Differential Geometry and Analysis for Elastic Rigidity and Flexibility

弹性刚柔度的微分几何与分析

• 批准号: 1813738
• 负责人: Mohammad Reza Pakzad
• 金额: $ 17.21万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1813738&HistoricalAwards=false
• 关键词: Differential; Geometry; Analysis; Elastic; Rigidity

The principal investigator pursues several directions of research in mathematics that are of both analytical and geometrical nature and could have applications in nonlinear elasticity and materials science. The questions under study lie at the interface of differential geometry and analysis. This project aims to establish new connections between these disciplines that will be of interest to mathematicians as well as members of the solid mechanics and materials science community. The work aims to advance materials science, particularly regarding the behavior of pre-strained elastic bodies or growing tissues. It is anticipated that the results will have many potential applications, from constructing shape-forming gels to detecting cancerous tumors. This research project further explores the flexibility vs. rigidity dichotomy, particularly in the context of isometric immersions and solutions to Monge-Ampere equations of low regularity. The rigidity and flexibility of solutions to various classes of partial differential equations in geometry and continuum physics has been a longstanding mathematical problem. Recent developments in fluid dynamics have interested mathematicians and materials scientists in analytical and geometrical aspects of these problems. Meanwhile, a rigorous understanding of nonlinear elastic phenomena (including elastic rigidity and flexibility) requires a deep engagement with mathematical disciplines such as differential geometry and nonlinear analysis. The principal investigator plans to use techniques including convex integration, the theory of nonlinear partial differential equations, geometric function theory, and geometric measure theory to tackle these questions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

主要研究者从事数学研究的几个方向，这些方向具有分析和几何性质，并可能在非线性弹性和材料科学中应用。所研究的问题处于微分几何和分析的交界处。该项目旨在建立这些学科之间的新联系，这将是数学家以及固体力学和材料科学界的成员感兴趣的。这项工作旨在推进材料科学，特别是关于预应变弹性体或生长组织的行为。预计这些结果将有许多潜在的应用，从构建形状形成凝胶到检测癌性肿瘤。这个研究项目进一步探讨了灵活性与刚性二分法，特别是在等距浸入和解决方案的低规律性的蒙格-安培方程的背景下。几何和连续介质物理中各类偏微分方程解的刚性和柔性是一个长期存在的数学问题。流体动力学的最新发展使数学家和材料科学家对这些问题的分析和几何方面感兴趣。同时，对非线性弹性现象（包括弹性刚度和柔性）的严格理解需要深入接触微分几何和非线性分析等数学学科。主要研究者计划使用包括凸积分、非线性偏微分方程理论、几何函数理论和几何测度理论在内的技术来解决这些问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。