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Optimal rotary control of the cylinder wake using POD Reduced Order Model Michel Bergmann, Laurent Cordier & Jean-Pierre Brancher Laurent.Cordier@ensem.inpl-nancy.fr Laboratoire d Energ etique et de M ecanique Th eorique et

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Optimal rotary control of the cylinder wake using POD Reduced Order Model

Michel Bergmann, Laurent Cordier & Jean-Pierre Brancher

Laurent.Cordier@ensem.inpl-nancy.fr

Laboratoire d’ ´ Energ´ etique et de M´ ecanique Th´ eorique et Appliqu´ ee (LEMTA) UMR 7563 (CNRS - INPL - UHP) ENSEM - 2, avenue de la Forˆ et de Haye BP 160 - 54504 Vandoeuvre Cedex, France

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Outline

I - Flow configuration and numerical methods II - Optimal control approach III - Proper Orthogonal Decomposition (POD) IV - Reduced Order Model of the cylinder wake (ROM) V - Optimal control formulation applied to the ROM VI - Results of POD ROM VII - Nelder-Mead Simplex method Conclusions and perspectives

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Motivations Cylinder wake flow ?

Prototype configuration of separated flow Experimental study of Tokumaru and Dimotakis (JFM 1991)Re = 15000 ◮ Unforced flow ◮ Forced flow = ⇒ 80% of drag reduction

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I - Configuration and numerical method

Two dimensional flow around a circular cylinder at Re = 200 Viscous, incompressible and Newtonian fluid ∇ · u = 0 ; ∂u ∂t + (u · ∇) u = −∇P + 1 Re∆ u Cylinder oscillation with a tangential velocity γ(t)

  • ✁
✁ ✁ ✁ ✁ ✁

Γe Γsup Γs Γinf Γc D γ(t) x y U ∞

Control parameter : α(t) = γ(t) U∞ = R ˙ θ(t) U∞ = Tangential velocity Upstream velocity

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I - Configuration and numerical method

Fractional step method in time (pressure correction) Finite Element Method (FEM) in space (P1, P1) Numerical domain Ω = {−10 ≤ x ≤ 20 ; −10 ≤ y ≤ 10} ; D = 1 Mesh : 25042 triangles, 12686 vertices ◮ Numerical code written by M.Braza (IMFT-EMT2) & D.Ruiz (ENSEEIHT)

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I - Configuration and numerical method

Iso pressure at t = 100.

80 90 100 110 120

  • 0.5

0.5 1 1.5

time units CD, CL CD CL

Aerodynamic coefficients. Iso vorticity at t = 100. Authors St CD Braza et al. (1986) 0.2000 1.4000 Henderson et al. (1997) 0.1971 1.3412 He et al. (2000) 0.1978 1.3560 this study 0.1983 1.3972 Strouhal number and drag coefficient.

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II - Optimal control Definition

Mathematical method allowing to determine without a priori knowledge a control law based on the optimization of a cost functional. State equations F(φ, c) = 0 ; (Navier-Stokes + I.C. + B.C.) Control variables c ; (Blowing/suction, design parameters ...) Cost functional J (φ, c). (Drag, lift, target function, ...) Find a control law c and state variables φ such that the cost functional J (φ, c) reach an extremum under the constraint F(φ, c) = 0.

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II - Optimal control Lagrange multipliers

Constrained optimization ⇒ unconstrained optimization ◮ Introduction of Lagrange multipliers ξ (adjoint variables). ◮ Lagrange functional : L(φ, c, ξ) = J (φ, c)− < F(φ, c), ξ > ◮ Force L to be stationary ⇒ look for δL = 0 : δL = ∂L ∂φ δφ + ∂L ∂c δc + ∂L ∂ξ δξ = 0 ◮ Hypothesis : φ, c and ξ assumed to be independent of each other : ∂L ∂φ δφ = ∂L ∂c δc = ∂L ∂ξ δξ = 0 where ∂L ∂x = lim

ǫ− →0

L(x + ǫδx) − L(x) ǫ = 0 ∀δx (Fréchet derivative)

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II - Optimal control Optimality system

