The document discusses uncertainty quantification and robust design approaches for aircraft design. It compares using a polynomial chaos expansion with an adaptive sparse grid to represent input uncertainties and the objective function. This allows solving the robust optimization problem with reduced computational cost compared to evaluating on a full tensor grid. The methodology is demonstrated on a transonic airfoil design test case with geometrical uncertainties, comparing different robust measures of performance.

1. 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 (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 1 / 1

2. Test case RAE2822
Test case RAE2822
Transonic Euler flow
M = 0.73, α = 2◦
Target lift C0
L = 0.816
21 design variables
193 × 33 grid,
129 surface points
Deterministic shape
optimization problem
min
y,p
f(y, p)
s.t. c(y, p) = 0
h(y, p) ≥ 0
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 2 / 1

3. Test case RAE2822
Geometrical uncertainties
Transformed 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−1
s(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])
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 3 / 1

4. Test case RAE2822
Geometrical uncertainties
Transformed 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 distribution
s (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 1
0
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 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Cumulative distribution function
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 3 / 1

5. 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
3
x 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
3
x 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
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 3 / 1

6. Test case RAE2822
transformed Gaussian random field s : Γ × O → R
Gaussian field ψ:
E (ψ (x, ζ)) = 0 , cov (x, y) = σ(x) σ(y) exp − x−y 2
l2
Transformation of ψ to marginal sine-shaped distribution
s (x, ζ) = c(x) · arccos (1 − 2Φ (ψ (x, ζ)))
Karhunen-Lo`eve expansion ψ9 (x, ζ) =
9
i=1
λKL
i zKL
i (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 80 100 120
0
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
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 3 / 1

7. Robust formulation
Robust optimization problem
min
y(s(ζ)),p
R(f(y(s(ζ)), p, s(ζ)))
s.t. c(y(s(ζ)), p, s(ζ)) = 0 , ∀ζ ∈ O
H(y(s(ζ)), p, s(ζ)) ≥ 0
Expectation measure: min
y(s(ζ)),p
E(f(y(s(ζ)), p, s(ζ)))
Mean-risk approach:
Mean-variance : min
y(s(ζ)),p
E(f(y(s(ζ)), p, s(ζ))) + cV(f(y(s(ζ)), p, s(ζ)))
Expected excess: min
y(s(ζ)),p
E(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
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 4 / 1

8. Summary of the methodology
Quantification One-shot optimization
Approximate the input random field in
a finite number of random variables
(→ goal-oriented KL expansion)
sd (x, ζ) =
d
i=1
λKL
i ˜zKL
i (x) Xi (ζ)
Represent the objective function
using the non-intrusive PC approach
+ discretize the probability space to
compute the expansion
(→ adaptive sparse grid)
RN(f(p, sd)) =
N
i=1
f(p, si
d) ωi
Solve the lower level problem by a
discretization (reduction) approach
s0 = argmin
sd
˜h(p, sd)
Use the generalized version of the
one-shot algorithm to solve the
resulting robust optimization problem
min
yi,p
N
i=1
f(yi, p, si
d) ωi
s.t. c(yi, p, si
d) = 0
h(y0, p, s0
d) ≥ 0
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 5 / 1

9. Numerical results of the robust optimization
Test case: RAE2822, transonic Euler flow (2D probability space)
Robust optimization problem
min
y(s(X1,X2)),p
R(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=0
˜fi(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 5
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 6 / 1

10. Numerical results of the robust optimization
Test case: RAE2822, transonic Euler flow (2D probability space)
Robust optimization problem
min
y(s(X1,X2)),p
R(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=0
˜fi(p) · Φi (s(ζ))
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
2
3
4
5
6
7
8
x 10
−3
1. eigenvector
polynomial chaos expansion of order 3
2. eigenvector
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 6 / 1

11. Numerical results of the robust optimization
Test case: RAE2822, transonic Euler flow (2D probability space)
Robust optimization problem
min
y(s(X1,X2)),p
R(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]
−1
−0.5
0
0.5
1
−1
0
1
4
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.eigenvector
locally refined sparse grid
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 6 / 1

12. Numerical results of the robust optimization
Test case: RAE2822, transonic Euler flow (2D probability space)
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+10
3
Var
E+10
4
Var
E+10
5
Var
EE η=0.00415
EE η=0.0044
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 7 / 1

13. Numerical results of the robust optimization
Test case: RAE2822, transonic Euler flow (9D probability space)
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
5.4
x 10
−3
perturbations
drag(objectivefunction)
1. eigenvector, indicator: 4.73 10
−04
2. eigenvector, indicator: 4.37 10
−04
3. eigenvector, indicator:−1.21 10
−04
4. eigenvector, indicator: 1.14 10−04
5. eigenvector, indicator:−1.25 10−04
6. eigenvector, indicator: 1.33 10
−05
7. eigenvector, indicator: 2.05 10−05
8. eigenvector, indicator:−1.33 10
−05
9. eigenvector, indicator: 6.23 10
−06
6. - 9. eigenvectors have no impact on the drag performance
→ reduction of the dimension
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 8 / 1

14. Numerical results of the robust optimization
Test case: RAE2822, transonic Euler flow (9D probability space)
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
4
x 10
−3
perturbations
lift(inequalityconstraint)
1. eigenvector, indicator: 4.60 10−3
2. eigenvector, indicator: −1.67 10
−3
3. eigenvector, indicator: 1.19 10
−3
4. eigenvector, indicator: −8.93 10
−4
5. eigenvector, indicator: 9.00 10
−4
6. eigenvector, indicator: −5.64 10
−4
7. eigenvector, indicator: −1.05 10
−4
8. eigenvector, indicator: 1.47 10
−4
9. eigenvector, indicator: −5.88 10
−5
6. - 9. eigenvectors have no impact on the drag performance
→ reduction of the dimension
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 8 / 1

