Quantifying the uncertainties introduced by dimension reduction in fluid dynamics

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Submitted on 26 Jun 2019

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Quantifying the uncertainties introduced by dimension reduction in fluid dynamics

Valentin Resseguier, Matheus Ladvig, Agustin Picard, Etienne Mémin, Reda Bouaida, Bertrand Chapron

To cite this version:

Valentin Resseguier, Matheus Ladvig, Agustin Picard, Etienne Mémin, Reda Bouaida, et al.. Quanti-fying the uncertainties introduced by dimension reduction in fluid dynamics. UNCECOMP 2019 - 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering, Jun 2019, Hersonissos, Greece. pp.1-21. �hal-02165809�

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D i g i t a l S y s t e m s

QUANTIFYING THE UNCERTAINTIES

INTRODUCED BY DIMENSION REDUCTION

IN FLUID DYNAMICS

Valentin Resseguier,

Matheus Ladvig, Agustin M Picard

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1. Context : observer for wind turbine application

2. Physics, data & reduced order model (ROM)

3. Simulation, measurements & data assimilation

4. Reduced order model under location uncertainty

5. Results

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PART I

CONTEXT :

OBSERVER FOR WIND

TURBINE APPLICATIONS

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CEN « Simulation » (~ 70 people)

R&D and engineering

Expertise:

• Radar, optronics, sonar •Geophysical fluid dyn. • Mechanical and thermal Business: • Scientific softwares • Simulations, HPC • VR & AR Lab (~ 15 peoples)

Research, R&T, R&D

Expertise:

•Geophysical fluid dyn. •Signal, data

assimilation

•Machine Learning •Multi-agents systems •Drones

Other Business Units

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Wind

Turbine

blade

Variable

blade lift

Controler

+

−

Desired

value

WIND TURBINE BLADE

LIFT CONTROL

Wind

_Damages

fluctuations

Simple model

Observer

Simple model Estimation and prediction: • Flow • Lift • …

Incomplete & noisy measurements

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PART II

PHYSICS, DATA

& REDUCED ORDER MODEL

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REDUCED ORDER MODEL (ROM)

Time Space

Parameters (if any)

Solution of an PDE with the form:

Full space

Reduced

space

Solution

coordinates

Dimension

!×# ~ 10

'

( ~ 10 − 100

Order of

magnitude

examples in CFD

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•

Principal Component Analysis (PCA) on a

dataset

to reduce the dimensionality:

Resolved modes

Snapshots Spatial modes

PCA

Off-line simulations

•

Approximation:

à ROM for very fast simulation of temporal modes

POD-GALERKIN

•

Projection of the “physics”

onto the spatial modes :

!" ( Physical equation (e.g. Navier-Stokes))

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PART III

SIMULATION,

MEASUREMENTS

& DATA ASSIMILATION

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Numerical

Simulation

(ROM)

à erroneous

On-line

measurements

à incomplete à possibly noisy

COMBINING SIMULATIONS AND MEASUREMENTS

3 ". $%& _{5 ". $}%& Velocity More accurate estimation globally in space ( ) * ∝ ( * ) (()) (()) ((*|)) Data assimilation ( particle filtering with tempering &

mutation )

Need for uncertainty / errors quantification à Random dynamics

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PART IV

REDUCED ORDER MODELS

UNDER LOCATION

UNCERTAINTY

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Randomized

Navier-Stokes model

__________________

• Good closure

• Good model error

quantification

for data assimilation

Residual

LOCATION UNCERTAINTY MODELS (LUM)

Assumed

time-uncorrelated

! = #

$%&

'

(

_$

)

_$

+

SALT LUM Memin, 2014 Resseguier et al. 2017 a, b, c, d Cai et al. 2017 Chapron et al. 2018 Yang & Memin 2019

Crisan et al., 2017

Gay-Balmaz & Holm 2017 Cotter and al. 2018 a, b Cotter and al. 2019 Holm, 2015 Holm and Tyranowski, 2016 Arnaudon et al. 2017 Mikulevicius & Rozovskii, 2004 Flandoli, 2011 References

:

Cotter and al. 2017 Resseguier et al. 2019 a, b

Resolved

modes

Randomized

ROM

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̇-Advection Diffusion

MODEL UNDER LOCATION UNCERTAINTY,

THE TRACER ADVECTION EXAMPLE

Drift

correction

Multiplicative

random

forcing

Balanced

energy

exchanges

!Θ

!#

= 0

Large scales: Small scales: Variance tensor:

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!"

_#

!$

= F

'

b + *

⋅#⋅

̇-

.

/

" + 0

_#⋅

̇-

_.

additive noise multiplicative noise n x M M x 1 n x 1 1 x M M x 1

(stochastic Navier-Stokes)

R E D U C E D M O D E L S U N D E R L O C AT I O N U N C E R TA I N T Y:

G A L E R K I N P R O J E C T I O N G I V E S S D E S F O R R E S O LV E D M O D E S

Large scales: Small scales: Variance tensor:

2

nd

_{order polynomial:}

coefficients given by physics,

and

1 = 2 ̇- 2 ̇-

/

3 Correlations to estimate

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SUMMARY

Off-line : Building ROM

On-line :

Simualtion & data assimialtion

Stochastic ROM Randomized Physics (LUM) Data DNS code Physics (Navier-Stokes) Stochastic ROM Flow ! = # $%& ' (_$)_$ Temporal modes (_$ Data assimilation (particle filtering) Measurements

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PART V

RESULTS :

UQ &

FAST OBSERVER

OF THE FLOW

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UNCERTAINTY QUANTIFICATION

u 2D Wake at Re 100 u 3D Wake at Re 300 6/ 26/ 19 Pr és ent at ion ... DNS POD-Galerkin DNS POD-Galerkin with fitted eddy viscosity (benchmark) Red. LUM RMSE Red. LUM bias Red. LUM ensemble minimal Red. LUM std

(no data assimilation)

u Red. LUM blindly describe unresolved triades

§ Stabilize the unstable modes

§ Maintains the variability of stables modes

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D ATA A S S I M I L AT I O N :

WA K E AT R E 1 0 0

Reference (DNS) 10#_{degrees of freedom} Our method (Red-LUM-based data-assimilation) 6degrees of freedom Theoretical bound

(Optimal from 6-d.o.f. linear decomposition)

6degrees of freedom

Benchmark

(POD-ROM (with eddy viscosity) + init. by obs.)

6degrees of freedom Vo rtic ity Vo rtic ity Vo rtic ity Vo rtic ity

Reduced order models with % = 6

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Reduced order models with ! = 6

and 2dB-SNR obs. assimilated every 5 sec

Reference (DNS) 10&_{degrees of} freedom Our method (Red-LUM-based data-assimilation) 6degrees of freedom Theoretical bound (optimal from 6-d.o.f. linear decomposition) 6degrees of freedom Benchmark (POD-ROM (with eddy viscosity) + init. by obs.) 6degrees of freedom

D ATA A S S I M I L AT I O N :

WA K E AT R E 3 0 0

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CONCLUSION

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CONCLUSION

u

Reduced order model (ROM) : for very fast and robust CFD

(10# → 6 degrees of freedom.)

§ Combine data & physics (built off-line)

§ Closure problem handled by LUM

u

Data assimilation : to correct the fast simulation on-line by incomplete/noisy measurements

§ Model error quantification handled by LUM

u

First results

§ Optimal unsteady flow estimation/prediction in the whole spatial domain (large-scale structures)

§ Robust far outside the learning period

NEXT STEPS

u

Increasing Reynolds

(reduced DNS à reduced LES)

u

Real measurements (PIV, TrimControl, …)

u