# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021022

## Optimal control of perfect plasticity part I: Stress tracking

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

* Corresponding author: Christian Meyer

Received  August 2020 Revised  November 2021 Published  March 2021

Fund Project: 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)

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.

Citation: Christian Meyer, Stephan Walther. Optimal control of perfect plasticity part I: Stress tracking. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021022
##### References:
 [1] S. Bartels, A. Mielke and T. Roubíček, Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM Journal on Numerical Analysis, 50 (2012), 951-976.  doi: 10.1137/100819205.  Google Scholar [2] H. Brézis, Opérateurs Maximaux Monotones, North-Holland, Amsterdam, 1973. Google Scholar [3] E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM Journal on Control and Optimization, 45 (2006), 1586-1611.  doi: 10.1137/050626600.  Google Scholar [4] S. Chowdhury, T. Gudi and A. K. Nandakumaran, Error bounds for a Dirichlet boundary control problem based on energy spaces, Mathematics of Computation, 86 (2017), 1103-1126.  doi: 10.1090/mcom/3125.  Google Scholar [5] G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Archive for Rational Mechanics and Analysis, 180 (2006), 237-291.  doi: 10.1007/s00205-005-0407-0.  Google Scholar [6] K. Gröger, A $W^{1, p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), 679-687.  doi: 10.1007/BF01442860.  Google Scholar [7] T. Gudi and R. Ch. Sau, Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem, ESAIM: Control, Optimisation and Calculus of Variations, 26 (2020), Paper No. 78, 19 pp. doi: 10.1051/cocv/2019068.  Google Scholar [8] W. Han and B. D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis, Second edition, Interdisciplinary Applied Mathematics, 9. Springer, New York, 2013. doi: 10.1007/978-1-4614-5940-8.  Google Scholar [9] R. Herzog and C. Meyer, Optimal control of static plasticity with linear kinematic hardening, ZAMM Z. Angew. Math. Mech., 91 (2011), 777-794.  doi: 10.1002/zamm.200900378.  Google Scholar [10] R. Herzog, C. Meyer and G. Wachsmuth, B- and strong stationarity for optimal control of static plasticity with hardening, SIAM Journal on Optimization, 23 (2013), 321-352.  doi: 10.1137/110821147.  Google Scholar [11] R. Herzog, C. Meyer and G. Wachsmuth, Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions, Journal of Mathematical Analysis and Applications, 382 (2011), 802-813.  doi: 10.1016/j.jmaa.2011.04.074.  Google Scholar [12] R. Herzog, C. Meyer and G. Wachsmuth, C-stationarity for optimal control of static plasticity with linear kinematic hardening, SIAM Journal on Control and Optimization, 50 (2012), 3052-3082.  doi: 10.1137/100809325.  Google Scholar [13] C. Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl., 55 (1976), 431-444.   Google Scholar [14] A. Maury, G. Allaire and F. Jouve, Elasto-plastic shape optimization using the level set method, SIAM Journal on Control and Optimization, 56 (2018), 556-581.  doi: 10.1137/17M1128940.  Google Scholar [15] S. May, R. Rannacher and B. Vexler, Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems, SIAM Journal on Control and Optimization, 51 (2013), 2585-2611.  doi: 10.1137/080735734.  Google Scholar [16] C. Meyer and S. Walther, Optimal control of perfect plasticity, part Ⅱ: Displacement tracking, preprint, 2020, arXiv: 2003.09619. Google Scholar [17] A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar [18] N. Ottosen and M. Ristinmaa, The Mechanics of Constitutive Modeling, Elsevier, Amsterdam, 2005. Google Scholar [19] J. C. Simo and T. J. R. Hughes, Computational Inelasticity, Interdisciplinary Applied Mathematics, 7. Springer-Verlag, New York, 1998.  Google Scholar [20] U. Stefanelli, D. Wachsmuth and G. Wachsmuth, Optimal control of a rate-independent evolution equation via viscous regularization, Discrete and Continuous Dynamical Systems. Series S, 10 (2017), 1467-1485.  doi: 10.3934/dcdss.2017076.  Google Scholar [21] P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, Journal de Mécanique, 20 (1981), 3–39.  Google Scholar [22] R. Temam, Mathematical Problems in Plasticity, Courier Dover Publications, 2018.  Google Scholar [23] G. Wachsmuth, Optimal Control of Quasistatic Plasticity, PhD thesis, TU Chemnitz, 2011. Google Scholar [24] G. Wachsmuth, Optimal control of quasi-static plasticity with linear kinematic hardening, Part Ⅰ: Existence and discretization in time, SIAM Journal on Control and Optimization, 50 (2012), 2836–2861 + loose erratum. doi: 10.1137/110839187.  Google Scholar [25] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening Ⅱ: Regularization and differentiability, Z. Anal. Anwend., 34 (2015), 391-418.  doi: 10.4171/ZAA/1546.  Google Scholar [26] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening Ⅲ: Optimality conditions, Z. Anal. Anwend., 35 (2016), 81-118.  doi: 10.4171/ZAA/1556.  Google Scholar [27] S. Walther, C. Meyer and H. Meinlschmidt, Optimal control of an abstract evolution variational inequality with application to homogenized plasticity, Journal of Nonsmooth Analysis and Optimization, 1. Google Scholar

