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, 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.