Analysis and computation of partial differential equations in Mechanics and related fields

力学及相关领域偏微分方程的分析与计算

• 批准号: 1312727
• 负责人: Anna Mazzucato
• 金额: $ 23.98万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312727&HistoricalAwards=false
• 关键词: Analysis; computation; partial; differential; equations

Mazzucato1312727 This project focuses on the analysis of partial differential equations arising in continuum mechanics and related fields, such as statistical mechanics and probability. Both theoretical and computational aspects are addressed. Its aim is to help advance our understanding of physical phenomena in many-particle systems and impact real-life applications. Three main areas of investigation are considered.(a) Incompressible fluid mechanics: boundary layers for linearized flows and helically-symmetric flows are studied.(b) Elasticity: the principal Investigator continues to model elasticity in polyhedral domains and to develop suitable numerical methods, in particular the Generalized Finite Element Method (GFEM); she is investigating how to obtain size estimates of inclusions from boundary measurements.(c) Solution methods for evolution equations: the investigator is further developing Green's function methods for parabolic equations, in particular Fokker-Planck equations; she also continues her work on a wave-packet solution method for variable-speed scattering problems, with applications to seismic imaging.Common themes, such as the investigation of the effect of boundaries and interfaces on continuum systems, and the use of specific techniques, such as scaling and localization, make the project a cohesive research program. Employing refined analytical tools, microlocal and harmonic analysis in particular, is warranted by the complexity of the problems studied, which feature nonlinearities in the partial differential equations, ill-posedness and instabilities, and singular geometries. The aim of this project is to advance our knowledge of complex phenomena occurring in the mechanics of fluids and elastic solids, by utilizing a rigorous analysis of the underlying mathematical models and by devising efficient, yet accurate, computational tools to simulate them. Some of these phenomena, such as turbulence in fluids, are a common occurrence, yet they still lack a thorough understanding. A relevant trait of the project is the interplay between theoretical and computational methods, with each providing its own avenue for investigation that can shed light on different aspects of the same phenomenon. Progress on each part of the project has the potential to impact real-life applications. Vorticity created by viscous flows at container walls enhances mixing and transport in fluids with applications for example to climate and environmental modeling, and industrial processes (part a) of the project). Helically-symmetric flows arise, for instance, in modeling of blood flow (part a)). Problems with interfaces appear naturally in a variety of applications, such as determining the elastic properties of composite materials and modeling of biological processes (part(b) of the project). Imaging by elastic waves is used as a non-invasive medical diagnostic tool and in probing the earth's interior, that is, in seismic imaging, for earthquake prediction (parts (b) and (c) of the project). Parabolic equations of the Fokker-Planck type arise in probability with applications, for example, to plasma physics and economics (part (c) of the project). The project provides training opportunities for both graduate and undergraduate students.

Mazzucato1312727 该项目重点分析连续介质力学和相关领域（例如统计力学和概率）中出现的偏微分方程。 理论和计算方面都得到了解决。 其目的是帮助增进我们对多粒子系统中物理现象的理解并影响现实生活中的应用。 考虑三个主要研究领域。(a) 不可压缩流体力学：研究线性流和螺旋对称流的边界层。(b) 弹性：主要研究者继续对多面体域中的弹性进行建模，并开发合适的数值方法，特别是广义有限元法 (GFEM)；她正在研究如何从边界测量中获得夹杂物的尺寸估计。(c) 演化方程的求解方法：研究人员正在进一步开发抛物线方程的格林函数方法，特别是福克-普朗克方程；她还继续研究变速散射问题的波包求解方法，并将其应用于地震成像。共同的主题，例如边界和界面对连续介质系统的影响的研究，以及特定技术的使用，例如缩放和定位，使该项目成为一个有凝聚力的研究项目。 由于所研究问题的复杂性，需要采用精细的分析工具，特别是微局域分析和调和分析，这些问题的特点是偏微分方程中的非线性、不适定性和不稳定性以及奇异几何。 该项目的目的是通过对基础数学模型的严格分析并设计高效而准确的计算工具来模拟它们，从而增进我们对流体和弹性固体力学中发生的复杂现象的了解。 其中一些现象，例如流体中的湍流，是常见的现象，但仍然缺乏透彻的理解。 该项目的一个相关特征是理论方法和计算方法之间的相互作用，每种方法都提供了自己的研究途径，可以揭示同一现象的不同方面。 该项目每个部分的进展都有可能影响现实生活中的应用。 容器壁上的粘性流产生的涡流增强了流体的混合和传输，适用于气候和环境建模以及工业过程（项目的 a 部分）等应用。 例如，在血流建模中会出现螺旋对称流（第 a 部分）。 界面问题自然出现在各种应用中，例如确定复合材料的弹性特性和生物过程建模（项目的（b）部分）。 弹性波成像被用作非侵入性医疗诊断工具，并用于探测地球内部，即地震成像，用于地震预测（项目的(b)和(c)部分）。 福克-普朗克类型的抛物线方程以概率的形式出现，例如应用于等离子体物理学和经济学（该项目的（c）部分）。 该项目为研究生和本科生提供培训机会。