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A non-invasive heuristic approach to shape optimization in forming

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Abstract

The aim is to determine—relating to a given forming process—the optimal material (undeformed) configuration of a workpiece when knowing the target spatial (deformed) configuration. Therefore, the nodal positions of a discretized setting based on the finite element method (FEM) are the discrete free parameters of the form finding problem. As a verification, inputting the determined optimal material nodal positions, a subsequent re-computation of the forming process should then result in exactly the target spatial nodal positions. A new, non-invasive iterative algorithm, which is purely based on the nodal data of each iteration, is proposed to determine the discretized optimal material configuration. Specifically, the \(L^2\)-smoothed deformation gradient at each discretization node is used to update the discretized material configuration by a transformation of the difference vectors between the currently computed and the target spatial nodal positions. The iterative strategy can be easily coupled in a non-invasive fashion via subroutines with arbitrary external FEM software. Since only the computed positions of the discretization nodes are required for an update step within the form finding algorithm, the procedure does not depend on the specific material modelling and is moreover applicable to arbitrary element types, e. g. solid- or solid-shell-elements. Furthermore the convergence rate for solving the form finding problem is nearly linear. This is demonstrated by examples that are realized by a coupling of Matlab (iterative update procedure) and MSC.Marc (external FEM software). Solving the form finding problem to determine an optimum workpiece design is of great interest especially for metal forming applications.

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Acknowledgments

This work is part of the collaborative research project Manufacturing of complex functional components with variants by using a new metal forming process - Sheet-Bulk metal forming (SFB/TR73: www.tr-73.de).

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  1. Chair of Applied Mechanics, Friedrich-Alexander-University of Erlangen-Nuremberg, Egerlandstrasse 5, 91058, Erlangen, Germany

    P. Landkammer & P. Steinmann

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  1. P. Landkammer
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  2. P. Steinmann
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Correspondence to P. Steinmann.

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Landkammer, P., Steinmann, P. A non-invasive heuristic approach to shape optimization in forming. Comput Mech 57, 169–191 (2016). https://doi.org/10.1007/s00466-015-1226-2

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  • DOI: https://doi.org/10.1007/s00466-015-1226-2

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