## mpccdist2 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Elastoplasticity:
• nonlinear elliptic - VI operator of order 2.
• Defined on a 2-dim domain in 2-dim space
• No time dependence.

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2014-04-23 by Thomas Betz. Published on 2014-05-27

## mpccdist2 description:

### Introduction

We consider the optimal control of static elastoplasticity with linear kinematic hardening. This leads to an optimal control problem governed by an elliptic variational inequality of ﬁrst kind in mixed form or, equivalently, an MPCC in function space.
The objective functional tracks the state both on the entire domain and additionally on a submanifold of the domain. The control cost is taken into account by a tracking type term as well. There are no constraints on the control, which acts in a distributed way on the domain. The Dirichlet boundary is the entire boundary of the domain.
A locally optimal control is known, whose corresponding state has a bi-active set with a positive measure.

The problem and its solution are taken from [Betz et al.2014, Section 6.1].

### Variables & Notation

#### Unknowns

The state is composed of the stress $\sigma$, back stress $\chi$, displacement $u$ and plastic multiplier $\lambda$.

#### Given Data

desired displacement in the domain:

 ${u}_{\Omega }\left(x\right)=\left\{\begin{array}{cc}\left(\begin{array}{c}\hfill U\left(\right\parallel x{\parallel }^{2}\left)\right+\left(\right8{x}_{1}^{2}+4{x}_{2}^{2}+4{x}_{1}{x}_{2}\left)\right{U}^{\prime \prime }\left(\right\parallel x{\parallel }^{2}\left)\right+6{U}^{\prime }\left(\right\parallel x{\parallel }^{2}\left)\right\hfill \\ \hfill U\left(\right\parallel x{\parallel }^{2}\left)\right+\left(\right4{x}_{1}^{2}+8{x}_{2}^{2}+4{x}_{1}{x}_{2}\left)\right{U}^{\prime \prime }\left(\right\parallel x{\parallel }^{2}\left)\right+6{U}^{\prime }\left(\right\parallel x{\parallel }^{2}\left)\right\hfill \end{array}\right),\phantom{\rule{1em}{0ex}}\hfill & x\in B\hfill \\ \left(\begin{array}{c}\hfill U\left(\right\parallel x{\parallel }^{2}\left)\right+\left(\right6{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}\left)\right{U}^{\prime \prime }\left(\right\parallel x{\parallel }^{2}\left)\right+4{U}^{\prime }\left(\right\parallel x{\parallel }^{2}\left)\right\hfill \\ \hfill U\left(\right\parallel x{\parallel }^{2}\left)\right+\left(\right2{x}_{1}^{2}+6{x}_{2}^{2}+4{x}_{1}{x}_{2}\left)\right{U}^{\prime \prime }\left(\right\parallel x{\parallel }^{2}\left)\right+4{U}^{\prime }\left(\right\parallel x{\parallel }^{2}\left)\right\hfill \end{array}\right),\phantom{\rule{1em}{0ex}}\hfill & x\in R\hfill \end{array}\right\$

with

 $\begin{array}{ccc}\hfill U\left(t\right)& =\left\{\begin{array}{cc}-{\sigma }_{0}\phantom{\rule{0.3em}{0ex}}{t}^{2}+\frac{3}{2}{\sigma }_{0}\phantom{\rule{0.3em}{0ex}}t-\frac{13}{16}{\sigma }_{0},\phantom{\rule{1em}{0ex}}\hfill & t<\frac{1}{4}\hfill \\ {\sigma }_{0}\phantom{\rule{0.3em}{0ex}}\sqrt{t}-{\sigma }_{0},\phantom{\rule{1em}{0ex}}\hfill & t\ge \frac{1}{4}\hfill \end{array}\right\\hfill & \hfill \\ \hfill {U}^{\prime }\left(t\right)& =\left\{\begin{array}{cc}-2\phantom{\rule{0.3em}{0ex}}{\sigma }_{0}t+\frac{3}{2}{\sigma }_{0},\phantom{\rule{1em}{0ex}}\hfill & t<\frac{1}{4}\hfill \\ 0.5\phantom{\rule{0.3em}{0ex}}{\sigma }_{0}\phantom{\rule{0.3em}{0ex}}{t}^{-1∕2},\phantom{\rule{1em}{0ex}}\hfill & t\ge \frac{1}{4}\hfill \end{array}\right\\hfill \\ \hfill {U}^{″}\left(t\right)& =\left\{\begin{array}{cc}-2\phantom{\rule{0.3em}{0ex}}{\sigma }_{0},\phantom{\rule{1em}{0ex}}\hfill & t<\frac{1}{4}\hfill \\ -0.25\phantom{\rule{0.3em}{0ex}}{\sigma }_{0}\phantom{\rule{0.3em}{0ex}}{t}^{-3∕2},\phantom{\rule{1em}{0ex}}\hfill & t\ge \frac{1}{4}\hfill \end{array}\right\\hfill \\ \hfill \end{array}$

desired displacement on the submanifold $\partial B$:

