Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip

Ishiwata, Tetsuya; Ohtsuka, Tsutomu

doi:10.3934/dcdsb.2019058

Cited by 6 publications

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“…(Note that we do not assume the symmetry of this metric.) For the evolution of a pinned spiral, the authors of this paper introduced a discrete model by an ODE system of the facet lengths in [15], due to the ideas in [3,23,14], see also [13] for details.…”

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“…Recently, Tsai and the second author extended this numerical algorithm to the motion of spirals; see [19,20]. The aim of this paper is to show the numerical difference between the spirals calculated by the discrete model due to [15] and the level set method due to [17]. To measure the difference between these spirals, we calculate the area of an interposed region by their spirals.…”

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“…Thus, we shall give a way to construct θ D in §3.2. Note that the discrete model in [15] is constructed from W γ . Then, we shall give a way to construct γ from γ • in §3.1 to obtain the level set equation corresponding to the discrete model.…”

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“…Numerical methods. In this section, we recall the discrete model due to [15] and the level set method due to [17]. To compare the evolving spiral curves from these models, we have to give a Wulff shape W γ for the discrete model and a corresponding surface energy density γ for the level set method.…”

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confidence: 99%

“…We denote a polygonal spiral curve evolving by (1) by Γ D (t) = k j=0 L j (t). According to [15], we here consider the evolution of a positive convex polygonal spiral. Assume that the j-th facet L j (t) is given as…”

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Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow

Ishiwata

Ohtsuka

2021

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered from two viewpoints; a discrete model consisting of an ODE system describing facet lengths and another using level set method. We investigate the difference of these models numerically by calculating the area of an interposed region by their spiral curves. The area difference is calculated by the normalized L 1 norm of the difference of step-like functions which are branches of arg(x) whose discontinuities are on the spirals. We find that the differences in the numerical results are small, even though the model equations around the center and the farthest facet are slightly different.

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Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow

Ishiwata

Ohtsuka

2021

Discrete &Amp; Continuous Dynamical Systems - S

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Crystalline flow starting from a general polygon

Giga¹,

Giga²,

Kuroda³

et al. 2022

DCDS

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This paper solves a singular initial value problem for a system of ordinary differential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.

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Tracking the critical points of curves evolving under planar curvature flows

Fehér¹,

Domokos²,

Krauskopf³

2021

JCD

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We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function <inline-formula><tex-math id="M1">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function <inline-formula><tex-math id="M2">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> and of the curvature <inline-formula><tex-math id="M3">\begin{document}$ \kappa(\varphi) $\end{document}</tex-math></inline-formula> (characterized by <inline-formula><tex-math id="M4">\begin{document}$ dr/d\varphi = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ d\kappa /d\varphi = 0 $\end{document}</tex-math></inline-formula>, respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically — in the spirit of experimental mathematics — we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.

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Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip

Cited by 6 publications

References 0 publications

Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow

Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow

Crystalline flow starting from a general polygon

Tracking the critical points of curves evolving under planar curvature flows

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