Efficient, Adaptive, and Convergent Numerical Methods for Phase Field and Phase Field Crystal Equations with Applications

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

This research focuses on the design and practical application of efficient computational methods for two classes of mathematical models. The first class is comprised of the phase field crystal equation, the dynamic density functional theory equation, and several related models. Broadly, these equations describe crystal growth and dynamics in various materials science applications, modeling solidification; grain boundary dynamics; crack propagation; and vacancy transport to name a few. The second class is comprised of models that describe two-phase fluid flows. A prototypical two-phase fluid system is oil and vinegar; the constituent fluids do not mix and can form complex multi-phase structures that are impossible to predict without numerical simulation. Specifically, the principal investigator shall develop efficient computational methods for Cahn-Hillard-Navier-Stokes and related equations that model two-phase flows in various technologically important contexts. Such models can be used to describe micro-fluids; tumor growth; two-phase polymer flows used to create organic photovoltaic devices; and flows in porous media such as encountered in oil and natural gas recovery. The physical phenomena describe technologically important phenomena in several scientific areas. Gaining a more thorough physical understanding of the modeled processes through scientific computing is fundamental to designing better cancer treatments; faster, more reliable semiconducting and photovoltaic devices; and more durable polymer components, to mention a few applications. Through this research project, graduate students will be exposed to cutting-edge scientific computing technologies, to traditional techniques of rigorous mathematical analysis, and to applications in real-world problems of materials fabrication. The principal investigator shall conduct some computational and modeling work for Phase Field Crystal-type equations in materials science and for the Cahn-Hillard-Navier-Stokes-type equations for two-phase flows. The equations under study are coupled systems of highly nonlinear, high-order partial differential and integro-partial differential equations. Because of this, designing efficient and reliable numerical methods that give rise to convergent approximations is a non-trivial task. This project will design unconditionally energy stable, 1st and 2nd-order-in-time approximations based on the convex splitting framework. PI will rigorously prove that some of the schemes are optimally convergent. PI will implement optimally or nearly-optimally efficient solvers that take advantage of the variational structure of the proposed schemes. With his materials science collaborators, the PI will develop new PFC-type models to describe complex phase transformations in novel materials for energy applications. The work will present some new ideas for implementing and rigorously analyzing efficient and stable approximation schemes for the equations. The overarching objectives of the project are to design stable and convergent schemes for the proposed models, to implement efficient solvers for the schemes, and to apply these to real-world phenomena in materials and fluids science. The PI will leverage his experience designing fast adaptive multigrid solvers, using finite element and finite difference methods in space, and energy stable convex-splitting methods to discretize time. The codes that the PI develops, including improved versions of the PI's BSAM software package and new planned adaptive finite element packages, will be made publicly available through the PI's website, to the benefit of the wider scientific community.