Research Group Numerical Mathematics (Partial Differential Equations)






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Research Group Numerical Mathematics
(Partial Differential Equations)

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

General Information

Keywords

  • Elastoplasticity
  • Optimal Control of Variational Inequalities
  • Quasilinear Systems
  • Necessary and Sufficient Optimality Conditions

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state, the overall loading process is no longer reversible and permanent deformations remain even after the loads are withdrawn.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.

Related Publications

Related Talks

  • Herzog Optimal Control of Elastoplastic Processes, ISMP, Berlin, August 2012
  • Wachsmuth: Optimal Control of Quasistatic Plasticity, ISMP, Berlin, August 2012
  • Wachsmuth: Optimal Control of Quasistatic Plasticity - An MPCC in Function Space, Dissertation Defense, Chemnitz, December 2011
  • Wachsmuth: Optimality Conditions for Optimal Control of Quasistatic Plasticity, Workshop on Control and Optimization of PDEs, Graz, October 2011
  • Herzog: Optimality Conditions in Optimal Control of Elastoplasticity, Workshop on Control and Optimization of PDEs, Graz, October 2011
  • Meyer: On Optimal Control of Elasto-Plastic Problems, SIAM Conference on Optimization, Darmstadt, May 2011
  • Wachsmuth: C-Stationarity for Optimal Control of Static Plasticity, Annual Meeting of the DFG Priority Program 1253 (Optimization with Partial Differential Equations), Freising, Germany, September 2010
  • Herzog: Regularization and C-Stationarity for an Optimal Control Problem in Static Plasticity, Workshop on Optimal Control and Partial Differential Equations, Greifswald, Germany, August 2010
  • Wachsmuth: C-stationarity for optimal control of static plasticity, Workshop on Analysis and Numerics of PDE Constrained Optimization, Lambrecht, Germany, July 2010
  • Wachsmuth: Regularity of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions, Berliner Oberseminar Nichtlineare partielle Differentialgleichungen (Langenbach-Seminar), Berlin, Germany, April 2010
  • Wachsmuth: Regularity of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions, Zuse Institute Berlin, Germany, April 2010
  • Meyer: Optimal Control of Static Plasticity with Linear Kinematic Hardening, ISIMM Workshop on Mathematical Problems of Solid Mechanics, Darmstadt, Germany, October 2009
  • Herzog: Optimal Control of Variational Inequalities in Plasticity, Annual Meeting of the DFG Priority Program 1253 (Optimization with Partial Differential Equations), Bad Staffelstein, Germany, October 2009
  • Griesse: Optimal Control of Static Plasticity, Fourth German Polish Conference on Optimization, Moritzburg, Germany, March 2009