The setting of the problem is almost the same as in 3.1, except that the crack is planar and the plane on which S lies (we take the coordinate system so that this plane is given by ) is assumed to be known, as in the cases of interfacial cracks. The inverse problem determines other geometrical information of the crack from the far fields produced by known incident waves.
As always we shall start with the direct problem in which S is known. We are interested in determining a displacement field which satisfies (19) -- (23). The solution to this problem is given by (24). The boundary condition on S yields a boundary integral equation (25) which determines the crack opening displacement. One solves this equation numerically with collocation using piecewise constant spatial shape functions and piecewise linear time shape functions. With this choice one can compute all the relevant integrals analytically when the crack is planar. As is obtained the far field is computed with (28) and .
We next consider the inverse problem, which we present in a discretised form. Suppose that S is illuminated with N known incident elastic waves and the far fields are measured in the M directions at . The cost function is then defined as
and the crack S is determined as the minimiser of .
In the numerical analysis the crack is assumed to be star shaped. The shape parameters we have used for the present application are based on the polar coordinate representation of the edge of S given by
Then the coordinates of the `origin' and the coefficients of the Fourier series for r are taken as the shape parameters. Note that the series for r begins with n=2 thus avoiding non-uniqueness of the parametrisation caused by the arbitrariness of . In the numerical example to follow we truncate the series and consider 5 parameters (). The derivatives of J with respect to are computed directly.