In this paper we present a novel algorithm for simulating geometrical flows, and in particular the Willmore flow, with conservation of volume and area. The idea is to adapt the class of diffusion-redistanciation algorithms to the Willmore flow in both two and three dimensions. These algorithms rely on alternating diffusions of the signed distance function to the interface and a redistanciation step, and with careful choice of the applied diffusions, end up moving the zero level-set of the distance function by some geometrical quantity without resorting to any explicit transport equation. The constraints are enforced between the diffusion and redistanciation steps via a simple rescaling method. The energy globally decreases at the end of each global step. The algorithms feature the computational efficiency of thresholding methods without requiring any adaptive remeshing thanks to the use of a signed distance function to describe the interface. This opens their application to dynamic fluid-structure simulations for large and realistic cases. The methodology is validated by computing the equilibrium shapes of two- and three-dimensional vesicles, as well as the Clifford torus.

在本文中，我们提出了一种新颖的算法，用于模拟几何流量，尤其是Willmore流量，并节省了体积和面积。这个想法是使扩散再扩散算法的类别在二维和三维上都适应于Willmore流。这些算法依赖于有符号距离函数到界面的交替扩散以及重新协调步骤，并且通过仔细选择所应用的扩散，最终使距离函数的零级集移动了一定的几何量，而无需借助任何显式传输方程。通过简单的缩放方法，可以在扩散步骤和重新调整步骤之间强制执行约束。在每个全局步骤结束时，全局能量都会减少。由于使用了有符号的距离函数来描述接口，因此该算法的特征在于阈值方法的计算效率，而无需任何自适应的重新网格化。这为大型和现实情况打开了它们在动态流体结构模拟中的应用。通过计算二维和三维囊泡以及Clifford圆环的平衡形状来验证该方法。