Management of uncertainties in numerical aerodynamics

Uncertainty quantification and treatment in aircraft design -comparison of approaches

Dishi Liu, Alexander Litvinenko, Claudia Schillings, Volker Schulz

MUNA Final Workshop

October 25, 2012 - DLR Braunschweig

research supported by BMWI within the collaborative project MUNA

V. Schulz (Universitat Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 1 / 1

Test case RAE2822

Test case RAE2822Transonic Euler flowM = 0.73, α = 2◦

Target lift C0L = 0.816

21 design variables193× 33 grid,129 surface pointsDeterministic shapeoptimization problem

miny,p

f (y, p)

s.t. c(y, p) = 0

h(y, p) ≥ 0


Test case RAE2822

Geometrical uncertaintiesTransformed Gaussian random field s : Γ×O → R

Assumption:sl ≤ s (x, ζ) ≤ su

Transformation of Gaussian field ψ:

s (x, ζ) = Θ (x, ψ (x, ζ)) = F−1s(x) (Φ (ψ (x, ζ)))

with ψ determined by the mean E (ψ (x, ζ)) = ψ0 (x) and the covariance cov (x, y)

Perturbed geometry:v (x, ζ) = x + s (x, ζ) · n (x) , ∀x ∈ Γ, ζ ∈ O

(cf. e.g. [Matthies, Keese 2003])


Test case RAE2822Geometrical uncertaintiesTransformed Gaussian random field s : Γ×O → R

Gaussian field ψ with

E (ψ (x, ζ)) = 0 , cov (x, y) = σ(x) σ(y) exp(−‖x−y‖2

l2

)with σ(x) = (0.8− x1)0.75 and l = 0.005

Transformation of ψ to marginal sine-shaped distributions (x, ζ) = c(x) · arccos (1− 2Φ (ψ (x, ζ)))

with c(x) = (0.8− x1)0.75 ·√

0.00002

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

Probability density function

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Cumulative distribution function


Test case RAE2822

Realizations of the Gaussian random field ψ (x, ζ)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

1.5

x

Upper part

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1

1.5

x

Lower part

Realizations of the transformed random field s (x, ζ)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1

0

1

2

3x 10

−3

x

Upper part

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1

0

1

2

3x 10

−3

x

Lower part

Resulting perturbed shapes v (x, ζ) = x + s (x, ζ) · n (x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

x

y


Test case RAE2822transformed Gaussian random field s : Γ×O → R

I Gaussian field ψ:E (ψ (x, ζ)) = 0 , cov (x, y) = σ(x) σ(y) exp

(−‖x−y‖2

l2

)I Transformation of ψ to marginal sine-shaped distribution

s (x, ζ) = c(x) · arccos (1− 2Φ (ψ (x, ζ)))

Karhunen-Loeve expansion ψ9 (x, ζ) =9∑

i=1

√λKL

i zKLi (x) Xi (ζ)

0 20 40 60 80 100 120

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

1. eigenvector

2. eigenvector

3. eigenvector

4. eigenvector

5. eigenvector

6. eigenvector

7. eigenvector

8. eigenvector

9. eigenvector

1. eigenvalue

2. eigenvalue

3. eigenvalue

4. eigenvalue

5. eigenvalue

6. eigenvalue

7. eigenvalue

8. eigenvalue

9. eigenvalue

remaining eigenvalues


Robust formulation

Robust optimization problem

miny(s(ζ)),p

R(f (y(s(ζ)), p, s(ζ)))

s.t. c(y(s(ζ)), p, s(ζ)) = 0 , ∀ζ ∈ OH(y(s(ζ)), p, s(ζ)) ≥ 0

Expectation measure: miny(s(ζ)),p

E(f (y(s(ζ)), p, s(ζ)))

Mean-risk approach:

I Mean-variance : miny(s(ζ)),p

E(f (y(s(ζ)), p, s(ζ))) + cV(f (y(s(ζ)), p, s(ζ)))I Expected excess: min

y(s(ζ)),pE(max{f (y(s(ζ)), p, s(ζ))− η, 0})

Worst-case constraints: h(y, p, s(ζ)) ≥ 0 , ∀ζ

Robust optimization: h(y, p, s(ζ)) ≥ 0 , ∀ζ ∈ H


Summary of the methodology

Quantification One-shot optimization

Approximate the input random field ina finite number of random variables(→ goal-oriented KL expansion)

sd (x, ζ) =d∑

i=1

√λKL

i zKLi (x) Xi (ζ)

Represent the objective functionusing the non-intrusive PC approach+ discretize the probability space tocompute the expansion(→ adaptive sparse grid)

RN(f (p, sd)) =N∑

i=1f (p, si

d)ωi

Solve the lower level problem by adiscretization (reduction) approach

s0 = argminsd

h(p, sd)

Use the generalized version of theone-shot algorithm to solve theresulting robust optimization problem

minyi,p

N∑i=1

f (yi, p, sid)ωi

s.t. c(yi, p, sid) = 0

h(y0, p, s0d) ≥ 0


Numerical results of the robust optimizationTest case: RAE2822, transonic Euler flow (2D probability space)

