Optimal Dirichlet boundary control problems of high-lift configurations with control and integral state constraints [Elektronische Ressource] / Christian John. Betreuer: Fredi Tröltzsch

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Optimal Dirichlet boundary controlproblems of high-lift configurations withcontrol and integral state constraintsvorgelegt vonDiplom-WirtschaftsmathematikerDipl.-Math.oec. Christian Johnaus BerlinVon der Fakultät II - Mathematik und Naturwissenschaftender Technischen Universität Berlinzur Erlangung des akademischen GradesDoktor der NaturwissenschaftenDr. rer. nat.genehmigte DissertationPromotionsausschuss:Vorsitzender: Prof. Dr. John M. SullivanGutachter: Prof. Dr. Fredi TröltzschGutachter: Prof. Dr. Arnd RöschTag der wissenschaftlichen Aussprache: 13.07.2011Berlin 2011D 83iiAbstractThis thesis investigates optimal control problems related to Navier-Stokesequations. We investigate two control problems related to the aerodynamicoptimization of flows around airfoils in high-lift configurations.The first issue is the steady state maximization of lift subject to restric-tions on the drag. This leads to a Dirichlet boundary control problem forthestationaryNavier-Stokesequationswithconstrainedcontrolfunctionsbe-2 2longing to L under an integral state constraint. The control space L makesit necessary to deal with very weak solutions of the Navier-Stokes equationsandbecauseofthelowregularityofcontrolandstate, wereformulatethecostfunctional and the integral state constraint.

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Optimal Dirichlet boundary control
problems of high-lift configurations with
control and integral state constraints
vorgelegt von
Diplom-Wirtschaftsmathematiker
Dipl.-Math.oec. Christian John
aus Berlin
Von der Fakultät II - Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. John M. Sullivan
Gutachter: Prof. Dr. Fredi Tröltzsch
Gutachter: Prof. Dr. Arnd Rösch
Tag der wissenschaftlichen Aussprache: 13.07.2011
Berlin 2011
D 83iiAbstract
This thesis investigates optimal control problems related to Navier-Stokes
equations. We investigate two control problems related to the aerodynamic
optimization of flows around airfoils in high-lift configurations.
The first issue is the steady state maximization of lift subject to restric-
tions on the drag. This leads to a Dirichlet boundary control problem for
thestationaryNavier-Stokesequationswithconstrainedcontrolfunctionsbe-
2 2
longing to L under an integral state constraint. The control space L makes
it necessary to deal with very weak solutions of the Navier-Stokes equations
andbecauseofthelowregularityofcontrolandstate, wereformulatethecost
functional and the integral state constraint. We derive first-order necessary
and second-order sufficient optimality conditions and treat the problem nu-
merically by direct solution of the associated nonsmooth optimality system
and additionally by an SQP-method, which convergence we proved.
The second part is based on ak-!-Wilcox98 turbulence model, describ-
ing the nonstationary behavior of the fluid closer to the reality. To deal with
thecurseofdimension, wediscussareduced-ordermodel(ROM)byadapting
a small system of ODEs to solutions computed with the full model. Based
on this ROM, we investigate an optimal control problem theoretically and
numerically.
Acknowledgment
I think it is impossible to write a doctoral dissertation without some kind of
support. I am grateful to the DFG (SFB 557 ’Control of complex turbulent
shear flows’) supporting me in the first three years financially and to the
FAZIT-Stiftung for their financial support in the last 9 months.
My first and biggest gratitude goes to my supervisor Prof. Dr. Fredi
Tröltzsch for the interesting topic, his helpful advice and his personality in
general. I thank Dr. Daniel Wachsmuth, who worked with me together for
about 2 years when i started my project at the TU Berlin, for introducing me
intothetopicofoptimalcontrolofNavier-Stokesequations, hisinterestinmy
work and many helpful inspirations. Many thanks go to my colleagues in the
research group ’Optimization on PDEs’ at the TU Berlin, especially Kristof
Altmann for introducing me into COMSOL Multiphysics, and to Prof. Dr.
Arnd Rösch for his willingness to review this thesis.
iiiFurthermore, my gratitude to B.R. Noack and M. Schlegel for the good
and successful cooperation. They introduced me into the topic of proper or-
thogonal decomposition and reduced-order modeling and supported me with
both words and deeds. I also want to thank M. Luchtenburg for explaining
me his reduced-order model, M. Nestler for his helpful assistance and the
group of Prof. Thiele, especially B. Günther and A. Carnarius, for providing
me with simulations of the URANS system.
