We consider the time-discretized problem of the quasi-static evolution problem in perfect plasticity posed in a non-reflexive Banach space. Based on a novel equivalent reformulation in a reflexive Banach space, the primal problem is characterized as a Fenchel dual problem of the classical incremental stress problem. This allows to obtain necessary and sufficient optimality conditions for the time-discrete problems of perfect plasticity. Furthermore, the consistency of a primal-dual stabilization scheme is proven. As a consequence, not only stresses, but also displacements and strains are shown to converge to a solution of the original problem in a suitable topology. The corresponding dual problem has a simpler structure and turns out to be well-suited for numerical purposes. For the resulting subproblems an efficient algorithmic approach in the infinite-dimensional setting based on the semismooth Newton method is proposed.
Keywords: Perfect plasticity, Prandtl–Reuss plasticity, small-strain, Fenchel duality, semismooth Newton
@article{COCV_2021__27_S1_A2_0,
author = {Hinterm\"uller, M. and R\"osel, S.},
title = {Duality results and regularization schemes for {Prandtl{\textendash}Reuss} perfect plasticity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2018004},
mrnumber = {4222147},
zbl = {1468.74007},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018004/}
}
TY - JOUR AU - Hintermüller, M. AU - Rösel, S. TI - Duality results and regularization schemes for Prandtl–Reuss perfect plasticity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018004/ DO - 10.1051/cocv/2018004 LA - en ID - COCV_2021__27_S1_A2_0 ER -
%0 Journal Article %A Hintermüller, M. %A Rösel, S. %T Duality results and regularization schemes for Prandtl–Reuss perfect plasticity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018004/ %R 10.1051/cocv/2018004 %G en %F COCV_2021__27_S1_A2_0
Hintermüller, M.; Rösel, S. Duality results and regularization schemes for Prandtl–Reuss perfect plasticity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. S1. doi: 10.1051/cocv/2018004
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Cité par Sources :
This research was carried out in the framework of Matheon supported by the Einstein Foundation Berlin within the ECMath projects OT1, SE5 and SE15 as well as project A-AP24. The authors further gratefully acknowledge the support of the DFG through the DFG-SPP 1962: Priority Programme “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” within Projects 10, 11 and 13.
