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

SLIDE 1

Improvement of Reduced Order Modeling based

n 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

SLIDE 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

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 2

SLIDE 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

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 3

SLIDE 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
L

H D U(y) U = 0 U = 0 Ω ⊲ 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

SLIDE 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 : max

Φ |(U, Φ)|2,

Φ2 = 1. ⊲ Optimal convergence in L2 norm de Φ(X) ⇒ Dynamical reduction possible.

riginal axis
riginal axis

Φ1 Φ2 U(X) U(X) − Um(X) 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

SLIDE 6

I - A pressure extended Reduced Order Model

⊲ Equivalent with Fredholm equation : Z

D

Rij(X, X′)Φ(j)

n (X′) dX′ = λnΦ(i) n (X)

n = 1, .., Ns ֒ → R(X, X′) : Space-time correlation tensor ⊲ Snapshots method, Sirovich (1987) : Z

T

C(t, t′)an(t′) dt′ = λnan(t) ֒ → C(t, t′) : Temporal correlations ⊲ Φ(X) flow basis : U(x, t) =

Ns

X

n=1

an(t)Φn(x).

Temps

Space C

r

r e l a t i

n

Average on space X X′ 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

SLIDE 7

I - A pressure extended Reduced Order Model

Truncation of the POD basis to keep 99% of the Relative Information Content RIC(M) =

M

X

k=1

λk . Ns X

k=1

λk Example at Re = 200 for U = (u, p)T with Ns = 200

2 4 6 8 10 12 14 16 18 20 10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

10

6

10

7

λn index of POD modes

Fig. : POD spectrum.

5 10 15 20 50 60 70 80 90 100

RIC (%) index of POD modes

Fig. : RIC(M), M nb modes POD retenus.

Nr = arg min

M RIC(M) s.t. RIC(Nr) > 99%

⇒ Nr = 5 !

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 7

SLIDE 8

I - A pressure extended Reduced Order Model

The POD basis is Φ = (φ, ψ)T

∇∧φ1 ψ1 ∇∧φ3 ψ3 ∇∧φ5 ψ5

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

SLIDE 9

I - A pressure extended Reduced Order Model

◮ Momentum conservation Detailled model (exact) ∂u ∂t + (u · ∇)u = −∇p + 1 Re ∆u Galerkin projection using e u(x, t) =

Nr

X

i=1

ai(t)φi(x) and e p(x, t) =

Nr

X

i=1

ai(t)ψi(x) : @φi,

Nr

X

j=1

φj

daj dt +

Nr

X

j=1 Nr

X

k=1

(φj · ∇) φk ajak +

Nr

X

j=1

∇ψj aj − 1 Re

Nr

X

j=1

∆φj aj 1 A

Ω

= 0. The Reduced Order Model is then :

Nr

X

j=1

L(m)

ij

daj dt =

Nr

X

j=1

B(m)

ij

aj +

Nr

X

j=1 Nr

X

k=1

C(m)

ijk ajak

֒ → 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

SLIDE 10

I - A pressure extended Reduced Order Model

◮ Mass conservation Detailled model ∇ · u = 0 Projection onto the POD basis

Nr

X

j=1

aj∇ · φj = 0 Minimizing residuals in a least squares sense, we obtain :

Nr

X

j=1

B(c)

ij aj = 0,

where B(c)

ij

= (∇ · φj)T ∇ · φj The modified ROM that satisfies both momentum and continuity equation writes :

Nr

X

j=1

L(m)

ij

daj dt =

Nr

X

j=1

“ B(m)

ij

+ αB(c)

ij

” aj +

Nr

X

j=1 Nr

X

k=1

C(m)

ijk ajak

֒ → The ROM has moreover to satisfy the flow rate conservation.

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 10

SLIDE 11

I - A pressure extended Reduced Order Model

◮ Flow rate conservation For the 2-D confined flow : Z

S

u ds = c, For each slides Sl :

Nr

X

i=1

aj(t) Z

Sl

φu

j ds = c, Nr

X

j=1

daj dt

Z

Sl

φu

j ds = 0.

In a least square sense :

Nr

X

j=1

L(r)

ij

daj dt = 0.

Finally, the ROM writes

Nr

X

j=1

“ L(m)

ij

+ βL(r)

ij

” daj

dt =

Nr

X

j=1

“ B(m)

ij

+ αB(c)

ij

” aj +

Nr

X

j=1 Nr

X

k=1

C(m)

ijk ajak,

with initial conditions ai(0) = (U(x, 0), Φi(x))Ω i = 1, · · · , Nr.

