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Computational Approaches

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.





next up previous
Next: A BIE Approach Up: LAPLACE CASES Previous: Mathematical Studies



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