The expression for can further be simplified through the introduction of the adjoint variables. To see this we start with the following observation, which follows from Green's identity: Let and satisfy

for any which vanishes at the tips of **S**, where **p** and **q**
are given functions. Then one has

Using this result, (9) and integration by parts, one rewrites (8) as

where is a solution of

satisfying at the tips.
The quantity is called the adjoint variable. With
(15) one solves the discretised version of
(16) **N** times, while with
(8) one solves the discretised version of
(9) **N**
(number of shape parameters for **S**) times. Hence the use of
(15) is considered superior to that of (8) when
the size of the inverse problem is big. However, the improvement of
the efficiency may not be very pronounced in small problems since reusing
results obtained in the calculation of **J** already makes the
computation of very efficient. As regards the accuracy, our
experience in the straight crack case tells that both (8)
and (15) yield equally accurate results.

One may say that the expression in (15) does not clearly
show that the cost stays constant for an which does
not change **S**. A careless repetition of the procedure used in the
derivation of (12), however, will lead to an
erroneous result, due to the near-tip singularities of . As a
matter of fact, One needs a lemma, which was stated without proof in
Nishimura & Kobayashi[29], to rewrite (15) into
a clearer form.

Proof (sketch). This lemma follows from an identity given by

where is a sufficiently smooth function. This identity is obtained by a direct calculation.

With this lemma, one rewrites (15) into

This formula clearly states that the cost **J** does not vary for a
purely tangential move unless one of the tips changes the
location. We have not so far tested numerical implementations based
on the formula in (18), but the performance of this
approach is expected to be similar to that based on (15) because
these formulae are interrelated via identities. Finally, we note that
the 3D counterpart of (18) has not been obtained since
the 3D version of the lemma is not available.

Thu Feb 19 01:36:51 JST 1998