Advanced Search
Article Contents
Article Contents

Area preserving geodesic curvature driven flow of closed curves on a surface

  • * Corresponding author: Miroslav Kolář

    * Corresponding author: Miroslav Kolář 
The first author is supported by the grant No. 14-36566G of the Czech Science Foundation and by the grant No. 15-27178A of Ministry of Health of the Czech Republic.
Abstract Full Text(HTML) Figure(6) / Table(5) Related Papers Cited by
  • We investigate a non-local geometric flow preserving surface area enclosed by a curve on a given surface evolved in the normal direction by the geodesic curvature and the external force. We show how such a flow of surface curves can be projected into a flow of planar curves with the non-local normal velocity. We prove that the surface area preserving flow decreases the length of the evolved surface curves. Local existence and continuation of classical smooth solutions to the governing system of partial differential equations is analysed as well. Furthermore, we propose a numerical method of flowing finite volume for spatial discretization in combination with the Runge-Kutta method for solving the resulting system. Several computational examples demonstrate variety of evolution of surface curves and the order of convergence.

    Mathematics Subject Classification: Primary:35K57, 35K65, 65N40, 65M08;Secondary:53C80.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Illustration of a curve $\mathcal{G}_t$ on a given surface $\mathcal{M}$ and its projection $\Gamma_t$ to plane

    Figure 2.  Discretization of a segment of a curve by flowing finite volumes

    Figure 3.  Left: the initial curve $\mathcal{G}_{ini}$ (dashed) and the final curve $\mathcal{G}_T$ at $T = 10$ (solid) and several intermediate curves $\mathcal{G}_t$ (dotted). The underlying surface $\mathcal{M}$ is plotted in gray color. Right: time evolution of the projected planar curves $\Gamma_t$ (see Example 1)

    Figure 4.  Left: the initial curve $\mathcal{G}_{ini}$ (dashed) and the final curve $\mathcal{G}_T$ at $T = 30$ (solid). The underlying surface $\mathcal{M}$ is plotted in gray color. Right: time evolution of the projected planar curves $\Gamma_t$ (see Example 2)

    Figure 5.  Left: the initial curve $\mathcal{G}_{ini}$ (dashed) and the final curve $\mathcal{G}_T$ at $T = 8$ (solid) are presented. The surface $\mathcal{M}$ is plotted in gray color. Right: time evolution of the projected planar curves $\Gamma_t$ (see Example 3)

    Figure 6.  Left: the initial curve $\mathcal{G}_{ini}$ (dashed) and the final curve $\mathcal{G}_T$ at $T = 15$ (solid) are shown. The surface $\mathcal{M}$ is plotted in gray. Right: Time evolution of the projected planar curves $\Gamma_t$ (see Example 4)

    Table 1.  Settings of computational examples

    Ex. $\mathbf{X}_{ini}, u \in [0,1]$ $\varphi$
    1 $\mathbf{X}_{ini} = (\frac14 + r(u) \cos(2 \pi u), -\frac14 + r(u) \sin(2 \pi u))^T$ $\varphi(x,y) = \sqrt{4 - x^2 - y^2}$
    2 $\mathbf{X}_{ini} = (\cos(2 \pi u), \frac1{10} + \sin(2 \pi u))^T$ $\varphi(x,y) = y^2$
    3 $\mathbf{X}_{ini} = (\cos(2 \pi u), \frac15 + \sin(2 \pi u))^T$ $\varphi(x,y) = \sin(\pi y)$
    4 $\mathbf{X}_{ini} = (\frac12 \cos(2 \pi u), \sin(2 \pi u))^T$ $\varphi(x,y) = x^2 - y^4$
     | Show Table
    DownLoad: CSV

    Table 2.  Table of EOCs for Example 1

    $M$ $error_{max}$EOC $error_{L1}$EOC
    100 $3.2397 \cdot 10^{-2}$- $3.2516 \cdot 10^{-2}$-
    200 $8.2467 \cdot 10^{-3}$1.97408.2767 $\cdot 10^{-3}$1.9740
    300 $3.6408 \cdot 10^{-3}$2.01653.6542 $\cdot 10^{-3}$2.0164
    400 $2.0411 \cdot 10^{-3}$2.01182.0485 $\cdot 10^{-3}$2.0117
    500 $1.3033 \cdot 10^{-3}$2.01031.3081 $\cdot 10^{-3}$2.0102
     | Show Table
    DownLoad: CSV

    Table 3.  Table of EOCs for Example 2

    $M$ $error_{max}$EOC $error_{L1}$EOC
    100 $1.4812 \cdot 10^{-3}$- $1.4839 \cdot 10^{-3}$-
    200 $3.7049 \cdot 10^{-4}$1.9993 $3.7092 \cdot 10^{-4}$2.0002
    300 $1.6453 \cdot 10^{-4}$2.0019 $1.6471 \cdot 10^{-4}$2.0022
    400 $8.2431 \cdot 10^{-5}$2.0045 $9.2525 \cdot 10^{-5}$2.0046
    500 $5.9055 \cdot 10^{-5}$2.0077 $5.9114 \cdot 10^{-5}$2.0077
     | Show Table
    DownLoad: CSV

    Table 4.  Table of EOCs for Example 3

    $M$ $error_{max}$EOC $error_{L1}$EOC
    100 $4.3505 \cdot 10^{-3}$- $4.7156 \cdot 10^{-3}$-
    200 $9.4649 \cdot 10^{-4}$2.2005 $9.5944 \cdot 10^{-4}$2.2972
    300 $4.1813 \cdot 10^{-4}$2.0149 $4.2481 \cdot 10^{-4}$2.0082
    400 $2.3506 \cdot 10^{-4}$2.0021 $2.3885 \cdot 10^{-4}$2.0015
    500 $1.5050 \cdot 10^{-4}$1.9980 $1.5293 \cdot 10^{-4}$1.9980
     | Show Table
    DownLoad: CSV