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 11

SLIDE 12

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

50

a2

50

50

a3

20
10

10 20

a4

10 20 30 40

20
10

10 20

t a5

Fig. : Temporal evolution of the POD ROM

coefficients over 25 vortex shedding periods

60
40
20

20 40 60

60
40
20

20 40 60

a3 a2

20
10

10 20

60
40
20

20 40 60

a4 a2

20
10

10 20

60
40
20

20 40 60

a5 a2

20
10

10 20

20
10

10 20

a5 a4

Fig. : Limit cycles of the POD ROM coefficients
ver 25 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 12

SLIDE 13

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

50

a2

50

50

a3

20
10

10 20

a4

20
10

10 20

a5

Fig. : Temporal evolution of the POD ROM

coefficients over 25 vortex shedding periods

60
40
20

20 40 60

60
40
20

20 40 60

a3 a2

20
10

10 20

60
40
20

20 40 60

a4 a2

20
10

10 20

60
40
20

20 40 60

a5 a2

20
10

10 20

20
10

10 20

a5 a4

Fig. : Limit cycles of the POD ROM coefficients
ver 25 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 12

SLIDE 14

I - A pressure extended Reduced Order Model

◮ Drawbaks same as usual, i.e. lack of dissipation... Re = 200, 3 modes ⇒ exponential divergence

5 10 15

200
100

100 200

t a2

Fig. : Temporal evolution of the POD ROM

coefficients over 25 vortex shedding periods

200
150
100
50

50 100 150 200

200
150
100
50

50 100 150 200

a3 a2

Fig. : Limit cycles of the POD ROM coefficients
ver 25 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 12

SLIDE 15

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

SLIDE 16

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(e u(x, t), e p(x, t)) = R(x, t), where e u and e p obtained using POD and L is the NSE operator Model A[Nr], unstable POD ROM built with Nr basis functions Φi(x) . Algorithm 1. Integrate the ROM to obtain ai(t) and extract Ns snapshots ai(tk), k = 1, . . . , Ns. 2. Compute e u(x, tk) =

Nr

X

i=1

ai(tk)φi(x), e p(x, tk) =

Nr

X

i=1

ai(tk)ψi(x), and R(x, tk). 3. Compute the POD modes Ψ(x) of the NSE residuals. 4. Add the K first residual modes Ψ(x) to the existing POD basis Φi(x) and build a new ROM (here the mass and flow rate constraints are important). Model B[Nr;K], PODRES ROM built with Nr POD basis functions Φi(x) + K RES basis functions Ψi(x)

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 14

SLIDE 17

II - POD ROM stabilization

◮ SUPG and VMS methods ⇒ Idea approximate the fine scales using the NSE residuals R = (RM, RC)T u′(x, t) = τM RM(x, t) and p′(x, t) = τC RC(x, t) ֒ → Class of penalty methods, i.e.

Nr

X

j=1

Lij daj dt =

Nr

X

j=1

Bijaj +

Nr

X

j=1 Nr

X

k=1

Cijkajak + Fi(t) Model C[Nr], SUPG method F SUP G

i

(t) = (e u · ∇Φi + ∇Ψi, τM RM(x, t))Ω + (∇ · Φi, τC RC(x, t))Ω Model D[Nr], VMS method F V MS

i

(t) = F SUP G

i

(t) + (e u · (∇Φi)T , τM RM(x, t))Ω − (∇Φi, τM RM(x, t) ⊗ τM RM(x, t))Ω ֒ → Parameters τM and τC are determined using adjoint based minimization method

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 15

SLIDE 18

II - POD ROM stabilization

◮ Re = 200 and Nr = 5 POD basis function → erroneous limit cylcles

10 20 30 40 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

R Number of Vortex Shedding periods

A[5]

Fig. : temporal evolution of the L2 norm of the

POD-NSE residuals

60
40
20

20 40 60

60
40
20

20 40 60

a3 a2

20
10

10 20

60
40
20

20 40 60

a4 a2

20
10

10 20

60
40
20

20 40 60

a5 a2

20
10

10 20

20
10

10 20

a5 a4

Fig. : Limit cycles of the POD ROM coefficients
ver 20 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 16

SLIDE 19

II - POD ROM stabilization

◮ Re = 200 and Nr = 5 POD basis function → erroneous limit cylcles

10 20 30 40 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

R Number of Vortex Shedding periods

A[5] B[5;2]

Fig. : temporal evolution of the L2 norm of the

POD-NSE residuals

60
40
20

20 40 60

60
40
20

20 40 60

a3 a2

20
10

10 20

60
40
20

20 40 60

a4 a2

20
10

10 20

60
40
20

20 40 60

a5 a2

20
10

10 20

20
10

10 20

a5 a4

Fig. : Limit cycles of the POD ROM coefficients
ver 20 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 16

