论文标题
用于区域保存的广义平均曲率流的结构保存参数有限元法
A structure-preserving parametric finite element method for area-conserved generalized mean curvature flow
论文作者
论文摘要
我们提出并分析了一个具有结构的参数有限元法（SP-PFEM），以模拟由面积保存的广义平均曲率流动在二维（2D）中受到的封闭曲线的运动。我们首先提出了一种变分的配方，并严格证明它保留了流量的两个基本几何结构，即（a）封闭曲线包围的面积的保护； （b）曲线周长的减小。然后，通过使用空间中的分段线性参数有限元元素来开发半差异方案来近似变化公式。借助离散的Cauchy的不平等和离散功率平均不平等，显示了半混凝土方案的区域保护和周长降低性质。在此基础上，通过将向后的Euler方法与单位正常矢量的适当近似结合在一起，成功构建了结构的完全离散方案，该方案可以成功地保留在离散级别上同时保留两个必需的几何结构。最后，数值实验测试了收敛速率，区域保护，周边降低和网状质量，并描绘了曲线的演变。数值结果表明，所提出的SP-PFEM为2D中面积保存的广义平均曲率流提供了可靠且强大的工具。
We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) to simulate the motion of closed curves governed by area-conserved generalized mean curvature flow in two dimensions (2D). We first present a variational formulation and rigorously prove that it preserves two fundamental geometric structures of the flows, i.e., (a) the conservation of the area enclosed by the closed curve; (b) the decrease of the perimeter of the curve. Then the variational formulation is approximated by using piecewise linear parametric finite elements in space to develop the semi-discrete scheme. With the help of the discrete Cauchy's inequality and discrete power mean inequality, the area conservation and perimeter decrease properties of the semi-discrete scheme are shown. On this basis, by combining the backward Euler method in time and a proper approximation of the unit normal vector, a structure-preserving fully discrete scheme is constructed successfully, which can preserve the two essential geometric structures simultaneously at the discrete level. Finally, numerical experiments test the convergence rate, area conservation, perimeter decrease and mesh quality, and depict the evolution of curves. Numerical results indicate that the proposed SP-PFEM provides a reliable and powerful tool for the simulation of area-conserved generalized mean curvature flow in 2D.
