Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition.

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Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition.

Laurent CORDIER

Michel BERGMANN & Jean-Pierre BRANCHER

[email protected]

Laboratoire d’ Énerg étique et de M écanique Th éorique et Appliqu ée UMR 7563 (CNRS - INPL - UHP)

ENSEM - 2, avenue de la For ˆet de Haye BP 160 - 54504 Vandoeuvre Cedex, France

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Introduction Configuration and numerical method

Two dimensional flow around a circular cylinder at Re = 200 Viscous, incompressible and Newtonian fluid

Cylinder oscillation with a tangential sinusoidal velocity γ(t) γ(t) = VT

U^∞ = Asin(2πStft)

000000

111111 θ x y

0 Γe

Γsup

Γs

Γinf

Γc

Ω

U^∞

D VT(t)

Find the control parameters c = (A, St

_f

)

^T

such that the mean drag coefficient is minimized

hCDiT = 1 T

Z T

0

Z 2π

0

2p nx R dθ dt − 1 T

Z T

0

Z 2π

0

2 Re

∂u

∂x nx + ∂u

∂y ny

R dθ dt,

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Introduction Parametric study

0.9929 6.9876

1.3928

1.3878

1.3928

1.3878 1.3905

1.1728

1.0320 3.6306

2.5516

1.0529

1.3526 1.2327

0 0.2 0.4 0.6 0.8 1

0 1 2 3 4 5 6

Amplitude

Strouhal

Variation of the mean drag coefficient with A and St_f.

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Introduction Mean drag coefficient & steady unstable base flow

25 50 75 100 125 150 175 200

0.75 1 1.25 1.5 1.75 2 2.25 2.5

hCDiT

Re

hCDiT

C_D^base

C_D⁰ Natural flow

Base 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 laminar regime. Phys. Fluids, 14(2), pp. 810-826.

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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

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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 h|(u,Φ)|²i s.t. kΦk² = 1.

◮ Snapshots method, Sirovich (1987) :

Z

T

C(t, t^′)a⁽ⁿ⁾(t^′)dt^′ = λ⁽ⁿ⁾a⁽ⁿ⁾(t).

◮ Optimal convergence in L² norm (energy)of Φ(X)

⇒ Dynamical order reduction is possible.

Coordinate axis

Coordinateaxis Φ₁

Φ₂

Original point cloud

Point cloud, mean shifted (centered around origin)

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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.

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ROM Non-equilibrium modes (Noack et al. 2004)

◮ Necessity for a given reference flow to introduce new modes : either new operating conditions or shift-modes

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00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000

11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111

Controlled space Dynamics I

Dynamics II φ^I₀

φ^I₂

φ^II₁ φ^II₂

φ^I₁

φ^II₀ 0

φ^I_neq^→^II

Fig. : Schematic representation of a dynamical transition with a non-equilibrium mode

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ROM A robust POD surrogate for the drag coefficient

◮ POD approximations consistent with our approach :

U(x, t) = (u, v, p)^T =

N

X

i=0

ai(t)φi(x)

| {z }

Galerkin modes +

N+M

X

i=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 modes

Dynamics of the reference flow I

i = 1

POD ROM

Temporal dynamics of the modes (eventually, the mode i = 0 is solved then a0 ≡ a0(t))

i = 2

· · · i = N non-equilibrium modes

Inclusion of new operating conditions II, III, IV,· · ·

i = N + 1

· · ·

i = N + M

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ROM Galerkin projection

◮ Galerkin projection of NSE onto the POD basis :

φ_i, ∂u

∂t + ∇ · (u ⊗ u)

=

φ_i, −∇p + 1

Re∆u

.

◮ Reduced order dynamical system where only (N + M + 1) (≪ NP OD) modes are retained (state equations) :

d ai(t) d t =

N+M

X

j=0

Bij aj(t) +

N+M

X

j=0

N+M

X

k=0

Cijk aj(t)ak(t)

+ Di

d γ

d t + Ei +

N+M

X

j=0

Fij aj(t)

!

γ(c, t) + Giγ²(c, t), ai(0) = (U(x, 0), φ_i(x)).

Bij, Cijk, Di, Ei, Fij and Gi depend on φ_i, U_c and Re.

