In the inverse problem one carries out N experiments on the boundary of the domain, and thus knows N pairs of () on , where N is taken bigger than the dimensionality of the problem. If the crack S is known, the corresponding crack opening displacement satisfies
Note that the hypersingular integral equation in (5) is equivalent to a variational equation given by
where is an arbitrary function (test function) which vanishes at the tips of S, and s is the arc length of S with which the normal vector to S is computed as . This variational equation is obtained from an identity given by
which follows from Laplace's equation for G. Also, note that is a known harmonic function in D. The solution to (5) should satisfy (3) on . For an arbitrary S which may not be the true one, however, the solution to (5) does not necessarily satisfy (3) with . Motivated by this observation one introduces a cost function defined by
where is a solution to (5). The cost J is nonnegative, and vanishes for the ture crack S. Hence it is reasonable to determine the most plausible crack as the minimiser of the cost .
A practical way of minimising is to discretise S by using finite number of shape parameters for S, denoted by , and use nonlinear programming technique to minimise the resulting nonlinear function of . Therefore, it is worth while to prepare formulae for the derivatives (variations) of with respect to the change of S, since some nonlinear programming algorithms use gradients of the cost. To do this, one has to make it precise what one means by the word `derivatives with respect to the change of S'. The approach used by Nishimura & Kobayashi uses a `reference crack' denoted by , and indicates a point on by greek letters , etc. A point on S is now considered as an image of via a mapping . The change of S is then taken as the variation of , or a derivative of with respect to the shape parameters . Also, and are defined to be functions of . With these preparation, the LHS of (6), for example, reads
to be precise. The derivative of J with respect to , denoted by , is now computed as
where indicates the derivative with respect to the th argument, is short for with (), and stands for the alternating tensor. The sensitivity is obtained from a variational equation for , which is derived as one takes the derivative of (6) as follows:
In the derivation of this formula, one assumes that is a fixed function of (and hence ), and uses the fact that satisfies Laplace's equation in D. Notice that vanishes at the tips of S by definition. This guarantees the uniqueness of the solution of (9) given the RHS. Also notice that is interpreted as the variation of if the dimension of is infinite.
One is now ready to proceed to numerical calculations.