Improvement of Reduced Order Modeling based on Proper Orthogonal Decomposition

Transcription

1 ICCFD5, Seoul, Korea, July 7-11, 28 p. 1 Improvement of Reduced Order Modeling based on Proper Orthogonal Decomposition Michel Bergmann, Charles-Henri Bruneau & Angelo Iollo Michel.Bergmann@inria.fr bergmann/ INRIA Bordeaux Sud-Ouest Institut de Mathématiques de Bordeaux 351 cours de la Libération 3345 TALENCE cedex, France

2 ICCFD5, Seoul, Korea, July 7-11, 28 p. 2 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

3 ICCFD5, Seoul, Korea, July 7-11, 28 p. 3 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

4 ICCFD5, Seoul, Korea, July 7-11, 28 p. 4 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 = U(y) D Ω H U = L Numerical methods Penalization method for the square cylinder Multigrids V-cycles method in space C.-H. Bruneau solver Gear method in time

5 ICCFD5, Seoul, Korea, July 7-11, 28 p. 5 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 U(X). (X = (x, t) D = Ω R + ) Φ(X) solution of problem : original axis Φ 2 Φ 1 U(X) max Φ (U,Φ) 2, Φ 2 = 1. original axis Optimal convergence in L 2 norm de Φ(X) Dynamical reduction possible. U(X) U m (X) Lumley J.L. (1967) : The structure of inhomogeneous turbulence. Atmospheric Turbulence and Wave Propagation, ed. A.M. Yaglom & V.I. Tatarski, pp

6 ICCFD5, Seoul, Korea, July 7-11, 28 p. 6 I - A pressure extended Reduced Order Model Equivalent with Fredholm equation : Z R ij (X, X )Φ n (j) (X ) dx = λ n Φ (i) n (X) D R(X, X ) : Space-time correlation tensor n = 1,.., N s Snapshots method, Sirovich (1987) : Temps Z T C(t, t )a n (t ) dt = λ n a n (t) C(t, t ) : Temporal correlations Space X Correlation 1 1 Φ(X) flow basis : U(x, t) = N s X n=1 a n (t)φ n (x). Average on space X Sirovich L. (1987) : Turbulence and the dynamics of coherent structures. Part 1,2,3 Quarterly of Applied Mathematics, XLV N 3, pp

7 ICCFD5, Seoul, Korea, July 7-11, 28 p. 7 I - A pressure extended Reduced Order Model Truncation of the POD basis to keep 99% of the Relative Information Content MX. XN s RIC(M) = λ k k=1 k=1 Example at Re = 2 for U = (u, p) T with N s = 2 λ k λn RIC (%) index of POD modes Fig. : POD spectrum index of POD modes Fig. : RIC(M), M nb modes POD retenus. N r = arg min M RIC(M) s.t. RIC(N r) > 99% N r = 5!

8 ICCFD5, Seoul, Korea, July 7-11, 28 p. 8 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

9 ICCFD5, Seoul, Korea, July 7-11, 28 p. 9 I - A pressure extended Reduced Order Model Momentum conservation Detailled model (exact) u t + (u )u = p + 1 Re u Galerkin projection using eu(x, t) i, j=1 φ j da j dt + j=1 k=1 i=1 (φ j ) φ k a j a k + a i (t)φ i (x) and ep(x, t) = j=1 ψ j a j 1 Re i=1 j=1 a i (t)ψ i (x) : 1 φ j a j A Ω =. The Reduced Order Model is then : j=1 L (m) ij da j dt = j=1 B (m) ij a j + j=1 k=1 C (m) ijk a ja k The ROM does not satisfy a priori the mass conservation (for non divergence free modes, as NSE-Residual modes)

10 ICCFD5, Seoul, Korea, July 7-11, 28 p. 1 I - A pressure extended Reduced Order Model Mass conservation Detailled model u = Projection onto the POD basis j=1 a j φ j = Minimizing residuals in a least squares sense, we obtain : j=1 B (c) ij a j =, where B (c) ij = ( φ j ) T φ j The modified ROM that satisfies both momentum and continuity equation writes : j=1 L (m) ij da j dt = j=1 B (m) ij + αb (c) ij a j + j=1 k=1 C (m) ijk a ja k The ROM has moreover to satisfy the flow rate conservation.

