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Inverse Problem

The inverse problem is formulated as follows. Let S be a curved surface in whose location and shape are unknown. Suppose that the far fields produced by N known incident waves are observed in directions in the time intervals given by , where C is a set on the unit sphere . The problem is to find the most plausible crack from the experimental data and an a priori information that there exists only one crack. As far as the present author knows there is no known uniqueness results of the solution to this problem.

As in the Laplace case the crack S is determined as the minimiser of the following cost function:

 

where and stand for the far fields computed assuming that the crack is given by S, and observed experimentally, respectively.

In the numerical analysis one introduces simplifying assumptions regarding the shape of S and represents S with the `material' coordinate and finite number of shape parameters as

For example one can describe a penny shaped crack S with 6 shape parameters (see (32)). Also the cost function in (29) is replaced by the following discretised version:

where and are the time to make measurements, and appropriate numbers, respectively. Note that the set C is now taken to be a discrete one. Since the cost J is just a nonlinear function of , it can be minimised with NLP, with our choice being Powell's variable metric method, as in the Laplace case.

Once again one needs to compute derivatives of J with respect to , denoted by , in order to use quasi Newton methods.

The computation of this derivative, however, is carried out almost in the same manner as in the Laplace case. Indeed, one has

Also, the last factor in the above expression can be written in terms of known quantities and as

 

Finally, by taking the derivative of both sides of (26), one obtains the following variational equation for :

 

The LHS of the above equation has the same form as that of (26), and the RHS is known once () is. Hence one can solve this variational equation for in the same manner as (26). We thus see that all the quantities needed for the computation of are readily available.



next up previous
Next: Numerical Techniques Up: Galerkin Approach Previous: Direct Problem



N. Nishimura
Thu Feb 19 01:36:51 JST 1998