Variational Methods for Materials Science and Mechanics

这个项目的中心主题是现代应用数学在工程、材料科学和力学中的应用。主要的科学目标之一是从数学上解决锂离子电池充放电动力学中遇到的许多问题。了解这些现象对于克服可充电锂离子电池的缺点和开发新材料是至关重要的。其他关注的主题领域是固体和流体中的相变以及材料中的缺陷。调查员通过几项将研究与教育相结合的活动，包括专题课程、暑期学校、工作组、课程设计讲习班和职业发展讲习班，继续培训本科生和研究生以及博士后研究员。研究生和本科生参与了该项目的工作。该项目的主要科学进展是：1)锂离子电池的变分模型。高性能的正负极材料在锂化和脱氢过程中会发生相变。这里的目标是专注于足够丰富的变分模型，以捕捉在锂离子电池充放电动力学中遇到的许多问题。这位研究者和他的合作者所得到的能量估计最近被用来给出高维的保质量Allen-Cahn方程和Cahn-Hilliard方程的解的界面的第一个慢动作估计，而不需要对初始数据和区域形式的结构假设。目标是将这些结果推广到局部极小化和其他泛函。3)自由边界问题。这里的目标是专注于自由边界问题的变分模型，其解表现出奇异性。研究生和本科生参与该项目的工作。

1714088Leoni The central theme of this project is the application of modern applied mathematics to engineering, materials science, and mechanics. One of the main scientific goals is to address mathematically many of the issues encountered in the charge-discharge dynamics of Lithium-Ion batteries. The understanding of the these phenomena is paramount to overcome the shortcomings of rechargeable Lithium-Ion batteries and to develop novel materials. Other thematic areas of focus are phase transitions in solids and fluids and defects in materials. The investigator continues the training of undergraduate and graduate students and postdoctoral fellows through several activities that integrate research and education, including topics courses, summer schools, working groups, course design workshops, and career-building workshops. Graduate and undergraduate students participate in the work of the project. The main scientific thrusts of the project are:1) Variational Models for Lithium-Ion Batteries. High performance cathode and anode materials undergo phase transformations upon lithiation and delithiation. The goal here is to focus on variational models that are sufficiently rich to capture many of the issues encountered in the charge-discharge dynamics of Lithium-Ion batteries.2) Slow motion of interfaces. Sharp energy estimates obtained by the investigator and his collaborators have recently been used to give the first slow motion estimates of the interface of solutions of the mass-preserving Allen-Cahn equations and of the Cahn-Hilliard equations in higher dimensions and without structural assumptions on the initial data and on the form of the domain. The goal is to extend these results to local minimizers and to other functionals.3) Free boundary problems. The goal here is to focus on variational models for free boundary problems whose solutions exhibit singularities.Graduate and undergraduate students participate in the work of the project.