基于阈值动力学方法的复杂自由界面问题的建模与模拟

本项目在理论、算法及应用的角度进一步发展阈值动力学方法，基于区域特征函数这一锐利界面的表示方式，从基于点云数据结构的表面重建这一算法本质为自由界面非均匀运动的问题出发，推导基于区域特征函数的非均匀的能量泛函近似方式并发展基于阈值动力学方法的表面重建算法。基于此，本项目将推广阈值动力学方法至一系列复杂自由界面运动问题的建模与模拟，发展一系列各向异性平均曲率流、曲率表面扩散流、Willmore流等问题的高效数值算法，并应用于固态去湿及含表面活性剂的液体浸润问题等。理论上，本项目将对新的近似能量泛函的收敛性，算法的收敛性、稳定性及精度，以及在时间步长趋于零时的界面动力学的收敛做严格分析。本项目还将深度结合阈值动力学框架与深度神经网络，开发数据驱动且可解释的深度学习方法，实现实时表面重建及学习与预测界面自由界面运动等任务。

The proposal aims at the development in the theory, algorithms and applications of the threshold dynamics method. Based on the sharp interface representation using characteristic functions, this proposal starts with surface reconstruction from point cloud which can be viewed as an inhomogeneous free interface problem, derives a new approximation to the inhomogeneous energy functional, and develop a new threshold dynamics type method. Based on this, this proposal will generalize the threshold dynamics method to modeling and simulation of a class of free interface problems to develop efficient numerical method for anisotropic mean curvature flow, surface diffusion of curvature flow, Willmore flow and so on. These methods will be applied into solid de-wetting problems, wetting problems with surfactant and so on. Theoretically, this proposal will rigorously prove the convergence of the approximate energy functional and the convergence, stability and accuracy of the numerical methods and use multi-scale analysis to study the motion law of interface as time step goes to zero. In addition, this proposal will combine the framework of threshold dynamics methods and deep neural network to develop data driven and explainable deep learning method to instantly reconstruct surface and predict interface motions.