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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.
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  • 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.


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  • 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
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    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
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