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Contrôle optimal par réduction de modèle PODet méthode à région de confiance du sillage laminaire d’un
cylindre circulaire.
Michel Bergmann
Laurent Cordier & Jean-Pierre Brancher
Laboratoire d’Energetique et de Mecanique Theorique et Appliquee
UMR 7563 (CNRS - INPL - UHP)
ENSEM - 2, avenue de la Foret de Haye
BP 160 - 54504 Vandoeuvre Cedex, France
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.1/22
Introduction Configuration and numerical method
Two dimensional flow around a circular cylinder at Re = 200Viscous, incompressible and Newtonian fluidCylinder oscillation with a tangential sinusoidal velocity γ(t)
γ(t) =VT
U∞
= A sin(2πStf t)
θ
x
y
0
Γe
Γsup
Γs
Γinf
Γc
Ω
U∞
D
VT (t)
Find the control parameters c = (A, Stf )T such that the mean
drag coefficient is minimized
〈CD〉T =1
T
Z T
0
Z 2π
0
2 p nx R dθ dt −1
T
Z T
0
Z 2π
0
2
Re
∂u
∂xnx +
∂u
∂yny
R dθ dt,
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.2/22
Introduction Parametric study
0.9929
6.9876
1.3928
1.3878
1.3928
1.38781.3905
1.1728
1.03203.6306
2.55161.0529
1.35261.2327
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
Am
plitu
de
StrouhalVariation of the mean drag coefficient with A and Stf .Numerical minimum
Amin, Stfmin
= (4.3, 0.74).
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.3/22
Introduction Mean drag coefficient & steady unstable base flow
25 50 75 100 125 150 175 2000.75
1
1.25
1.5
1.75
2
2.25
2.5〈C
D〉 T
Re
〈CD〉T
CbaseD
C0D
Natural flowBase flow
Fig. : Variation with the Reynolds number of the mean drag coefficient. Contributions and
corresponding flow patterns of the base flow and unsteady flow.
Protas, B. et Wesfreid, J.E. (2002) : Drag force in the open-loop control of the cylinder wake in the laminarregime. Phys. Fluids, 14(2), pp. 810-826.
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.4/22
Reduced Order Model (ROM) and optimization problems
x∆
Initialization
Optimization
Optimization on simplified model
a(x), grad a(x)
f(x), grad f(x)
Recourse to detailed model (TRPOD)High−fidelity model
Approximation model
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.5/22
Reduced Order Model (ROM) Proper Orthogonal Decomposition (POD)
Introduced in turbulence by Lumley (1967).
Method of information compression
Look for a realization Φ(X) which is clo-ser, in an average sense, to realizations u(X).(X = (x, t) ∈ D = Ω × R
+)
Φ(X) solution of the problem :
maxΦ
〈|(u,Φ)|2〉 s.t. ‖Φ‖2 = 1.
Snapshots method, Sirovich (1987) :ZT
C(t, t′)a(n)(t′) dt′ = λ(n)a(n)(t).
Optimal convergence in L2 norm (energy)ofΦ(X)⇒ Dynamical order reduction is possible.
Coordinate axis
Coo
rdin
ate
axis
Φ1
Φ2
Original point cloud
Point cloud, mean shifted(centered around origin)
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.6/22
ROM Parameter sampling in an optimization setting
General configuration. Ideal sampling.
Unsuitable sampling. Unsuitable sampling.
Discussion of parameter sampling in an optimization setting (from Gunzburger, 2004).−−−− path to optimizer using full system, initial values, optimal values, and •
parameter values used for snapshot generation.
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.7/22
ROM A simple configuration, a rich dynamical behavior
Stf = 0, 1 CD = 4, 25. Stf = 0, 2 CD = 2, 24.
Stf = 0, 3 CD = 1, 57. Stf = 0, 4 CD = 1, 25.
Stf = 0, 5 CD = 1, 09. Stf = 0, 6 CD = 1, 02.
Stf = 0, 7 CD = 1, 03. Stf = 0, 8 CD = 1, 07.
Stf = 0, 9 CD = 1, 13. Stf = 1, 0 CD = 1, 18.