◮ State equations (∂L ∂ξ δξ = 0) : F(φ, c) = 0 ◮ Co-state (adjoint) equations (∂L ∂φ δφ = 0) : ∂F ∂φ ∗ ξ = ∂J ∂φ ∗ ◮ Optimality condition (∂L ∂c δc = 0) : ∂J ∂c ∗ = ∂F ∂c ∗ ξ ⇒ Expensive method in CPU time and storage memory for large system ! Bewley et al. (2000) : 108 grid points ⇒ Ensure only a local (generally not global) minimum

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II - Optimal control Iterative method

◮ c(0) given ; for n = 0, 1, 2, ... and while a convergence criterium is not satisfied, do :

  • 1. From t = 0 to t = T solve the state equations with c(n) ;

֒ → state variables φ(n)

  • 2. From t = T to t = 0 solve the co-state equations with φ(n) ;

֒ → co-state variables ξ(n)

  • 3. Solve the optimality condition with φ(n) and ξ(n) ;

֒ → objective gradient δc(n)

  • 4. New control law ֒

→ c(n+1) = c(n) + ω(n) δc(n) ◮ End do.

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II - Optimal control Reduced Order Model (ROM)

"without an inexpensive method for reducing the cost of flow computation, it is unlikely that the solution of optimization problems involving the three dimensional unsteady Navier-Stokes system will become routine"

  • M. Gunzburger, 2000

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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III - Proper Orthogonal Decomposition (POD)

◮ Introduced in fluid mechanics (turbulence context) by Lumley (1967). ◮ Look for a realization φ(X) which is closer, in an average sense, to the realizations u(X). (X = (x, t) ∈ D = Ω × R+) ◮ φ(X) solution of the problem : max φ |(u, φ|2 s.t. φ2 = 1. ◮ Snapshots method, Sirovich (1987) :

  • T

C(t, t′)a(n)(t′) dt′ = λ(n)a(n)(t). ◮ Optimal convergence L2 norm (energy) of φ(X) ⇒ Dynamical order reduction is possible. ◮ Decomposition of the velocity field : u(x, t) =

NP OD

  • i=1

a(i)(t)φ(i)(x).

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III - POD POD modes : uncontrolled flow (γ = 0)

First POD mode. Third POD mode. Second POD mode. Fourth POD mode.

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III - Reduced Order Model of the cylinder wake (ROM)

◮ Galerkin projection of NSE on the POD basis :

  • φ(i), ∂u

∂t + (u · ∇)u

  • =
  • φ(i), −∇p + 1

Re∆u

  • .

◮ Integration by parts (Green’s formula) leads :

  • φ(i), ∂u

∂t + (u · ∇)u

  • =
  • p, ∇ · φ(i)

− 1 Re

  • (∇ ⊗ φ(i))T , ∇ ⊗ u
  • − [p φ(i)] + 1

Re[(∇ ⊗ u)φ(i)]. with [a] =

  • Γ

a · n dΓ and (A, B) =

  • Ω

A : B dΩ =

  • i, j
  • Ω

AijBji dΩ.

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III - Reduced Order Model of the cylinder wake (ROM)

◮ Velocity decomposition with NP OD modes : u(x, t) = um(x) + γ(t) uc(x) +

NP OD

  • k=1

a(k)(t)φ(k)(x). ◮ Reduced order dynamical system where only Ngal (≪ NP OD) modes are retained (state equations) : d a(i)(t) d t =Ai +

Ngal

  • j=1

Bij a(j)(t) +

Ngal

  • j=1

Ngal

  • k=1

Cijk a(j)(t)a(k)(t) + Di d γ d t +  Ei +

Ngal

  • j=1

Fij a(j)(t)   γ + Giγ2 a(i)(0) = (u(x, 0), φ(i)(x)). Ai, Bij, Cijk, Di, Ei, Fij and Gi depend on φ, um, uc and Re.

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IV - Reduced Order Model of the cylinder wake Stabilization

Integration and "optimal" stabilization of the POD ROM for γ = A sin(2πStt), A = 2 and St = 0.5. POD reconstruction errors ⇒ temporal modes amplification

5 10

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

a(n)

time units

Temporal evolution of the first 6 POD temporal modes.

◮ Reasons : Extraction by POD only of the large energetic eddies Dissipation takes place in small eddies ◮ Solution : Addition of an optimal artificial viscosity on each POD mode projection (Navier-Stokes) prediction before stabilization (POD ROM) prediction after stabilization (POD ROM).

