Simultaneous reconstruction of shape and impedance in corrosion detection

Corrosion detection can be modelled by the Laplace equation for an electric or a heat potential in a simply connected planar domain D with a homogeneous impedance boundary condition on a non-accessible part of the boundary ∂D. We consider the inverse problem to simultaneously recover the non-accessible part of the boundary and the impedance function from two pairs of Cauchy data on the accessible part of the boundary. Our approach extends the method proposed by Kress and Rundell [16] for the corresponding problem to recover the interior boundary curve of a doubly connected planar domain and is based on our previous work on reconstruction of the impedance function for a known shape or the shape for a known impedance function [4, 5]. Based either on a potential approach or on a Green’s integral formulation the inverse problem is equivalent to a system of nonlinear and ill-posed integral equations that can be solved iteratively by linearization. We will present the mathematical foundation of the method and, in particular, establish injectivity for the linearized system at the exact solution. Numerical reconstructions will show the feasibility of the method.