11 ICCFD5, Seoul, Korea, July 7-11, 28 p. 11 I - A pressure extended Reduced Order Model Flow rate conservation For the 2-D confined flow : For each slides S l : In a least square sense : Finally, the ROM writes j=1 L (m) ij + βl (r) daj ij dt i=1 N r X j=1 = Z S Z a j (t) da j dt j=1 j=1 u ds = c, Z L (r) ij S l φ u j S l φ u j da j dt B (m) ij ds = c, ds =. =. + αb (c) ij a j + j=1 k=1 C (m) ijk a ja k, with initial conditions a i () = (U(x, ), Φ i (x)) Ω i = 1,, N r.

12 I - A pressure extended Reduced Order Model Advantage no modelisation of the pressure term Re = 2, 11 modes convergence towards the exact limit cycles (= DNS) a a3 5 a4 a a a a a a5 a a t 3 Fig. : Temporal evolution of the POD ROM coefficients over 25 vortex shedding periods -1 a Fig. : Limit cycles of the POD ROM coefficients over 25 vortex shedding periods ICCFD5, Seoul, Korea, July 7-11, 28 p. 12

13 I - A pressure extended Reduced Order Model Drawbaks same as usual, i.e. lack of dissipation... Re = 2, 5 modes convergence towards an erroneous limit cycles (6= DNS) a a3 5 a4 a a a a a a5 a a a Fig. : Temporal evolution of the POD ROM coefficients over 25 vortex shedding periods Fig. : Limit cycles of the POD ROM coefficients over 25 vortex shedding periods ICCFD5, Seoul, Korea, July 7-11, 28 p. 12

14 ICCFD5, Seoul, Korea, July 7-11, 28 p. 12 I - A pressure extended Reduced Order Model Drawbaks same as usual, i.e. lack of dissipation... Re = 2, 3 modes exponential divergence a2 a Fig. : Temporal evolution of the POD ROM coefficients over 25 vortex shedding periods t a 2 Fig. : Limit cycles of the POD ROM coefficients over 25 vortex shedding periods

15 ICCFD5, Seoul, Korea, July 7-11, 28 p. 13 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

16 ICCFD5, Seoul, Korea, July 7-11, 28 p. 14 II - POD ROM stabilization Residuals based stabilization method Idea add dominant POD-NSE residual modes to the existing basis The POD-NSE residuals are L(eu(x, t), ep(x, t)) = R(x, t), where eu and ep obtained using POD and L is the NSE operator Model A [N r], unstable POD ROM built with N r basis functions Φ i (x). Algorithm 1. Integrate the ROM to obtain a i (t) and extract N s snapshots a i (t k ), k = 1,..., N s. 2. Compute eu(x, t k ) = i=1 a i (t k )φ i (x), ep(x, t k ) = i=1 3. Compute the POD modes Ψ(x) of the NSE residuals. a i (t k )ψ i (x), and R(x, t k ). 4. Add the K first residual modes Ψ(x) to the existing POD basis Φ i (x) and build a new ROM (here the mass and flow rate constraints are important). Model B [N r;k], PODRES ROM built with N r POD basis functions Φ i (x) + K RES basis functions Ψ i (x)

17 ICCFD5, Seoul, Korea, July 7-11, 28 p. 15 II - POD ROM stabilization SUPG and VMS methods Idea approximate the fine scales using the NSE residuals R = (R M, R C ) T u (x, t) = τ M R M (x, t) and p (x, t) = τ C R C (x, t) Class of penalty methods, i.e. j=1 L ij da j dt = Model C [N r], SUPG method j=1 B ij a j + j=1 k=1 C ijk a j a k + F i (t) Fi SUP G (t) = (eu Φ i + Ψ i, τ M R M (x, t)) Ω + ( Φ i, τ C R C (x, t)) Ω Model D [N r], VMS method F V MS i (t) = Fi SUP G (t) + (eu ( Φ i ) T, τ M R M (x, t)) Ω ( Φ i, τ M R M (x, t) τ M R M (x, t)) Ω Parameters τ M and τ C are determined using adjoint based minimization method

18 ICCFD5, Seoul, Korea, July 7-11, 28 p. 16 II - POD ROM stabilization Re = 2 and N r = 5 POD basis function erroneous limit cylcles A [5] a3 4 2 a R a 2 a a5 2 a Number of Vortex Shedding periods Fig. : temporal evolution of the L 2 norm of the POD-NSE residuals a 2 a Fig. : Limit cycles of the POD ROM coefficients over 2 vortex shedding periods -1-2

19 ICCFD5, Seoul, Korea, July 7-11, 28 p. 16 II - POD ROM stabilization Re = 2 and N r = 5 POD basis function erroneous limit cylcles A [5] B [5;2] a3 4 2 a R a 2 a a5 2 a Number of Vortex Shedding periods Fig. : temporal evolution of the L 2 norm of the POD-NSE residuals a 2 a Fig. : Limit cycles of the POD ROM coefficients over 2 vortex shedding periods -1-2

