Optimal Control in Elastoplasticity - Analysis, Algorithms, Numerical Analysis and Applications


Start: 2009-10-01
End: 2012-09-30
Principal Investigators: Roland Herzog, Christian Meyer
Staff: Gerd Wachsmuth
Funded by: DFG within the Priority Program funding scheme
Part of: Optimization with Partial Differential Equations (SPP 1253)

Project Description

Solid bodies depart from their rest shape under the influence of applied loads. In case the applied loads or stresses are sufficiently small, many solids exhibit a linearly elastic and reversible behavior. If, however, the stress induced by the applied loads exceeds a certain threshold (the yield stress), the material behavior switches from the elastic to the so-called plastic regime. In this state, the overall loading process is no longer reversible and permanent deformations remain even after the loads are withdrawn. Mathematically, this leads to a description involving variational inequalities.

Plastic deformation is desired for instance as an industrial shaping technique of metal workpieces, as e.g. by deep-drawing of body sheets in the automotive industry. The task of finding appropriate time-dependent loads which effect a desired final deformation leads to optimal control problems for elastoplasticity systems. These are also motivated by the desire to reduce the amount of springback, i.e., the partial reversal of the final material deformation due to a release of the stored elastic energy once the loads are removed.

The proposed project targets optimal control problems for static and quasi-static models of infinitesimal elastoplasticity with hardening. Its main goals are to investigate these optimization problems, to quantify the error due to discretization, and to develop fast algorithms for their solution. Models of elastoplasticity involve non-smooth features due to their description by variational inequalities and pointwise projections. The mathematical treatment of associated optimal control problems is therefore highly challenging and it requires a substantial extension of the established techniques.

