Phase Field Computational Methods for Materials Science Problems

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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.