弹性薄膜表面浸润的建模和计算

弹性材料表面的浸润问题近年来成为交叉学科领域的一个热门研究课题。它与材料断裂、尖点形成和接触线粘滑运动等重要物理现象有密切联系。研究弹性材料的表面浸润理论还可以为一些新材料的设计提供理论指导。本项目针对弹性薄膜的表面浸润问题，利用能量变分原理进行数学建模，分别提出弹性薄膜表面浸润的静力学和动力学模型，并且应用到趋弯(bendotaxis)现象的研究中。我们将从分析和数值两个方面开展对模型的研究：分析方面使用渐进展开分析和均匀化理论结合Onsager变分原理对模型进行极限行为分析；数值方面耦合浸没界面和参数化有限元方法，发展能量稳定的数值格式，实现对接触线运动和流场的高精度计算。我们将利用所建立的模型研究尖点形成和接触线粘滑运动等物理现象，为弹性材料表面浸润问题及其工业应用提供数学理论基础。

Wetting on elastic materials has become a hot topic in the interdisciplinary research. A lot of important physical phenomena, such as fracture, cusp formation, and stick-slip motion of contact line, have been found in wetting process on elastic materials. The development of the related theory also help design and manufacture new materials. This project aims to the study of wetting on elastic membrane. We propose to model this system by variational energetics from both static and dynamic perspectives, and apply the resulting model to study an interesting phenomenon called bendotaxis. The research will be conducted both analytically and numerically: we will study the limiting behaviors of the models by asymptotic analysis, homogenization theory, and Onsager principle; numerically we will develop a hybrid immersed interface - parametric finite element method, which is energy stable and captures the contact line motion and flow field up to high order accuracy. We will also study the cusp formation and the stick-slip motion of the contact line by using the models that we develop, and establish mathematical theory for the potential industrial applications.