 ${u}_{\partial B}\left(x\right)=\left(\begin{array}{c}\hfill -{\sigma }_{0}\hfill \\ \hfill -{\sigma }_{0}\hfill \end{array}\right)$

desired control:

 ${f}_{\Omega }\left(x\right)=\left(\begin{array}{c}\hfill \frac{2}{\alpha }U\left(\right\parallel x{\parallel }^{2}\left)\right-\left(\right4{x}_{1}^{2}+2{x}_{2}^{2}+2{x}_{1}{x}_{2}\left)\right{U}^{\prime \prime }\left(\right\parallel x{\parallel }^{2}\left)\right-3{U}^{\prime }\left(\right\parallel x{\parallel }^{2}\left)\right\hfill \\ \hfill \frac{2}{\alpha }U\left(\right\parallel x{\parallel }^{2}\left)\right-\left(\right2{x}_{1}^{2}+4{x}_{2}^{2}+2{x}_{1}{x}_{2}\left)\right{U}^{\prime \prime }\left(\right\parallel x{\parallel }^{2}\left)\right-3{U}^{\prime }\left(\right\parallel x{\parallel }^{2}\left)\right\hfill \end{array}\right)$

### Problem Description

With the given Lamé coeﬃcients, the inverse elasticity and hardening tensors read

 $\begin{array}{ccc}\hfill {ℂ}^{-1}\sigma & =\frac{1}{2{\mu }_{L}}\sigma -\frac{{\lambda }_{L}}{2{\mu }_{L}\left(2{\mu }_{L}+2{\lambda }_{L}\right)}trace\left(\sigma \right)I=\sigma ,\hfill & \hfill \\ \hfill {ℍ}^{-1}\chi & =\frac{1}{{k}_{1}}\chi =\chi ,\hfill \end{array}$

where $I:{ℝ}_{\text{sym}}^{2×2}\to {ℝ}_{\text{sym}}^{2×2}$ is the identity mapping and

The yield function is given by

 $\varphi \left(\sigma ,\chi \right)=\frac{\parallel {\sigma }^{D}+{\chi }^{D}{\parallel }_{F}^{2}-{\sigma }_{0}^{2}}{2},$

where $\parallel \cdot {\parallel }_{F}$ denotes the Frobenius norm of a matrix. The corresponding inner product $trace\left({A}^{\top }B\right)$ is used in the calculation of ${\left(\cdot ,\cdot \right)}_{{L}^{2}\left(\Omega ;{ℝ}_{\text{sym}}^{2×2}\right)}$.

### Optimality System

According to [Betz and Meyer2015, Theorem 4.6] (Theorem 4.4 in the Preprint) the following conditions are suﬃcient for local optimality of the control $f$ with associated state $\left(\right\sigma ,\chi ,u,\lambda \left)\right$.
There exist adjoint variables $\left(\right\zeta ,\psi ,w\left)\right\in {L}^{\infty }\left(\right\Omega ;{ℝ}_{\text{sym}}^{2×2}{\left)\right}^{2}×{H}_{0}^{1}\left(\right\Omega ;{ℝ}^{2}\left)\right$ and multipliers $\left(\mu ,𝜃\right)\in {L}^{\infty }\left(\Omega \right)×{L}^{\infty }\left(\Omega \right)$ satisfying

1. the optimality system
2. the second-order condition
There exists $\kappa >0$ such that
 ${\partial }_{\left(\sigma ,\chi ,u,\lambda ,f\right)}^{2}\mathsc{ℒ}\left(\right\sigma ,\chi ,u,\lambda ,f,\zeta ,\psi ,w,\mu ,𝜃\left)\right\left(\right{\sigma }^{\prime },{\chi }^{\prime },{u}^{\prime },{\lambda }^{\prime },h{\left)\right}^{2}\ge \kappa \parallel h{\parallel }_{U}^{2}$

holds for all $h\in {L}^{2}\left(\right\Omega ;{ℝ}^{2}\left)\right$ and $\left(\right{\sigma }^{\prime },{\chi }^{\prime },{u}^{\prime },{\lambda }^{\prime }\left)\right$ solving