Fig. : Iso-values of the vorticity fields ωz for A = 3
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.8/22
ROM Non-equilibrium modes (Noack et al. 2004)
Necessity for a given reference flow to introduce new modes : either newoperating conditions or shift-modes
Controlled space
Dynamics I
Dynamics IIφI
0
φI2
φII1
φII2
φI1
φII0
0
φI→IIneq
Fig. : Schematic representation of a dynamical transition with a non-equilibrium mode
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.9/22
ROM A robust POD surrogate for the drag coefficient
POD approximations consistent with our approach :
U (x, t) = (u, v, p)T =
NX
i=0
ai(t) φi(x)| z Galerkin modes
+
N+MXi=N+1
ai(t) φi(x)| z non-equilibrium modes
+ γ(c, t) Uc(x)| z control function
Physical aspects Modes Dynamical aspects
actuation mode Uc predetermined dynamics
mean flow mode Um, i = 0 a0 = 1
Galerkin modesDynamics of the referenceflow I
i = 1
POD ROMTemporal dynamics of themodes (eventually, themode i = 0 is solved thena0 ≡ a0(t))
i = 2
· · ·
i = N
non-equilibrium modesInclusion of new operatingconditions II, III, IV, · · ·
i = N + 1
· · ·
i = N + M
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.10/22
ROM Galerkin projection
Galerkin projection of NSE onto the POD basis :
φi,∂u
∂t+ ∇ · (u ⊗ u)
=
φi, −∇p +
1
Re∆u
.
Reduced order dynamical system where only (N + M + 1) (≪ NPOD)
modes are retained (state equations) :
d ai(t)
d t=
N+MXj=0
Bij aj(t) +
N+MXj=0
N+MXk=0
Cijk aj(t)ak(t)
+ Di
d γ
d t+
Ei +
N+MXj=0
Fij aj(t)
!γ(c, t) + Giγ
2(c, t),
ai(0) = (U (x, 0), φi(x)).
Bij , Cijk, Di, Ei, Fij and Gi depend on φi, U c and Re.
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.11/22
Surrogate drag function and model objective functionGeneralities
Drag operator :
CD : R3 → R
u 7→ 2
Z 2π
0
u3nx −
1
Re
∂u1
∂xnx −
1
Re
∂u1
∂yny
R dθ,
(1)
Surrogate drag function :fCD(t) = a0(t)N0 +
N+MXi=N+1
ai(t)Ni| z evolution of the mean drag
+
NXi=1
ai(t)Ni| z fluctuations C′
D(t)
with Ni = CD(φi).
Model objective function :
m = 〈fCD(t)〉T =1
T
Z T
0
a0(t)N0 +
N+MXi=N+1
ai(t)Ni
!dt.
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.12/22
Surrogate drag function Test case A = 2 and St = 0.5
Comparison of real drag coefficient CD (symbols) and model function fCD
(lines) at the design parameters.
0 5 10 151.045
1.05
1.055
1.06
1.065
1.07
1.075
1.08
1.085
1.09
1.095
1.1
t
CD
&
f C D
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.13/22
Robustness of the model objective functionTest case A = 2 and St = 0.5
0.4 0.5 0.61.5
2
2.51.293
1.275
1.257
1.239
1.221
1.203
1.185
1.167
1.149
1.131
1.113
1.095
1.077
1.059
1.041
A
St(a) Real objective function J
0.4 0.5 0.61.5
2
2.51.221
1.209
1.197
1.185
1.173
1.161
1.149
1.137
1.125
1.113
1.101
1.089
1.077
1.065
1.053
A
St(b) Model objective function m
Fig. : Comparison of the real and the model objective functions associated to the mean drag coefficient.
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.14/22
Trust-Region Proper Orthogonal DecompositionGeneralities
Range of validity of the POD ROM restricted to a vicinity of the design parameters
Objective : Use ROMs to solve large-scale optimization problems with assu-rance of :
1. Automatic restriction of therange of validity
2. Global convergence
Solution
Embed the POD technique into the concept of trust-region methods :Trust-Region Proper Orthogonal Decomposition (Fahl, 2000)
Conn, A.R., Gould, N.I.M. et Toint, P.L. (2000) : Trust-region methods. SIAM, Philadelphia.
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.15/22
Trust-Region Proper Orthogonal Decomposition (TRPOD)Algorithm
Initialization : c0, Navier-Stokes resolution, J0. k = 0.