20 II - POD ROM stabilization Re = 2 and Nr = 5 POD basis function erroneous limit cylcles.5 A[5] B [5;2] C [5].35.3 R a3.4 a a a a5.1 a Number of Vortex Shedding periods Fig. : temporal evolution of the L2 norm of the POD-NSE residuals a a4 Fig. : Limit cycles of the POD ROM coefficients over 2 vortex shedding periods ICCFD5, Seoul, Korea, July 7-11, 28 p. 16

21 II - POD ROM stabilization Re = 2 and Nr = 5 POD basis function erroneous limit cylcles.5 A[5] B [5;2] C [5] D[5].35 R a3.4 a a a a5.1 a Number of Vortex Shedding periods Fig. : temporal evolution of the L2 norm of the POD-NSE residuals a a4 Fig. : Limit cycles of the POD ROM coefficients over 2 vortex shedding periods ICCFD5, Seoul, Korea, July 7-11, 28 p. 16

22 ICCFD5, Seoul, Korea, July 7-11, 28 p. 16 II - POD ROM stabilization Re = 2 and N r = 3 POD basis function divergence A [3] 1 5 R 1 1 a Number of Vortex Shedding periods Fig. : temporal evolution of the L 2 norm of the POD-NSE residuals a 2 Fig. : Limit cycle of the POD ROM coefficients over 2 vortex shedding periods

23 ICCFD5, Seoul, Korea, July 7-11, 28 p. 16 II - POD ROM stabilization Re = 2 and N r = 3 POD basis function divergence A [3] B [3;2] 1 5 R 1 1 a Number of Vortex Shedding periods Fig. : temporal evolution of the L 2 norm of the POD-NSE residuals a 2 Fig. : Limit cycle of the POD ROM coefficients over 2 vortex shedding periods

24 ICCFD5, Seoul, Korea, July 7-11, 28 p. 16 II - POD ROM stabilization Re = 2 and N r = 3 POD basis function divergence A [3] B [3;2] C [3] 1 5 R 1 1 a Number of Vortex Shedding periods Fig. : temporal evolution of the L 2 norm of the POD-NSE residuals a 2 Fig. : Limit cycle of the POD ROM coefficients over 2 vortex shedding periods

25 ICCFD5, Seoul, Korea, July 7-11, 28 p. 16 II - POD ROM stabilization Re = 2 and N r = 3 POD basis function divergence A [3] B [3;2] C [3] D [3] 1 5 R 1 1 a Number of Vortex Shedding periods Fig. : temporal evolution of the L 2 norm of the POD-NSE residuals a 2 Fig. : Limit cycle of the POD ROM coefficients over 2 vortex shedding periods

26 ICCFD5, Seoul, Korea, July 7-11, 28 p. 17 III - Improvement of the functional subspace Functional subspace drawbacks, Φ n (x) : lack of representativity of 3D flows outside the database Reconstructed energy (in percents) N r =4 N =8 r N r =2 N = r Non domensional time Figures results from Buffoni etal. Journal of Fluid Mech. 569 (26) Problems for 3D flow control Erroneous turbulence properties (spectrum, etc) Goal : determine Φ n (x) at Re 2 starting from Φ n (x) at Re 1 for low numerical costs.

27 ICCFD5, Seoul, Korea, July 7-11, 28 p. 18 III - Improvement of the functional subspace Method 1 : Krylov-like method to improve the functional subspace Φ n (x) Use of the POD-NSE residuals : L(eu(x, t), ep(x, t)) = R(x, t), eu and ep are POD fields, L is the NSE operator Algorithm Start with the POD basis to be improved, Φ i with i = 1,..., N r. Let N = N r. 1. Build and solve the corresponding ROM to obtain a i (t) and extract N s snapshots a i (t k ) with i = 1,..., N r and k = 1,..., N s. 2. Compute eu(x, t k ) = i=1 a i (t k )φ i (x), ep(x, t k ) = i=1 3. Compute the POD modes Ψ(x) of the NSE residuals. a i (t k )ψ i (x), and R(x, t k ). 4. Add the K first residual modes Ψ(x) to the existing POD basis Φ i (x) Φ Φ + Ψ N r N r + K If N r is below than a threshold, return to 1. Else, go to Perform a new POD compression with N r = N. If convergence is satisfied, stop. Else, return to 1.

