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Mathematics  2004 

An unconditionally convergent iterative scheme for initial shape identification in small deformations

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Abstract:

The question of interest in the present study is: ``Given a body subject to mechanical loads, how to define the initial geometry so that the deformed one matches precisely a prescribed shape?'' This question is particularly relevant in forming processes where the tolerated mismatch between the deformed and desired geometries may be lower than a Micron. The method proposed here uses as a first ``guess'' to the required initial geometry the desired one, then it updates iteratively the locations of a set of boundary points so that their locations in the deformed configuration come closer and closer to the desired ones. The scheme is shown to converge unconditionally for small deformations in the sense that arbitrarily small mismatch between the deformed and desired shape can be achieved. Moreover, since it is based entirely on geometric considerations, the convergence should not be affected by the nature of material, i.e. it is independent of the constitutive law. The success of the method is illustrated by considering an example.

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