing 

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References

  1. 1.
    Bank, R.E., Smith, R.K.: Mesh smoothing using a posteriori error estimates. SIAM Journal on Numerical Analysis 34(3), 979–997 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bank, R.E., Xu, J.: An algorithm for coarsening unstructured meshes. Numerische Mathematik 73(1), 1–36 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Delaunay, B.: Sur la sphère vide. Izvestia Akademii Nauk SSSR. Otdelenie Matematicheskikh i Estestvennykh Nauk 7, 793–800 (1934)Google Scholar
  4. 4.
    George, A., Liu, J.W.H.: An implementation of a pseudoperipheral node finder. ACM Transactions on Mathematical Software 5(3), 284–295 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    George, J.A.: Computer implementation of the finite element method. PhD thesis, Computer Science Department, Stanford University, CA (1971)Google Scholar
  6. 6.
    Gonzaga de Oliveira, S.L.: A review on delaunay refinement techniques. In: Murgante, B., Gervasi, O., Misra, S., Nedjah, N., Rocha, A.M.A.C., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2012, Part I. LNCS, vol. 7333, pp. 172–187. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  7. 7.
    Green, P.J., Sibson, R.: Computing Dirichlet tessellations in the plane. The Computer Journal 21(2), 168–173 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 49(36), 409–436 (1952)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Huang, W., Russell, R.: Adaptive moving mesh methods, 1st edn. Applied mathematical sciences. Springer, New York (2011)Google Scholar
  10. 10.
    Kobbelt, L., Campagna, S., Vorsatz, J., Seidel, H.-P.: Interactive multi-resolution modeling on arbitrary meshes. In: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, ACM SIGGRAPH 1998, pp. 105–114. ACM, New York (1998)Google Scholar
  11. 11.
    Lanczos, C.: Solutions of systems of linear equations by minimized iterations. Journal of Research of the National Bureau of Standards 49(3), 33–53 (1952)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lawson, C.L.: Software for C\(^1\) surface interpolation. In: Rice, J.R. (ed.) Matematical Software III, pp. 161–194. Academic Press, Orlando (1977)Google Scholar
  13. 13.
    Oliveira, F.S.: Numerical solutions of partial differential equations with discretization by finite volume and adaptively refined and moving meshes (in Portuguese). Master’s thesis, Departamento de Ciência da Computação - Universidade Federal de Lavras, Lavras, Brazil (2014)Google Scholar
  14. 14.
    Oliveira, F.S., de Oliveira, S.L.G., Kischinhevsky, M., Tavares, J.M.R.S.: Moving mesh methods for numerical solution of partial differential equations (in Portuguese). Revista de Sistemas de Informação da Faculdade Salesiana Maria Auxiliadora-FSMA, 11:11–16 (June 2013)Google Scholar
  15. 15.
    Ruppert, J.: A Delaunay refinement algorithm for quality 2-dimensional mesh generation. Journal of Algorithms 18(3), 548–585 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Taubin, G.: Curve and surface smoothing without shrinkage. In: Proceedings of the Fifth International Conference on Computer Vision, ICCV 1995, vol. 5, pp. 852–857. IEEE Computer Society, Washington, DC (1995)Google Scholar
  17. 17.
    Taubin, G.: A signal processing approach to fair surface design. In: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1995, pp. 351–358. ACM, New York (1995)Google Scholar
  18. 18.
    Taubin, G., Zhang, T., Golub, G.: Optimal surface smoothing as filter design. In: Buxton, B.F., Cipolla, R. (eds.) ECCV 1996. LNCS, vol. 1064, pp. 283–292. Springer, Heidelberg (1996) CrossRefGoogle Scholar
  19. 19.
    Thompson, J.F., Soni, B.K., Weatherhill, N.P.: Handbook of Grid Generation. CRC Press, New York (1999) zbMATHGoogle Scholar
  20. 20.
    Üngör, A.: Off-Centers: a new type of steiner points for computing size-optimal quality-guaranteed delaunay triangulations. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 152–161. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  21. 21.
    Üngör, A.: Off-centers: A new type of Steiner points for computing size-optimal quality-guaranteed Delaunay triangulations. Computational Geometry 42(2), 109–118 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Voronoi, G.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. Journal für die reine und angewandte Mathematik (Crelles Journal) 1908(134), 198–287 (1908)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sanderson L. Gonzaga de Oliveira
    • 1
    Email author
  • Frederico Santos de Oliveira
    • 1
  • Guilherme Oliveira Chagas
    • 1
  1. 1.Universidade Federal de LavrasLavrasBrazil

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