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Keywords: analytic solution
Global classification: nonlinear-quadratic
Functional: convex quadratic
Geometry: easy, fixed
Design: coupled via volume data
- quasi-linear other operator of order 2.
- Defined on a 2-dim domain in 2-dim space
- Time dependent.
Submitted on 2016-07-01 by Marita Holtmannspötter. Published on 2016-12-23
We study the optimal control of a particular gradient enhanced damage model. The damage model is based on Dimitrijevic and Hackl , the optimization problem is currently unpublished and provided by M. Holtmannspötter.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.
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 operator in the evolution law of the local damage variable. To describe the area where diﬀerentiability is critical, we deﬁne three sets:
We use a standard tracking type functional:
The above minimization is subject to the constraints
The functions , , and in the system are inserted to allow the construction of a known solution for the optimization problem. The function can be interpreted as a given (uncontrolled) load and as initial local damage.
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 is a set of measure zero a.e. in , 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 are given by the existence of , , such that the following adjoint system
Moreover, the gradient equation
holds a.e. in .
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
In virtue of this construction, the unique global optimum is , , , and and consequently the adjoint state is in all three cases. Clearly, the corresponding value of the objective is zero.
Notice that in this case, the function .
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.