We consider a two-dimensional sharp-interface model for solid-state dewetting of thin films with anisotropic surface energies on curved substrates, where the film/vapor interface and substrate surface are represented by an evolving and a static curve, respectively. The model is governed by the anisotropic surface diffusion for the evolving curve, with appropriate boundary conditions at the contact points where the two curves meet. The continuum model obeys an energy decay law and preserves the enclosed area between the two curves. We introduce an arclength parameterization for the substrate curve, which plays a crucial role in a structure-preserving approximation as it straightens the curved substrate and tracks length changes between contact points. Based on this insight, we introduce a symmetrized weak formulation which leads to an unconditional energy stable parametric approximation in terms of the discrete energy. We also provide an error estimate of the enclosed area, which depends on the substrate profile and can be zero in the case of a flat substrate. Furthermore, we introduce a correction to the discrete normals to enable an exact area preservation for general curved substrates. The resulting nonlinear system is efficiently solved using a hybrid iterative algorithm which combines both Picard and Newton's methods. Numerical results are presented to show the robustness and good properties of the introduced method for simulating solid-state dewetting on various curved substrates.

翻译：我们考虑一种二维尖锐界面模型，用于描述薄膜在曲面基底上的各向异性表面能固态去湿过程，其中薄膜/蒸气界面和基底表面分别由一条演化曲线和一条静态曲线表示。该模型由演化曲线的各向异性表面扩散控制，并在两条曲线相交的接触点处满足适当的边界条件。该连续模型遵循能量衰减定律，并保持两条曲线之间的封闭面积不变。我们为基底曲线引入了弧长参数化，这在结构保持近似中起着关键作用，因为它将弯曲基底展平并追踪接触点之间的长度变化。基于这一见解，我们提出了一种对称弱形式，该形式在离散能量意义下导出了一个无条件能量稳定的参数近似。我们还给出了封闭面积的误差估计，该估计依赖于基底轮廓，在平坦基底情况下误差可为零。此外，我们引入了对离散法向量的修正，以实现对一般曲面基底的精确面积保持。所得非线性系统采用一种结合了Picard法和牛顿法的混合迭代算法高效求解。数值结果展示了所提方法在模拟各种曲面基底上固态去湿过程时的鲁棒性和优良特性。