Skip to main content
SpringerLink
Log in

On Surface Area and Length Preserving Flows of Closed Curves on a Given Surface

  • Conference paper
  • First Online:
Numerical Mathematics and Advanced Applications ENUMATH 2017 (ENUMATH 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 126))

Included in the following conference series:

  • 1614 Accesses

Abstract

In this paper we investigate two non-local geometric geodesic curvature driven flows of closed curves preserving either their enclosed surface area or their total length on a given two-dimensional surface. The method is based on projection of evolved curves on a surface to the underlying plane. For such a projected flow we construct the normal velocity and the external nonlocal force. The evolving family of curves is parametrized by a solution to the fully nonlinear parabolic equation for which we derive a flowing finite volume approximation numerical scheme. Finally, we present various computational examples of evolution of the surface area and length preserving flows of surface curves. We furthermore analyse the experimental order of convergence. It turns out that the numerical scheme is of the second order of convergence.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Kolář, M. Beneš, D. Ševčovič, Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete Contin. Dyn. Syst. - Ser. B 22(10), 367–3689 (2017)

    MathSciNet  MATH  Google Scholar 

  2. M. Kolář, M. Beneš, D. Ševčovič, Computational analysis of the conserved curvature driven flow for open curves in the plane. Math. Comput. Simul. 126, 1–13 (2016)

    Article  MathSciNet  Google Scholar 

  3. 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)

    Article  MathSciNet  Google Scholar 

  4. M. Gage, On an area-preserving evolution equation for plane curves. Contemp. Math. 51, 51–62 (1986)

    Article  MathSciNet  Google Scholar 

  5. C. Kublik, S. Esedoḡlu, J.A. Fessler, Algorithms for area preserving flows. SIAM J. Sci. Comput. 33, 2382–2401 (2011)

    Google Scholar 

  6. J. McCoy, The surface area preserving mean curvature flow. Asian J. Math. 7, 7–30 (2003)

    Article  MathSciNet  Google Scholar 

  7. D. Ševčovič, S. Yazaki, On a gradient flow of plane curves minimizing the anisoperimetric ratio, IAENG Int. J. Appl. Math. 43(3), 160–171 (2013)

    MathSciNet  Google Scholar 

  8. J. Rubinstein, P. Sternberg, Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264 (1992)

    Article  MathSciNet  Google Scholar 

  9. M. Beneš, S. Yazaki, M. Kimura, Computational studies of non-local anisotropic Allen-Cahn equation. Math. Bohem. 136, 429–437 (2011)

    MathSciNet  MATH  Google Scholar 

  10. J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. III. Nucleation of a two-component incompressible fluid. J. Chem. Phys. 31, 688–699 (1959)

    Google Scholar 

  11. S. Allen, J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Article  Google Scholar 

  12. I.V. Markov, Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy, 2nd edn. (World Scientific Publishing Company, Springer, 2004)

    Google Scholar 

  13. K. Deckelnick, Parametric mean curvature evolution with a Dirichlet boundary condition. J. Reine Angew. Math. 459, 37–60 (1995)

    MathSciNet  MATH  Google Scholar 

  14. K. Mikula, D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force. Comput. Vis. Sci. 6, 211–225 (2004)

    Article  MathSciNet  Google Scholar 

  15. K. Mikula, D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. Math. Methods Appl. Sci. 27, 1545–1565 (2004)

    Article  MathSciNet  Google Scholar 

  16. D. Ševčovič, K. Mikula, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math. 61, 1473–1501 (2001)

    Article  MathSciNet  Google Scholar 

  17. M. Beneš, J. Kratochvíl, J. Křišt’an, V. Minárik, P. Pauš, A parametric simulation method for discrete dislocation dynamics. Eur. Phys. J. ST 177, 177–192 (2009)

    Article  Google Scholar 

  18. M. Beneš, M. Kimura, P. Pauš, D. Ševčovič, T. Tsujikawa, S. Yazaki, Application of a curvature adjusted method in image segmentation. Bull. Inst. Math. Acad. Sinica (N. S.) 3, 509–523 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Miroslav Kolář and Michal Beneš were partly supported by the project No. 14-36566G of the Czech Science Foundation and by the project No. OHK4-001/17 2017-19 of the Student Grant Agency of the Czech Technical University in Prague. The third author was supported by the VEGA grant 1/0062/18.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Miroslav Kolář & Michal Beneš

  2. Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská Dolina, Bratislava, Slovakia

    Daniel Ševčovič

Authors
  1. Miroslav Kolář
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Beneš
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Ševčovič
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miroslav Kolář .

Editor information

Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Florin Adrian Radu

  2. University of Bergen, Bergen, Norway

    Kundan Kumar

  3. University of Bergen, Bergen, Norway

    Inga Berre

  4. University of Bergen, Bergen, Norway

    Jan Martin Nordbotten

  5. Hasselt University, Diepenbeek, Belgium

    Iuliu Sorin Pop

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kolář, M., Beneš, M., Ševčovič, D. (2019). On Surface Area and Length Preserving Flows of Closed Curves on a Given Surface. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_24

Download citation

Publish with us

Policies and ethics

Search

Navigation