Inverse problem methodology and finite elements in the identification of cracks, sources, materials, and their geometry in inaccessible locations

Inverse problem methodology is extended, through the more difficult geometric differentiation of finite-element matrices, to identify the location, material, and value of unknown sources within an inaccessible region using exterior measurements. This is done through the definition of an object function that vanishes at its minimum when the externally measured electric field matches the electric field given by an assumed configuration that is optimized to match measurements. The method is demonstrated by identifying the shape, permittivity, charge, and location of an electrostatic source through exterior measurement. The procedure is then extended to eddy current problems for the identification of the location and shape of cracks in metallic structures. An example demonstrates that when dealing with eddy current problems the least squares object function used by others has multiple local minima so that gradient methods have to be combined with search methods to identify the one absolute minimum. Procedures are also given for handling situations with no cracks and overdescribed cracks