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IV - Reduced Order Model of the cylinder wake Stabilization

2 4 6 8 10 12 14 10

  • 2

10

  • 1

10

|a(n)| POD modes index

Comparison of energetic spectrum.

5 10 0.001 0.002 0.003 0.004 0.005 0.006 0.007

||a(n)| − |a(n)

proj||

POD modes index

Comparison of absolute errors.

◮ Good agreements between POD ROM spectrum and DNS spectrum ◮ Reduction of the reconstruction error between predicted (POD ROM) and projected (DNS) modes ⇒ Validation of the POD ROM

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V - Optimal control formulation based on ROM

◮ Objective functional : J (a, γ(t)) = T J(a, γ(t)) dt = T  

Ngal

  • i=1

a(i)2 + α 2 γ(t)2   dt. α : regularization parameter (penalization). ◮ Co-state equations : d ξ(i)(t) dt = −

Ngal

  • j=1

 Bji + γ Fji +

Ngal

  • k=1

(Cjik + Cjki) a(k)   ξ(j)(t) − 2a(i) ξ(i)(T) = 0. ◮ Optimality condition (gradient) : δγ(t) = −

Ngal

  • i=1

Di dξ(i) dt +

Ngal

  • i=1

 Ei +

Ngal

  • j=1

Fija(j) + 2Giγ(t)   ξ(i) + αγ.

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VI - Results of POD ROM Generalities

◮ No reactualization of the POD basis. ◮ The energetic representativity is a priori different to the dynamical

  • ne :

֒ → possible inconvenient for control, ֒ → a POD dynamical system represents a priori only the dynamics (and its vicinity) used to build the low dynamical model. ◮ Construction of a POD basis representative of a large range of dynamics : ֒ →excitation of a great number of degrees of freedom scanning γ(t) in amplitudes and frequencies.

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VI - Results of POD ROM Excitation

10 20 30 40 50

  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 5

γe

time units

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500

St

γe(t) = A1 sin(2πSt1 t) × sin(2πSt2 t − A2 sin(2πSt3 t)) with A1 = 4, A2 = 18, St1 = 1/120, St2 = 1/3 and St3 = 1/60. ◮ 0 ≤ amplitudes ≤ 4 and Fourier analysis ⇒ 0 ≤ frequencies ≤ 0.65 ◮ γe initial control law in the iterative process.

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VI - Results of POD ROM Energy

10 20 30 40 50 10

  • 7

10

  • 6

10

  • 5

10

  • 4

10

  • 3

10

  • 2

10

  • 1

10 10

1

γ = 0 γ = γe

λ(i) POD modes index

10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

relative Ec POD modes index

◮ Stationary cylinder γ = 0 : ֒ → 2 modes out of 100 are sufficient to restore 97% of the kinetic energy. ◮ Controlled cylinder γ = γe : ֒ → 40 modes out of 100 are then necessary to restore 97% of the kinetic energy ⇒ Improvement of the POD ROM robustness to dynamical evolutions.

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VI - Results of POD ROM Optimal control

10 20 30

  • 3
  • 2
  • 1

1 2 3

γopt

time units

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 2500 3000

St

◮ Reduction of the wake instationarity. γopt ≃ A sin(2πStt) with A = 2.2 and St = 0.53 J (γe) = 9.81 = ⇒ J (γopt) = 5.63. ◮ The control is optimal for the reduced order model based on POD. ◮ Is it also optimal for the Navier-Stokes model ?

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VI - Results of POD ROM Comparison of wakes’ organization

◮ No mathematical proof concerning the Navier Stokes optimality.

no control γ = 0

  • ptimal control γ = γopt

Isocontours of vorticity ωz.

◮ no control : γ = 0 ⇒ Asymmetric flow. ֒ → Large and energetic eddies. ◮ optimal control : γ = γopt ⇒ Symmetrization of the (near) wake. ֒ → Smaller and lower energetic eddies.

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VI - Results of POD ROM Aerodynamic coefficients

10 20 30 40 50 60 70 1 1.1 1.2 1.3 1.4 1.5

CD

time units

50 100 150

  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8

CL

time units

◮ Important drag reduction : CD0 = 1.40 for γ = 0 and CD = 1.04 for γ = γopt CD/CD0 = 0.74 ⇒ more than 25%. ◮ Decrease of the lift amplitude : CL = 0.68 for γ = 0 and CL = 0.13 for γ = γopt.

