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.

NSF奖摘要-DMS -0072556女性科学：部分微分方程的边界价值问题的分析和计算研究：直接和反向问题abstract0072556 Vogeliusthis Thisthis Project使用分析和计算技术的混合物来实现建模和非源性测试，以解决各种方法的方法。该研究研究了使用磁性和电数据来识别本来已知的介质内的小物体（或缺陷）。还研究了与腐蚀建模有关的非线性边界价值问题解决方案的定性和定量行为。 目的是开发成像技术，以有效评估（无法访问）腐蚀损伤。还研究了施加的电沉积电流的优化。关于反问题的工作包括对与磁性水力动力学有关的半线性边界值问题出现的非线性电流密度的可识别性研究。 在复合材料中遇到的（有效）边界层行为的表征将继续进行工作；重点将首先放在具有非理性斜率的多边形域上，但是预计那里开发的技术最终将导致对任意域的边界层的更深入了解。 另一个重要的活动将是研究具有极其紧密接口的（层压或纤维增强）复合材料（层压或纤维增强）复合材料的定性和定量行为。这项研究的目标之一是通过将有关在存在各种缺陷和不均匀性的情况下的数学算法信息中纳入有关相关场的行为的数学算法信息，从而显着提高电气和电磁成像技术的有效性。此类缺陷和不均匀性的例子范围从机械组件中的裂缝或管道内的腐蚀点，一直到埋在田间的抗人地雷。 研究的第二个主要领域是对复合材料的研究，该研究通过其强调应力浓度和边界层的重视，预计将更好地理解微观现象与宏观失败之间的关系。