相场模型的高精度算法设计及应用

The phase-field models originate from the mathematical description of the interface performance in the multi-phase materials and are widely used to study the interface problems in fluid dynamics, environmental science and mechanics of materials. However, it is still quite a challenging interdisciplinary project to study the application of phase-field models in efficient simulations of the interface evolution in complex physical process. The typical applications include the synthesis of advanced composite material, the environmental change, the complex multi-phase fluid, the abnormal diffusion phenomenon, etc. For the complex multi-phase problems in these fields, we need to further study the current models and algorithms, construct the phase-field models which are consistent with the physical laws and develop the associated highly efficient and high-order numerical schemes. The main goals of the project are three folds: to derive suitable energetic variational phase-field models for the multiphase complex material system that couples the hydrodynamics, microstructure and interfacial dynamics; to develop efficient, easy-to-implement, energy stable numerical schemes to accurately capture the dynamics of interface singularities as well as the microstructures for the derived multiphase complex material systems; and to perform numerical simulations to validate the models and numerical schemes, and to study some physically or biologically motivated problems using the developed predictive tools, consisting of models, numerical schemes and simulations.

相场模型来源于多相材料界面行为的数学描述，被广泛应用于研究流体力学、环境科学和材料力学的界面问题。但如何用相场模型更有效的模拟复杂物理过程中的界面演化过程依然是一个具有高度挑战性的跨学科课题，其典型应用包括：先进复合材料合成、环境变化、多相复杂流体、反常扩散现象研究等。针对这些领域中的复杂多相问题，需要对现有模型和算法进行集成、提炼，构造与物理定律相容的相场模型及其相应的高效高精度数值算法。研究重点是建立合适的能量相场模型来模拟金属合金和其它多相复杂材料系统的微观结构和界面动力学；构造高效、简单易行、能量稳定的数值格式，使其能够准确捕捉多相系统中的界面奇性；通过数值模拟检验所发展的模型和数值格式，用由模型，数值格式和仿真组成的预测工具来研究材料科学的前沿问题。

本项目的主要目的是构造新颖的高精度方法来求解一类具有高度挑战性的问题，包括高维偏微分方程、声波和电磁波散射、非线性薛定谔方程，以及流体力学和材料力学中的多相复杂流体问题。具体内容包括：(1)构造求解高维偏微分方程的高阶、可行的自适应谱方法；(2)对计算流体/材料力学中常用的相场模型，构造稳健、有效的数值方法；(3)对一类具有实际应用和数值求解上有高度挑战性的非线性偏微分方程，构造快速数值方法；(4)利用高阶及能量稳定的数值方法来模拟一些多相复杂流体问题。