Singular Problems in Continuum Mechanics

连续介质力学中的奇异问题

基本信息

批准号：
1615457
负责人：
Anna Mazzucato
金额：
$ 28.52万
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：
2016
资助国家：
美国
起止时间：
2016-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615457&HistoricalAwards=false
关键词：
Singular Problems Continuum Mechanics

项目摘要

The investigator studies several problems in fluid mechanics and mechanics of elastic materials that are characterized by a singular or nearly singular behavior. In the first part of the project, the investigator focuses on understanding and modeling the behavior of complex fluid flows under the requirement of incompressibility, that is, preserving the volume occupied by the fluid. Many fluids are approximately incompressible, including water and under some conditions even air. The complex behavior that is investigated is due to friction between the fluid and rigid walls, and to stirring of the fluid itself. Both are major phenomena in fluid mechanics and are still not fully understood rigorously. Wall friction creates drag and produces swirls in the flow that contribute to the onset of turbulence. Quantifying the effect of stirring and mixing in fluid flows has direct impact on many physical processes, from extrusion and molding, to efficient combustion, to pollutant dispersal. In the second part of the project, the investigator studies the propagation of time-periodic waves in elastic materials. Waves can be used to remotely probe materials by recording their elastic response under an applied disturbance. This is a so-called inverse problem, as material properties are inferred from measurements, and it is highly sensitive to errors. Mathematical algorithms are used to improve the reliability of the reconstruction. The type of inverse problem that is studied finds applications in geological exploration, earthquake prediction, and ground-penetrating radar. The project provides training opportunities for both graduate and undergraduate students.The investigator uses analytical and computational techniques to study problems in incompressible fluid mechanics and in mechanics of deformable solids. The project has two main parts: I. Incompressible Fluid Mechanics: I.a. Vanishing viscosity limit and boundary layer analysis; I.b. Optimal mixing and irregular transport. II. Elasticity: Inverse boundary problem for time-harmonic waves in elastic media. These problems are characterized by the presence of singularities, in the form of singular problems for partial differential equations (PDE), such as the zero-viscosity limit for the Navier-Stokes equations, or in the form of singular coefficients in the PDE, such as in mixing problems under non-Lipschitz flows, or singular domains for the PDE, such as domains with corners. The project addresses some fundamental open questions in fluid mechanics concerning the behavior of incompressible fluids at high Reynolds numbers. It seeks to shed light on the mechanism for boundary layer separation and the validity of Prandtl approximation through the rigorous analysis of special flows. It also seeks to quantify mixing properties of flows under the strong divergence-free constraint from the point of view of transport equations and geometric analysis. In the second part, the project addresses the stability and performance of reconstruction algorithms in inverse boundary problems, which have had a major impact on non-invasive imaging techniques. The appearance of singularities and the interplay between analysis and geometry are recurring themes of the project. A variety of techniques, in many cases combined in a novel way, such as in mixing problems, are used to carry out the work. The project provides training opportunities for both graduate and undergraduate students.

研究者研究了弹性材料的流体力学和力学问题，这些问题的特征是单数或几乎奇异的行为。在项目的第一部分中，研究人员专注于理解和建模在不可压缩性的要求下，即保留流体占据的体积的行为。许多液体大致不可压缩，包括水，甚至在某些条件下甚至空气。所研究的复杂行为是由于流体和刚性壁之间的摩擦以及流体本身搅动。两者都是流体力学中的主要现象，并且仍然不完全理解。壁摩擦会在流动中产生阻力并产生漩涡，从而导致湍流发作。量化流体流中搅拌和混合的效果对从挤出和成型到有效燃烧到污染物分散的许多物理过程有直接影响。在项目的第二部分中，研究人员研究了弹性材料中时间周期波的传播。波可以通过记录其在应用的干扰下记录其弹性响应来远程探测材料。这是一个所谓的反问题，因为从测量值中推断出材料特性，并且对错误高度敏感。数学算法用于提高重建的可靠性。研究的反问题类型在地质探索，地震预测和地面渗透雷达中发现了应用。该项目为研究生和本科生提供了培训机会。该项目有两个主要部分：I。不可压缩的流体力学：I.A。消失的粘度极限和边界层分析； I.B.最佳混合和不规则运输。 ii。弹性：弹性介质中时间谐波波的逆边界问题。这些问题的特征是存在奇异性的特征，以部分微分方程（PDE）的奇异问题形式，例如Navier-Stokes方程的零粘度限制，或以PDE中的奇异系数的形式，例如在非lipschitz Flow下混合问题的奇异系数，或与PDE的非lipschitular domains Corners等界面，例如cornerers cornerers corners，例如cornerers。该项目解决了有关高雷诺数字上不可压缩流体的行为的流体机制中的一些基本开放问题。它试图通过对特殊流的严格分析来阐明边界层分离的机制和prandtl近似的有效性。它还试图从运输方程的角度和几何分析的角度量化强大差异约束下的流量的混合特性。在第二部分中，该项目解决了反向边界问题中重建算法的稳定性和性能，这对非侵入性成像技术产生了重大影响。奇点的出现以及分析与几何形状之间的相互作用是该项目的重复主题。在许多情况下，以一种新颖的方式（例如在混合问题中）进行了多种技术，用于执行工作。该项目为研究生和本科生提供了培训机会。