    Table 5.  Table of EOCs for Example 4

    $M$ $error_{max}$EOC $error_{L1}$EOC
    100 $1.8882 \cdot 10^{-3}$- $1.9422 \cdot 10^{-3}$-
    200 $4.7176 \cdot 10^{-4}$2.0009 $4.8494 \cdot 10^{-4}$2.0018
    300 $2.0979 \cdot 10^{-4}$1.9986 $2.1563 \cdot 10^{-4}$1.9988
    400 $1.1808 \cdot 10^{-4}$1.9978 $1.2136 \cdot 10^{-4}$1.9980
    500 $7.5628 \cdot 10^{-5}$1.9966 $7.7728 \cdot 10^{-5}$1.9968
     | Show Table
    DownLoad: CSV
  • [1] S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.
    [2] S. Angenent, Nonlinear analytic semiflows, Proc. R. Soc. Edinb., Sect. A, 115 (1990), 91-107.  doi: 10.1017/S0308210500024598.
    [3] M. Beneš, Diffuse-interface treatment of the anisotropic mean-curvature flow, Appl. Math., 48 (2003), 437-453.  doi: 10.1023/B:APOM.0000024485.24886.b9.
    [4] M. BenešM. KimuraP. PaušD. ŠevčovičT. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bull. Inst. Math. Acad. Sinica (N. S.), 3 (2008), 509-523. 
    [5] M. BenešJ. KratochvílJ. Křišt'anV. Minárik and P. Pauš, A parametric simulation method for discrete dislocation dynamics, Eur. Phys. J. ST, 177 (2009), 177-192. 
    [6] M. BenešS. Yazaki and M. Kimura, Computational studies of non-local anisotropic Allen-Cahn equation, Math. Bohemica, 136 (2011), 429-437. 
    [7] L. Bronsard and B. Stoth, Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807.  doi: 10.1137/S0036141094279279.
    [8] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅲ. Nucleation of a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699.  doi: 10.1002/9781118788295.ch5.
    [9] M. C. Dallaston and S. W. McCue, A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area Proc. R. Soc. A 472 (2016), 20150629, 15 pp. doi: 10.1098/rspa.2015.0629.
    [10] K. Deckelnick, Parametric mean curvature evolution with a Dirichlet boundary condition, J. Reine Angew. Math., 459 (1995), 37-60.  doi: 10.1515/crll.1995.459.37.
    [11] I. C. DolcettaS. F. Vita and R. March, Area preserving curve shortening flows: From phase separation to image processing, Interfaces Free Bound., 4 (2002), 325-343.  doi: 10.4171/IFB/64.
    [12] J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3.
    [13] S. EsedoḡluS. Ruuth and R. Tsai, Threshold dynamics for high order geometric motions, Interfaces Free Bound., 10 (2008), 263-282.  doi: 10.4171/IFB/189.
    [14] M. Gage, On an area-preserving evolution equation for plane curves, Contemp. Math., 51 (1986), 51-62.  doi: 10.1090/conm/051/848933.
    [15] M. HenryD. Hilhorst and M. Mimura, A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 125-154.  doi: 10.3934/dcdss.2011.4.125.
    [16] M. KolářM. Beneš and D. Ševčovič, Computational analysis of the conserved curvature driven flow for open curves in the plane, Math. Comput. Simulation, 126 (2016), 1-13. 
    [17] C. KublikS. Esedoḡlu and J. A. Fessler, Algorithms for area preserving flows, SIAM J. Sci. Comput., 33 (2011), 2382-2401.  doi: 10.1137/100815542.
    [18] A. Lunardi, Abstract quasilinear parabolic equations, Math. Ann., 267 (1984), 395-416.  doi: 10.1007/BF01456097.
    [19] I. V. MarkovCrystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy, 2 edition, World Scientific Publishing Company, 2004. 
    [20] J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.  doi: 10.4310/AJM.2003.v7.n1.a2.
    [21] K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), 1473-1501.  doi: 10.1137/S0036139999359288.
    [22] K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Vis. Sci., 6 (2004), 211-225.  doi: 10.1007/s00791-004-0131-6.
    [23] K. Mikula and D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Math. Methods Appl. Sci., 27 (2004), 1545-1565.  doi: 10.1002/mma.514.
    [24] P. PaušM. BenešM. Kolář and J. Kratochvíl, Dynamics of dislocations described as evolving curves interacting with obstacles, Model. Simul. Mater. Sc., 24 (2016), 035003. 
    [25] J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264.  doi: 10.1093/imamat/48.3.249.
    [26] D. Ševčovič, Qualitative and quantitative aspects of curvature driven flows of planar curves, in Topics on Partial Differential Equations, Jindřich Nečas Cent. Math. Model. Lect. Notes, 2, Matfyzpress, Prague, 2007, 55-119.
    [27] D. Ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity, Japan. J. Ind. Appl. Math., 28 (2011), 413-442.  doi: 10.1007/s13160-011-0046-9.
    [28] D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Math. Methods Appl. Sci., 35 (2012), 1784-1798.  doi: 10.1002/mma.2554.
    [29] S. Yazaki, On the tangential velocity arising in a crystalline approximation of evolving plane curves, Kybernetika, 43 (2007), 913-918. 
  • 加载中




Article Metrics

HTML views(632) PDF downloads(329) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint