C-Stationarity for Optimal Control of Static Plasticity with Linear Kinematic Hardening

Abstract

Abstract. An optimal control problem is considered for the variational inequality representing the stress-based (dual) formulation of static elastoplasticity. The linear kinematic hardening model and the von Mises yield condition are used. Existence and uniqueness of the plastic multiplier is rigorously proved, which allows for the re-formulation of the forward system using a complementarity condition.In order to derive necessary optimality conditions, a family of regularized optimal control problems is analyzed, wherein the static plasticity problems are replaced by their viscoplastic approximations. By passing to the limit in the optimality conditions for the regularized problems, necessary optimality conditions of C-stationarity type are obtained.

Elasto-plastic Shape Optimization Using the Level Set Method

“…For the existence and uniqueness, the proof is the same as the one for proposition 2.8 in [26] with hardening. …”

“…The regularity of u is given by theorem 1.1 of [25]. The Fréchet differentiability of f γ is done in [26].…”

“…The solution we choose, which was largely investigated in the framework of control theory for problems with hardening but not in the framework of shape optimization, is the use of a regularized penalized problem to get rid of the variational inequality. On this issue we mention, for the static case, [24], [26], [27], [11] (using a primal formulation), [28], [5] (for a second order optimality condition), for the quasi-static case [69] and for other plastic models [10] and [36].…”

Optimal Control of Elastoplastic Processes: Analysis, Algorithms, Numerical Analysis and Applications

“…Here, γ > 0 is the penalty parameter. In [Herzog et al, 2012, Section 2.2] we obtained the following smoothed version of the optimality condition (2.5):…”

Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization

“…These problems are often referred to as mathematical programs with complementarity constraints (MPCC) in the literature, cf. [32,37,38] in a finite dimensional setting, or [14,15,17] for problems posed in function space. Using penalty and regularization techniques, much work was done in the 1970s and 1980s concerning the optimal control of elliptic VIs as can be seen in the monograph by Barbu [2], in which the 'adapted penalty' approach of Lions [30,31] and Yvon [43] is generalized to a larger setting.…”