An operator-splitting optimization approach for phase-field simulation of equilibrium shapes of crystals
报告专家：张振（南方科技大学）
报告时间：2024年4月17日 下午16:00-17:00
报告地点：数学学院东409
课程介绍：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 anisotropic strength is very large (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 satisfying accuracy.Finally, we present numerous numerical results to demonstrate that the ESC can be well simulated under different anisotropies in surface energies, confirming the effectiveness and efficiency of the proposed method.
专家简介：张振博士现任南方科技大学数学系长聘副教授。他本科毕业于中国科学技术大学数学系，之后进入香港科技大学学习并于2013年获得应用数学博士学位，师从王筱平教授。他于2013年至2015年在新加坡国立大学从事计算数学的博士后研究，并在2015年加入了南方科技大学数学系。他主要从事应用数学方面的研究，目前在M3AS、SISC、JFM、JASA等期刊发表20余篇论文。他曾主持多项省部级基金项目，包括国自然项目、科技部重点专项的课题负责人。
邀请人：贺巧琳
