Optimal rotary control of the cylinder wake using POD reduced order - PowerPoint PPT Presentation

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

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

✁ � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � I - Configuration and numerical method Γ sup Two dimensional flow around a y circular cylinder at R e = 200 Γ e Γ c Viscous, incompressible and γ ( t ) Γ s Newtonian fluid U ∞ x Cylinder oscillation with a tan- D gential velocity γ ( t ) 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

I - Configuration and numerical method Iso pressure at t = 100 . Iso vorticity at t = 100 . 1.5 Authors S t C D C D 1 C L Braza et al. (1986) 0.2000 1.4000 C D , C L Henderson et al. (1997) 0.1971 1.3412 0.5 He et al. (2000) 0.1978 1.3560 0 this study 0.1983 1.3972 -0.5 Strouhal number and drag coefficient. 80 90 100 110 120 time units Aerodynamics coefficients. Optimal rotary control of the cylinder wake using POD reduced order model – p. 4/30

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

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 δ L = 0 : δ L = ∂ 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 Optimal rotary control of the cylinder wake using POD reduced order model – p. 6/30

II - Optimal control optimal system ◮ State equation ( ∂ L ∂ξ δξ = 0 ) : F ( φ, c ) = 0 ◮ Co-sate equation ( ∂ L ∂φ δφ = 0 ) : � ∗ � ∗ � ∂ F � ∂ J ξ = ∂φ ∂φ ◮ Optimality condition ( ∂ L ∂c δc = 0 ) : � ∗ � ∗ � ∂ J � ∂ F = ξ ∂c ∂c ⇒ 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

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

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 � 2 = 1 . �| ( u , φ | 2 � ◮ φ ( X ) solution of the problem : max s.t. φ ◮ 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 : N P OD � a ( i ) ( t ) φ ( i ) ( x ) . u ( x , t ) = i =1 Optimal rotary control of the cylinder wake using POD reduced order model – p. 9/30

III - Reduced Order Model of the cylinder wake (ROM) ◮ Galerkin’s projection of NSE on the POD basis : � � � � φ ( i ) , ∂ u φ ( i ) , − ∇ p + 1 ∂t + ( u · ∇ ) u = Re ∆ u . ◮ Integration by parts (Green’s formula) leads : � � φ ( i ) , ∂ u − 1 � p, ∇ · φ ( i ) � � ( ∇ ⊗ φ ( i ) ) T , ∇ ⊗ u � ∂t + ( u · ∇ ) u = Re − [ p φ ( i ) ] + 1 Re [( ∇ ⊗ u ) φ ( i ) ] . � � � � with [ a ] = a · n d Γ and ( A, B ) = A : B d Ω = A ij B ji d Ω . Γ Ω Ω i, j Optimal rotary control of the cylinder wake using POD reduced order model – p. 10/30

III - Reduced Order Model of the cylinder wake (ROM) ◮ Velocity decomposition with N P OD modes : N P OD � a ( k ) ( t ) φ ( k ) ( x ) . u ( x , t ) = u m ( x ) + γ ( t ) u c ( x ) + k =1 ◮ Reduced order dynamical system where only N gal ( ≪ N P OD ) modes are retained (state’s equation) :  N gal N gal N gal d a ( i ) ( t ) � � �  B ij a ( j ) ( t ) + C ijk a ( j ) ( t ) a ( k ) ( t ) = A i +    d t   j =1 j =1 k =1      N gal d γ �  γ + G i γ 2 F ij a ( j ) ( t ) + D i d t +  E i +     j =1    a ( i ) (0) = ( u ( x , 0) , φ ( i ) ( x )) .   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

IV - Reduced Order Model of the cylinder wake Stabilization Integration and (optimal) stabilization of the POD ROM for γ = A sin(2 πS t t ) , A = 2 et S t = 0 . 5 . POD reconstruction errors ⇒ temporal modes amplification 1.5 ◮ Causes : 1 Extraction of large energetic eddies 0.5 a ( n ) Dissipation takes place in small 0 eddies -0.5 ◮ Solution : -1 Optimal addition of artificial vis- -1.5 0 5 10 time units cosity on each POD mode Temporal evolution of the first 6 POD temporal modes. 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

IV - Reduced Order Model of the cylinder wake Stabilization 0 10 0.007 0.006 proj |�| |�| a ( n ) |� − �| a ( n ) 0.005 �| a ( n ) |� 0.004 -1 10 0.003 0.002 0.001 -2 10 0 0 2 4 6 8 10 12 14 5 10 POD modes index POD modes index Comparison of energetic spectrum. 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

IV - Optimal control formulation based on reduced order model ◮ Objective functional :   � T � T N gal a ( i )2 + α � 2 γ ( t ) 2  dt. J ( a , γ ( t )) = J ( a , γ ( t )) dt =  0 0 i =1 α : regularization parameter (penalization). ◮ Co-state’s equations :    N gal N gal d ξ ( i ) ( t )  � � ( C jik + C jki ) a ( k )  ξ ( j ) ( t ) − 2 a ( i )  = −  B ji + γ F ji +  dt j =1 k =1   ξ ( i ) ( T ) = 0 .  ◮ Optimality condition (gradient) :   N gal N gal N gal dξ ( i ) F ij a ( j ) + 2 G i γ ( t )  ξ ( i ) + αγ. � � � δγ ( t ) = − D i +  E i + dt i =1 i =1 j =1 Optimal rotary control of the cylinder wake using POD reduced order model – p. 14/30

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 ) ֒ → γ ( n +1) ( t ) = γ ( n ) ( t ) + ω ( n ) δγ ( n ) ( t ) 4. New control law ֒ ◮ End do. Optimal rotary control of the cylinder wake using POD reduced order model – p. 15/30