二维和三维约束Willmore流的扩散-再定距格式：应用于囊泡的平衡形状
Diffusion-redistanciation schemes for 2D and 3D constrained Willmore flow: application to the equilibrium shapes of vesicles
摘要：
在本文中，我们提出了一种新的算法来模拟几何流，特别是Willmore流，与体积和面积守恒。我们的想法是适应类的扩散redistanciation算法的Willmore流在二维和三维。这些算法依赖于交替扩散的有符号的距离函数的接口和redistanciation步骤，并仔细选择所应用的扩散，最终移动零水平集的距离函数的一些几何量，而不诉诸任何明确的传输方程。通过一个简单的重新缩放方法之间的扩散和redistanciation步骤的约束。能量在每个全局步骤结束时全局减小。该算法的特点是计算效率的阈值方法，而不需要任何自适应重新网格感谢使用一个有符号的距离函数来描述的接口。这打开了他们的应用程序的动态流体结构模拟大型和现实的情况。该方法通过计算二维和三维囊泡的平衡形状以及Clifford环面来验证。
英文摘要：
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.
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