Two Step Time Discretization of Willmore Flow

  • N. Olischläger
  • M. Rumpf
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5654)

Abstract

Based on a natural approach for the time discretization of gradient flows a new time discretization for discrete Willmore flow of polygonal curves and triangulated surfaces is proposed. The approach is variational and takes into account an approximation of the L 2-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approximation of the Willmore functional at the new time step. To evaluate the Willmore energy on the unknown surface of the next time step, we first ask for the solution of a inner, secondary variational problem describing a time step of mean curvature motion. The time discrete velocity deduced from the solution of the latter problem is regarded as an approximation of the mean curvature vector and enters the approximation of the actual Willmore functional. To solve the resulting nested variational problem in each time step numerically relaxation theory from PDE constraint optimization are taken into account. The approach is applied to polygonal curves and triangular surfaces and is independent of the co-dimension. Various numerical examples underline the stability of the new scheme, which enables time steps of the order of the spatial grid size.

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References

  1. 1.
    Willmore, T.: Riemannian Geometry. Claredon Press, Oxford (1993)MATHGoogle Scholar
  2. 2.
    Simonett, G.: The Willmore Flow near spheres. Diff. and Integral Eq. 14(8), 1005–1014 (2001)MathSciNetMATHGoogle Scholar
  3. 3.
    Kuwert, E., Schätzle, R.: The Willmore flow with small initial energy. J. Differential Geom. 57(3), 409–441 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Nitzberg, M., Mumford, D., Shiota, T.: Filtering, Segmentation and Depth. LNCS, vol. 662. Springer, Heidelberg (1993)CrossRefMATHGoogle Scholar
  5. 5.
    Mumford, D.: Elastica and computer vision. In: Bajaj, C. (ed.) Algebraic Geometry and Its Applications, pp. 491–506. Springer, New York (1994)CrossRefGoogle Scholar
  6. 6.
    Yoshizawa, S., Belyaev, A.G.: Fair triangle mesh generation with discrete elastica. In: Proceedings of the Geometric Modeling and Processing; Theory and Applications (GMP 2002), Washington, DC, USA, pp. 119–123. IEEE Computer Society, Los Alamitos (2002)CrossRefGoogle Scholar
  7. 7.
    Chan, T.F., Kang, S.H., Shen, J.: Euler’s elastica and curvature-based inpainting. SIAM Appl. Math. 63(2), 564–592 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Polden, A.: Closed Curves of Least Total Curvature. SFB 382 Tübingen, Preprint 13 (1995)Google Scholar
  9. 9.
    Polden, A.: Curves and Surfaces of Least Total Curvature and Fourth-Order Flows. Dissertation, Universität Tübingen (1996)Google Scholar
  10. 10.
    Kuwert, E., Schätzle, R.: Removability of Point Singularities of Willmore Surfaces. Preprint SFB 611, Bonn (2002)Google Scholar
  11. 11.
    Kuwert, E., Schätzle, R.: Gradient flow for the Willmore functional. Comm. Anal. Geom. 10(5), 1228–1245 (2002) (electronic)MathSciNetMATHGoogle Scholar
  12. 12.
    Dziuk, G., Kuwert, E., Schätzle, R.: Evolution of elastic curves in ℝn: existence and computation. SIAM J. Math. Anal. 33(5), 1228–1245 (2002) (electronic)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Droske, M., Rumpf, M.: A level set formulation for Willmore flow. Interfaces and Free Boundaries 6(3), 361–378 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Deckelnick, K., Dziuk, G.: Error analysis of a finite element method for the Willmore flow of graphs. Interfaces and Free Boundaries 8, 21–46 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Barrett, J.W., Garcke, H., Nürnberg, R.: A parametric finite element method for fourth order geometric evolution equations. J. Comp. Phys. 222, 441–467 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bobenko, A., Schröder, P.: Discrete Willmore flow. In: SIGGRAPH (Courses). ACM Press, New York (2005)Google Scholar
  17. 17.
    Dziuk, G.: Computational parametric Willmore flow. Preprint Fakultät für Mathematik und Physik, Universität Freiburg 13-07 (2007)Google Scholar
  18. 18.
    Welch, W., Witkin, A.: Variational surface modeling. In: SIGGFRAPH Computer Graphics, vol. 26, pp. 157–166 (1992)Google Scholar
  19. 19.
    Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proc. of SIGGRAPH 2000, New Orleans, USA, pp. 417–424 (2000)Google Scholar
  20. 20.
    Bertalmio, M., Bertozzi, A., Sapiro, G.: Navier-stokes, fluid dynamics, and image and video inpainting. In: IEEE Proceedings of the International Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 355–362 (2001)Google Scholar
  21. 21.
    Clarenz, U., Diewald, U., Dziuk, G., Rumpf, M., Rusu, R.: A finite element method for surface restoration with smooth boundary conditions. Computer Aided Geometric Design 21(5), 427–445 (2004)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rane, S.D., Remus, J., Sapiro, G.: Wavelet-domain reconstruction of lost blocks in wireless image transmission and packet-switched networks. In: 2002 International Conference on Image Processing. Proceedings, September 22-25, vol. 1 (2002)Google Scholar
  23. 23.
    Xu, G., Pan, Q.: G1 surface modelling using fourth order geometric flows. Computer-Aided Design 38(4), 392–403 (2006)CrossRefGoogle Scholar
  24. 24.
    Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. 3, 253–271 (1995)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Chambolle, A.: An algorithm for mean curvature motion. Interfaces and free Boundaries 6, 195–218 (2004)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Bellettini, G., Caselles, V., Chambolle, A., Novaga, M.: Crystalline mean curvature flow of convex sets. Technical Report 7641, UMR CNRS (2004)Google Scholar
  27. 27.
    Chambolle, A., Novaga, M.: Convergence of an algorithm for anisotropic mean curvature motion. SIAM J. Math. Anal. 37, 1978–1987 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58, 603–611 (1991)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Hildebrandt, S., Leis, R. (eds.) Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematics, vol. 1357, pp. 142–155. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  30. 30.
    Thomée, V.: Galerkin finite element methods for parabolic problems, 2nd edn. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin (2006)MATHGoogle Scholar
  31. 31.
    Diewald, U., Morigi, S., Rumpf, M.: A cascadic geometric filtering approach to subdivision. Computer Aided Geometric Design 19, 675–694 (2002)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Deckelnick, K., Dziuk, G.: Error analysis for the elastic flow of parametrized curves. Preprint Fakultät für Mathematik und Physik, Universität Freiburg 14-07 (2007) (to appear in Math. Comp.)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • N. Olischläger
    • 1
  • M. Rumpf
    • 1
  1. 1.Institut für Numerische SimulationRheinische Friedrich-Wilhelms-Universität BonnGermany

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