We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy $\gamma(\theta)$ -- anisotropic surface diffusion -- in two dimensions, while $\theta$ is the angle between the outward unit normal vector and the vertical axis. By introducing a positive definite surface energy (density) matrix $G(\theta)$, we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized in space by the parametric finite element method and area/mass conservation and energy dissipation are established for the semi-discretization. Then the problem is further discretized in time by a (semi-implicit) backward Euler method so that only a linear system is to be solved at each time step for the full-discretization and thus it is efficient. We establish well-posedness of the full-discretization and identify some simple conditions on $\gamma(\theta)$ such that the full-discretization keeps energy dissipation and thus it is unconditionally energy-stable. Finally the ES-PFEM is applied to simulate solid-state dewetting of thin films with anisotropic surface energies, i.e. the motion of an open curve under anisotropic surface diffusion with proper boundary conditions at the two triple points moving along the horizontal substrate. Numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed ES-PFEM.
翻译:我们建议一种能感度参数定值元素法(ES-PFEM),将表面扩散下的封闭曲线运动分解为两个维度,以厌异度表面能量 $\gamma(theta) $(theta) $(theta) 美元 -- -- 厌异度表面扩散 -- -- 厌异度表面能量 美元 -- -- 反异度表层扩散 -- -- 以两个维度两个维度,而$(theta) 美元是外向单元正常矢量矢量矢量和垂直轴之间的角。通过引入一种积极的表面能量(密度) 矩阵(g(theta) 美元) 后向 Euler 法进一步分解问题,因此在全异度表面扩散的每次步骤中只能解决一个线性系统,从而证明它满足了面积/质量的节能保护和能量消散。我们通过偏差元素的偏差元素偏移法,在全度表面的轨道流度流度流度和精确度轨道上建立精确的流化状态,从而显示完全的静态的能量递化结果。