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VI - Results of POD ROM Numerical costs

◮ Optimal control of NSE by He et al. (2000) : ֒ → harmonic control law with A = 3 and St = 0.75. ⇒ 30% drag reduction. ◮ Optimal control POD ROM (this study) : ֒ → harmonic control law with A = 2.2 and St = 0.53. ⇒ 25% drag reduction. Reduction costs using POD ROM compared to NSE : CPU time : 100 Memory storage : 600 ֒ → "Optimal" control of 3D flows becomes possible ! ◮ Does the POD ROM control law correspond to the global minimum ?

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VI - Results of POD ROM Numerical optimization

1.02121 0.989091 0.796364 0.712919 0.709227 0.892727 1.34242 1.6101 1.70646 1.13899 0.839192 0.95697 0.967677 0.935556 0.828485 0.764242

0.25 0.5 0.75 1 1.25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

A St

Iso-relative- drag coefficient CD(A, St)/CD0 in (A, St) plan.

Observations ◮ Minimum is located in a smooth valley ֒ → Global minimum :

around A = 4.4 and St = 0.76

◮ Maximum is located in a sharp peak ֒ → Global maximum :

near St = 0.2, the natural frequency : lock-on flow

Finding the global minimum with an optimization algorithm may be difficult due to the smooth valley

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VI - Results of POD ROM Local versus global minimum

◮ POD ROM control law does not correspond to the global minimum ֒ → POD ROM parameters : A = 2.2 and St = 0.53 ⇒ CD = 1.04 ֒ → Global minimum parameters : A = 4.4 and St = 0.76 ⇒ CD = 0.98 ◮ Results in (A, St) quite different but not so far in terms of CD ֒ → The smooth valley is reached ◮ Improvement : coupling to the POD ROM approach an efficient new

  • ptimization algorithm for smooth fonctions

֒ → Take results obtained by POD ROM as initial conditions

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VII - Nelder-Mead Simplex method Generalities

Advantages ◮ Numerical simplicities ◮ Adaptive topology ◮ Free gradient optimization method ◮ Good results with smooth functions Drawbacks ◮ No proof of optimality for simplex dimensions greater than two ◮ Need to fix free parameters ◮ Maybe more iterations than gradient based optimisation algorithms...

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VII - Nelder-Mead Simplex method Results

1.02121 0.989091 0.796364 0.712919 0.892727 1.34242 1.6101 1.70646 1.13899 0.839192 0.95697 0.967677 0.935556 0.828485 0.764242

0.25 0.5 0.75 1 1.25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

A St

Iso-relative- drag coefficient CD(A, St)/CD0 in (A, St) plan.

Observations ◮ Topology adaptation function of the curve of the valley ◮ Minimum found by the simplex method : A = 4.5 and St = 0.76 ֒ → Seems to be the global mini- mum ◮ 30 NSE resolutions ⇒ 5% ad- ditive drag reduction compared to POD ROM Relative drag reduction by POD ROM : 25% (1 NSE resolution) Usefulness of coupling a new algorithm ?

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Conclusions

Important drag reduction obtained by POD ROM : more than 25% of relative drag reduction This solution is not the global minimum of the drag function POD ROM compared to NSE ⇒ important reduction of numerical costs : ֒ → Reduction factor of the CPU time : 100 ֒ → Reduction factor of the memory storage : 600 "OPTIMAL" CONTROL OF 3D FLOWS POSSIBLE BY POD ROM Coupling POD ROM with the Nelder-Mead simplex method leads a

priori to the global minimum of the drag function

But the gain on the drag function is quite small (5%) compared to results obtained by POD ROM

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Perspectives

Improve the representativity of the POD ROM ֒ → "Optimize" the temporal excitation γe ֒ → Mix snapshots corresponding to different dynamics (temporal excitations) ֒ → Introduction of shift-mode ? Look for harmonic control γ(t) = A sin(2π St t) with POD basis reactualization (closed loop on NSE and not only on POD ROM) Coupling the POD ROM approach with Trust Region POD method (TRPOD) = ⇒ proof of convergence under weak conditions Introducing the pressure into the POD dynamical system ֒ → pressure contribution to drag coefficient : 80% Optimal control of the Navier-Stokes equations

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