Suppose that we have a structural member whose mechanical behaviour can well be modelled as elastic. Also assume that this member is known to contain a crack in its interior, but the geometry of this crack is unknown. To determine this crack we illuminate it by known elastic waves and measure the resulting scattered waves at several observation points as functions of time. Our interest is to determine the location and the shape of the crack from the data thus obtained. This inverse problem is supposed to model engineering practices of ultrasonic flaw detection.
Not much have been investigated concerning inverse problems of this type except in a few numerical works. Nishimura used the Galerkin method as the direct problem solver in 2D and 3D analyses of the problem stated above. If the size of the corresponding direct problem is small, however, Galerkin's method may not be the most effective choice. In view of this, Kobayashi & Nishimura tested the use of collocation in the determination of the shape of a crack which is known to lie on a given plane. We shall see some details of these analyses in the next section. In Oishi et al. these authors used neural networks and FEM to determine a crack in a 2D specimen. In addition to these treatments in time domain, we can mention the work by Tanaka et al. who solved a similar inverse problem in frequency domain (at a fixed frequency) with conventional BEM. Kress also considered a frequency domain crack determination in 2D, where he established a uniqueness result when the boundary condition on the crack is of homogeneous Dirichlet type and the complete far fields for all the plane incident waves are known.
The examples we have so far seen belong to the so called nonlinear inversion which reduces the original inverse problem to a nonlinear parameter estimation problem, which necessarily includes iterative computations to deal with nonlinearity. Some authors try to simplify the analysis by introducing linearisations such as the Born or Kirchhoff approximations. Hirose, for example, used these approximations in a frequency domain analysis of flaw identification to find out that the difference of the Born and Kirchhoff inversion results can be used to tell if a defect is a void or a crack.
Examples of real data inversion in elastodynamic crack determination seem to be scarce. One of the reason for this is that the elastic waves produced by ultrasonic transducers are usually unknown. Recently, Nishimura & Kobayashi tried to estimate the elastic wave forms from transducers by using real data obtained in a velocity measurement using a laser interferometer.