Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity
Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity
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DOI：
10.1002/mma.2554
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发表时间：
2007-11
影响因子：
2.9
通讯作者：
D. Ševčovič;S. Yazaki
中科院分区：
文献类型：
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作者：
D. Ševčovič;S. Yazaki
在本文中，我们研究了一系列与时间相关的平面闭合 Jordan 曲线，其沿法线方向演化，速度假设为曲线曲率、切向角和位置向量的函数。我们遵循直接方法并分析相关几何量的偏微分方程的控制系统。我们关注一类所谓的曲率调整切向速度，用于计算平面闭合曲线的曲率驱动流。这种曲率调整的切向速度取决于曲率模量及其曲线平均值。利用抽象抛物方程的理论，我们证明了控制方程组经典解的局部存在性、唯一性和连续性。我们还分析了几何流动，其中法向速度可能取决于全局曲线量，例如曲线的长度、封闭面积或总弹性能。我们还提出了基于流动有限体积法的稳定数值近似方案。本文还给出了各种非局部几何流的几个计算示例。版权所有 © 2012 约翰·威利父子有限公司
In this paper, we investigate a time‐dependent family of plane closed Jordan curves evolving in the normal direction with a velocity that is assumed to be a function of the curvature, tangential angle, and position vector of a curve. We follow the direct approach and analyze the system of governing PDEs for relevant geometric quantities. We focus on a class of the so‐called curvature adjusted tangential velocities for computation of the curvature driven flow of plane closed curves. Such a curvature adjusted tangential velocity depends on the modulus of the curvature and its curve average. Using the theory of abstract parabolic equations, we prove local existence, uniqueness, and continuation of classical solutions to the system of governing equations. We furthermore analyze geometric flows for which normal velocity may depend on global curve quantities such as the length, enclosed area, or total elastic energy of a curve. We also propose a stable numerical approximation scheme on the basis of the flowing finite volume method. Several computational examples of various nonlocal geometric flows are also presented in this paper. Copyright © 2012 John Wiley & Sons, Ltd.