(untitled) (Advances in Applied Mathematics and Mechanics, 2025)
This paper explores the formation of round and square crystals on curved dynamic substrates, employing the evolving surface phase field model as the mathematical framework and utilizing the evolving surface finite element method as the computational approach. The investigation focuses on the physical process of crystal growth on uneven, dynamic ultra-thin materials and biology thin films. The curved dynamic substrate is represented by the evolving surface. On evolving surfaces, two models, the classical phase field crystal model and the square phase field crystal model, are established and their respective physical properties are examined. Additionally, an efficient numerical simulation technique is devised by integrating the evolving surface finite element method with a time semi-implicit scheme. To validate the models, a series of numerical simulations are conducted to examine the physical properties and dynamical behaviors of the crystals formed under various conditions.