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

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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.