Fixed Point Iterative Schemes for Initial Shape Identification

The question of interest in the present study is ”given a work-piece subject to prescribed loads, how to define its initial shape such that the work-piece matches a prescribed geometry after deformation?”. This question is particularly relevant in forming processes where the tolerated mismatch between the deformed and desired geometries may be lower than a Micron. To tackle this optimal shape design problem, a range of fixed point iterative schemes is proposed, i.e. the next initial shape is deduced from the previous one by subtracting the error in the previous final shape possibly corrected by an additional term. The required form of this additional, corrective term is revealed through a convergence analysis of the schemes. The schemes are applied to a test problem and their performance compared. The problem consists in designing the hole in a membrane such that its contour matches a prescribed shape when the membrane is stretched by a given load.