Improvement of Reduced Order Modeling based on Proper Orthogonal - PowerPoint PPT Presentation

Improvement of Reduced Order Modeling based on Proper Orthogonal Decomposition Michel Bergmann, Charles-Henri Bruneau & Angelo Iollo Michel.Bergmann@inria.fr http://www.math.u-bordeaux.fr/˜bergmann/ INRIA Bordeaux Sud-Ouest Institut de Math´ ematiques de Bordeaux 351 cours de la Lib´ eration 33405 TALENCE cedex, France ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 1

Summary Context and flow configuration I - A pressure extended Reduced Order Model based on POD ◮ Proper Orthogonal Decomposition (POD) ◮ Reduced Order Model (ROM) II - Stabilization of reduced order models ◮ Residuals based stabilization method ◮ Classical SUPG and VMS methods III - Improvement of the functional subspace ◮ Krylov like method ◮ An hybrid DNS/POD ROM method (Database modification) Conclusions ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 2

Context and flow configuration Context ⊲ ◦ Need of reduced order model for flow control purpose → ֒ To reduce the CPU time → To reduce the memory storage during adjoint-based minimization process ֒ ◦ Optimization + POD ROM methods ֒ → Generalized basis, no POD basis actualization : fast but no "convergence" proof ֒ → Trust Region POD (TRPOD), POD basis actualization : proof of convergence! ◦ Drawbacks → Need to stabilize POD ROM (lack of dissipation, numerical issues, pressure term) ֒ ֒ → Basis actualization : DNS → high numerical costs ! ◦ Solutions ֒ → Efficient ROM & stabilization → Low costs functional subspace adaptation during optimization process ֒ ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 3

Context and flow configuration Flow Configuration ⊲ ◦ 2-D Confined flow past a square cylinder in laminar regime ◦ Viscous fluid, incompressible and newtonian ◦ No control U = 0 Ω ��� ��� U ( y ) ��� ��� H ��� ��� ��� ��� ��� ��� ��� ��� D U = 0 L Numerical methods ⊲ ◦ Penalization method for the square cylinder ◦ Multigrids V-cycles method in space C.-H. Bruneau solver ◦ Gear method in time ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 4

I - A pressure extended Reduced Order Model Proper Orthogonal Decomposition (POD), Lumley (1967) ⊲ Look for the flow realization Φ ( X ) that is "the closest" in an average sense to realizations Φ 2 original axis U ( X ) . Φ 1 ( X = ( x , t ) ∈ D = Ω × R + ) U ( X ) ⊲ Φ ( X ) solution of problem : � Φ � 2 = 1 . Φ �| ( U , Φ ) | 2 � , max original axis ⊲ Optimal convergence in L 2 norm de Φ ( X ) U ( X ) − U m ( X ) ⇒ Dynamical reduction possible. Lumley J.L. (1967) : The structure of inhomogeneous turbulence. Atmospheric Turbulence and Wave Propagation , ed. A.M. Yaglom & V.I. Tatarski, pp. 166-178. ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 5

I - A pressure extended Reduced Order Model ⊲ Equivalent with Fredholm equation : Z n ( X ′ ) d X ′ = λ n Φ ( i ) R ij ( X , X ′ )Φ ( j ) n ( X ) n = 1 , .., N s D → R ( X , X ′ ) : Space-time correlation tensor ֒ Temps ⊲ Snapshots method, Sirovich (1987) : Z � � n � � C ( t, t ′ ) a n ( t ′ ) dt ′ = λ n a n ( t ) o i t a l � � e � � T X ′ r r o C �� �� Space �� �� → C ( t, t ′ ) : Temporal correlations ֒ �� �� �� �� ⊲ Φ ( X ) flow basis : N s X U ( x , t ) = a n ( t ) Φ n ( x ) . X Average on space n =1 Sirovich L. (1987) : Turbulence and the dynamics of coherent structures. Part 1,2,3 Quarterly of Applied Mathematics , XLV N ◦ 3, pp. 561–571. ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 6

I - A pressure extended Reduced Order Model Truncation of the POD basis to keep 99% of the Relative Information Content . N s M X X RIC ( M ) = λ k λ k k =1 k =1 Example at Re = 200 for U = ( u , p ) T with N s = 200 7 10 100 6 10 5 10 90 4 10 RIC (%) 80 3 10 λ n 2 10 70 1 10 60 0 10 -1 10 50 -2 10 0 2 4 6 8 10 12 14 16 18 20 0 5 10 15 20 index of POD modes index of POD modes Fig. : POD spectrum. Fig. : RIC(M), M nb modes POD retenus. N r = arg min M RIC ( M ) s.t. RIC ( N r ) > 99% ⇒ N r = 5 ! ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 7

I - A pressure extended Reduced Order Model The POD basis is Φ = ( φ , ψ ) T ∇ ∧ φ 1 ψ 1 ∇ ∧ φ 3 ψ 3 ∇ ∧ φ 5 ψ 5 Fig. : Representation of some POD modes. Iso-vorticity (left) and isobars (right). Dashed lines represent negative values (the pressure reference is arbitrarily chosen to be zero ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 8

