## mpccdist3 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Damage:
• quasi-linear other operator of order 2.
• Defined on a 2-dim domain in 2-dim space
• Time dependent.

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2016-07-01 by Marita Holtmannspötter. Published on 2016-12-23

## mpccdist3 description:

### Introduction

We study the optimal control of a particular gradient enhanced damage model. The damage model is based on Dimitrijevic and Hackl [2008], the optimization problem is currently unpublished and provided by M. Holtmannspötter.

The presented damage model features two damage variables, one with higher spatial regularity and one which carries the evolution of damage in time. Their diﬀerence is penalized in the free energy functional. The evolution of damage in time is modeled by a nonsmooth operator ODE. Therefore the control-to-state operator is not diﬀerentiable whenever there are biactive points present.

This example features three sets of data, such that the known global optimum has either only inactive points, only strongly active points, or only biactive points.

### Variables & Notation

#### Unknowns

In contrast to the original damage model in Dimitrijevic and Hackl [2008], the situation is simpliﬁed by considering only scalar valued functions. The unknown functions are

#### Given Data

For the construction of analytically known solutions, the following functions need to be speciﬁed:

These data will be speciﬁed below in the supplementary materials section, such that globally optimal solutions with the desired properties are known. In addition, the following data are needed to specify all variables in the problem:

In this example, the diﬃculty lies in the potential non-diﬀerentiability of the $max\left\{0,\cdot \right\}$ operator in the evolution law of the local damage variable. To describe the area where diﬀerentiability is critical, we deﬁne three sets:

### Problem Description

We use a standard tracking type functional:

$\begin{array}{llll}\hfill \text{Minimize}\phantom{\rule{1em}{0ex}}& \frac{1}{2}\parallel u-{u}_{d}{\parallel }_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}+\frac{1}{2}\parallel \phi -{\phi }_{d}{\parallel }_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{1em}{0ex}}+\frac{1}{2}\parallel d-{d}_{d}{\parallel }_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}+\frac{1}{2}\parallel \ell -{\ell }_{d}{\parallel }_{{L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\right)}^{2}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The above minimization is subject to the constraints

The functions ${e}_{1}$, ${e}_{2}$, and ${d}_{0}$ in the system are inserted to allow the construction of a known solution for the optimization problem. The function ${e}_{1}$ can be interpreted as a given (uncontrolled) load and ${d}_{0}$ as initial local damage.

### Optimality System

The control-to-state operator is, in general, not diﬀerentiable. Consequently, standard methods for the derivation of necessary optimality conditions using adjoint techniques fail. If, however, the biactive set $B\left(t\right)$ is a set of measure zero a.e. in $\left(0,T\right)$, then the directional derivative of the control-to-state map is linear, and adjoint states can be introduced. In this case, ﬁrst order necessary optimality conditions for the above problem in a point $\left(u,\phi ,d,\ell \right)$ are given by the existence of ${p}_{1}$, ${p}_{2}$, ${p}_{3}$ such that the following adjoint system

$\begin{array}{lll}\hfill {p}_{1}+\ell -{\ell }_{d}=0& \phantom{\rule{2em}{0ex}}& \hfill \end{array}$

holds a.e. in $\left(0,T\right)×\Omega$.

### Supplementary Material

In this section, we provide three diﬀerent sets of data, leading to the three distinct cases featuring only inactive points, only active points, and only biactive points. For all three settings, we deﬁne the auxiliary functions

$\begin{array}{llll}\hfill {e}_{1}\left(x,t\right)& =-g\left({\phi }_{d}\left(x,t\right)\right)\phantom{\rule{0.3em}{0ex}}△{u}_{d}\left(x,t\right)-{g}^{\prime }\left({\phi }_{d}\left(x,t\right)\right)\phantom{\rule{0.3em}{0ex}}\nabla {u}_{d}\left(x,t\right)\cdot \nabla {\phi }_{d}\left(x,t\right)-{\ell }_{d}\left(x,t\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {e}_{2}\left(x,t\right)& =-\alpha \phantom{\rule{0.3em}{0ex}}△{\phi }_{d}\left(x,t\right)+\beta \phantom{\rule{0.3em}{0ex}}\left({\phi }_{d}\left(x,t\right)-{d}_{d}\left(x,t\right)\right)+\frac{1}{2}{g}^{\prime }\left({\phi }_{d}\left(x,t\right)\right)\phantom{\rule{0.3em}{0ex}}|\nabla {u}_{d}\left(x,t\right){|}^{2},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {d}_{0}\left(x\right)& ={d}_{d}\left(x,0\right).\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

In virtue of this construction, the unique global optimum is $u={u}_{d}$, $\phi ={\phi }_{d}$, $d={d}_{d}$, and $\ell ={\ell }_{d}$ and consequently the adjoint state is $\left({p}_{1},{p}_{2},{p}_{3}\right)\equiv \left(0,0,0\right)$ in all three cases. Clearly, the corresponding value of the objective is zero.

#### Case 1: Only inactive points ($I\left(t\right)=\Omega$)

$\begin{array}{llll}\hfill {\ell }_{d}\left(x,t\right)& =sin\left(\pi {x}_{1}\right)sin\left(\pi {x}_{2}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {u}_{d}\left(x,t\right)& =\frac{1}{2{\pi }^{2}}sin\left(\pi {x}_{1}\right)sin\left(\pi {x}_{2}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\phi }_{d}\left(x,t\right)& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill d\left(x,t\right)& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

#### Case 2: Only strongly active points ($A\left(t\right)=\Omega$)

$\begin{array}{llll}\hfill {\ell }_{d}\left(x,t\right)& =sin\left(\pi {x}_{1}\right)sin\left(\pi {x}_{2}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {u}_{d}\left(x,t\right)& =\frac{1}{2{\pi }^{2}}sin\left(\pi {x}_{1}\right)sin\left(\pi {x}_{2}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\phi }_{d}\left(x,t\right)& =\frac{1}{2}\left({x}_{1}^{2}+{x}_{2}^{2}\right)-\frac{1}{3}\left({x}_{1}^{3}+{x}_{2}^{3}\right)+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {d}_{d}\left(x,t\right)& =\left(\right{\phi }_{d}\left(x,t\right)-\frac{r}{\beta }\left)\right\phantom{\rule{0.3em}{0ex}}\left(\right1-{e}^{-\frac{\beta }{\delta }t}\left)\right\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ Notice that in this case, the function ${e}^{-\frac{\beta }{\delta }t}={e}^{-1{0}^{4}t}\approx 0$.

#### Case 3: Only biactive points ($B\left(t\right)=\Omega$)

$\begin{array}{llll}\hfill {\ell }_{d}\left(x,t\right)& =sin\left(\pi {x}_{1}\right)sin\left(\pi {x}_{2}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {u}_{d}\left(x,t\right)& =\frac{1}{2{\pi }^{2}}sin\left(\pi {x}_{1}\right)sin\left(\pi {x}_{2}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\phi }_{d}\left(x,t\right)& =\frac{1}{2{\pi }^{2}}cos\left(\pi {x}_{1}\right)cos\left(\pi {x}_{2}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {d}_{d}\left(x,t\right)& ={\phi }_{d}\left(x,t\right)-\frac{r}{\beta }\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

### References

B. J. Dimitrijevic and K. Hackl. A method for gradient enhancement of continuum damage models. Technische Mechanik, 28(1):43–52, 2008. URL http://www.uni-magdeburg.de/ifme/zeitschrift_tm/2008_Heft1/05_Dimitrievich_Hackl.pdf.