15. Numerical results of the robust optimization
Test case: RAE2822, transonic Euler flow (9D probability space)
Robust optimization problem
min
y(s(X1,...,X9)),p
E(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
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 9 / 1

16. Numerical results of the robust optimization
Test case: RAE2822, transonic Euler flow (9D probability space)
Robust optimization problem
min
y(s(X1,...,X9)),p
E(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)
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 9 / 1

17. Uncertainty quantification - objective
We estimate some statistics of CL and CD under the uncertainty of
airfoil geometry.
Target statistics:
Mean, µ , µd
Standard deviations, σ , σd
Exceedance probabilities,
P ,κ = Pro{CL < µ − κ · σ } and
Pd,κ = Pro{CD > µd − κ · σd} with κ = 2, 3.
Objective:
To identify an efficient method for estimating the statistics in this
kind of problem.
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 10 / 1

18. Uncertainty quantification - comparison of methods
To compare the efficiency of methods in estimating the statistics.
Methods:
gradient-employing
gradient-assisted point-collocation polynomial chaos (GAPC)
gradient-assisted radial basis function (GARBF)
gradient-enhanced Kriging (GEK)
not gradient-employing
quasi-Monte Carlo quadrature (QMC), with low discrepancy sequence
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 M
M = 2N for gradient-employing method
M = N for others
Accuracy: judged by comparing with a reference statistics obtained by a QMC of
N = 5 × 105
.
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 11 / 1

19. Uncertainty quantification - numerical results
50 100 150 200
1e−06
1e−05
1e−04
1e−03
1e−02
1e−01
1e+00
M
Error
On estimating the mean of CL
QMC
PC
GEK
GAPC
GARBF
50 100 150 200
1e−06
1e−04
1e−02
1e+00
M
Error
On estimating the stdv of CL
QMC
PC
GEK
GAPC
GARBF
Abbildung: Comparison on estimating the mean and standard deviation of CL
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 12 / 1

20. Uncertainty quantification - numerical results
50 100 150 200
1e−04
1e−03
1e−02
1e−01
1e+00
M
Error
On estimating Pl,2
QMC
PC
GEK
GAPC
GARBF
50 100 150 200
1e−04
1e−03
1e−02
1e−01
1e+00
M
Error
On estimating Pl,3
QMC
PC
GEK
GAPC
GARBF
Abbildung: Comparison on estimating the exceedance probabilities of CL
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 13 / 1

21. Uncertainty quantification - numerical results
50 100 150 200
1e−08
1e−06
1e−04
1e−02
M
Error
On estimating the mean of CD
QMC
PC
GEK
GAPC
GARBF
50 100 150 200
1e−06
1e−04
1e−02
M
On estimating the stdv of CD
QMC
PC
GEK
GAPC
GARBF
Abbildung: Comparison on estimating the mean and standard deviation of CD
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 14 / 1

22. Uncertainty quantification - numerical results
50 100 150 200
1e−04
1e−03
1e−02
1e−01
M
On estimating Pd,2
QMC
PC
GEK
GAPC
GARBF
50 100 150 200
1e−04
1e−03
1e−02
1e−01
1e+00
M
On estimating Pd,3
QMC
PC
GEK
GAPC
GARBF
Abbildung: Comparison on estimating the exceedance probabilities of CD
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 15 / 1

23. Uncertainty quantification - numerical results
0.79 0.8 0.81 0.82 0.83
0
20
40
60
80
100
C
L
pdf
pdf of C
L
by GEK, M=50
reference pdf
pdf by GEK
0.79 0.8 0.81 0.82 0.83
0
20
40
60
80
100
C
L
pdf
pdf of C
L
by GAPC, M=50
reference pdf
pdf by GAPC
0.79 0.8 0.81 0.82 0.83
0
20
40
60
80
100
C
L
pdf
pdf of C
L
by GARBF, M=50
reference pdf
pdf by GARBF
Abbildung: Comparison on estimating the pdf of CL at M = 50
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 16 / 1

24. Uncertainty quantification - numerical results
0.003 0.004 0.005 0.006 0.007 0.008 0.009
0
100
200
300
400
500
600
CD
pdf
pdf of C
D
by GEK, M=50
reference pdf
pdf by GEK
0.003 0.004 0.005 0.006 0.007 0.008 0.009
0
100
200
300
400
500
600
CD
pdf
pdf of C
D
by GAPC, M=50
reference pdf
pdf by GAPC
0.003 0.004 0.005 0.006 0.007 0.008 0.009
0
100
200
300
400
500
600
CD
pdf
pdf of C
D
by GARBF, M=50
reference pdf
pdf by GARBF
Abbildung: Comparison on estimating the pdf of CD at M = 50
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 17 / 1

25. Uncertainty quantification - numerical results
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.
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 18 / 1

26. Uncertainty quantification - conclusion
Conclusion:
Gradient-employing surrogate methods outperform the others
GEK seems more efficient than other gradient-employing
methods, 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 this
comparison may not indicate its asymptotic capacity.
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 19 / 1

27. Summary
Summary:
An aerodynamic testcase of geometric uncertainties is setup for
joint research on UQ and RD.
Karhunen-Lo`eve expansion proved an effect tool to reduce the
number of variables.
Robust design is implemented on the testcase with different
robust measures and numerical methods.
Uncertainty quantification methods are applied on the testcase
and their efficiency compared.
V. Schulz (Universit¨at Trier) Uncertainty Quantification and Robust Design MUNA - October 25, 2012 20 / 1