Finally, i thank my friends and my family.
ivContents
1 Introduction 1
2 The steady-state Navier-Stokes equation 7
2.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Stokes equations . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Very weak formulation . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 More regular solutions . . . . . . . . . . . . . . . . . . 17
2.4.2 Regularity assumption . . . . . . . . . . . . . . . . . . 19
3 The optimal control problem 21
3.1 Reformulation of the boundary integrals . . . . . . . . . . . . 21
3.2 Further reformulation of the boundary integrals . . . . . . . . 22
3.3 The optimal control problem . . . . . . . . . . . . . . . . . . . 24
3.3.1 Existence of solutions . . . . . . . . . . . . . . . . . . . 24
4 Optimality conditions 29
4.1 First order necessary optimality conditions . . . . . . . . . . . 29
4.2 Second-order sufficient optimality condition . . . . . . . . . . 33
4.3 Finite-dimensional control set . . . . . . . . . . . . . . . . . . 39
4.4 sufficient optimality conditions for the finite-
dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Numerical investigations 47
5.1 One-shot approach . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1.1 Numerical results . . . . . . . . . . . . . . . . . . . . . 49
5.2 SQP-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 The SQP-method for our problem . . . . . . . . . . . . 58
5.2.2 Gradient-projection method . . . . . . . . . . . . . . . 65
5.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 66
vCONTENTS
6 Convergence of the SQP-method 69
6.1 Generalized equations . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Perturbed optimization problem . . . . . . . . . . . . . . . . . 76
6.3 A modified problem . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.1 Existence of a solution . . . . . . . . . . . . . . . . . . 79
6.3.2 Lipschitz stability . . . . . . . . . . . . . . . . . . . . . 80
6.4 Strong regularity of the original perturbed problem . . . . . . 85
7 The nonstationary case 89
7.1 Model reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 Proper orthogonal decomposition POD . . . . . . . . . . . . . 92
8 Reduced-order model (ROM) 103
8.1 A generalized model . . . . . . . . . . . . . . . . . . . . . . . 106
8.1.1 Mean-field theory . . . . . . . . . . . . . . . . . . . . . 106
8.1.2 Galerkin model . . . . . . . . . . . . . . . . 110
8.2 Modifications on the reduced-order model . . . . . . . . . . . 117
8.2.1 Filtering of the POD modes and coefficients . . . . . . 117
8.2.2 Parameter calibration . . . . . . . . . . . . . . . . . . . 124
8.3 Computation of lift . . . . . . . . . . . . . . . . . . . . . . . . 129
8.4 Numerical investigation . . . . . . . . . . . . . . . . . . . . . . 130
9 The optimal control problem 135
9.1 First-order necessary optimality conditions . . . . . . . . . . . 136
9.2 Numerical investigation . . . . . . . . . . . . . . . . . . . . . . 139
10 Conclusion 145
11 Zusammenfassung 147
viChapter 1
Introduction
In this thesis, we study optimal control problems related to Navier-Stokes
equations, describing the motion of fluid. We investigate minimizations of
functionals subject to state equations. The objective functionals depend on
the velocity field u, the pressure p and the control function g:
Our main concern is maximization the lift of an airplane, while drag
remains beyond a given threshold. Therefore, we consider an objective func-
tional J(u;p;g) characterizing the lift, the Navier-Stokes equations as state
equations and a constraint on the drag.
In given literature, there are two different approaches to get influence on
the flow around a body. The first one is the possibility of passive control.
There are several possibilities of passive control, e.g. passive blowing,
roughness and shaping. Passive noise control devices include shields of rigid
and compliant walls, mufflers, silencers, resonators and absorbent materials,
see [43] for more details. The idea behind most of them is to reduce vortices
and make the airstream around the wing smoother.
The second ansatz is active flow control, which was investigated in partic-
ular by the SFB 557 ’Control of complex turbulent shear flows’. Here, little
slits are installed on a part of the wing, where suction and blowing of air is
possible to reduce vortices.
Generally, flow control is a research field gaining a lot of interest in both
academic research and industry. It is researched by engineers (experimen-
tal and computation fluid dynamics), mathematicians (control theory and
optimization) and physicists.
In this work, the following optimal flow control problem is considered: ac-
tive control of the flow of a fluid around an aircraft by means of suction and
blowing on the wing to influence the resulting lift and drag. The associated
background of applications in fluid mechanics, active separation control, was
the subject of various papers written from an engineering point of view and
1CHAPTER 1. INTRODUCTION
has been proven to be effective in experiments as well as simulations. We
only mention [17, 19, 87, 88, 89, 112], whose considerations are close to our
setting, see Chapter 7 to 9.
The first part of this thesis deals with the steady-state problem. Here, we
assume a low Reynolds number so that we avoid the discussion of turbulence.
Furthermore, we consider a simplified control model, which is composed of
thecostfunctional, thesteady-stateNavier-Stokesequations, andconstraints
on the control function as well as the state, for a mathematical investigation.
First, the steady-state Navier-Stokes equations, describing the motion of the
fluid around the wing, are investigated and we clarify the following questions.
1. What is the best (suitable) definition of a solution of the state equation
for the formulated problem?
2. What are the requirements such that a (unique) solution for the state
equations exists?
3. What preliminary results can be found in the existing literature?
4. What regularity assumptions are needed?
5. What are the requirements such that a solution for the stated opti-
mization problem exists?
6. How are the necessary and sufficient optimality conditions formulated?
7. What is an appropriate numerical optimization method and does it
converge?
We will characterize optimality of control strategies for our setting by
necessary and sufficienty conditions.