SLIDE 20

II - POD ROM stabilization

◮ Re = 200 and Nr = 5 POD basis function → erroneous limit cylcles

10 20 30 40 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

R Number of Vortex Shedding periods

A[5] B[5;2] C[5]

Fig. : temporal evolution of the L2 norm of the

POD-NSE residuals

60
40
20

20 40 60

60
40
20

20 40 60

a3 a2

20
10

10 20

60
40
20

20 40 60

a4 a2

20
10

10 20

60
40
20

20 40 60

a5 a2

20
10

10 20

20
10

10 20

a5 a4

Fig. : Limit cycles of the POD ROM coefficients
ver 20 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 16

SLIDE 21

II - POD ROM stabilization

◮ Re = 200 and Nr = 5 POD basis function → erroneous limit cylcles

10 20 30 40 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

R Number of Vortex Shedding periods

A[5] B[5;2] C[5] D[5]

Fig. : temporal evolution of the L2 norm of the

POD-NSE residuals

60
40
20

20 40 60

60
40
20

20 40 60

a3 a2

20
10

10 20

60
40
20

20 40 60

a4 a2

20
10

10 20

60
40
20

20 40 60

a5 a2

20
10

10 20

20
10

10 20

a5 a4

Fig. : Limit cycles of the POD ROM coefficients
ver 20 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 16

SLIDE 22

II - POD ROM stabilization

◮ Re = 200 and Nr = 3 POD basis function → divergence

10 20 30 40 10

3

10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

R Number of Vortex Shedding periods

A[3]

Fig. : temporal evolution of the L2 norm of the

POD-NSE residuals

150
100
50

50 100 150

150
100
50

50 100 150

a3 a2

Fig. : Limit cycle of the POD ROM coefficients over

20 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 16

SLIDE 23

II - POD ROM stabilization

◮ Re = 200 and Nr = 3 POD basis function → divergence

10 20 30 40 10

3

10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

R Number of Vortex Shedding periods

A[3] B[3;2]

Fig. : temporal evolution of the L2 norm of the

POD-NSE residuals

150
100
50

50 100 150

150
100
50

50 100 150

a3 a2

Fig. : Limit cycle of the POD ROM coefficients over

20 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 16

SLIDE 24

II - POD ROM stabilization

◮ Re = 200 and Nr = 3 POD basis function → divergence

10 20 30 40 10

3

10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

R Number of Vortex Shedding periods

A[3] B[3;2] C[3]

Fig. : temporal evolution of the L2 norm of the

POD-NSE residuals

150
100
50

50 100 150

150
100
50

50 100 150

a3 a2

Fig. : Limit cycle of the POD ROM coefficients over

20 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 16

SLIDE 25

II - POD ROM stabilization

◮ Re = 200 and Nr = 3 POD basis function → divergence

10 20 30 40 10

3

10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

R Number of Vortex Shedding periods

A[3] B[3;2] C[3] D[3]

Fig. : temporal evolution of the L2 norm of the

POD-NSE residuals

150
100
50

50 100 150

150
100
50

50 100 150

a3 a2

Fig. : Limit cycle of the POD ROM coefficients over

20 vortex shedding periods

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 16

SLIDE 26

III - Improvement of the functional subspace

◮ Functional subspace drawbacks, Φn(x) : lack of representativity of 3D flows

utside the database

360 380 400 420 440 50 60 70 80 90 100 Non−domensional time Reconstructed energy (in percents) Nr=4 Nr=8 Nr=20 Nr=60

Figures results from Buffoni etal. Journal of Fluid Mech. 569 (2006)

Problems for 3D flow control
Erroneous turbulence properties (spectrum, etc)

⇒ Goal : determine Φn(x) at Re2 starting from Φn(x) at Re1 for low numerical costs.

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 17

SLIDE 27

III - Improvement of the functional subspace

⊲ Method 1 : Krylov-like method to improve the functional subspace Φn(x)

Use of the POD-NSE residuals : L(e

u(x, t), e p(x, t)) = R(x, t), e u and e p are POD fields, L is the NSE operator Algorithm Start with the POD basis to be improved, Φi with i = 1, . . . , Nr. Let N0 = Nr. 1. Build and solve the corresponding ROM to obtain ai(t) and extract Ns snapshots ai(tk) with i = 1, . . . , Nr and k = 1, . . . , Ns. 2. Compute e u(x, tk) =

Nr

X

i=1

ai(tk)φi(x), e p(x, tk) =

Nr

X

i=1

ai(tk)ψi(x), and R(x, tk). 3. Compute the POD modes Ψ(x) of the NSE residuals. 4. Add the K first residual modes Ψ(x) to the existing POD basis Φi(x)

Φ ← Φ + Ψ
Nr ← Nr + K
If Nr is below than a threshold, return to 1. Else, go to 5.

