- It is obvious that the crack determination problems is not
restricted to the Laplace or elastodynamic cases. For example,
extensions of the techniques discussed in this paper to Helmholtz'
equation[23] and wave equation[27] are found in
literature. Kress also considered some uniqueness issues in the 2
dimensional Helmholtz case with homogeneous Dirichlet boundary
conditions on the crack, in addition to presenting a numerical
method[15,16]. His conclusion is that the unique
determination of a crack in a infinite plane is possible with one
incident plane wave[16]. We can also mention the paper by
Andrieux[2] where time dependent problems governed by
the heat equation etc. are considered.
- It appears to the present author that more efforts to verify the
numerical methods for crack determination problems with real data
are necessary. This is particularly true in elastodynamics where
measuring physically clear-cut quantities such as displacement or
velocity is difficult. More collaboration of researchers in
numerical and experimental fields seems to be desirable.
- In reality the shape of cracks are not linear or circular, and
the shape of the structure containing the crack is not as simple as
could be approximated by the whole space. If one wishes to solve
more realistic inverse problems, one would have to consider bigger
problems than have been treated so far. It is, however, extremely
difficult to solve large inverse problems because of the
ill-posedness of the problem and the computational load. Although
the development of computer may solve the latter problem, the
ill-posedness cannot be easily conquered unless one has enough
experimental data, or one finds a better setting of the problem.
Use of improved mathematical programming methods, for
example, will not solve the problem. To get more
information about the crack, one may for example think of increasing
the number of observation points etc. to obtain literally more data.
Or one may improve the measuring devices to get data with higher S/N
ratio. Or one may change the arrangement of measuring devices to get
better resolution. One may also think of combining different
experiments such as ultrasonic and electromagnetic tests together to
have more redundancy. From the viewpoint of the numerical analysis
the last two are of interest, and more efforts need to be directed
to these possibilities.

Thu Feb 19 01:36:51 JST 1998