Robust optimization problemmin

y(s(X1,X2)),pR(f (y(s(X1,X2)), p, s(X1,X2)))

s.t. c(y(s(X1,X2)), p, s(X1,X2)) = 0, ∀X1,X2

h(y(s(X1,X2)), p, s(X1,X2)) ≥ 0, X1,2 ∈ [−1, 1]

fPC(p, s(ζ)) =3∑

i=0fi(p) · Φi (s(ζ))

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

tensor grid

tensor grid: 225 points−8 −6 −4 −2 0 2 4 6 8

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

adaptive sparse grid (PC)

adaptive sparse grid: 41 points

→ reduction factor 5V. Schulz (Universitat Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 6 / 1




s.t. c(y(s(X1,X2)), p, s(X1,X2)) = 0, ∀X1,X2

h(y(s(X1,X2)), p, s(X1,X2)) ≥ 0, X1,2 ∈ [−1, 1]

fPC(p, s(ζ)) =3∑

i=0fi(p) · Φi (s(ζ))

−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

32

3

4

5

6

7

8

x 10−3

1. eigenvector

polynomial chaos expansion of order 3

2. eigenvector





s.t. c(y(s(X1,X2)), p, s(X1,X2)) = 0, ∀X1,X2

h(y(s(X1,X2)), p, s(X1,X2)) ≥ 0, X1,2 ∈ [−1, 1]

−1−0.5

00.5

1

−1

0

14

4.5

5

5.5

6

x 10−3

1. eigenvector

lift performance [−1,1]x[−1,1]

2. eigenvector−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1. eigenvector

2. eig

envecto

r

locally refined sparse grid



Measure of robustness Statistics

fnominal E Var EEE η=4.15·10−3 EEE η=4.4·10−3

single-setpoint 3.45 · 10−3 4.47 · 10−3 1.13 · 10−6 5.20 · 10−4 4.03 · 10−4

E 3.75 · 10−3 4.15 · 10−3 3.57 · 10−7 1.94 · 10−4 1.19 · 10−4

E + 103 · Var 3.81 · 10−3 4.10 · 10−3 2.55 · 10−7 1.73 · 10−4 9.90 · 10−5

E + 104 Var 4.04 · 10−3 4.33 · 10−3 8.91 · 10−8 2.18 · 10−4 9.13 · 10−5

E + 105 · Var 5.93 · 10−3 6.06 · 10−3 2.22 · 10−8 1.91 · 10−3 1.66 · 10−3

EEE η=4.15·10−3 3.77 · 10−3 4.13 · 10−3 1.26 · 10−7 1.33 · 10−4 6.41 · 10−5

EEE η=4.40·10−3 3.81 · 10−3 4.16 · 10−3 1.09 · 10−7 1.33 · 10−4 6.09 · 10−5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x

y

single−setpoint

E

E+103 Var

E+104 Var

E+105 Var

EE η=0.00415

EE η=0.0044



Adaptive selection of the KL basis

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

4.4

4.6

4.8

5

5.2

5.4x 10

−3

perturbations

dra

g (

obje

ctive function)

1. eigenvector, indicator: 4.73 10−04


3. eigenvector, indicator:−1.21 10−04







6. - 9. eigenvectors have no impact on the drag performance

→ reduction of the dimension



Adaptive selection of the KL basis

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5

−4

−3

−2

−1

0

1

2

3

4x 10

−3

perturbations

lift (inequalit

y c

onstr

ain

t)


2. eigenvector, indicator: −1.67 10−3








6. - 9. eigenvectors have no impact on the drag performance

→ reduction of the dimension




y(s(X1,...,X9)),pE(f (y(s(X1, . . . ,X9)), p, s(X1, . . . ,X9)))

s.t. c(y(s(X1, . . . ,X9)), p, s(X1,X2)) = 0, ∀X1, . . . ,X9

h(y(s(X1, . . . ,X9)), p, s(X1, . . . ,X9)) ≥ 0, X1,...,9 ∈ [−1, 1]

goal-oriented KL basis 9→ 5

39 collocation points (drag)

84 grid points (lift)

Measure of robustness Expected Excess

fnominal E

single-setpoint 3.45 · 10−3 4.04 · 10−3

E (5D) 3.74 · 10−3 3.95 · 10−3

E (9D) 3.79 · 10−3 3.99 · 10−3




y(s(X1,...,X9)),pE(f (y(s(X1, . . . ,X9)), p, s(X1, . . . ,X9)))

s.t. c(y(s(X1, . . . ,X9)), p, s(X1,X2)) = 0, ∀X1, . . . ,X9

h(y(s(X1, . . . ,X9)), p, s(X1, . . . ,X9)) ≥ 0, X1,...,9 ∈ [−1, 1]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x

y

robust optimization (5D)robust optimization (9D)


Uncertainty quantification - objective

We estimate some statistics of CL and CD under the uncertainty ofairfoil geometry.

