OPTPDE - A Collection of Problems in PDE Constrained Optimization

Welcome to the OPTPDE Problem Collection

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Keywords: analytic solution

Global classification: nonlinear-quadratic

Functional: convex quadratic

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

Description (pdf)
Bibliography (bib)
Problemdata as MATLAB files (tar.gz)

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

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 The damage model is based on Dimitrijevic and Hackl [2008], the optimization problem is currently unpublished and provided by M. Holtmannspötter.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

\begin{aligned} ℓ & \in L^{2} ((0, T), L^{2} (Ω)) & u & \in L^{2} ((0, T), H_{0}^{1} (Ω)) & state variable (displacement) φ & \in L^{2} ((0, T), H^{1} (Ω)) & state variable (nonlocal damage) d & \in H^{1} ((0, T), L^{2} (Ω)) & state variable (local damage) . \end{aligned}

Given Data

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

\begin{aligned} ℓ_{d} & \in L^{2} ((0, T), L^{2} (Ω)) & desired control \\ u_{d} & \in L^{2} ((0, T), L^{2} (Ω)) & desired displacement \\ φ_{d} & \in L^{2} ((0, T), L^{2} (Ω)) & desired nonlocal damage \\ d_{d} & \in L^{2} ((0, T), L^{2} (Ω)) & desired local damage \\ e_{1} & \in L^{2} ((0, T), L^{2} (Ω)) & e_{2} & \in L^{2} ((0, T), H^{1} (Ω)) & auxiliary forcing in the nonlocal damage d_{0} & \in L^{2} (Ω) & initial local damage . \end{aligned}

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:

\begin{aligned} T & = 1 & ﬁnal time \\ Ω & = {(0, 1)}^{2} & computational domain \\ α & = 1 & nonlocal damage parameter \\ β & = 1 0^{4} & penalty parameter for coupling of d and φ \\ δ & = 1 & viscosity parameter in the damage evolution \\ r & = 1 & fracture toughness \\ η & = 1 0^{- 2} & regularization parameter to avoid material degeneration \\ g (z) & = (1 - η) e^{- z} + η & C^{2} function satisfying g (0) = 1 and \lim_{z \to \infty} g (z) = η \\ g^{'} (z) & = η e^{- z} & derivative of the above function \\ g^{″} (z) & = - η e^{- z} & derivative of the above function \end{aligned}

Additional Notation

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

\begin{array}{l} A (t) & = {x \in Ω : - β (d (x, t) - φ (x, t)) - r > 0} & the strongly active set \\ B (t) & = {x \in Ω : - β (d (x, t) - φ (x, t)) - r = 0} & the biactive set \\ I (t) & = {x \in Ω : - β (d (x, t) - φ (x, t)) - r < 0} & the inactive set \end{array}

Problem Description

We use a standard tracking type functional:

\begin{array}{l} Minimize & \frac{1}{2} ∥ u - u_{d} ∥_{L^{2} (0, T; L^{2} (Ω))}^{2} + \frac{1}{2} ∥ φ - φ_{d} ∥_{L^{2} (0, T; L^{2} (Ω))}^{2} \\ + \frac{1}{2} ∥ d - d_{d} ∥_{L^{2} (0, T; L^{2} (Ω))}^{2} + \frac{1}{2} ∥ ℓ - ℓ_{d} ∥_{L^{2} (0, T; L^{2} (Ω))}^{2} . \end{array}

