An inverse problem in electromagnetic crack detection

In this work, the inverse problem of predicting the shape and size of a surfacebreaking crack in a nonferrous metal sheet is examined. The crack is interrogated by a uniform stream of alternating current and thin-skin electromagnetic theory is used. The initial data is assumed to be the distribution of the normal component of the magnetic field along the top of the crack. The inverse problem is formulated using the potential for the surface magnetic field on one crack face and its conjugate function as the independent variables. An application of Green's theorem leads to a Fredholm integral equation of the first kind for the shape of the lower edge of the crack. The behaviour of the shape function near the crack tips is determined by examining the complex potential and its inverse function for a circular arc shaped crack. With the use of this information, the integral equation is solved numerically for general initial data, using a minimization technique. In addition, an explicit solution to the equation is found for one particular class of initial data. Finally, the numerical procedure is tested using potential distributions from cracks of known shape.