Reconstruction of Elliptic Cracks in Three-Dimensional Ultrasonic NDT

Abstract

The paper is devoted to a reconstruction of the thin crack of elliptic configuration in the 3d acoustic medium. The geometry of the crack is defined by five parameters, two direction angles for orientation of the unit normal to its plane and three parameters defining the geometry of the crack in the plane of its disposition. The input data is a measured back-scattered far-field diagram (echo-method) known at a fixed frequency for all directions of irradiation. We firstly formulate a respective direct problem giving the basic 2d integral equation valid on the face of the crack, both for acoustically soft and acoustically hard boundary conditions. In the former case the kernel of the basic integral equation possesses a weak singularity. In the latter case this possesses a hyper-singular singularity. Then for both the cases we develop quadrature formulas appropriate for the elliptic domain, to solve numerically the arising integral equations. Further, there is given a formulation for respective inverse reconstruction problem, which is to define the geometry of the crack from the known measured far-field back-scattered wave pattern of amplitude versus incident angles. This problem is reduced to an optimization problem for the discrepancy functional. In the considered reconstruction of elliptic cracks this functional is in fact a function of the introduced five geometrical parameters. The solution of the optimization problem is attained by a global random search. Then we give some examples of the reconstruction. Finally, we investigate the influence of the error in the recorded input data on the precision of the reconstruction.