The above minimization is subject to the constraints

\begin{array}{l} \int_{Ω} g (φ (t)) \nabla u (t) \cdot \nabla v d x = \int_{Ω} (ℓ (t) + e_{1} (t)) \forall v \in H_{0}^{1} (Ω) \int_{Ω} α \nabla φ (t) \cdot \nabla ψ + β & + \frac{1}{2} g^{'} (φ (t)) \nabla u (t) \cdot \nabla u (t) = \int_{Ω} β & \forall ψ & \in H^{1} (Ω) ḋ (t) & = \frac{1}{δ} \max {- β (d (t) - φ (t)) - r, 0} & a.e. in Ω for almost all t \in (0, T), as well as d (0) & = d_{0} & a.e. in Ω . 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 (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, ℓ) are given by the
existence of p_{1}, p_{2}, p_{3} such
that the following adjoint system \begin{array}{l} \int_{Ω} g (φ (t)) \nabla p_{1} (t) \cdot \nabla v d x + \int_{Ω} g^{'} (φ (t)) = \int_{Ω} (u (t) - u_{d} (t)) & \forall v & \in H_{0}^{1} (Ω) \int_{Ω} α \nabla p_{2} (t) \cdot \nabla ψ + β & - \frac{1}{δ} χ_{A (t)} + \int_{Ω} \frac{1}{2} g^{″} (φ (t)) = \int_{Ω} (φ (t) - φ_{d} (t)) & \forall ψ & \in H^{1} (Ω) - \dot{p_{3}} (t) & = β & a.e. in Ω for almost all t \in (0, T), as well as p_{3} (T) & = 0 & a.e. in Ω . Moreover, the gradient equation \begin{array}{l} p_{1} + ℓ - ℓ_{d} = 0 \end{array} holds a.e. in (0, T) \times 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 \begin{array}{l} e_{1} (x, t) & = - g (φ_{d} (x, t)) △ u_{d} (x, t) - g^{'} (φ_{d} (x, t)) \nabla u_{d} (x, t) \cdot \nabla φ_{d} (x, t) - ℓ_{d} (x, t), \\ e_{2} (x, t) & = - α △ φ_{d} (x, t) + β (φ_{d} (x, t) - d_{d} (x, t)) + \frac{1}{2} g^{'} (φ_{d} (x, t)) | \nabla u_{d} (x, t) |^{2}, \\ d_{0} (x) & = d_{d} (x, 0) . \end{array} In virtue of this construction, the unique global optimum is u = u_{d}, φ = φ_{d}, d = d_{d}, and ℓ = ℓ_{d} and consequently
the adjoint state is (p_{1}, p_{2}, p_{3}) \equiv (0, 0, 0) in all three cases. Clearly, the corresponding value of the objective is zero. Case 1: Only inactive points (I (t) = Ω) \begin{array}{l} ℓ_{d} (x, t) & = \sin (π x_{1}) \sin (π x_{2}) \\ u_{d} (x, t) & = \frac{1}{2 π^{2}} \sin (π x_{1}) \sin (π x_{2}) \\ φ_{d} (x, t) & = 0 \\ d (x, t) & = 0 \end{array} Case 2: Only strongly active points (A (t) = Ω) \begin{array}{l} ℓ_{d} (x, t) & = \sin (π x_{1}) \sin (π x_{2}) \\ u_{d} (x, t) & = \frac{1}{2 π^{2}} \sin (π x_{1}) \sin (π x_{2}) \\ φ_{d} (x, t) & = \frac{1}{2} (x_{1}^{2} + x_{2}^{2}) - \frac{1}{3} (x_{1}^{3} + x_{2}^{3}) + 1 \\ d_{d} (x, t) & = (φ_{d} (x, t) - \frac{r}{β}) (1 - e^{- \frac{β}{δ} t}) \end{array} Notice that in this case, the function e^{- \frac{β}{δ} t} = e^{- 1 0^{4} t} \approx 0 . Case 3: Only biactive points (B (t) = Ω) \begin{array}{l} ℓ_{d} (x, t) & = \sin (π x_{1}) \sin (π x_{2}) \\ u_{d} (x, t) & = \frac{1}{2 π^{2}} \sin (π x_{1}) \sin (π x_{2}) \\ φ_{d} (x, t) & = \frac{1}{2 π^{2}} \cos (π x_{1}) \cos (π x_{2}) \\ d_{d} (x, t) & = φ_{d} (x, t) - \frac{r}{β} \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 . Legal notice Privacy statement \end{array} \end{array}

Departmentof Mathematics

Welcome to the OPTPDE Problem Collection

mpccdist3 details:

mpccdist3 description:

Introduction

Variables & Notation

Unknowns

Given Data

Additional Notation

Problem Description

Optimality System

Supplementary Material

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

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

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

References

Departmentof Mathematics

Welcome to the OPTPDE Problem Collection

mpccdist3 details:

mpccdist3 description:

Introduction

Variables & Notation

Unknowns

Given Data

Additional Notation

Problem Description

Optimality System

Supplementary Material

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

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

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

References

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

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

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