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

相场和相场晶体方程的高效、自适应和收敛数值方法及其应用

• 批准号: 1418692
• 负责人: Steven Wise
• 金额: $ 10.5万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418692&HistoricalAwards=false
• 关键词: Efficient; Adaptive; Convergent; Numerical; Methods

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.

本研究主要针对两类数学模型，探讨其有效计算方法的设计与实际应用。第一类包括相场晶体方程、动力学密度泛函理论方程和几个相关的模型。概括地说，这些方程描述了各种材料科学应用中的晶体生长和动力学，建模凝固;晶界动力学;裂纹扩展;和空位传输等等。 第二类由描述两相流体流动的模型组成。 典型的两相流体系统是油和醋;组成流体不混合，可以形成复杂的多相结构，如果没有数值模拟，这些结构是不可能预测的。具体而言，主要研究者应开发有效的计算方法，用于Cahn-Hillard-Navier-Stokes方程和相关方程，用于在各种重要的技术背景下模拟两相流。这种模型可用于描述微流体;肿瘤生长;用于创建有机光伏器件的两相聚合物流;以及在石油和天然气开采中遇到的多孔介质中的流动。物理现象描述了几个科学领域中的重要技术现象。通过科学计算对建模过程获得更透彻的物理理解是设计更好的癌症治疗方法的基础;更快，更可靠的半导体和光伏器件;以及更耐用的聚合物组件，仅举几例应用。通过这个研究项目，研究生将接触到尖端的科学计算技术，严格的数学分析的传统技术，并在材料制造的实际问题的应用。主要研究者应进行材料科学中的相场晶体型方程和两相流的Cahn-Hillard-Navier-Stokes型方程的计算和建模工作。 所研究的方程是高度非线性、高阶偏微分和积分-偏微分方程的耦合系统。 正因为如此，设计有效和可靠的数值方法，产生收敛近似是一个不平凡的任务。该项目将基于凸分裂框架设计无条件能量稳定的一阶和二阶时间近似。PI将严格证明某些方案是最优收敛的。PI将实现最优或接近最优的有效求解器，利用所提出的方案的变分结构。PI将与他的材料科学合作者一起开发新的PFC型模型，以描述用于能源应用的新型材料中的复杂相变。这项工作将提出一些新的想法，实施和严格分析有效和稳定的近似计划的方程。该项目的总体目标是为拟议的模型设计稳定和收敛的方案，为这些方案实施有效的求解器，并将其应用于材料和流体科学中的真实现象。PI将利用他的经验设计快速自适应多重网格求解器，在空间中使用有限元和有限差分方法，以及能量稳定的凸分裂方法来离散时间。PI开发的代码，包括PI的BSAM软件包的改进版本和新计划的自适应有限元软件包，将通过PI的网站公开提供，以使更广泛的科学界受益。