Analytical and Computational Studies of Boundary Value Problems for PDE's. Direct and Inverse Problems

偏微分方程边值问题的分析和计算研究。

• 批准号: 0072556
• 负责人: Michael Vogelius
• 金额: $ 14.11万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072556&HistoricalAwards=false
• 关键词: Analytical; Computational; Studies; Boundary; Value

NSF Award Abstract - DMS-0072556Mathematical Sciences: Analytical and Computational Studies of Boundary Value Problems for Partial Differential Equations: Direct and Inverse ProblemsAbstract0072556 VogeliusThis project uses a mixture of analytical and computational techniques to carry out modeling and nondestructive inspection for various problems of continuum mechanics. The research investigates the use of magnetic as well as electric data to identify small objects (or defects) inside an otherwise known medium. The qualitative and quantitative behavior of solutions to nonlinear boundary value problems that arise in connection with corrosion modeling is also investigated. The goal is to develop imaging techniques that permit effective assessment of (inaccessible) corrosion damage. Optimization of the imposed currents for electrodeposition is also under study. The work on inverse problems includes a study of the identifiability of nonlinear current densities that appear in semilinear boundary value problems related to magnetohydrodynamics. Work will continue on characterization of the (effective) boundary layer behavior encountered in composite materials; the focus will first be on polygonal domains with irrational slopes, but it is expected that the techniques developed there will ultimately lead to a deeper understanding of boundary layers for arbitrary domains. Another important activity will be the study of the qualitative and quantitative behavior of stresses in (laminated or fiber-reinforced) composites with extremely close interfaces. One goal of this research is to significantly increase the effectiveness of electric and electromagnetic imaging techniques by incorporating into the mathematical algorithms information about the behavior of the associated fields in the presence of various defects and inhomogeneities. Examples of such defects and inhomogeneities range from cracks in a mechanical component, or corrosion spots inside a pipe, all the way to anti-personnel mines buried in a field. The second main area of research is study of composite materials, which, through its emphasis on stress concentrations and boundary layers, is expected to lead to a better understanding of the relationship between microscopic phenomena and macroscopic failures.

数学科学：偏微分方程边值问题的解析与计算研究：正问题与反问题[Abstract] 0072556 vogelius本项目采用分析与计算相结合的方法，对连续介质力学的各种问题进行建模和无损检测。这项研究调查了利用磁和电数据来识别已知介质中的小物体（或缺陷）。本文还研究了与腐蚀建模有关的非线性边值问题解的定性和定量行为。目标是开发成像技术，以有效评估（难以接近的）腐蚀损伤。电沉积施加电流的优化也在研究中。反问题的工作包括研究出现在与磁流体力学有关的半线性边值问题中的非线性电流密度的可辨识性。在复合材料中遇到的（有效）边界层行为的表征工作将继续进行；重点将首先放在具有非理性斜率的多边形域上，但预计在那里开发的技术最终将导致对任意域的边界层有更深入的理解。另一项重要活动将是研究具有极紧密界面的（层压或纤维增强）复合材料的定性和定量应力行为。本研究的一个目标是通过将有关存在各种缺陷和不均匀性的相关场的行为的信息纳入数学算法来显着提高电和电磁成像技术的有效性。这种缺陷和不均匀性的例子从机械部件的裂缝，或管道内的腐蚀点，一直到埋在野外的杀伤人员地雷。第二个主要研究领域是复合材料的研究，它通过强调应力集中和边界层，有望更好地理解微观现象和宏观失效之间的关系。