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Optimal rotary control of the cylinder wake using POD reduced order model Michel Bergmann, Laurent Cordier & Jean-Pierre Brancher Michel.Bergmann@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

Michel.Bergmann@ensem.inpl-nancy.fr

Laboratoire d’ ´ Energ´ etique et de M´ ecanique Th´ eorique et Appliqu´ ee 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 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 - General observations VIII - Nelder Mead Simplex method Conclusions and perspectives

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I - Configuration and numerical method

Two dimensional flow around a circular cylinder at Re = 200 Viscous, incompressible and Newtonian fluid Cylinder oscillation with a tan- gential velocity γ(t)

  • ✁
✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁

Γe Γsup Γs Γinf Γc D γ(t) x y U ∞

Fractional steps method in time Finite Elements Method (FEM) in space ◮ 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

Aerodynamics 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 empiricism a control law starting from the optimization of a cost functional. State equation F(φ, c) = 0 ; (Navier-Stokes + C.I. + C.L.) Control variables c ; (Blowing/suction, design parameters ...) Cost functional J (φ, c). (Drag, lift, ...) Find a control law c and state variable φ such that the cost functional J (φ, c) reach an extremum under the constrain F(φ, c) = 0.

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II - Optimal control Lagrange multipliers

Constrained optimization ⇒ unconstrained optimization ◮ Introduction of Lagrange multipliers ξ. ◮ 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 ◮ Suppose φ, c and ξ independant each other : ∂L ∂φ δφ = ∂L ∂c δc = ∂L ∂ξ δξ = 0

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II - Optimal control optimal system

◮ State equation (∂L ∂ξ δξ = 0) : F(φ, c) = 0 ◮ Co-sate equation (∂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 ! ⇒ Ensure only a local (generally not global) minimum

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

◮ Galerkin’s 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’s equation) :                    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 et 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.

◮ Causes : Extraction of large energetic eddies Dissipation takes place in small eddies ◮ Solution : Optimal addition of artificial vis- cosity on each POD mode projection (Navier Stokes) prediction before stabilisation (POD ROM) prediction after stabilisation (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 and DNS spectrum ◮ Reduced reconstruction error between predicted (POD) and projected (DNS) modes ⇒ Validation of the POD ROM

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IV - Optimal control formulation based on reduced order model

◮ 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’s 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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IV - Optimal control formulation based on reduced order model

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

  • 1. From t = 0 to t = T solve the state’s equations with γ(n)(t) ;

֒ → state’s variables a(n)(t)

  • 2. From t = T to t = 0 solve the co-state’s equations with a(n)(t) ;

֒ → co-state’s variables ξ(n)(t)

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

֒ → objective gradient δγ(n)(t)

  • 4. New control law ֒

→ γ(n+1)(t) = γ(n)(t) + ω(n) δγ(n)(t) ◮ End do.

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V - Closed loop results 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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V - Closed loop results 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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V - Closed loop results 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 suffi cient to restitute 97% of the kinetic energy. ◮ Controlled cylinder γ = γe : ֒ → 40 modes out of 100 are then necessary to restitute 97% of the kinetic energy ⇒ Robustness evolution with dynamical evolutions.

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V - Closed loop results 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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V - Closed loop results 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 ⇒ Asymmetrical flow. ֒ → Large and energetic eddies. ◮ optimal control : γ = γopt ⇒ Symmetrization of the (near) wake. ֒ → Smaller and lower energetic eddies.

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V - Closed loop results Aerodynamics 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

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

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V - Closed loop results 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. Less energetic costs (greater energetic gain ?) Calculus time costs : 100 less using POD ROM than NSE ! (for co-states equations and optimality conditions too) Memory storage : 600 less variables using POD ROM than NSE ! ֒ → "Optimal" control of 3D flows becomes possible !

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VI - General observations Numerical experimentation

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 pic ֒ → Global maximum :

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

Finding the global minimum with an optimization algorithm may leads difficulties because of the smooth valley

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VI - General observations Maximal rotation angle

30 60 90 120 150 180 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

St = 0.2 St = 0.3 St = 0.4 St = 0.5 St = 0.6 St = 0.7 St = 0.8 St = 0.9 St = 1.0 St = 0.53

CD(θ)/CD0 θ

Relative drag coefficient for different Strouhal numbers vs. maximum rotation angle.

◮ maximum rotation angle : θ = A/(πSt) Observations ◮ No drag reduction possible near natural frequency ◮ Maximum drag reduction around θ = θopt = 95◦ ֒ → For all frequencies g.t. natural fre-

quency

֒ → Minimum drag : CDmin = 0.71CD0 = 0.98 Existence of an "optimal" maximal rotation angle.

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VI - General observations Maximal rotation angle

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6 0.7 0.8 0.9 1 1.1 1.2

CDmin CDθopt CD St CD0 StNat

Comparison between CDmin and CDθopt.

Notations CDmin(St) = min

A∈R CD(θ, St)

CDθopt(St) = CD(95◦, St) Observations ◮ Good agreements between CDmin and CDθopt for St > StNat ◮ θopt is not optimal for St < StNat In order to minimise the drag one couldn’t choose A and St

  • independently. It seems that A/St = 5.2 (θopt = 95◦).

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VI - General observations Discussion

◮ Control law obtain by POD ROM is not optimal for drag minimisation ֒ → Parameters obtain : A = 2.2 and St = 0.53 (θmax = 76◦) ⇒ CD = 1.04 ֒ → Optimal parameters : A = 4.4 and St = 0.76 (θmax = 105◦ = θopt) ⇒ CD = 0.98 ◮ Results in (A, St) quite different but not so far in CD terms ֒ → The smooth valley is reached ◮ Couple an efficient new optimisation algorithm for smooth fonctions ֒ → Take results obtain by POD ROM as initial conditions

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VII - Nelder Mead Simplex method Generalities

Avantages ◮ Numerical simplicities ◮ Adaptive topology ◮ Gradients calculations not necessary ◮ Good results with smooth fonctions Disadvantages ◮ 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 Nelder-Mead simplex method : A = 4.5 and St = 0.76 ⇒ θ = 107◦ ֒ → Seems to be the global mini- mum ◮ 30 NSE resolutions ⇒ 5% ad- ditive drag reduction compare to POD ROM Relative drag reduction by POD ROM : 25% (1 NSE resolution) Utility of coupling a new algorithm ?

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Conclusions

Significative drag reduction minimizing the wake instationnarity of the reduced order model : more than 25% of the relative drag But this is not a minimum (global a local) of the drag function Low calculus cost ⇒ numerical cost negligible. ֒ → Calculus time costs : 100 less using POD ROM than NSE ! (for co-states equations and optimality conditions too) ֒ → Memory storage : 600 less variables using POD ROM than NSE ! "OPTIMAL" CONTROL OF 3D FLOWS POSSIBLE BY POD ROM Existence of a optimal maximal rotation angle for effective drag reduction, θ = 105◦ Couple POP ROM to Nelder Mead simplex method leads a priori the global minimum of the drag functional But the gain on the drag function is quite small compare to result

  • btain by POD ROM

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Perspectives

Improve the representativity of the low order model ֒ → "Optimize" the temporal excitation γe ֒ → Mix snapshots corresponding to several differents dynamics (temporal excitations) Look for harmonic control γ(t) = A sin(2π St t) with POD basis reactualization (close loop on NSE and not only on POD ROM) Couple this optimal system with trust region methods (TRPOD) = ⇒ proof of convergence under weak conditions Couple pressure with the POD dynamical system ֒ → pressure contribution to drag coefficient : 80% Optimal control of the Navier Stokes equations

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