Phase Field Computational Methods for Materials Science Problems

Cahn-Hillard（CH）模型已经被物理和计算科学家以及数学分析家广泛研究。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的研究跨越了纯数值分析，有效的方法实现，以及材料科学和其他领域的应用。我的工作，无论是过去的还是现在提出的，都属于这一范围的中间，得到了两边同事的支持。我提出了三个相关的项目，这将是有价值的更大的社区。性能指标评测问题：在文献中，有一个系统的误解的优点隐式时间推进和能量稳定的格式的缺点。根本的竞争是在亚稳定动力学中的全隐式方案的增加的准确性和能量稳定方案的计算效率之间，能量稳定方案是不准确的，但只有简单的条款（线性，常系数或凸）隐式处理。在NIST发布的CH中有一个基准问题，可以用来量化两种方法的相对性能。我将进一步开发和验证CH的基准问题和一些新的六阶功能化Cahn Hillard（FCH）和相场晶体models.Vector FCH方法：最近有兴趣在矢量FCH类型的模型。这些具有丰富的分叉结构，并导致高度变化的溶液形态。它们可以模拟许多应用中的现象，包括不同活化聚合物材料在溶剂中的相互作用。这些是高度非线性的六阶模型，因此计算困难。解决方案中的空间和时间尺度范围广泛，使标准计算软件包无法实现。全隐式时间步进的准确性对于这些模型是必要的，但是求解所产生的非线性系统的计算复杂性是良好分辨的2D和3D计算的主要障碍。我将探索提高这些解决方案的性能的选项（稀疏，直接解决方案;完全非线性多重网格;使用适当的策略移动到更大规模的计算环境）。数值分析：我一直在与同事的数值分析一类能量稳定的计划CH和相关的模式。我的一个观察是，与合适的自适应时间步进策略，全隐式时间步进不会导致CH计算中的能量增加，只要自适应地选择时间步长以匹配动态的时间尺度。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证明的，对于这些情况，精确的大时间步长是合适的。这将是一个工作与学生，新宇成，共同监督董丽。

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.