An alternating iterative algorithm for the reconstruction of internal cracks in athree-dimensional solid body

Abstract

This paper deals with the inverse problem of determining cracks lying on an apriori known internal surface inside a three-dimensional elastic body fromoverdetermined elastostatic boundary data on the outer surface. A newalternating iterative algorithm for the reconstruction of such internal cracks isproposed with two variants: for the first variant the complete Cauchy data of onemeasurement have to be given on the whole outer boundary, whereas the othervariant only needs overdetermined boundary data on some part of the outerboundary. With this approach, the case of multiple and generally shaped crackslying on a curved internal surface inside an anisotropic material can also easily betreated. Numerical examples are given for the case of a planar elliptic crack in an isotropicelastic body, demonstrating the fast convergence and good regularizingproperties of the proposed algorithm. Due to the general character ofthe algorithm, it can also easily be extended to other elliptic operators.