We propose a threshold-type algorithm to the $L^2$-gradient flow of the Canham-Helfrich functional generalized to $\mathbb{R}^N$. The algorithm to the Willmore flow is derived as a special case in $\mathbb{R}^2$ or $\mathbb{R}^3$. This algorithm is constructed based on an asymptotic expansion of the solution to the initial value problem for a fourth order linear parabolic partial differential equation whose initial data is the indicator function on the compact set $\Omega_0$. The crucial points are to prove that the boundary $\partial\Omega_1$ of the new set $\Omega_1$ generated by our algorithm is included in $O(t)$-neighborhood from $\partial\Omega_0$ for small time $t>0$ and to show that the derivative of the threshold function in the normal direction for $\partial\Omega_0$ is far from zero in the small time interval. Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our threshold-type algorithm.

翻译：我们提出了一种阈值型算法，用于求解推广至$\mathbb{R}^N$的Canham-Helfrich泛函的$L^2$-梯度流。该算法在$\mathbb{R}^2$或$\mathbb{R}^3$中退化成为Willmore流的一种特殊情况。该算法基于一个四阶线性抛物型偏微分方程初值问题解的渐近展开构建，其中初值数据为紧集$\Omega_0$上的指示函数。关键点在于：证明由算法生成的新集合$\Omega_1$的边界$\partial\Omega_1$在微小时间$t>0$下包含于$\partial\Omega_0$的$O(t)$邻域内，并表明在微小时间区间内，$\partial\Omega_0$法线方向上的阈值函数导数远离零。最后，我们利用该阈值型算法提供了受Willmore流控制的平面曲线数值算例。