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：正问题和反问题摘要0072556 Vogelius这个项目使用分析和计算技术的混合物来进行建模和无损检测的各种问题的连续介质力学。该研究调查了使用磁和电数据来识别已知介质内的小物体（或缺陷）。定性和定量的行为的解决方案，在腐蚀建模中出现的非线性边值问题也进行了研究。 目标是开发成像技术，允许有效评估（无法访问）腐蚀损坏。电沉积施加电流的优化也在研究之中。反问题的工作包括研究非线性电流密度的可识别性，出现在半线性边界值问题有关的磁流体力学。 工作将继续在复合材料中遇到的（有效的）边界层行为的表征;重点将首先在多边形域与不合理的斜率，但预计开发的技术将最终导致边界层的任意域的更深入的理解。 另一项重要活动将是研究具有非常紧密界面的（层压或纤维增强）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显着提高电和电磁成像技术的有效性，通过纳入到数学算法的信息的行为相关领域的各种缺陷和不均匀性的存在。这种缺陷和不均匀性的例子包括机械部件上的裂纹或管道内的腐蚀点，一直到埋在地里的杀伤人员地雷。 第二个主要研究领域是复合材料的研究，通过强调应力集中和边界层，预计将导致更好地理解微观现象和宏观故障之间的关系。

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.