Let us now describe the setting of our optimization problem in detail.
n
Here,
is an open bounded domain ofR ,n = 2; 3 with boundary , which
is assumed to be sufficiently smooth, more details later. The velocity field of
the fluid is denoted by u and the pressure by p. The control is a boundary
velocity field denoted byg and the viscosity parameter = 1=Re is a positive
number. Let us denote the surface measure by ds(x) or short ds. The
termr denotes the gradient and the Laplace operator, which is applied
componentwise. The resulting force of the fluid on the wing embedded in the
fluid in direction~e is given as the boundary integral
Z
@y
F = pn ~e ds; (1.0.1)~e w
@nww
2where is the boundary of the wing with its outer normal n . Since nw w w
points into the fluid, the normal n is the negative of the outer normal of thew
fluid domain
, n = n. Let the vectors ~e and ~e indicate the directionsw l d
of lift and drag. Then, we are able to calculate the lift and the drag with
the boundary integral (1.0.1) where~e or~e have to be inserted instead of~e.l d
Here, ~e is the normalized vector directed opposite to the gravity, and ~e isd l
the normalized vector in the direction opposing the main flow field.
The optimization problem is then formulated as follows: Find a control
2 nu in L ( ) that maximizes the lift
Z
@u
pn ~e ds (1.0.2)l
@n
w
subject to the steady state Navier-Stokes equations describing the motion of
the fluid
u + (ur)u +rp = 0 in

divu = 0 in

(1.0.3)
u =g on ;c
u = 0 on n ;c
the convex control constraints
g(x)2G a.e. on ; (1.0.4)c
and the maximal drag constraint
Z
@u
pn ~e dsD : (1.0.5)d 0
@n
w
The boundary is a curve satisfyingw
Z
n ds = 0: (1.0.6)
w
As shown later, the pressure is only unique up to a constant. The constraint
(1.0.6) avoids that this constant changes the objective functional arbitrar-
ily. The control acts on a part of the boundary of the body andc w
homogeneous Dirichlet boundary conditions are prescribed on the boundary
n .c
The set of admissible controls G is a bounded, convex, closed, and non-
n
empty subset of R . Furthermore, we assume 02 G, which gives us the
option to turn off the control admissible in the optimization problem. For a
more detailed discussion of such convex control constraints, we refer to [110].
3CHAPTER 1. INTRODUCTION
Let us shortly review available literature on analysis of optimal control
problems for the Navier-Stokes equations. Starting with Abergel and Temam
[2]thereisanevergrowinglistofcontributions. Letusonlymentionthework
by Gunzburger, Hou, and Svobodny [51], Gunzburger and Manservisi [52],
Hinze and Kunisch [54, 55], Kunisch and de los Reyes [28], de los Reyes and
Yousept [30], de los Reyes and Tröltzsch [29], Abergel and Casas [1], Casas
[24, 23], Tröltzsch and Roubiček [84] and Wachsmuth [103]. Finite-element
error estimates can be found in the work of Casas, Mateos and Raymond
[20]. Optimal flow control problems with state constraints were studied by
Griesse and Reyes [50], Reyes and Kunisch [80].
Thenoveltyofthefirstpartofthisthesisisthatitcombinestheuseofvery
2
low regular boundary controls, i.e. in L ( ) , and integral state constraints.
There are only a few contributions to optimal control theory using Dirichlet
2
controlsinL ,seeforinstanceKunischandVexler[61]andCasas,Mateosand
Raymond [20]. In the context of steady-state Navier-Stokes equations this
2
is a new and promising approach, since the use of L -controls yields localiz-
1=2
able optimality conditions, whereas the use of, for instance, H ( ) -controls
yields optimality conditions containing non-local boundary operators.
2
In view of the low L -regularity of the controls, the boundary integrals
(1.0.2) and (1.0.5) are no longer well-defined, since the velocity field u is not
regular enough to admit traces on the boundary. Therefore, we transform the
boundaryintegralsintovolumeintegralsleadingincaseofthedragconstraint
to a non-standard mixed control-state constraint, see Section 3.2 below.
As it is well-known, the steady-state Navier-Stokes equations are solvable
in suitable spaces. If the data and/or the Reynolds number 1= are small
enough then the solution will be unique. To judge whether this condition is
fulfilled in a concrete application is a delicate issue in particular in the case of
inhomogeneous boundary conditions, see the discussion in the monograph of
Galdi [44]. Hence, instead of assuming smallness of the data, we assume non-
singularity at the optimal control, which is equivalent to unique solvability
of a certain linearized equation, see Section 2.4.2.
By assuming the existence of a linearized Slater point to the state con-
straint (1.0.5), we are able to prove first-order necessary optimality condi-
tions, see Section 4.1. For the special case of smooth controls, the resulting
optimality system simplifies considerably, see Section 4.3. Furthermore, we
state a second-order sufficient optimality condition for the problem under
consideration. The first part of this thesis is complemented by numerical
experiments on a high-lift configuration. One numerical approach was to
solve the associated nonsmooth first-order optimality system of two coupled
Navier-Stokes equations. At the end of this topic, we also implemented an
SQP method with a penalty term in the cost functional to handle the integral
4