Optimal control of perfect plasticity part I: Stress tracking

Christian Meyer; Stephan Walther

doi:10.3934/mcrf.2021022

Article Contents

2022, Volume 12, Issue 2: 275-301. Doi: 10.3934/mcrf.2021022

This issue Previous Article Next Article Solvable approximations of 3-dimensional almost-Riemannian structures

Optimal control of perfect plasticity part I: Stress tracking

Christian Meyer^, and
Stephan Walther

TU Dortmund, Faculty of Mathematics, Vogelpothsweg 87, 44227 Dortmund, Germany

^* Corresponding author: Christian Meyer
^* Corresponding author: Christian Meyer

Received: August 2020

Revised: November 2021

Early access: March 2021

Published: June 2022

This research was supported by the German Research Foundation (DFG) under grant number ME 3281/9-1 within the priority program Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization (SPP 1962)

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Abstract

The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is to optimize the stress field by controlling the displacement at prescribed parts of the boundary. The control thus enters the system in the Dirichlet boundary conditions. Therefore, the safe load condition is automatically fulfilled so that the system admits a solution, whose stress field is unique. This gives rise to a well defined control-to-state operator, which is continuous but not Gâteaux differentiable. The control-to-state map is therefore regularized, first by means of the Yosida regularization resp. viscous approximation and then by a second smoothing in order to obtain a smooth problem. The approximation of global minimizers of the original non-smooth optimal control problem is shown and optimality conditions for the regularized problem are established. A numerical example illustrates the feasibility of the smoothing approach.

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Mathematics Subject Classification: 49J20, 49K20, 65K10, 74C05, 74C10.

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Figure 1. Legend; values in $ \big[{ \rm{GPa}}\big] $

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Figure 2. Evolution of $ |\sigma(x,t)|_F $

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Figure 3. Zoom to the left part of the beam from the left column of Figure 2

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Table 1. Comparison of the numerical results for different values of $ \lambda $

$ \lambda $	iteration	$ \langle {g_h} , {-g_h} \rangle_{H^1_0( \mathcal{X}_c)} $	$ \frac{F_\delta(\ell_h - \tau \,g_h) - F_\delta(\ell_h)}{\tau} $	err	$ \operatorname{dist}_ \mathcal{K} $
0.001	100	-4.7174e-07	-4.8520e-07	0.027751	0.00048
0.01	25	-2.0089e-07	-2.0869e-07	0.037369	0.00192
0.1	33	-2.4687e-07	-2.5552e-07	0.033854	0.01781
1	58	-2.1643e-07	-2.1790e-07	0.006773	0.13652
10	100	-2.0106e-06	-2.0122e-06	0.000833	0.62584
100	62	-2.4884e-07	-2.4876e-07	0.000338	5.31148

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Table 2. Comparison of the numerical results for different numbers of time steps

$ n_t $	iteration	$ \langle {g_h} , {-g_h} \rangle_{H^1_0( \mathcal{X}_c)} $	$ \frac{F_\delta(\ell_h - \tau \,g_h) - F_\delta(\ell_h)}{\tau} $	err	$ \operatorname{dist}_ \mathcal{K} $
8	51	-2.3590e-07	-2.8903e-07	0.183828	0.0478
32	45	-2.4318e-07	-2.5225e-07	0.035941	0.1066
128	58	-2.1643e-07	-2.1790e-07	0.006773	0.1365
512	48	-2.2542e-07	-2.2541e-07	0.000045	0.1318
2048	41	-2.3150e-07	-2.3165e-07	0.000662	0.1339

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