28 ICCFD5, Seoul, Korea, July 7-11, 28 p. 19 III - Improvement of the functional subspace {Φ i } N i=1 for Re 1. Let N r = N and T = [, T] Build and solve the ROM on T Extract N s snapshots a i (t k ) Compute eu(x, t k ) and ep(x, t k ) Coarse scales (Φ) if N r < N max if N r N max Compute the residuals R(x, t k ) from eu(x, t k ) and ep(x, t k ) NS operator evaluated for Re 2 Missing scales (u = τr) Perform a new POD using eu(x, t k ) and ep(x, t k ) Extract first N modes Φ i Let N r = N Add {Ψ i } K i=1 to {Φ i} N r i=1 Let Φ Φ + Ψ Let N r N r + K Perform a POD from R(x, t k ) Extract first K modes Ψ i

29 ICCFD5, Seoul, Korea, July 7-11, 28 p. 2 III - Improvement of the functional subspace First test case : 1D burgers equation Re 1 = 5 Re 2 = 3 (Φ Re 1 Φ Re 2.5) Φ1 Φ x x Φ4 Φ x x

30 ICCFD5, Seoul, Korea, July 7-11, 28 p. 2 III - Improvement of the functional subspace First test case : 1D burgers equation Re 1 = 5 Re 2 = 3 (Φ Re 1 Φ Re 2.5) 1.8 Φ Φ Re Iterations number Only 6 ROM integrations (on T = 1) are necessary to converge (no DNS!!)

31 ICCFD5, Seoul, Korea, July 7-11, 28 p. 21 III - Improvement of the functional subspace Second test case : 2D NSE equations Re 1 = 1 Re 2 = 2 (Φ Re 1 Φ Re 2.5) Initial basis, Re 1 = 1 "Target" basis, Re 2 = 2 φ 1 φ 1 φ 2 φ 2 φ 4 φ 4 φ 6 φ 6

32 ICCFD5, Seoul, Korea, July 7-11, 28 p. 21 III - Improvement of the functional subspace Second test case : 2D NSE equations Re 1 = 1 Re 2 = 2 (Φ Re 1 Φ Re 2.5).8.7 Φ Φ Re Iterations number No convergence...

33 ICCFD5, Seoul, Korea, July 7-11, 28 p. 22 III - Improvement of the functional subspace Observations the decomposition U (x, t) = τr(x, t) is used to stabilize the ROM Very good results for NSE the decomposition U (x, t) = τr(x, t) is used to improve POD basis Very good results for Burgers, quite bad results for NSE Possible explanation the decomposition U (x, t) = τr(x, t) is only valid for Small values of U (x, t) (for instance, non resolved POD modes). Can we find a good approximation τ of the elementary Green s function? Not sure... Future works Look for an other decomposition for the missing scales U (x, t) U (x, t) = M(t)R(x, t), where M R 3 3

34 ICCFD5, Seoul, Korea, July 7-11, 28 p. 23 III - Improvement of the functional subspace Method 2 : hybrid ROM-DNS method to adapt the functional subspace Φ n (x) Database modification : statistics evolution ϕ : Φ (k) Φ (k+1) ROM DNS Re 2 ROM DNS Re 2 ROM DNS Re 2 Time U(x, t k 1 ) U(x, t k ) U(x, t k+1 ) 1. Database modification [U(x, t 1 ) U(x, t 2 )... U(x, t Nr ) ] eu [1,,N r] (x, t k ) = n=1 One snapshot modification using few DNS iterations a n (t k )φ n (x), U(x, t s ) = e U [1,,N r] (x, t s ) + U s (x, t s ). In a general way eu(x, t k ) = e U [1,,N r] (x, t k ) + δ ks U (x, t s ),

35 ICCFD5, Seoul, Korea, July 7-11, 28 p. 24 III - Improvement of the functional subspace 2 Modification temporal correlations tensor C(t k, t l ) = (U(x, t k ), U(x, t l )) Ω XN r a i (t k )φ i (x) + U (x, t k ), = + i=1 i=1 j=1 i=1 j=1 1 a j (t l )φ j (x) + U (x, t l ) A a i (t k )a j (t l )(φ i (x), φ j (x)) Ω + U (x, t k ), U (x, t l ) {z } Ω =δ ij a i (t k ) φ i (x), U (x, t l ) Ω {z } = Final approximation XN r Z C(t k, t l ) = a i (t k )a i (t l ) + δ ks δ ls i=1 + j=1 n c X Ω i=1 Ω a lj U T (x, t k ), φ j (x). Ω {z } = U i (x, t s )U i (x, t s ) dx.

