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Concluding Remarks

 

  1. 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.

  2. 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.

  3. 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.



next up previous
Next: References Up: No Title Previous: Numerical Example



N. Nishimura
Thu Feb 19 01:36:51 JST 1998