The sets are deﬁned as follows,

and the Lagrangian $\mathsc{ℒ}$ is deﬁned by

 $\begin{array}{ccc}\hfill & \mathsc{ℒ}\left(\sigma ,\chi ,u,\lambda ,f,\zeta ,\psi ,w,\mu ,𝜃\right)\hfill & \hfill \\ \hfill & \phantom{\rule{1em}{0ex}}=J\left(u,f\right)+\left(\right\sigma -𝜀\left(u\right)+\lambda \left(\right{\sigma }^{D}+{\chi }^{D}\left)\right,\zeta {\left)\right}_{{L}^{2}\left(\Omega ;{ℝ}_{\text{sym}}^{2×2}\right)}\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}+\left(\right\chi +\lambda \left(\right{\sigma }^{D}+{\chi }^{D}\left)\right,\psi {\left)\right}_{{L}^{2}\left(\Omega ;{ℝ}_{\text{sym}}^{2×2}\right)}-\left(\right\sigma ,𝜀\left(w\right){\left)\right}_{{L}^{2}\left(\Omega ;{ℝ}_{\text{sym}}^{2×2}\right)}\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}+{\left(f,w\right)}_{{L}^{2}\left(\Omega ;{ℝ}^{2}\right)}-{\left(\lambda ,\mu \right)}_{{L}^{2}\left(\Omega \right)}+{\left(\varphi \left(\sigma ,\chi \right),𝜃\right)}_{{L}^{2}\left(\Omega \right)}.\hfill \end{array}$

Consequently ${\partial }_{\left(\sigma ,\chi ,u,\lambda ,f\right)}^{2}\mathsc{ℒ}\left(\right\sigma ,\chi ,u,\lambda ,f,\zeta ,\psi ,w,\mu ,𝜃\left)\right\left(\right{\sigma }^{\prime },{\chi }^{\prime },{u}^{\prime },{\lambda }^{\prime },h{\left)\right}^{2}$ is given as follows:

 $\begin{array}{ccc}\hfill {\partial }_{\left(\sigma ,\chi ,u,\lambda ,f\right)}^{2}& \mathsc{ℒ}\left(\right\sigma ,\chi ,u,\lambda ,f,\zeta ,\psi ,w,\mu ,𝜃\left)\right\left(\right{\sigma }^{\prime },{\chi }^{\prime },{u}^{\prime },{\lambda }^{\prime },h{\left)\right}^{2}\hfill & \hfill \\ \hfill & ={\left({u}^{\prime },{u}^{\prime }\right)}_{{L}^{2}\left(\Omega ;{ℝ}^{2}\right)}+{\left({u}^{\prime },{u}^{\prime }\right)}_{{L}^{2}\left(\partial B;{ℝ}^{2}\right)}+\alpha {\left(h,h\right)}_{{L}^{2}\left(\Omega ;{ℝ}^{2}\right)}\hfill \\ \hfill & \phantom{\rule{1em}{0ex}}+2\left(\right{\lambda }^{\prime }\left(\right{\left({\sigma }^{\prime }\right)}^{D}+{\left({\chi }^{\prime }\right)}^{D}\left)\right,\phantom{\rule{0.3em}{0ex}}{\zeta }^{D}+{\psi }^{D}{\left)\right}_{{L}^{2}\left(\Omega ;{ℝ}_{\text{sym}}^{2×2}\right)}\hfill \\ \hfill & \phantom{\rule{1em}{0ex}}+\left(\right∥{\left({\sigma }^{\prime }\right)}^{D}+{\left({\chi }^{\prime }\right)}^{D}{∥}_{F}^{2},\phantom{\rule{0.3em}{0ex}}𝜃{\left)\right}_{{L}^{2}\left(\Omega \right)}.\hfill \end{array}$

### Supplementary Material

Locally optimal control, state, adjoint state and multipliers are known analytically:

The magnitude of the optimal state and control for the values $\alpha =1$ and ${\sigma }_{0}=2$ as given in Betz et al.  are depicted in Figure 0.1. The reference value for the objective,

 $J\approx 156.448\phantom{\rule{1em}{0ex}}738\phantom{\rule{1em}{0ex}}479\phantom{\rule{1em}{0ex}}265\phantom{\rule{1em}{0ex}}980\phantom{\rule{1em}{0ex}}83$

corresponding to these values of $\alpha$ and ${\sigma }_{0}$ is given in Betz et al. .  Figure 0.1: Analytical values of the optimal state $|u|$ (left) and control $|f|$ (right) for parameters $\alpha =1$ and ${\sigma }_{0}=2$. Figure courtesy of K. Rosin

### Revision History

• 2015–04–22: Updated reference to the published version.
• 2014–11–29: Explicit formulas for the second-order derivative of the Lagrangian added. Formulas for optimal $\sigma$ and $f$ added. Pictures of the optimal solution added. Reference for objective value added. Matlab code added.
• 2014–04–23: problem added to the collection

### References

T. Betz and C. Meyer. Second-order suﬃcient optimality conditions for optimal control of static elastoplasticity with hardening. ESAIM: Control, Optimisation and Calculus of Variations, 21(1):271–300, 2015. doi: 10.1051/cocv/2014024.

T. Betz, C. Meyer, A. Rademacher, and K. Rosin. Adaptive optimal control of elastoplastic contact problems. Technical report, Fakultät für Mathematik, TU Dortmund, May 2014. URL http://www.mathematik.tu-dortmund.de/papers/BetzMeyerRademacherRosin2014.pdf. Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 496.