The mean curvature flow describes the evolution of a surface(a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical method for mean curvature flows by using the Onsager principle as an approximation tool. We first show that the mean curvature flow can be derived naturally from the Onsager variational principle. Then we consider a piecewisely linear approximation of the curve and derive a discrete geometric flow. The discrete flow is described by a system of ordinary differential equations for the nodes of the discrete curve. We prove that the discrete system preserve the energy dissipation structure in the framework of the Onsager principle and this implies the energy decreasing property. The ODE system can be solved by an improved Euler scheme and this leads to an efficient fully discrete scheme. We first consider the method for a simple mean curvature flow and then extend it to volume preserving mean curvature flow and also a wetting problem on substrates. Numerical examples show that the method has optimal convergence rate and works well for all the three problems.

翻译：平均曲率流描述了曲面（曲线）以正比于局部平均曲率的法向速度演化的过程，在数学、科学与工程领域具有广泛应用。本文利用昂萨格原理作为近似工具，提出一种求解平均曲率流的数值方法。首先，我们证明平均曲率流可自然地由昂萨格变分原理推导得出。随后，对曲线进行分片线性近似，推导出离散几何流方程。该离散流由描述离散曲线节点的一阶常微分方程组表示。我们证明该离散系统在昂萨格原理框架下保持能量耗散结构，这确保了能量递减性质。该常微分方程组可采用改进欧拉格式求解，从而得到高效的完全离散格式。首先将该方法应用于简单平均曲率流，随后拓展至保体积平均曲率流及基底润湿问题。数值实验表明，该方法具有最优收敛阶，且对三类问题均具有良好的求解效果。