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

NSF奖摘要-DMS-0072556数学科学：偏微分方程边值问题的分析和计算研究：正问题和逆问题Vogelius这个项目使用分析和计算技术的混合来对连续介质力学的各种问题进行建模和无损检测。这项研究调查了使用磁和电数据来识别其他已知介质中的小物体(或缺陷)。文中还研究了与腐蚀模拟有关的非线性边值问题解的定性和定量行为。其目标是开发能够有效评估(难以接近的)腐蚀损害的成像技术。电沉积外加电流的优化也在研究中。反问题的工作包括研究与磁流体力学有关的半线性边值问题中出现的非线性电流密度的可辨识性。将继续对复合材料中遇到的(有效)边界层行为进行表征；重点将首先放在具有无理斜率的多边形域上，但预计那里开发的技术最终将导致对任意域边界层的更深入理解。另一项重要的活动将是研究具有非常紧密界面的(叠层或纤维增强的)复合材料中应力的定性和定量行为。这项研究的一个目标是通过将有关存在各种缺陷和不均匀的关联场的行为的信息合并到数学算法中来显著提高电和电磁成像技术的有效性。这种缺陷和不均匀的例子从机械部件的裂缝或管道内的腐蚀点，一直到埋在田野中的杀伤人员地雷。第二个主要研究领域是复合材料研究，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破坏之间的关系。

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.