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Surrogate drag function and model objective function

◮ Drag operator :

C^D :R³ → R

u 7→ 2

Z 2π

0

u3nx − 1 Re

∂u1

∂x nx − 1 Re

∂u1

∂y ny

R dθ,

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◮ Surrogate drag function :

CfD(t) = a0(t)N0 +

N+M

X

i=N+1

ai(t)Ni

| {z }

evolution of the mean drag +

N

X

i=1

ai(t)Ni

| {z }

fluctuations CD^′ (t)

with Ni = C^D(φ_i).

◮ Model objective function :

m = hC^fD(t)iT = 1 T

Z T

0

a₀(t)N₀ +

N+M

X

i=N+1

ai(t)Ni

!

dt.

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Surrogate drag function ^{Test case} A = 2 ^and St = 0.5

◮ Comparison of real drag coefficient CD (symbols) and model function C^fD

(lines) at the design parameters.

0 5 10 15

1.045 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 CD

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Robustness of the model objective function ^{Test case} A = 2 ^and St = 0.5

0.4 0.5 0.6

1.5 2 2.5

1.293 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 0.6

1.5 2 2.5

1.221 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.

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Trust-Region Proper Orthogonal Decomposition Generalities

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 assurance of :

1. Automatic restriction of the range of validity

2. Global convergence

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00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000 0000000

1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111 1111111

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.

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Trust-Region Proper Orthogonal Decomposition (TRPOD) ^Algorithm

Initialization :c₀, Navier-Stokes resolution,J0. k = 0.

∆0

Construction of the POD ROM and evaluation of the model

objective function mk

Solve the optimality system based on the POD ROM under the constraints ∆k

c_k+1 andmk+1

Solve the

Navier-Stokes equations and Jk+1 Evaluation of the performance (J − J )/(m −m )

poor medium good

∆k+1 <∆k

∆k+1 . ∆k

∆k+1 > ∆k

k = k+ 1 k =k+ 1 c₀

c_k c_k+1 c_k+1

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TRPOD Numerical results

Initial control parameters : A = 1.0 et St = 0.2

0 0.2 0.4 0.6 0.8 1

0 1 2 3 4 5 6

A

St

0 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

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TRPOD Numerical results

0 0.2 0.4 0.6 0.8 1

0 1 2 3 4 5 6

A

St

0 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

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TRPOD Numerical results

0 0.2 0.4 0.6 0.8 1

0 1 2 3 4 5 6

A

St

0 5 10 15

0.975 1 1.025 1.05

Jk

4 resolutions of the Navier-Stokes equations

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TRPOD Numerical results

0 0.2 0.4 0.6 0.8 1

0 1 2 3 4 5 6

A

St

0 5 10 15

0.9 1 1.1 1.2 1.3 1.4

Jk

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TRPOD Drag coefficient

◮ Optimal control law : γopt(t) = Asin(2πSt t) avec A = 4.25 et St = 0.74

0 25 50 75 100

0.8 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)

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TRPOD Vorticity contour plots

Uncontrolled flow, γ = 0.

Controlled flow,γ = γ_opt. Fig. : Iso-values of vorticity ωz.

Controlled flow : near wake strongly unsteady, far wake (after 5 diameters) steady and symmetric → steady unstable base flow

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Numerical costs ^Discussion

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 reactualization of the POD ROM :

⇒ 25% drag reduction for A = 2.2 and St = 0.53.

Reduction costs compared to NSE : CPU time : 100

Memory 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 : 4

Memory storage : 400 but global convergence.

֒→ "Optimal" control of 3D flows becomes possible !

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Conclusions and perspectives

Conclusions on TRPOD

Important relative drag reduction : more than 30% of relative drag reduction

Global convergence : mathematical assurance that the solution is identical 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 Perspectives

Test mode interpolations for controlled flows (Morzynski and Tadmor’s talks, GAMM 2006)

Test other reduced basis method than classical POD

Centroidal Voronoi Tessellations (Gunzburguer, 2004) : "intelligent"

sampling in the control parameter space

Model-based POD (Willcox, 2005) : modify the definitions of the POD modes

Control of the cylinder wake in the laminar regime by Trust-Region Proper Orthogonal Decomposition.