36 ICCFD5, Seoul, Korea, July 7-11, 28 p. 25 III - Improvement of the functional subspace eu [1,,5] 1 6 eu [1,,11] 1 5 U U e [1,,2] 1 5 U U e [1,,2] 1 4 U contribution λn λn U contribution N r = N r = index of POD modes index of POD modes N r = 5 N r = 11 Fig. : Comparison of the temporal correlation tensor eigenvalues evaluated from the exact field, U, and from the N r -modes approximated one, e U [1,,N r]. Very good approximation, and very low costs method!

37 ICCFD5, Seoul, Korea, July 7-11, 28 p. 26 III - Improvement of the functional subspace 3 Functional subspace adaptation φ (n+1) k (x) = 1 λ (n+1) k j=1 eu (n) (x, t j )a (n+1) k (t j ) φ (n+1) k (x) = 1 λ (n+1) k i=1 j=1 a (n+1) k (t j )a (n) i (t j ) φ (n) i (x)+ 1 U (n) (x, t s )a (n+1) k (t s ). λ k φ (n+1) k (x) = i=1 K (n+1) ki φ (n) i (x) + S (n+1) k (x). Taken S (n+1) with elements S (n+1) ij = S j (n+1) i, the actualized basis is obtained using the linear application ϕ : R n R n R n R n defined as Incrementation n = n + 1. ϕ : φ (n) φ (n+1) = φ (n) K (n+1) + S (n+1)

38 ICCFD5, Seoul, Korea, July 7-11, 28 p. 27 III - Improvement of the functional subspace Results for a dynamical evolution from Re 1 = 1 to Re 2 = % DNS.8 Φ Φ Re n vs Fig. : temporal evolution of the POD basis

39 ICCFD5, Seoul, Korea, July 7-11, 28 p. 27 III - Improvement of the functional subspace Results for a dynamical evolution from Re 1 = 1 to Re 2 = % DNS 9% DNS.8 Φ Φ Re n vs Fig. : temporal evolution of the POD basis

40 ICCFD5, Seoul, Korea, July 7-11, 28 p. 27 III - Improvement of the functional subspace Results for a dynamical evolution from Re 1 = 1 to Re 2 = % DNS 9% DNS 8% DNS Φ Φ Re n vs Fig. : temporal evolution of the POD basis

41 ICCFD5, Seoul, Korea, July 7-11, 28 p. 27 III - Improvement of the functional subspace Results for a dynamical evolution from Re 1 = 1 to Re 2 = 2 Φ Φ Re % DNS 9% DNS 8% DNS 7% DNS n vs Fig. : temporal evolution of the POD basis

42 ICCFD5, Seoul, Korea, July 7-11, 28 p. 27 III - Improvement of the functional subspace Results for a dynamical evolution from Re 1 = 1 to Re 2 = 2 Φ Φ Re % DNS 9% DNS 8% DNS 7% DNS % DNS n vs Fig. : temporal evolution of the POD basis

43 ICCFD5, Seoul, Korea, July 7-11, 28 p. 28 III - Improvement of the functional subspace Observations Results are very good if a sufficient amount of DNS is performed Good for a percentage Possible explanation DNS DNS+P ODROM greater than 7% convergence N P exact "target" N N + P > 7% POD ROM DNS

44 ICCFD5, Seoul, Korea, July 7-11, 28 p. 28 III - Improvement of the functional subspace Observations Results are very good if a sufficient amount of DNS is performed Good for a percentage Possible explanation DNS DNS+P ODROM greater than 7% convergence N P exact "target" N N + P = 7% POD ROM DNS

45 ICCFD5, Seoul, Korea, July 7-11, 28 p. 28 III - Improvement of the functional subspace Observations Results are very good if a sufficient amount of DNS is performed Good for a percentage Possible explanation DNS DNS+P ODROM greater than 7% divergence... N P exact "target" N N + P 7% POD ROM DNS

46 ICCFD5, Seoul, Korea, July 7-11, 28 p. 29 Conclusions A pressure extended Reduced Order Model The pressure is naturally included in the ROM no modelisation of pressure term but need of modelisation interaction with non resolved modes (dissipation) Stabilization of Reduced Order Models based on POD Add some residual modes Good results SUPG and VMS methods Very good results Try to improve the functional subspace Improvement using POD-NSE residuals (Krylov like method) Very good for 1D burgers equation but quite poor results for 2D NSE equations Problem with the continuity equation? Missing scales "fine scales" approximation U (x, t) = τr(x, t) not good! Database modification : an hybrid DNS/ROM method Fast evaluation of temporal correlations tensor Linear actualization of the POD basis DNS must correct ROM good results for amount of DNS greater than 7%