Optimal rotary control of the cylinder wake using POD reduced order model p. 1/30

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1 Optimal rotary control of the cylinder wake using POD reduced order model Michel Bergmann, Laurent Cordier & Jean-Pierre Brancher Laboratoire d Énergétique et de Mécanique Théorique et Appliquée UMR 7563 (CNRS - INPL - UHP) ENSEM - 2, avenue de la Forêt de Haye BP Vandoeuvre Cedex, France Optimal rotary control of the cylinder wake using POD reduced order model p. 1/30

2 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 Optimal rotary control of the cylinder wake using POD reduced order model p. 2/30

3 PSfrag replacements I - Configuration and numerical method Two dimensional flow around a circular cylinder at R e = 200 Viscous, incompressible and Newtonian fluid Cylinder oscillation with a tangential velocity γ(t) Γ e U y γ(t) D Γ c x Γ sup Γ s Γ inf Fractional steps method in time Finite Elements Method (FEM) in space Numerical code written by M.Braza (IMFT-EMT2) & D.Ruiz (ENSEEIHT) Optimal rotary control of the cylinder wake using POD reduced order model p. 3/30

4 I - Configuration and numerical method Iso pressure at t = 100. Iso vorticity at t = placements CD, CL C D C L time units Aerodynamics coefficients. Authors S t C D Braza et al. (1986) Henderson et al. (1997) He et al. (2000) this study Strouhal number and drag coefficient. Optimal rotary control of the cylinder wake using POD reduced order model p. 4/30

5 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. Optimal rotary control of the cylinder wake using POD reduced order model p. 5/30

6 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 L L δφ + δc + φ c ξ δξ = 0 Suppose φ, c and ξ independant each other : L L L δφ = δc = φ c ξ δξ = 0 Optimal rotary control of the cylinder wake using POD reduced order model p. 6/30

7 II - Optimal control optimal system State equation ( L ξ δξ = 0) : F(φ, c) = 0 Co-sate equation ( L δφ = 0) : φ ( ) F ξ = φ Optimality condition ( L δc = 0) : c ( ) J = c ( J φ ( F c ) ) ξ Expensive method in CPU time and storage memory for large system! Ensure only a local (generally not global) minimum Optimal rotary control of the cylinder wake using POD reduced order model p. 7/30

8 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" Initialization M. Gunzburger, 2000 Recourse to detailed model (TRPOD) High fidelity model f(x), grad f(x) Approximation model a(x), grad a(x) x Optimization Optimization on simplified model Optimal rotary control of the cylinder wake using POD reduced order model p. 8/30

9 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. φ 2 = 1. Snapshots method, Sirovich (1987) : C(t, t )a (n) (t ) dt = λ (n) a (n) (t). T Optimal convergence L 2 norm (energy) of φ(x) Dynamical order reduction is possible. Decomposition of the velocity field : u(x, t) = N P OD i=1 a (i) (t)φ (i) (x). Optimal rotary control of the cylinder wake using POD reduced order model p. 9/30

10 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 + 1Re ) u. Integration by parts (Green s formula) leads : ( φ (i), u t ) + (u )u = ( p, φ (i)) 1 Re [p φ (i) ] + 1 Re [( u)φ(i) ]. ( ) ( φ (i) ) T, u with [a] = Γ a n dγ and (A, B) = Ω A : B dω = i, j Ω A ij B ji dω. Optimal rotary control of the cylinder wake using POD reduced order model p. 10/30

11 III - Reduced Order Model of the cylinder wake (ROM) Velocity decomposition with N P OD modes : u(x, t) = u m (x) + γ(t) u c (x) + N P OD k=1 a (k) (t)φ (k) (x). Reduced order dynamical system where only N gal ( N P OD ) modes are retained (state s equation) : d a (i) (t) d t =A i + N gal j=1 + D i d γ d t + B ij a (j) (t) + E i + a (i) (0) = (u(x, 0), φ (i) (x)). N gal j=1 N gal j=1 N gal k=1 C ijk a (j) (t)a (k) (t) F ij a (j) (t) γ + G i γ 2 A i, B ij, C ijk, D i, E i, F ij and G i depend on φ, u m, u c and Re. Optimal rotary control of the cylinder wake using POD reduced order model p. 11/30

12 IV - Reduced Order Model of the cylinder wake Stabilization Integration and (optimal) stabilization of the POD ROM for γ = A sin(2πs t t), A = 2 et S t = 0.5. POD reconstruction errors temporal modes amplification ments 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 viscosity on each POD mode projection (Navier Stokes) prediction before stabilisation (POD ROM) prediction after stabilisation (POD ROM). Optimal rotary control of the cylinder wake using POD reduced order model p. 12/30

13 IV - Reduced Order Model of the cylinder wake Stabilization s j d b a (n) 10-1 PSfrag replacements a (n) a (n) proj POD modes index Comparison of energetic spectrum 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 Optimal rotary control of the cylinder wake using POD reduced order model p. 13/30

14 IV - Optimal control formulation based on reduced order model Objective functional : J (a, γ(t)) = T 0 J(a, γ(t)) dt = T 0 N gal a (i)2 + α 2 γ(t)2 dt. i=1 α : regularization parameter (penalization). Co-state s equations : d ξ (i) N (t) gal = B ji + γ F ji + dt j=1 ξ (i) (T ) = 0. N gal k=1 (C jik + C jki ) a (k) ξ (j) (t) 2a (i) Optimality condition (gradient) : N gal dξ (i) N gal δγ(t) = D i + E i + dt i=1 i=1 N gal j=1 F ij a (j) + 2G i γ(t) ξ (i) + αγ. Optimal rotary control of the cylinder wake using POD reduced order model p. 14/30

