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).
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