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