Phase Field Computational Methods for Materials Science Problems

卡恩-希拉德(CH)模型已被物理和计算科学家以及数学分析师广泛研究。这是一个关于相分离和扩散混合之间竞争的通用材料科学模型。计算CH的研究涵盖纯数值分析、高效方法实现以及材料科学和其他领域的应用。我的工作，无论是过去的还是提议的，都处于这一范围的中间，得到了双方同事的支持。我提出了三个对更大的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。基本的竞争是亚稳定动力学中全隐式格式提高的精度和能量稳定格式的计算效率之间的竞争，能量稳定格式不准确，但只隐式处理简单的项(线性、常系数或凸项)。NIST发布了一个针对CH的基准问题，可以用来量化这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，以及六阶泛函Cahn Hillard(FCH)和相场晶体模型的一些新的基准问题。矢量FCH方法：最近对矢量FCH类型的模型感兴趣。它们具有丰富的分叉结构，导致溶液形态变化很大。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，因此在计算上很困难。解决方案中广泛的空间和时间尺度使它们超出了标准计算程序包的范围。对于这些模型来说，全隐式时间步长的精度是必要的，但求解所产生的非线性系统的计算复杂性是良好地进行2D和3D计算的主要障碍。我将探索提高这些解的性能的选项(稀疏的、直接的解；完全非线性的多重网格；使用适当的策略转移到更大规模的计算环境)。数值分析：我一直在与同事们合作，对一类能量稳定的CH格式和相关模式进行数值分析。我的一个观察是，在适当的自适应时间步长策略下，完全隐式时间步长不会导致CH计算中能量的增加，只要时间步长被自适应地选择以匹配动力学的时间尺度。一些渐近的结果，我将在提案中详述，表明这在亚稳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。这将是与董力共同指导的学生程新宇的工作。

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.