Target statistics:

Mean, µ`, µd

Standard deviations, σ`, σd

Exceedance probabilities,

P`,κ = Pro{CL < µ` − κ · σ`} andPd,κ = Pro{CD > µd − κ · σd} with κ = 2, 3.

Objective:To identify an efficient method for estimating the statistics in thiskind of problem.


Uncertainty quantification - comparison of methodsTo compare the efficiency of methods in estimating the statistics.

Methods:

gradient-employingI gradient-assisted point-collocation polynomial chaos (GAPC)I gradient-assisted radial basis function (GARBF)I gradient-enhanced Kriging (GEK)

not gradient-employingI quasi-Monte Carlo quadrature (QMC), with low discrepancy sequenceI polynomial chaos (PC), with coefficients estimated by sparse grid quadrature

Criteria: computation cost for a certain accuracy in the statistics.

Cost: measured in elapse time-penalized sample number MM = 2N for gradient-employing methodM = N for others

Accuracy: judged by comparing with a reference statistics obtained by a QMC ofN = 5× 105.


Uncertainty quantification - numerical results

50 100 150 2001e−06

1e−05

1e−04

1e−03

1e−02

1e−01

1e+00

M

Err

or

On estimating the mean of CL

QMCPCGEKGAPCGARBF

50 100 150 2001e−06

1e−04

1e−02

1e+00

ME

rro

r

On estimating the stdv of CL

QMCPCGEKGAPCGARBF

Abbildung: Comparison on estimating the mean and standard deviation of CL



50 100 150 2001e−04

1e−03

1e−02

1e−01

1e+00

M

Err

or

On estimating Pl,2

QMCPCGEKGAPCGARBF

50 100 150 2001e−04

1e−03

1e−02

1e−01

1e+00

ME

rro

r

On estimating Pl,3

QMCPCGEKGAPCGARBF

Abbildung: Comparison on estimating the exceedance probabilities of CL



50 100 150 2001e−08

1e−06

1e−04

1e−02

M

Err

or

On estimating the mean of CD

QMCPCGEKGAPCGARBF

50 100 150 200

1e−06

1e−04

1e−02

M

On estimating the stdv of CD

QMCPCGEKGAPCGARBF

Abbildung: Comparison on estimating the mean and standard deviation of CD



50 100 150 200

1e−04

1e−03

1e−02

1e−01

M

On estimating Pd,2

QMCPCGEKGAPCGARBF

50 100 150 2001e−04

1e−03

1e−02

1e−01

1e+00

M

On estimating Pd,3

QMCPCGEKGAPCGARBF

Abbildung: Comparison on estimating the exceedance probabilities of CD



0.79 0.8 0.81 0.82 0.830

20

40

60

80

100

CL

pdf

pdf of CL by GEK, M=50

reference pdfpdf by GEK

0.79 0.8 0.81 0.82 0.830

20

40

60

80

100

CL

pdf

pdf of CL by GAPC, M=50

reference pdfpdf by GAPC

0.79 0.8 0.81 0.82 0.830

20

40

60

80

100

CL

pdf

pdf of CL by GARBF, M=50

reference pdfpdf by GARBF

Abbildung: Comparison on estimating the pdf of CL at M = 50



0.003 0.004 0.005 0.006 0.007 0.008 0.0090

100

200

300

400

500

600

CD

pdf

pdf of CD

by GEK, M=50

reference pdfpdf by GEK

0.003 0.004 0.005 0.006 0.007 0.008 0.0090

100

200

300

400

500

600

CD

pdf

pdf of CD

by GAPC, M=50

reference pdfpdf by GAPC

0.003 0.004 0.005 0.006 0.007 0.008 0.0090

100

200

300

400

500

600

CD

pdf

pdf of CD

by GARBF, M=50

reference pdfpdf by GARBF

Abbildung: Comparison on estimating the pdf of CD at M = 50



pdf of CL and CD by polynomial Chaos expansion (PC)PC expansion is computed on the sparse Gauss-Hermite grid (SGH)with 19 nodes ( polynomial order p = 1).

Abbildung: Comparison of pdf computed by PC and the reference pdf, p = 1,19 TAU evaluations.


Uncertainty quantification - conclusion

Conclusion:

Gradient-employing surrogate methods outperform the others

GEK seems more efficient than other gradient-employingmethods, especially when M is small, and when estimating the“far-end” exceedance probability and the pdf.

Since PC only has two data point, its performance in thiscomparison may not indicate its asymptotic capacity.


Summary

Summary:An aerodynamic testcase of geometric uncertainties is setup forjoint research on UQ and RD.

Karhunen-Loeve expansion proved an effect tool to reduce thenumber of variables.

Robust design is implemented on the testcase with differentrobust measures and numerical methods.

Uncertainty quantification methods are applied on the testcaseand their efficiency compared.


Management of uncertainties in numerical aerodynamics

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