Numerical Solution of Constrained Curvature Flow for Closed Planar Curves

  • Miroslav Kolář
  • Michal Beneš
  • Daniel Ševčovič
Conference paper
First Online:
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 112)

Abstract

This paper presents results of computational studies of the evolution law for the constrained mean curvature flow. The considered motion law originates in the theory of phase transitions in crystalline materials. It describes the evolution of closed embedded curves with constant enclosed area. In the paper, the motion law is treated by the parametric method, which leads into the system of degenerate parabolic equations for the parametric description of the curve. This system is numerically solved by means of the flowing finite volume method enhanced by tangential redistribution. Qualitative and quantitative results of computational studies are presented.

Keywords

Jordan Curve Planar Curf Degenerate Parabolic Equation Enclose Area Geometric Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgements

The first two authors were partly supported by the project

No. 14-36566G “Multidisciplinary research centre for advanced materials” of the Grant Agency of the Czech Republic.

The third author was supported by Slovak research agency grant VEGA 01/0780/15.

References

  1. 1.
    M. Gage, On an area-preserving evolution equation for plane curves. Contemp. Math. 51, 51–62 (1986)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    I.C. Dolcetta, S.F. Vita, R. March, Area preserving curve shortening flows: from phase separation to image processing. Interfaces Free Bound. 4, 325–343 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    C. Kublik, S. Esedoḡlu, J.A. Fessler, Algorithms for area preserving flows. SIAM J. Sci. Comput. 33, 2382–2401 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. McCoy, The surface area preserving mean curvature flow. Asian J. Math. 7, 7–30 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    M. Kolář, M. Beneš, D. Ševčovič, Computational studies of conserved mean-curvature flow. Mathematica Bohemica 139 (4), 677–684 (2014)MathSciNetMATHGoogle Scholar
  6. 6.
    D. Ševčovič, S. Yazaki, On a gradient flow of plane curves minimizing the anisoperimetric ratio. IAENG Int. J. Appl. Math. 43, 160–171 (2013)MathSciNetGoogle Scholar
  7. 7.
    J. Rubinstein, P. Sternberg, Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    M. Beneš, S. Yazaki, M. Kimura, Computational studies of non-local anisotropic Allen-Cahn equation. Mathematica Bohemica 134 (4), 429–437 (2011)MathSciNetMATHGoogle Scholar
  9. 9.
    J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system III. Nucleation of a two-component incompressible uid. J. Chem. Phys. 31, 688–699 (1959)Google Scholar
  10. 10.
    S. Allen, J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 27, 1084–1095 (1979)CrossRefGoogle Scholar
  11. 11.
    I.V. Markov, Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy, 2nd edn. (World Scientific Publishing Company, New Jersey, 2004)Google Scholar
  12. 12.
    M. Kolář, M. Beneš, D. Ševčovič, J. Kratochvíl, Mathematical model and computational studies of discrete dislocation dynamics. IAENG Int. J. Appl. Math. 45 (3), 198–207 (2015)MathSciNetGoogle Scholar
  13. 13.
    V. Minárik, M. Beneš, J. Kratochvíl, Simulation of dynamical interaction between dislocations and dipolar loops. J. Appl. Phys. 107, 061802 (2010)CrossRefGoogle Scholar
  14. 14.
    K. Deckelnick, Parametric mean curvature evolution with a dirichlet boundary condition. Journal für die reine und angewandte Mathematik 459, 37–60 (1995)MathSciNetMATHGoogle Scholar
  15. 15.
    K. Deckelnick, G.Dziuk, C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow. Acta Numerica 14, 139–232 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    J.W. Barrett, H. Garcke, R. Nürnberg, On the variational approximation of combined second and fourth order geometric evolution equations. SIAM J. Sci. Comput. 29 (3), 1006–1041 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    S. Osher, J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    M. Beneš, Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. Math. 48 (6), 437–453 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    P. Pauš, J. Kratochvíl, M. Beneš, A dislocation dynamics analysis of the critical cross-slip annihilation distance and the cyclic saturation stress in fcc single crystals at different temperatures. Acta Materialia 61, 7917–7923 (2013)CrossRefGoogle Scholar
  20. 20.
    D. Ševčovič, K. Mikula, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61 (3), 1473–1501 (2001)MathSciNetMATHGoogle Scholar
  21. 21.
    K. Mikula, D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force. Comput. Vis. Sci. 6 (3), 211–225 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    K. Deckelnick, G. Dziuk, On the approximation of the curve shortening flow. Pitman Res. Notes Math. Ser. 326, 100–108 (1995)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Miroslav Kolář
    • 1
  • Michal Beneš
    • 1
  • Daniel Ševčovič
    • 2
  1. 1.Faculty of Nuclear Sciences and Physical Engineering, Department of MathematicsCzech Technical University in PraguePrague 2Czech Republic
  2. 2.Faculty of Mathematics, Department of Applied Mathematics and StatisticsPhysics and Informatics, Comenius UniversityBratislavaSlovakia

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