In addition to these mathematical contributions, we can find many
numerical works to actually determine **S**. These methods are based on
similar principles. Namely they introduce finite number of shape
parameters to describe **S** and reduce the problem to one of the
parameter estimation. Kubo[18] and Sakagami et al.[36] solved
2D and 3D problems by solving many direct problems for candidate
cracks and by picking up the one that fits the experimental data best.
Their works are remarkable in that they try inversion of real data
also, while many other investigators use simulated data as the input.
Santosa & Vogelius[37] used FEM and Newton's method to determine
one straight line crack in 2D. Bryan & Vogelius[6] determined
more than one line cracks in 2D by using the following approach: they
start with a given number of cracks (typically one) and find the most
plausible arrangement of them with the help of integral equations and
Newton's method. Then, they divide some of these cracks into two and
restart the analysis. At a certain point in this algorithm the number
of cracks used in their computation exceeds the true number. As they
found, however, the excessive crack segments shrink to points as the
computation proceeds. A general approach in 2D and 3D determination
of one crack is found in Nishimura & Kobayashi[30], some detail
of which will be presented later. We here note that their computation
is considered to be one of the earliest attempts to apply variational
(Galerkin) methods to the solution of hypersingular integral equations
in 3D. Mellings & Aliabadi solved similar inverse problems in
2D[21] and 3D[22], respectively, with the help of the
conventional collocation BIEM. The paper by Andrieux and Ben
Abda[3] attempted to determine the plane on which a planar
crack lies with the help of Green's formula. Their method requires 1
pair of Dirichlet and Neumann data on the whole exterior boundary
. The analyses we have seen so far are related to linear
or planar cracks. Nishimura & Kobayashi[32] considered the
determination of a curved crack by using Tikhonov's regularisation to
stabilise the analysis. They could thus determine cracks having about
50 geometrical DOF. Recently Elcrat and Hu[8] proposed to use
Schwarz-Christoffel transformations to determine piecewise linear
cracks in 2D. They use a `give **u**, measure **f**' setting.

Thu Feb 19 01:36:51 JST 1998