∆0
Construction of the POD ROMand evaluation of the model
objective function mk
Solve the optimality systembased on the POD ROMunder the constraints ∆k
ck+1 and mk+1
Solve theNavier-Stokes equations andestimate a new POD basis
Jk+1Evaluation of the performance
(Jk+1 − Jk)/(mk+1 − mk)
poor medium good
∆k+1 < ∆k
∆k+1 . ∆k
∆k+1 > ∆k
k = k + 1
k = k + 1
c0
ck ck+1ck+1
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.16/22
TRPOD Numerical results
Initial control parameters : A = 1.0 et St = 0.2
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
A
St0 5 10 15
0.9
1
1.1
1.2
1.3
1.4
1.5
Jk
Iteration number k
Optimal control parameters : A = 4.25 et St = 0.74
Mean drag coefficient : J = 0.993
8 resolutions of the Navier-Stokes equations
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.17/22
TRPOD Numerical results
Initial control parameters : A = 6.0 et St = 0.2
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
A
St0 5 10 15
0.5
1
1.5
2
2.5
Jk
Iteration number k
Optimal control parameters : A = 4.25 and St = 0.74
Mean drag coefficient : J = 0.993
6 resolutions of the Navier-Stokes equations
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.17/22
TRPOD Numerical results
Initial control parameters : A = 1.0 et St = 1.0
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
A
St0 5 10 15
0.9
1
1.1
1.2
1.3
1.4
Jk
Iteration number k
Optimal control parameters : A = 4.25 and St = 0.74
Mean drag coefficient : J = 0.993
5 resolutions of the Navier-Stokes equations
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.17/22
TRPOD Numerical results
Initial control parameters : A = 6.0 et St = 1.0
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
A
St0 5 10 15
0.975
1
1.025
1.05
Jk
Iteration number k
Optimal control parameters : A = 4.25 and St = 0.74
Mean drag coefficient : J = 0.993
4 resolutions of the Navier-Stokes equations
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.17/22
TRPOD Drag coefficient
Optimal control law : γopt(t) = A sin(2πSt t) avec A = 4.25 et St = 0.74
0 25 50 75 1000.8
0.9
1
1.1
1.2
1.3
1.4
1.5
γ = 0
γ(t) = γopt(t)
Base flow
CD
Time units
Relative drag reduction of 30% (J0 = 1, 4 ⇒ Jopt = 0, 99)
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.18/22
TRPOD Vorticity contour plots
Uncontrolled flow, γ = 0.
Controlled flow, γ = γopt.
Fig. : Iso-values of vorticity ωz.
Controlled flow : near wake strongly unsteady, far wake (after 5 diameters) steadyand symmetric → steady unstable base flow
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.19/22
Numerical costsDiscussion
Optimal control of NSE by He et al. (2000) :⇒ 30% drag reduction for A = 3 and St = 0.75.
Optimal control POD ROM by Bergmann et al. (2005) with no reactualizationof the POD ROM :
⇒ 25% drag reduction for A = 2.2 and St = 0.53.Reduction costs compared to NSE :
CPU time : 100Memory storage : 600
but no mathematical proof concerning the Navier-Stokes optimality.
TRPOD (this study) :⇒ More than 30% of drag reduction for A = 4.25 and St = 0.738.
Reduction costs compared to NSE :CPU time : 4Memory storage : 400
but global convergence.
→ "Optimal" control of 3D flows becomes possible !
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.20/22
Conclusions and perspectives
Conclusions on TRPODImportant relative drag reduction : more than 30% of relative dragreduction
Global convergence : mathematical assurance that the solution isidentical to the one of the high-fidelity model
TRPOD compared to NSE ⇒ important reduction of numerical costs :→ Reduction factor of the CPU time : 4→ Reduction factor of the memory storage : 400
"OPTIMAL" CONTROL OF 3D FLOWS POSSIBLE BY POD ROM
PerspectivesOptimal control of the channel flow at Reτ = 180Test other reduced basis method than classical POD
Centroidal Voronoi Tessellations (Gunzburguer, 2004) : "intelligent"sampling in the control parameter spaceBalanced POD (Rowley, 2004)Model-based POD (Willcox, CDC-ECC 2005) : modify the definitions ofthe POD modes
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.21/22
Reverse von Karman flow
Contrôle partiel (3 paramètres de contrôle)
⇒ Effet propulsif, signe écoulement moyen inversé
Fig. : Contrôle sur la partie amont : −120˚ ≤ θ ≤ 120˚
Controle optimal par reduction de modele PODet methode a region de confiance du sillage laminaire d’un cylindre circulaire. – p.22/22