15 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. Optimal rotary control of the cylinder wake using POD reduced order model p. 15/30

16 V - Closed loop results Generalities No reactualization of the POD basis. The energetic representativity is a priori different to the dynamical one : 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. Optimal rotary control of the cylinder wake using POD reduced order model p. 16/30

17 V - Closed loop results Excitation γe s -3-4 PSfrag replacements time units S t γ e (t) = A 1 sin(2πs t 1 t) sin(2πs t 2 t A 2 sin(2πs t 3 t)) with A 1 = 4, A 2 = 18, S t 1 = 1/120, S t 2 = 1/3 and S t 3 = 1/60. 0 amplitudes 4 and Fourier analysis 0 frequencies 0.65 γ e initial control law in the iterative process. Optimal rotary control of the cylinder wake using POD reduced order model p. 17/30

18 V - Closed loop results Energy s γ = γ = γ e 10-5 PSfrag replacements c λ (i) POD modes index λ (i) relative Ec POD modes index Stationary cylinder γ = 0 : 2 modes out of 100 are sufficient to restitute 97% of the kinetic energy. Controlled cylinder γ = γ e : 40 modes out of 100 are then necessary to restitute 97% of the kinetic energy Robustness evolution with dynamical evolutions. Optimal rotary control of the cylinder wake using POD reduced order model p. 18/30

19 V - Closed loop results Optimal control γopt s -2 PSfrag replacements time units S t Reduction of the wake instationarity. γ opt A sin(2πs t t) with A = 2.2 and S t = 0.53 J (γ e ) = 9.81 = J (γ opt ) = The control is optimal for the reduced order model based on POD. Is it also optimal for the Navier Stokes model? Optimal rotary control of the cylinder wake using POD reduced order model p. 19/30

20 V - Closed loop results Comparison of wakes organization No mathematical proof concerning the Navier Stokes optimality. no control γ = 0 Isocontours of vorticity ω z. optimal control γ = γ opt no control : γ = 0 Asymmetrical flow. Large and energetic eddies. optimal control : γ = γ opt Symmetrization of the (near) wake. Smaller and lower energetic eddies. Optimal rotary control of the cylinder wake using POD reduced order model p. 20/30

21 V - Closed loop results Aerodynamics coefficients CD CL s 1.1 PSfrag replacements time units time units Very consequent drag reduction : C D 0 = 1.40 for γ = 0 et C D = 1.04 for γ = γ opt C D /C D 0 = 0.74 more than 25%. Decrease of the lift amplitude : C L = 0.68 for γ = 0 et C L = 0.13 for γ = γ opt. Optimal rotary control of the cylinder wake using POD reduced order model p. 21/30

22 V - Closed loop results Numerical costs Optimal control of NSE by He et al. (2000) : harmonic control law with A = 3 and S t = % drag reduction. Optimal control POD ROM (this study) : harmonic control law with A = 2.2 and S t = % 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! Optimal rotary control of the cylinder wake using POD reduced order model p. 22/30

23 VI - General observations Numerical experimentation A 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 St Iso-relative- drag coefficient C D (A, St)/C D 0 in (A, St) plan. 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 Optimal rotary control of the cylinder wake using POD reduced order model p. 23/30

24 VI - General observations Maximal rotation angle CD(θ)/CD 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 maximum rotation angle : θ = A/(πSt) Observations No drag reduction possible near natural frequency Maximum drag reduction around θ = θ opt = θ Relative drag coefficient for different Strouhal numbers vs. maximum rotation angle. For all frequencies g.t. natural frequency Minimum drag : C Dmin = 0.71C D 0 = 0.98 Existence of an "optimal" maximal rotation angle. Optimal rotary control of the cylinder wake using POD reduced order model p. 24/30

25 VI - General observations Maximal rotation angle C D min C D θ opt Notations C D min(st) = min A R C D(θ, St) C D θopt(st) = C D (95, St) CD 0.9 C D Observations 0.7 St Nat St Comparison between C D min and C D θopt. Good agreements between C D min and C D θopt for St > St Nat θopt is not optimal for St < St Nat In order to minimise the drag one couldn t choose A and St independently. It seems that A/St = 5.2 (θ opt = 95 ). Optimal rotary control of the cylinder wake using POD reduced order model p. 25/30

26 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 (θ max = 76 ) C D = 1.04 Optimal parameters : A = 4.4 and St = 0.76 (θ max = 105 θ opt ) C D = 0.98 Results in (A, St) quite different but not so far in C D terms The smooth valley is reached Couple an efficient new optimisation algorithm for smooth fonctions Take results obtain by POD ROM as initial conditions Optimal rotary control of the cylinder wake using POD reduced order model p. 26/30

27 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... Optimal rotary control of the cylinder wake using POD reduced order model p. 27/30

28 VII - Nelder Mead Simplex method Results A 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 minimum St Iso-relative- drag coefficient C D (A, St)/C D 0 in (A, St) plan. 30 NSE resolutions 5% additive drag reduction compare to POD ROM Relative drag reduction by POD ROM : 25% (1 NSE resolution) Utility of coupling a new algorithm? Optimal rotary control of the cylinder wake using POD reduced order model p. 28/30

29 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 obtain by POD ROM Optimal rotary control of the cylinder wake using POD reduced order model p. 29/30

30 Perspectives Improve the representativity of the low order model "Optimize" the temporal excitation γ e Mix snapshots corresponding to several differents dynamics (temporal excitations) Look for harmonic control γ(t) = A sin(2π S t 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 Optimal rotary control of the cylinder wake using POD reduced order model p. 30/30