5. Perform a new POD compression with Nr = N0.

If convergence is satisfied, stop. Else, return to 1.

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 18

SLIDE 28

III - Improvement of the functional subspace

{Φi}N0

i=1 for Re1.

Let Nr = N0 and T = [0, T] Build and solve the ROM on T Extract Ns snapshots ai(tk) Compute e u(x, tk) and e p(x, tk) ⇒ Coarse scales (Φ) Compute the residuals R(x, tk) from e u(x, tk) and e p(x, tk) NS operator evaluated for Re2 ⇒ Missing scales (u′ = τR) Perform a POD from R(x, tk) Extract first K modes Ψi Add {Ψi}K

i=1 to {Φi}Nr i=1

Let Φ ← Φ + Ψ Let Nr ← Nr + K Perform a new POD using e u(x, tk) and e p(x, tk) Extract first N0 modes Φi Let Nr = N0 if Nr < Nmax if Nr ≥ Nmax

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 19

SLIDE 29

III - Improvement of the functional subspace

◮ First test case : 1D burgers equation Re1 = 50 → Re2 = 300 (ΦRe1 · ΦRe2 ≈ 0.5)

0.2 0.4 0.6 0.8 1

0.1
0.05

0.05 0.1

Φ1 x

0.2 0.4 0.6 0.8 1

0.15
0.1
0.05

0.05 0.1 0.15

Φ2 x

0.2 0.4 0.6 0.8 1

0.2
0.15
0.1
0.05

0.05 0.1 0.15 0.2

Φ4 x

0.2 0.4 0.6 0.8 1

0.2
0.15
0.1
0.05

0.05 0.1 0.15 0.2

Φ8 x

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 20

SLIDE 30

III - Improvement of the functional subspace

◮ First test case : 1D burgers equation Re1 = 50 → Re2 = 300 (ΦRe1 · ΦRe2 ≈ 0.5)

2 4 6 8 10 0.2 0.4 0.6 0.8 1

Φ · ΦRe2 Iterations number

֒ → Only 6 ROM integrations (on T = 1) are necessary to converge (no DNS ! !)

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 20

SLIDE 31

III - Improvement of the functional subspace

◮ Second test case : 2D NSE equations Re1 = 100 → Re2 = 200 (ΦRe1 · ΦRe2 ≈ 0.5) Initial basis, Re1 = 100

∇∧φ1 ∇∧φ2 ∇∧φ4 ∇∧φ6

"Target" basis, Re2 = 200

∇∧φ1 ∇∧φ2 ∇∧φ4 ∇∧φ6

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 21

SLIDE 32

III - Improvement of the functional subspace

◮ Second test case : 2D NSE equations Re1 = 100 → Re2 = 200 (ΦRe1 · ΦRe2 ≈ 0.5)

20 40 60 80 0.3 0.4 0.5 0.6 0.7 0.8

Φ · ΦRe2 Iterations number

֒ → No convergence...

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 21

SLIDE 33

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 ∈ R3×3

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 22

SLIDE 34

III - Improvement of the functional subspace

⊲ Method 2 : hybrid ROM-DNS method to adapt the functional subspace Φn(x)

Database modification : statistics evolution ⇒ ϕ : Φ(k) → Φ(k+1)

ROM ROM ROM DNS Re2 DNS Re2 DNS Re2

U(x, tk−1) U(x, tk) U(x, tk+1)

Time

1. Database modification [U(x, t1) U(x, t2) . . . U(x, tNr) ] e U[1,··· ,Nr](x, tk) =

Nr

X

n=1

an(tk)φn(x), One snapshot modification using few DNS iterations U(x, ts) = e U[1,··· ,Nr](x, ts) + U⊥

s (x, ts).

In a general way e U(x, tk) = e U[1,··· ,Nr](x, tk) + δksU⊥(x, ts),

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 23

SLIDE 35

III - Improvement of the functional subspace

2 Modification temporal correlations tensor C(tk, tl) = (U(x, tk), U(x, tl))Ω = @

Nr

X

i=1

ai(tk)φi(x) + U⊥(x, tk),

Nr

X

j=1

aj(tl)φj(x) + U⊥(x, tl) 1 A

Ω

=

Nr

X

i=1 Nr

X

j=1

ai(tk)aj(tl) (φi(x), φj(x))Ω | {z }

=δij

+ “ U⊥(x, tk), U⊥(x, tl) ”

Ω

+

Nr

X

i=1

ai(tk) “ φi(x), U⊥(x, tl) ”

Ω

| {z }

=0

+

Nr

X

j=1

alj “ U⊥T (x, tk), φj(x) ”

Ω

| {z }

=0

. Final approximation C(tk, tl) =

Nr

X

i=1

ai(tk)ai(tl) + δksδls Z

Ω nc

X

i=1

U⊥i(x, ts)U⊥i(x, ts) dx.

