Computing equilibrium shapes of crystals (ESC) is a challenging problem in materials science that involves minimizing an orientation-dependent (i.e., anisotropic) surface energy functional subject to a prescribed mass constraint. The highly nonlinear and singular anisotropic terms in the problem make it very challenging from both the analytical and numerical aspects. Especially, when the strength of anisotropy is very strong (i.e., strongly anisotropic cases), the ESC will form some singular, sharp corners even if the surface energy function is smooth. Traditional numerical approaches, such as the $H^{-1}$ gradient flow, are unable to produce true sharp corners due to the necessary addition of a high-order regularization term that penalizes sharp corners and rounds them off. In this paper, we propose a new numerical method based on the Davis-Yin splitting (DYS) optimization algorithm to predict the ESC instead of using gradient flow approaches. We discretize the infinite-dimensional phase-field energy functional in the absence of regularization terms and transform it into a finite-dimensional constraint minimization problem. The resulting optimization problem is solved using the DYS method which automatically guarantees the mass-conservation and bound-preserving properties. We also prove the global convergence of the proposed algorithm. These desired properties are numerically observed. In particular, the proposed method can produce real sharp corners with satisfactory accuracy. Finally, we present numerous numerical results to demonstrate that the ESC can be well simulated under different types of anisotropic surface energies, which also confirms the effectiveness and efficiency of the proposed method.

翻译：计算晶体平衡形态（ESC）是材料科学中的一个挑战性问题，涉及在给定质量约束下最小化方向相关（即各向异性）的表面能泛函。该问题高度非线性和奇异性的各向异性项使其在分析与数值两方面都极具挑战性。尤其当各向异性强度很大时（即强各向异性情形），即使表面能函数光滑，ESC也会形成某些奇异尖角。传统数值方法（如$H^{-1}$梯度流）因需加入高阶正则化项以惩罚并圆化尖角，无法产生真正的尖锐角落。本文提出一种基于Davis-Yin分裂（DYS）优化算法的新数值方法来预测ESC，而非采用梯度流方法。我们对无正则化项的无限维相场能量泛函进行离散化，将其转化为有限维约束最小化问题。采用DYS方法求解所得优化问题，该方法自动保证质量守恒与保界性质。我们还证明了所提出算法的全局收敛性。数值实验验证了这些理想性质，尤其所提方法能以令人满意的精度产生真实尖角。最后，我们展示大量数值结果，证明在不同类型的各向异性表面能下均能良好模拟ESC，这进一步证实了所提方法的有效性与高效性。