Associated Publications

  • Juan Carlos de los Reyes, Roland Herzog and Christian Meyer
    Optimal control of static elastoplasticity in primal formulation
    SIAM Journal on Control and Optimization 54(6), p.3016-3039, 2016
    doi:10.1137/130920861
    bibtex 
    @ARTICLE{DeLosReyesHerzogMeyer:2016:1,
  AUTHOR = {de los Reyes, Juan Carlos and Herzog, Roland and Meyer, Christian},
  DATE = {2016},
  DOI = {10.1137/130920861},
  JOURNALTITLE = {SIAM Journal on Control and Optimization},
  NUMBER = {6},
  PAGES = {3016--3039},
  TITLE = {Optimal control of static elastoplasticity in primal formulation},
  VOLUME = {54},
}
  • Gerd Wachsmuth
    Optimal control of quasistatic plasticity with linear kinematic hardening III: optimality conditions
    Zeitschrift für Analysis und ihre Anwendungen 35(1), p.81-118, 2016
    doi:10.4171/ZAA/1556
    bibtex 
    @ARTICLE{Wachsmuth:2016:2,
  AUTHOR = {Wachsmuth, Gerd},
  DATE = {2016},
  DOI = {10.4171/ZAA/1556},
  JOURNALTITLE = {Zeitschrift für Analysis und ihre Anwendungen},
  NUMBER = {1},
  PAGES = {81--118},
  TITLE = {Optimal control of quasistatic plasticity with linear kinematic hardening III: optimality conditions},
  VOLUME = {35},
}
  • Gerd Wachsmuth
    Optimal control of quasistatic plasticity with linear kinematic hardening II: regularization and differentiability
    Zeitschrift für Analysis und ihre Anwendungen 34(4), p.391-418, 2015
    doi:10.4171/ZAA/1546
    bibtex 
    @ARTICLE{Wachsmuth:2015:1,
  AUTHOR = {Wachsmuth, Gerd},
  DATE = {2015},
  DOI = {10.4171/ZAA/1546},
  JOURNALTITLE = {Zeitschrift für Analysis und ihre Anwendungen},
  NUMBER = {4},
  PAGES = {391--418},
  TITLE = {Optimal control of quasistatic plasticity with linear kinematic hardening II: regularization and differentiability},
  VOLUME = {34},
}
  • Roland Herzog, Christian Meyer and Gerd Wachsmuth
    Optimal control of elastoplastic processes: analysis, algorithms, numerical analysis and applications
    Trends in PDE Constrained Optimization, International Series of Numerical Mathematics, p.27-41, 2014
    doi:10.1007/978-3-319-05083-6_4
    bibtex 
    @INCOLLECTION{HerzogMeyerWachsmuth:2014:1,
  AUTHOR = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd},
  PUBLISHER = {Springer},
  BOOKTITLE = {Trends in PDE Constrained Optimization},
  DATE = {2014},
  DOI = {10.1007/978-3-319-05083-6_4},
  PAGES = {27--41},
  SERIES = {International Series of Numerical Mathematics},
  TITLE = {Optimal control of elastoplastic processes: analysis, algorithms, numerical analysis and applications},
}
  • Gerd Wachsmuth
    Differentiability of implicit functions: beyond the implicit function theorem
    Journal of Mathematical Analysis and Applications 414(1), p.259-272, 2014
    doi:10.1016/j.jmaa.2014.01.007
    bibtex 
    @ARTICLE{Wachsmuth:2014:1,
  AUTHOR = {Wachsmuth, Gerd},
  DATE = {2014},
  DOI = {10.1016/j.jmaa.2014.01.007},
  JOURNALTITLE = {Journal of Mathematical Analysis and Applications},
  NUMBER = {1},
  PAGES = {259--272},
  TITLE = {Differentiability of implicit functions: beyond the implicit function theorem},
  VOLUME = {414},
}
  • Roland Herzog, Christian Meyer and Gerd Wachsmuth
    B- and strong stationarity for optimal control of static plasticity with hardening
    SIAM Journal on Optimization 23(1), p.321-352, 2013
    doi:10.1137/110821147
    bibtex 
    @ARTICLE{HerzogMeyerWachsmuth:2013:1,
  AUTHOR = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd},
  DATE = {2013},
  DOI = {10.1137/110821147},
  JOURNALTITLE = {SIAM Journal on Optimization},
  NUMBER = {1},
  PAGES = {321--352},
  TITLE = {B- and strong stationarity for optimal control of static plasticity with hardening},
  VOLUME = {23},
}
  • Roland Herzog, Christian Meyer and Gerd Wachsmuth
    C-stationarity for optimal control of static plasticity with linear kinematic hardening
    SIAM Journal on Control and Optimization 50(5), p.3052-3082, 2012
    doi:10.1137/100809325
    bibtex 
    @ARTICLE{HerzogMeyerWachsmuth:2012:1,
  AUTHOR = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd},
  DATE = {2012},
  DOI = {10.1137/100809325},
  JOURNALTITLE = {SIAM Journal on Control and Optimization},
  NUMBER = {5},
  PAGES = {3052--3082},
  TITLE = {C-stationarity for optimal control of static plasticity with linear kinematic hardening},
  VOLUME = {50},
}
  • Roland Herzog, Christian Meyer and Gerd Wachsmuth
    Optimale Steuerung in der Elastoplastizität
    GAMM-Rundbrief(2), p.16-20, 2012
    https://www.gamm-ev.de/wp-content/uploads/2020/06/RB_2012_02_weba.compressed.pdf
    bibtex 
    @ARTICLE{HerzogMeyerWachsmuth:2012:2,
  AUTHOR = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd},
  URL = {https://www.gamm-ev.de/wp-content/uploads/2020/06/RB_2012_02_weba.compressed.pdf},
  DATE = {2012},
  JOURNALTITLE = {GAMM-Rundbrief},
  NUMBER = {2},
  PAGES = {16--20},
  TITLE = {Optimale Steuerung in der Elastoplastizität},
}
  • Gerd Wachsmuth
    Optimal control of quasistatic plasticity with linear kinematic hardening, Part I: existence and discretization in time
    SIAM Journal on Control and Optimization 50(5), p.2836-2861, 2012
    doi:10.1137/110839187
    bibtex 
    @ARTICLE{Wachsmuth:2012:1,
  AUTHOR = {Wachsmuth, Gerd},
  DATE = {2012},
  DOI = {10.1137/110839187},
  JOURNALTITLE = {SIAM Journal on Control and Optimization},
  NUMBER = {5},
  PAGES = {2836--2861},
  TITLE = {Optimal control of quasistatic plasticity with linear kinematic hardening, Part I: existence and discretization in time},
  VOLUME = {50},
}
  • Roland Herzog and Christian Meyer
    Optimal control of static plasticity with linear kinematic hardening
    Journal of Applied Mathematics and Mechanics 91(10), p.777-794, 2011
    doi:10.1002/zamm.200900378
    bibtex 
    @ARTICLE{HerzogMeyer:2011:1,
  AUTHOR = {Herzog, Roland and Meyer, Christian},
  DATE = {2011},
  DOI = {10.1002/zamm.200900378},
  JOURNALTITLE = {Journal of Applied Mathematics and Mechanics},
  NUMBER = {10},
  PAGES = {777--794},
  TITLE = {Optimal control of static plasticity with linear kinematic hardening},
  VOLUME = {91},
}
  • Roland Herzog, Christian Meyer and Gerd Wachsmuth
    Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions
    Journal of Mathematical Analysis and Applications 382(2), p.802-813, 2011
    doi:10.1016/j.jmaa.2011.04.074
    bibtex 
    @ARTICLE{HerzogMeyerWachsmuth:2011:1,
  AUTHOR = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd},
  DATE = {2011},
  DOI = {10.1016/j.jmaa.2011.04.074},
  JOURNALTITLE = {Journal of Mathematical Analysis and Applications},
  NUMBER = {2},
  PAGES = {802--813},
  TITLE = {Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions},
  VOLUME = {382},
}
  • Roland Herzog, Christian Meyer and Gerd Wachsmuth
    Existence and regularity of the plastic multiplier in static and quasistatic plasticity
    GAMM Reports 34(1), p.39-44, 2011
    doi:10.1002/gamm.201110006
    bibtex 
    @ARTICLE{HerzogMeyerWachsmuth:2011:2,
  AUTHOR = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd},
  DATE = {2011},
  DOI = {10.1002/gamm.201110006},
  JOURNALTITLE = {GAMM Reports},
  NUMBER = {1},
  PAGES = {39--44},
  TITLE = {Existence and regularity of the plastic multiplier in static and quasistatic plasticity},
  VOLUME = {34},
}
  • Gerd Wachsmuth
    Optimal Control of Quasistatic Plasticity -- An MPCC in Function Space
    Ph.D. thesis, Technische Universität Chemnitz, Germany, 2011
    bibtex 
    @THESIS{Wachsmuth:2011:1,
  AUTHOR = {Wachsmuth, Gerd},
  INSTITUTION = {Technische Universität Chemnitz, Germany},
  DATE = {2011},
  TITLE = {Optimal Control of Quasistatic Plasticity -- An MPCC in Function Space},
  TYPE = {phdthesis},
}

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