I - A pressure extended Reduced Order Model ◮ Momentum conservation Detailled model (exact) ∂ u ∂t + ( u · ∇ ) u = − ∇ p + 1 Re ∆ u N r N r X X Galerkin projection using e u ( x , t ) = a i ( t ) φ i ( x ) and e p ( x , t ) = a i ( t ) ψ i ( x ) : i =1 i =1 0 1 N r N r N r N r N r X X X X X d a j ∇ ψ j a j − 1 @ φ i , A φ j d t + ( φ j · ∇ ) φ k a j a k + ∆ φ j a j = 0 . Re j =1 j =1 k =1 j =1 j =1 Ω The Reduced Order Model is then : N r N r N r N r X X X X d a j L ( m ) B ( m ) C ( m ) d t = a j + ijk a j a k ij ij j =1 j =1 j =1 k =1 → The ROM does not satisfy a priori the mass conservation ֒ (for non divergence free modes, as NSE-Residual modes) ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 9

I - A pressure extended Reduced Order Model ◮ Mass conservation Detailled model ∇ · u = 0 Projection onto the POD basis N r X a j ∇ · φ j = 0 j =1 Minimizing residuals in a least squares sense, we obtain : N r X B ( c ) B ( c ) = ( ∇ · φ j ) T ∇ · φ j ij a j = 0 , where ij j =1 The modified ROM that satisfies both momentum and continuity equation writes : N r N r “ ” N r N r X X X X d a j L ( m ) B ( m ) + αB ( c ) C ( m ) d t = a j + ijk a j a k ij ij ij j =1 j =1 j =1 k =1 → The ROM has moreover to satisfy the flow rate conservation. ֒ ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 10

I - A pressure extended Reduced Order Model ◮ Flow rate conservation Z u d s = c, For the 2-D confined flow : S Z N r X For each slides S l : φ u a j ( t ) j d s = c, S l i =1 Z N r X d a j φ u j d s = 0 . d t S l j =1 In a least square sense : N r X d a j L ( r ) d t = 0 . ij j =1 Finally, the ROM writes N r “ ” d a j N r “ ” N r N r X X X X L ( m ) + βL ( r ) B ( m ) + αB ( c ) C ( m ) d t = a j + ijk a j a k , ij ij ij ij j =1 j =1 j =1 k =1 with initial conditions a i (0) = ( U ( x , 0) , Φ i ( x )) Ω i = 1 , · · · , N r . ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 11

I - A pressure extended Reduced Order Model ◮ Advantage no modelisation of the pressure term Re = 200 , 11 modes ⇒ convergence towards the exact limit cycles ( = DNS) 50 60 60 a 2 0 40 40 -50 20 20 a 3 a 4 0 0 50 -20 -20 a 3 -40 -40 0 -60 -60 -50 a 2 -60 -40 -20 0 20 40 60 -20 -10 a 2 0 10 20 20 20 60 10 40 a 4 10 0 20 -10 a 5 a 5 0 0 -20 -20 20 -10 -40 10 a 5 -60 0 -20 -20 -10 a 2 0 10 20 -20 -10 a 4 0 10 20 -10 -20 0 10 20 30 40 t Fig. : Temporal evolution of the POD ROM Fig. : Limit cycles of the POD ROM coefficients coefficients over 25 vortex shedding periods over 25 vortex shedding periods ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 12

I - A pressure extended Reduced Order Model ◮ Drawbaks same as usual, i.e. lack of dissipation... Re = 200 , 5 modes ⇒ convergence towards an erroneous limit cycles ( � = DNS) 50 60 60 a 2 0 40 40 -50 20 20 a 3 a 4 0 0 50 -20 -20 a 3 -40 -40 0 -60 -60 -50 a 2 a 2 -60 -40 -20 0 20 40 60 -20 -10 0 10 20 20 20 60 10 40 a 4 10 0 20 -10 a 5 a 5 0 0 -20 -20 20 -10 -40 10 a 5 -60 0 -20 -20 -10 a 2 0 10 20 -20 -10 a 4 0 10 20 -10 -20 Fig. : Temporal evolution of the POD ROM Fig. : Limit cycles of the POD ROM coefficients coefficients over 25 vortex shedding periods over 25 vortex shedding periods ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 12

I - A pressure extended Reduced Order Model ◮ Drawbaks same as usual, i.e. lack of dissipation... Re = 200 , 3 modes ⇒ exponential divergence 200 150 200 100 100 50 a 3 0 0 a 2 -50 -100 -100 -200 -150 -200 -200 -150 -100 -50 0 50 100 150 200 a 2 5 10 15 t Fig. : Temporal evolution of the POD ROM Fig. : Limit cycles of the POD ROM coefficients coefficients over 25 vortex shedding periods over 25 vortex shedding periods ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 12

II - POD ROM stabilization ◮ Overview of stabilization methods (non-exhaustive) Eddy viscosity ֒ → Heisenberg viscosity → Spectral vanishing viscosity ֒ ֒ → Optimal viscosity Penalty method Calibration of POD ROM coefficients ◮ "New" stabilization methods in POD ROM context Residuals based stabilization method Streamline Upwind Petrov-Galerkin (SUPG) and Variational Multi-scale (VMS) methods ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 13