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.

Thu Feb 19 01:36:51 JST 1998