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

where

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.

Thu Feb 19 01:36:51 JST 1998