Center for UncertaintyQuantification

Center for UncertaintyQuantification

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UncertaintyQuantification inNumericalAerodynamicsAlexander Litvinenko, KAUST, joint project with H.Matthies (TU Braunschweig), D. Liu (DLR), C. Schillings (Uni

Mannheim), V. Schulz (Uni Trier)

UNCERTAINTY PROPAGATION IN AERODYNAMICS

Goal 1: Uncertainty quantification in aerodynamics, better decision making under uncertaintiesGoal 2: To compare efficiency of various methods in quantifying geometry-induced aerodynamicuncertainties, this work sets up a test case where geometry of an RAE2822 airfoil is perturbed bya Gaussian random field which is parameterized by 9 independent Gaussian variables througha Karhunen-Loève expansion.

Navier-Stokes equation: v · ∇v− 1Re∇

2v+∇p = g, and∇ · v = 0.+ b.c. and Wilcox-k-w turbulencemodel, RAE-2822 airfoil domain.TAU-solver (DLR) has more than 300 parameters! Many of them are or can be uncertain!What does it mean for the solution ?Uncertain Input: Parameters and variables (α, Ma, Re, ...), geometry of airfoilUncertain solution: statistical moments of (v, p, ρ)T exceedance probabilities P (v > v∗) in each point xprobability density functions of u position of shock.

α

v

v

u

u’

α’

v1

2

1-random α, Ma, 2,3-discretisation, 4,5-mean and variance of density, 6,7-mean densities withnumerical artifacts, 8-pdf of CL, 9-mean pressure, 10-random realizations of airfoil, 11,12 - MC

realizations (500) of pressure and friction coefficients, 13,14-5%-95% friction and pressure coefficients,15- Gauss-Hermite sparse grid with 281 points.

Example 1: Used PCE with sparse 2d-Gauss-Hermite grid with 13 points:Assume that input parameters α and Maare Gaussian RVs with

mean st. dev. σ σ/meanα 2.79 0.1 0.036Ma 0.734 0.005 0.007

Then uncertainties in the solution lift CL anddrag CD will be

mean st. dev. σ σ/meanCL 0.853 0.0174 0.02CD 0.0206 0.003 0.146

Uncertainties in geometry: Random boundary perturbations: ∂Dε(ω) = {x + εκ(x, ω)n(x) : x ∈∂D}, where κ(x, ω) is a random field.

COMPARISON OF VARIOUS METHODS

The mean and variance computed from rank-5approximation of the snapshot matrix, Case 9.

The mean computed from the rank-5 and rank-20approximations of the snapshot matrix, Case 1.

0.75 0.8 0.85 0.9 0.950

5

10

15

20

25Lift: Comparison of densities

0.005 0.01 0.015 0.02 0.025 0.03 0.0350

50

100

150Drag: Comparison of densities

0.75 0.8 0.85 0.9 0.950

0.2

0.4

0.6

0.8

1Lift: Comparison of distributions

0.005 0.01 0.015 0.02 0.025 0.03 0.0350

0.2

0.4

0.6

0.8

1Drag: Comparison of distributions

sgh13

sgh29

MC

PDFs and CDFs, computed by Monte Carlo and by PCEwith 13 and 29 GH sparse points,

Comparison of the gradient-enhanced kriging (GEK), implemented by SMART Toolbox (DLR),Quasi-Monte Carlo (QMC), gradient-enhanced radial basis function, polynomial chaos based + sparse

Gauss-Hermite quadrature (PC-SGH) and gradient-enhanced polynomial chaos (GEPC).

101

102

103

1e−06

1e−05

1e−04

1e−03

Nc

Err

or

On estimating µL

QMC

PC−SGH

GEK

GEPC

GERBF

101

102

103

1e−05

1e−04

1e−03

Nc

Err

or

On estimating σL

QMC

PC−SGH

GEK

GEPC

GERBF

101

102

103

1e−04

1e−03

1e−02

1e−01

Nc

Err

or

On estimating PL,2

QMC

PC−SGH

GEK

GEPC

GERBF

3ς1

101

102

103

1e−04

1e−03

1e−02

Nc

Err

or

On estimating PL,3

QMC

PC−SGH

GEK

GEPC

GERBF

3ς1

Absolute error in estimating mean, standard deviation and exceedance probabilities of CL

101

102

103

1e−06

1e−05

1e−04

Nc

Err

or

On estimating µD

QMC

PC−SGH

GEK

GEPC

GERBF

101

102

103

1e−06

1e−05

1e−04

Nc

Err

or

On estimating σD

QMC

PC−SGH

GEK

GEPC

GERBF

101

102

103

1e−04

1e−03

1e−02

Nc

Err

or

On estimating PD,2

QMC

PC−SGH

GEK

GEPC

GERBF

3ς1

101

102

103

1e−04

1e−03

1e−02

1e−01

Nc

Err

or

On estimating PD,3

QMC

PC−SGH

GEK

GEPC

GERBF

3ς1

Absolute error in estimating mean, standard deviation and exceedance probabilities of CD

Conclusion: QMC and four surrogate methods, polynomial chaos with coefficients determined bysparse grids, gradient-enhanced radial basis functions, gradient-enhanced polynomial chaos andgradient-enhanced Kriging, are compared in their efficiency. Gradient-employing surrogate methodsare more efficient. The advantage is due to the cheaper gradients obtained by using adjoint solver.

REFERENCES AND ACKNOWLEDGEMENTS

[1] D. Liu, A. Litvinenko, C. Schillings, and V. Schulz. Quantification of airfoil geometry-induced aerodynamic uncertainties-comparison of approaches. SIAM/ASA J. on Uncertainty Quantification 5 (1),334-352, 2017.

[2] A. LITVINENKO, H.G. MATTHIES, Sparse data formats and efficient numerical methods for uncertainties quantification in numerical aerodynamics, ECCM IV: Solids, Structures and Coupled Problems inEngineering, 2010

[3] B. N. KHOROMSKIJ, A. LITVINENKO, H. G. MATTHIES, Application of hierarchical matrices for computing the Karhunen-Loéve expan-sion, Computing, Vol. 84, Issue 1-2, pp 49-67, 2008.

[4] A. LITVINENKO , H. G. MATTHIES, Numerical Methods for Uncertainty Quantification and Bayesian Update in Aerodynamics MANAGEMENT AND MINIMISATION OF UNCERTAINTIES AND ERRORS INNUMERICAL AERODYNAMICS, VOL. 122, NOTES ON NUMERICAL FLUID MECHANICS AND MULTIDISCIPLINARY DESIGN, PP 265-282, SPRINGER, 2013

[5] A. LITVINENKO , H. G. MATTHIES, Uncertainty Quantification in numerical Aerodynamic via low-rank Response Surface, PAMM 12 (1), 781-784, 2012

This project MUNA was done under the framework of the German Luftfahrtforschungsprogramm funded by the Ministry of Economics (BMWA).