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该项目着重于在连续力学和相关领域（例如统计力学和概率）中出现的偏微分方程的分析。 理论方面和计算方面均已解决。 其目的是帮助我们促进我们对许多粒子系统中物理现象的理解，并影响现实生活中的应用。 考虑了三个主要的研究领域。（a）不可压缩的流体力学：研究线性流和螺旋对称流量的边界层。（b）弹性：主要研究者继续在多面体结构域中建模弹性，并开发合适的数值方法，特别是通用的有限元方法（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）进化方程的解决方案方法：研究者正在进一步开发Green的抛物线方程的功能方法，特别是Fokker-Planck方程；她还继续使用用于可变速度散射问题的波包解决方案方法的工作，并应用于地震成像。常见的主题，例如研究边界和接口对连续体系统的影响以及使用特定技术（例如缩放和本地化）的使用，使该项目成为凝聚力的研究计划。 尤其是使用精致的分析工具，特别是微局部和谐波分析，这些问题的复杂性是必要的，这些问题的复杂性具有偏微分方程中的非线性，不适合性和不稳定性和不稳定性以及奇异的几何形状。 该项目的目的是通过利用对基本数学模型的严格分析，并设计有效的，准确的计算工具来模拟它们，以促进我们对流体和弹性固体机制中发生的复杂现象的了解。 这些现象中的一些，例如流体中的湍流，是常见的情况，但它们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，可以阐明同一现象的不同方面。 项目的每个部分的进展都有可能影响现实生活中的应用。 在容器壁上的粘性流创造的涡度可以增强流体中的混合和运输与应用程序和环境建模以及项目的工业过程（A部分）的应用。 例如，在血流建模（A部分）时会出现螺旋对称流。 接口问题自然出现在各种应用中，例如确定复合材料的弹性和生物过程的建模（项目的（b）部分）。 弹性波的成像用作非侵入性医学诊断工具，并在探测地球内部，即在地震成像中，用于地震预测（项目的部分（b）和（c））。 Fokker-Planck类型的抛物线方程在应用程序的概率上出现，例如，该应用于等离子体物理和经济学（项目的（c）部分）。 该项目为研究生和本科生提供了培训机会。