We are concerned with nondestructive control issues, namely detection and recovery of cracks in a planar (2D) isotropic conductor from partial boundary measurements of a solution to the Laplace-Neumann problem. In the case of 2D-line segment (or 3D-planar) cracks, S. Andrieux and A. Ben Abda introduced in 1993 the reciprocity gap concept, and proved that such cracks can be completely determined, provided that complete data are available on the external boundary of the body. Moreover, they gave inversion formulae which determine explicitely the plane containing the cracks, and proved that the full reconstruction is possible. We show that the reciprocity gap principle is a relevent tool to reconstruct unknown planar cracks in various situations : The steady state heat conduction problem for a crack satisfying boundary conditions of Robin's type, the transient heat conduction problem, the elastostatic problem.
In the case of the Laplace equation, in 2D situation, we treat the case of partial boundary measurements. We first build an extension of that data to the whole boundary, using constructive approximation techniques, and then use localisation algorithm. This recovery procedure is almost explicit insofar as no resolution of the backward problem is required.
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