Phase Field Computational Methods for Materials Science Problems

Cahn-Hillard (CH) 模型已被物理和计算科学家以及数学分析师广泛研究。它是相分离和扩散混合之间竞争的通用材料科学模型。计算 CH 的研究涵盖了纯数值分析、高效方法实施以及材料科学和其他领域的应用。我的工作，无论是过去的还是提议的，都处于这个范围的中间，并得到了两边同事的支持。我提出了三个对更大社区有价值的相关项目。基准问题：在文献中，对隐式时间步进的优点和能量稳定方案的缺点存在系统性的误解。根本的竞争在于亚稳定动力学中完全隐式方案的准确性提高和能量稳定方案的计算效率之间的竞争，这些方案不准确，但仅隐式处理简单的项（线性、常系数或凸）。 NIST 发布了 CH 的基准问题，可用于量化两种方法的相对性能。我将开发和验证 CH 的进一步基准问题以及六阶功能化 Cahn Hillard (FCH) 和相场晶体模型的一些新颖问题。矢量 FCH 方法：最近人们对矢量 FCH 类型的模型感兴趣。它们具有丰富的分叉结构并导致溶液形态的高度变化。它们可以模拟许多应用中的现象，包括溶剂中不同活化聚合物材料的相互作用。这些是高度非线性的六阶模型，因此计算起来很困难。解决方案中广泛的空间和时间尺度使得它们超出了标准计算包的范围。完全隐式时间步进的精度对于这些模型来说是必要的，但求解所得非线性系统的计算复杂性是良好解析 2D 和 3D 计算的主要障碍。我将探索提高这些求解性能的选项（稀疏直接解；完全非线性多重网格；使用适当的策略转移到更大规模的计算环境）。数值分析：我一直在与同事合作对 CH 和相关模型的一类能量稳定方案进行数值分析。我的观察之一是，使用合适的自适应时间步长策略，只要自适应地选择时间步长以匹配动态的时间尺度，完全隐式时间步长不会导致 CH 计算的能量增加。我将在提案中详细介绍的一些渐近结果表明，这在亚稳态动力学的某些情况下是可以证明的，对于这种情况，准确的大时间步长是合适的。这将是与学生程新宇的合作，并由李东共同指导。

The Cahn-Hillard (CH) model has been extensively studied by physical and computational scientists as well as mathematical analysts. It is a generic material science model of the competition between phase separation and diffusive mixing. Research on computational CH spans the spectrum of pure numerical analysis, efficient method implementation, and applications to materials science and other fields. My work, past and proposed, is in the middle of this spectrum, supported by colleagues on either side. I propose three related projects that would be of value to the larger community. Benchmark Problems: In the literature, there is a systemic misunderstanding of the advantages of implicit time stepping and the disadvantages of energy stable schemes. The fundamental competition is between the increased accuracy of fully implicit schemes in meta-stable dynamics and the computational efficiency of energy stable schemes, which are inaccurate but have only simple terms (linear, constant coefficient - or convex) handled implicitly. There is a benchmark problem for CH posted at NIST that can be used to quantify the relative performance of the two approaches. I will develop and validate further benchmark problems for CH and some novel ones for sixth order Functionalized Cahn Hillard (FCH) and phase field crystal models.Vector FCH method: There is recent interest in models of vector FCH type. These have a rich bifurcation structure and lead to highly varied solution morphology. They can model phenomena in many applications, including the interaction of different activated polymer materials in a solvent. These are highly nonlinear, sixth order models, hence computationally difficult. The wide range of spatial and temporal scales in the solutions put them out of reach for standard computational packages. The accuracy of fully implicit time stepping is necessary for these models, but it is the computational complexity of solving the resulting nonlinear system that is the main obstacle to well-resolved 2D and 3D computations. I will explore options to improve the performance of these solves (sparse, direct solution; fully nonlinear multi-grid; moving to a larger scale computational environment with an appropriate strategy). Numerical Analysis: I have been working with colleagues on the numerical analysis of a class of energy stable schemes for CH and related models. One my observations is that with suitable adaptive time stepping strategies, fully implicit time stepping does not lead to an increase in energy in CH computations provided time steps are chosen adaptively to match the time scale of the dynamics. Some asymptotic results, which I will detail in the proposal, suggest that this is provable in certain cases of metastable dynamics, for which accurate, large time steps are appropriate. This will be work with a student, Xinyu Cheng, jointly supervised by Dong Li.