mpccdist3 details:

Keywords: analytic solution

Global classification: nonlinear-quadratic

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



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 The damage model is based on Dimitrijevic and Hackl [2008], the optimization problem is currently unpublished and provided by M. Holtmannspötter.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

L2((0,T),L2(Ω)) u L2((0,T),H 01(Ω))state variable (displacement) φ L2((0,T),H1(Ω)) state variable (nonlocal damage) d H1((0,T),L2(Ω)) state variable (local damage).

Given Data

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

d L2((0,T),L2(Ω)) desired control ud L2((0,T),L2(Ω)) desired displacement φd L2((0,T),L2(Ω)) desired nonlocal damage dd L2((0,T),L2(Ω)) desired local damage e1 L2((0,T),L2(Ω)) e2 L2((0,T),H1(Ω))auxiliary forcing in the nonlocal damage d0 L2(Ω) initial local damage.

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:

T = 1 final time Ω = (0,1)2 computational domain α = 1 nonlocal damage parameter  β = 104 penalty parameter for coupling of d and φ δ = 1 viscosity parameter in the damage evolution r = 1 fracture toughness η = 102 regularization parameter to avoid material degeneration g(z) = (1 η)ez + ηC2 function satisfying g(0) = 1 and  lim zg(z) = η g(z) = ηez derivative of the above function g(z) = ηez derivative of the above function

Additional Notation

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

A(t) = {x Ω : β(d(x,t) φ(x,t)) r > 0} the strongly active set B(t) = {x Ω : β(d(x,t) φ(x,t)) r = 0} the biactive set I(t) = {x Ω : β(d(x,t) φ(x,t)) r < 0} the inactive set

Problem Description

We use a standard tracking type functional:

Minimize 1 2u udL2(0,T;L2(Ω))2 + 1 2φ φdL2(0,T;L2(Ω))2 + 1 2d ddL2(0,T;L2(Ω))2 + 1 2 dL2(0,T;L2(Ω))2.

The above minimization is subject to the constraints

Ωg(φ(t))u(t) vdx =Ω((t) + e1(t))vdx v H01(Ω) Ωαφ(t) ψ + βφ(t)ψ + 1 2g(φ(t))u(t) u(t)ψdx =Ωβd(t)ψdx +Ωe2(t)ψdx ψ H1(Ω) (t) = 1 δmax{β(d(t) φ(t)) r,0} a.e. in Ω for almost all t (0,T), as well as d(0) = d0 a.e. in Ω.

The functions e1, e2, and d0 in the system are inserted to allow the construction of a known solution for the optimization problem. The function e1 can be interpreted as a given (uncontrolled) load and d0 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(t) is a set of measure zero a.e. in (0,T), 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 (u,φ,d,) are given by the existence of p1, p2, p3 such that the following adjoint system

Ωg(φ(t))p1(t) vdx +Ωg(φ(t))p 2(t)u(t) vdx =Ω(u(t) ud(t))vdx v H01(Ω) Ωαp2(t) ψ + βp2(t) 1 δχA(t)p3(t)ψ + g(φ(t))u(t) p 1(t)ψdx +Ω1 2g(φ(t))p2(t)u(t) u(t)ψdx =Ω(φ(t) φd(t))ψdx ψ H1(Ω) p3 ̇(t) = βp2(t) 1 δχA(t)p3(t) + d(t) dd(t) a.e. in Ω for almost all t (0,T), as well as p3(T) = 0 a.e. in Ω.

Moreover, the gradient equation

p1 + d = 0

holds a.e. in (0,T) × Ω.

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

e1(x,t) = g(φd(x,t))ud(x,t) g(φ d(x,t))ud(x,t) φd(x,t) d(x,t), e2(x,t) = αφd(x,t) + β(φd(x,t) dd(x,t)) + 1 2g(φ d(x,t))|ud(x,t)|2, d0(x) = dd(x,0).

In virtue of this construction, the unique global optimum is u = ud, φ = φd, d = dd, and = d and consequently the adjoint state is (p1,p2,p3) (0,0,0) in all three cases. Clearly, the corresponding value of the objective is zero.

Case 1: Only inactive points (I(t) = Ω)

d(x,t) = sin(πx1)sin(πx2) ud(x,t) = 1 2π2 sin(πx1)sin(πx2) φd(x,t) = 0 d(x,t) = 0

Case 2: Only strongly active points (A(t) = Ω)

d(x,t) = sin(πx1)sin(πx2) ud(x,t) = 1 2π2 sin(πx1)sin(πx2) φd(x,t) = 1 2(x12 + x 22) 1 3(x13 + x 23) + 1 dd(x,t) = φd(x,t) r β1 eβ δ t Notice that in this case, the function eβ δ t = e104t 0.

Case 3: Only biactive points (B(t) = Ω)

d(x,t) = sin(πx1)sin(πx2) ud(x,t) = 1 2π2 sin(πx1)sin(πx2) φd(x,t) = 1 2π2 cos(πx1)cos(πx2) dd(x,t) = φd(x,t) r β

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.