Mathematical problems from materials science

材料科学中的数学问题

• 批准号: 1311833
• 负责人: Robert Kohn
• 金额: $ 112.16万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-10-01 至 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311833&HistoricalAwards=false
• 关键词: Mathematical; problems; materials; science

Kohn1311833 This project has three main scientific thrusts: (a) The first and broadest thrust concerns elastic-energy-driven pattern formation in thin elastic sheets. The investigator studies stress-driven patterns involving wrinkles, folds, delamination, and other defects, with particular emphasis on situations where the energy-minimizing pattern develops fine-scale structure as the sheet thickness tends to zero. His approach is to focus on how the minimum energy scales with respect to the sheet thickness and other relevant physical parameters. (b) The second thrust concerns surface-energy-driven coarsening of two-phase mixtures. The investigator studies a family of nonlocal evolutions from the physics literature, which generalize the relatively well-understood model of "Cahn-Hilliard dynamics." The focus here is on understanding the large-time coarsening rate. (c) The third thrust concerns prediction with expert advice (a topic from the machine learning literature). The investigator's goal is a fresh perspective on some regret-minimization-based algorithms for prediction. He views regret minimization as a robust control problem and considers a suitable scaling limit in which the associated value function solves a differential equation. The investigator studies three interdisciplinary topics. The first two lie at the interface where mathematics meets physics and materials science, while the third lies at the interface with machine learning. In each area, challenges from applications drive the development of new mathematical methods. For example, the work on thin elastic sheets is helping develop a theory of energy-minimizing patterns, in much the same way that consideration of soap bubbles and soap films led to the theory of minimal surfaces a generation ago. It is of course a familiar fact that thin sheets often wrinkle or fold: our skin wrinkles and our clothes wrinkle; leaves, flowers, and hanging drapes have folds. Physical experiments in controlled settings can quantify such phenomena, and numerical simulations can demonstrate within a model how the patterns develop. But neither experiment nor simulation can tell us "why" a system chooses a particular pattern. The project provides a valuable complement to other methods, by showing that elastic energy minimization requires types of patterns. The project provides training opportunities for graduate students.

该项目有三个主要的科学推力：(a)第一个也是最广泛的推力涉及薄弹性薄片中弹性能驱动的图案形成。研究者研究了包括皱纹、褶皱、分层和其他缺陷在内的应力驱动模式，特别强调了当薄片厚度趋于零时，能量最小化模式发展为精细结构的情况。他的方法是关注最小能量与薄片厚度和其他相关物理参数的关系。(b)第二个推力涉及表面能驱动的两相混合物的粗化。研究者从物理学文献中研究了一个非局部进化家族，它推广了相对容易理解的“卡恩-希利亚德动力学”模型。这里的重点是理解大时间粗化率。(c)第三个推力涉及专家建议的预测（来自机器学习文献的主题）。研究者的目标是对一些基于遗憾最小化的预测算法的新视角。他将遗憾最小化视为一个鲁棒控制问题，并考虑了一个合适的缩放极限，在该极限下，相关的值函数求解微分方程。研究者研究了三个跨学科的课题。前两个位于数学与物理和材料科学的交汇处，而第三个位于与机器学习的交汇处。在每个领域，来自应用的挑战推动了新的数学方法的发展。例如，对薄弹性薄片的研究有助于发展能量最小化模式的理论，就像上一代对肥皂泡和肥皂膜的研究导致了最小表面理论一样。薄床单经常起皱，这当然是一个熟悉的事实：我们的皮肤会起皱，我们的衣服会起皱；树叶、花朵和悬挂的窗帘都有褶皱。在受控环境下的物理实验可以量化这些现象，而数值模拟可以在一个模型中证明这些模式是如何发展的。但是无论是实验还是模拟都不能告诉我们一个系统“为什么”会选择一个特定的模式。该项目通过显示弹性能量最小化需要各种类型的模式，为其他方法提供了有价值的补充。该项目为研究生提供了培训机会。