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 24

SLIDE 36

III - Improvement of the functional subspace

2 4 6 8 10 12 14 16 18 20 10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

10

6

10

7

λn index of POD modes e U[1,··· ,5] U ≡ e U[1,··· ,200] Nr = 5 U⊥ contribution

Nr = 5

2 4 6 8 10 12 14 16 18 20 10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

10

6

10

7

λn index of POD modes e U[1,··· ,11] U ≡ e U[1,··· ,200] Nr = 11 U⊥ contribution

Nr = 11

Fig. : Comparison of the temporal correlation tensor eigenvalues evaluated from the exact field, U, and

from the Nr-modes approximated one, e U [1,··· ,Nr].

֒ → Very good approximation, and very low costs method !

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 25

SLIDE 37

III - Improvement of the functional subspace

3 Functional subspace adaptation φ(n+1)

k

(x) = 1 λ(n+1)

k Nr

X

j=1

e U(n)(x, tj)a(n+1)

k

(tj) φ(n+1)

k

(x) = 1 λ(n+1)

k Nr

X

i=1 Nr

X

j=1

a(n+1)

k

(tj)a(n)

i

(tj) φ(n)

i

(x)+ 1 λk U⊥(n)(x, ts)a(n+1)

k

(ts). φ(n+1)

k

(x) =

Nr

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

= Sj

i (n+1), the actualized basis is obtained

using the linear application ϕ : Rn × Rn → Rn × Rn defined as ϕ : φ(n) → φ(n+1) = φ(n)K(n+1) + S(n+1) Incrementation n = n + 1.

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 26

SLIDE 38

III - Improvement of the functional subspace

◮ Results for a dynamical evolution from Re1 = 100 to Re2 = 200

5 10 15 20 25 30 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100% DNS Φ · ΦRe2 nvs

Fig. : temporal evolution of the POD basis

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 27

SLIDE 39

III - Improvement of the functional subspace

◮ Results for a dynamical evolution from Re1 = 100 to Re2 = 200

5 10 15 20 25 30 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100% DNS 90% DNS Φ · ΦRe2 nvs

Fig. : temporal evolution of the POD basis

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 27

SLIDE 40

III - Improvement of the functional subspace

◮ Results for a dynamical evolution from Re1 = 100 to Re2 = 200

5 10 15 20 25 30 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100% DNS 90% DNS 80% DNS Φ · ΦRe2 nvs

Fig. : temporal evolution of the POD basis

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 27

SLIDE 41

III - Improvement of the functional subspace

◮ Results for a dynamical evolution from Re1 = 100 to Re2 = 200

5 10 15 20 25 30 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100% DNS 90% DNS 80% DNS 70% DNS Φ · ΦRe2 nvs

Fig. : temporal evolution of the POD basis

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 27

SLIDE 42

III - Improvement of the functional subspace

◮ Results for a dynamical evolution from Re1 = 100 to Re2 = 200

5 10 15 20 25 30 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100% DNS 90% DNS 80% DNS 70% DNS 60% DNS Φ · ΦRe2 nvs

Fig. : temporal evolution of the POD basis

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 27

SLIDE 43

III - Improvement of the functional subspace

◮ Observations Results are very good if a sufficient amount of DNS is performed ֒ → Good for a percentage

DNS DNS+P ODROM greater than 70%

◮ Possible explanation POD ROM DNS exact "target" P N N N + P > 70% convergence

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 28

SLIDE 44

III - Improvement of the functional subspace

◮ Observations Results are very good if a sufficient amount of DNS is performed ֒ → Good for a percentage

DNS DNS+P ODROM greater than 70%

◮ Possible explanation POD ROM DNS exact "target" P N N N + P = 70% convergence

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 28

SLIDE 45

III - Improvement of the functional subspace

◮ Observations Results are very good if a sufficient amount of DNS is performed ֒ → Good for a percentage

DNS DNS+P ODROM greater than 70%

◮ Possible explanation POD ROM DNS exact "target" P N N N + P ≤ 70% divergence...

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 28

SLIDE 46

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 70%

ICCFD5, Seoul, Korea, July 7-11, 2008 – p. 29