We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points/lines, where three interfaces meet, and at the boundary points/lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.
翻译:我们从两个方面(2d)和三个方面(3d)来考虑曲线网络的演变情况。界面的动向通过地表扩散来描述,在三个界面相交的三重交叉点/线和边界点/线的边界条件来描述,在三个界面相交的三重交叉点/线和边界点/线上,一个界面与固定的平面边界相交;我们提出基于适当变式配方的参数限定要素方法;所构建的方法是半模糊的,能够满足每个封闭的泡泡和无条件的能量稳定性的体积保护,从而保持流动的两种基本几何结构。此外,该方法在网点的分布方面具有非常良好的特性,因此不需要网状平滑或正规化技术;还考虑对厌异地表能和非中性外部边界的情况采用一般办法;在异地和异地表能量和异地表表面能量的情况下,对二维曲线网络和三维表组的演变提出了数值结果。