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

该项目专注于设计和应用高效的计算算法来近似描述各种物理现象的模型的解，包括多相流体流动、组织生长、相变和聚合物加工。这位研究人员将研究适用于以下应用的模型：优化用于能量转换的有机光伏器件；预测癌症生长；模拟涉及内质网等亚细胞结构的复杂生物流动；以及设计耐用的半导体和能量存储材料。作为该项目的中心支柱，他将继续基于研究中开发的高效算法开发他的软件包BSAM。这个软件包，现在和将来都是免费和开源的，旨在解决二维和三维的广泛的非线性、多物理偏微分方程式。这个工具最终对许多学科的研究人员都很有用，并提供了有效模拟复杂现象的能力，而不需要重新发明算法或重新设计代码。主要研究人员将通过关注三个特定的项目目标来研究高阶、高度非线性的偏微分方程式。其中包括高阶能量稳定数值格式的设计和严格的数值分析，有效的二维和三维时空自适应建模和模拟的算法和软件的设计和分析，以及新型的、近乎最佳的复杂的预处理非线性解算器的设计和严格分析。该项目中研究的模型描述了许多物理过程，包括凝固；晶界动力学；裂纹扩展；肿瘤生长；两相聚合物流动；有机光伏处理；以及涉及脂质双层的复杂生物流动。由于所研究的方程是高度非线性的高阶偏微分方程组的耦合系统，因此分析其解并设计高效可靠的数值方法以获得收敛逼近是一项非常重要的任务。主要研究人员将设计无条件能量稳定的时间二阶和三阶近似，目的是严格证明这些格式是最优收敛的。他将实现最优或接近最优的高效求解器，利用所提议的格式的变分/凸性结构，产生复杂的数值软件。

This project focuses on the design and application of efficient computational algorithms for approximating the solutions to models describing various physical phenomena, including multi-phase fluid flow, tissue growth, phase transformations, and polymer processing. The investigator will study models that apply to applications that include optimizing organic photovoltaic devices for energy conversion; predicting cancerous tumor growth; simulating complex biological flows involving sub-cellular structures like endoplasmic reticulum; and designing durable semiconductor and energy storage materials. As a central pillar of this project, he will continue to develop his software package, BSAM, based on the efficient algorithms developed in the research. This software package, which is, and will always be, freely available and open source, is designed to solve a broad spectrum of nonlinear, multi-physics partial differential equations, in two and three dimensions. This tool is ultimately useful for researchers from many disciplines and provides the capability to efficiently simulate complex phenomena, without the need to reinvent algorithms or redesign code. The principal investigator will examine high-order, highly nonlinear partial differential equations through focus on three specific project goals. These include the design and rigorous numerical analysis of high-order energy stable numerical schemes, the design and analysis of algorithms and software for efficient two and three-dimensional time-space adaptive modeling and simulation, and the design and rigorous analysis of novel, nearly-optimally complex preconditioned nonlinear solvers.  The models under examination in the project describe a number of physical processes, including solidification; grain boundary dynamics; crack propagation; tumor growth; two-phase polymer flows; organic photovoltaic processing; and complex biological flows involving lipid bilayers. Because the equations under study are coupled systems of highly nonlinear, high-order partial differential equations, the analysis of their solutions and the design of efficient and reliable numerical methods that give rise to convergent approximations is a non-trivial task. The principal investigator will design unconditionally energy stable, second and third-order-in-time approximations and aims to rigorously prove that the schemes are optimally convergent. He will implement optimally or nearly-optimally efficient solvers that take advantage of the variational/convexity structure of the proposed schemes, producing sophisticated numerical software.