show all references

##### References:
 [1] S. Bartels, A. Mielke and T. Roubíček, Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM Journal on Numerical Analysis, 50 (2012), 951-976.  doi: 10.1137/100819205.  Google Scholar [2] H. Brézis, Opérateurs Maximaux Monotones, North-Holland, Amsterdam, 1973. Google Scholar [3] E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM Journal on Control and Optimization, 45 (2006), 1586-1611.  doi: 10.1137/050626600.  Google Scholar [4] S. Chowdhury, T. Gudi and A. K. Nandakumaran, Error bounds for a Dirichlet boundary control problem based on energy spaces, Mathematics of Computation, 86 (2017), 1103-1126.  doi: 10.1090/mcom/3125.  Google Scholar [5] G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Archive for Rational Mechanics and Analysis, 180 (2006), 237-291.  doi: 10.1007/s00205-005-0407-0.  Google Scholar [6] K. Gröger, A $W^{1, p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), 679-687.  doi: 10.1007/BF01442860.  Google Scholar [7] T. Gudi and R. Ch. Sau, Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem, ESAIM: Control, Optimisation and Calculus of Variations, 26 (2020), Paper No. 78, 19 pp. doi: 10.1051/cocv/2019068.  Google Scholar [8] W. Han and B. D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis, Second edition, Interdisciplinary Applied Mathematics, 9. Springer, New York, 2013. doi: 10.1007/978-1-4614-5940-8.  Google Scholar [9] R. Herzog and C. Meyer, Optimal control of static plasticity with linear kinematic hardening, ZAMM Z. Angew. Math. Mech., 91 (2011), 777-794.  doi: 10.1002/zamm.200900378.  Google Scholar [10] R. Herzog, C. Meyer and G. Wachsmuth, B- and strong stationarity for optimal control of static plasticity with hardening, SIAM Journal on Optimization, 23 (2013), 321-352.  doi: 10.1137/110821147.  Google Scholar [11] R. Herzog, C. Meyer and G. Wachsmuth, Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions, Journal of Mathematical Analysis and Applications, 382 (2011), 802-813.  doi: 10.1016/j.jmaa.2011.04.074.  Google Scholar [12] R. Herzog, C. Meyer and G. Wachsmuth, C-stationarity for optimal control of static plasticity with linear kinematic hardening, SIAM Journal on Control and Optimization, 50 (2012), 3052-3082.  doi: 10.1137/100809325.  Google Scholar [13] C. Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl., 55 (1976), 431-444.   Google Scholar [14] A. Maury, G. Allaire and F. Jouve, Elasto-plastic shape optimization using the level set method, SIAM Journal on Control and Optimization, 56 (2018), 556-581.  doi: 10.1137/17M1128940.  Google Scholar [15] S. May, R. Rannacher and B. Vexler, Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems, SIAM Journal on Control and Optimization, 51 (2013), 2585-2611.  doi: 10.1137/080735734.  Google Scholar [16] C. Meyer and S. Walther, Optimal control of perfect plasticity, part Ⅱ: Displacement tracking, preprint, 2020, arXiv: 2003.09619. Google Scholar [17] A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar [18] N. Ottosen and M. Ristinmaa, The Mechanics of Constitutive Modeling, Elsevier, Amsterdam, 2005. Google Scholar [19] J. C. Simo and T. J. R. Hughes, Computational Inelasticity, Interdisciplinary Applied Mathematics, 7. Springer-Verlag, New York, 1998.  Google Scholar [20] U. Stefanelli, D. Wachsmuth and G. Wachsmuth, Optimal control of a rate-independent evolution equation via viscous regularization, Discrete and Continuous Dynamical Systems. Series S, 10 (2017), 1467-1485.  doi: 10.3934/dcdss.2017076.  Google Scholar [21] P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, Journal de Mécanique, 20 (1981), 3–39.  Google Scholar [22] R. Temam, Mathematical Problems in Plasticity, Courier Dover Publications, 2018.  Google Scholar [23] G. Wachsmuth, Optimal Control of Quasistatic Plasticity, PhD thesis, TU Chemnitz, 2011. Google Scholar [24] G. Wachsmuth, Optimal control of quasi-static plasticity with linear kinematic hardening, Part Ⅰ: Existence and discretization in time, SIAM Journal on Control and Optimization, 50 (2012), 2836–2861 + loose erratum. doi: 10.1137/110839187.  Google Scholar [25] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening Ⅱ: Regularization and differentiability, Z. Anal. Anwend., 34 (2015), 391-418.  doi: 10.4171/ZAA/1546.  Google Scholar [26] G. Wachsmuth, Optimal control of quasistatic plasticity with linear kinematic hardening Ⅲ: Optimality conditions, Z. Anal. Anwend., 35 (2016), 81-118.  doi: 10.4171/ZAA/1556.  Google Scholar [27] S. Walther, C. Meyer and H. Meinlschmidt, Optimal control of an abstract evolution variational inequality with application to homogenized plasticity, Journal of Nonsmooth Analysis and Optimization, 1. Google Scholar
Legend; values in $\big[{ \rm{GPa}}\big]$
Evolution of $|\sigma(x,t)|_F$
">Figure 3.  Zoom to the left part of the beam from the left column of Figure 2
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
 $\lambda$ 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
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
 $n_t$ 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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