We developed a gPCE based surrogate. gPCE coefficients were computed with sparse Gauss-Hermite quadrature and compared with coefficients computed via MC and QMC methods

1. Stochastic methods for simulating
uncertainties in free stream turbulence and in
the geometry
Project MUNA: Final Workshop,
Alexander Litvinenko,
Institut f¨ur Wissenschaftliches Rechnen,
TU Braunschweig
0531-391-3008,
litvinen@tu-bs.de
March 22, 2010

2. Outline
Overview
Modelling of free stream turbulence
Numerics
Uncertainties in geometry
Numerics
Low-rank approximation of the solution
Numerics

3. Outline
Overview
Modelling of free stream turbulence
Numerics
Uncertainties in geometry
Numerics
Low-rank approximation of the solution
Numerics

4. Overview of uncertainties
Input:
1. Parameters (α, Ma, Re, ...)
2. Geometry
3. Parameters of turbulence
Uncertain output:
1. mean value and variance
2. exceedance probabilities P(u < u∗
)
3. probability density and distribution functions.

5. Our Aims
1. Sparse representation of the input data (random fields)
2. The whole computation process must be sparse and done in a
reasonable time
3. Changes in the deterministic solver so small as possible (use as
a black-box)
4. A sparse format for the solution

6. Stochastical Methods overview
1. Monte Carlo Simulations (easy to implement, parallelisable,
expensive, dim. indepen.).
2. Stoch. collocation methods with global or local polynomials (easy
to implement, parallelisable, cheaper than MC, dim. depen.).
3. Stochastic Galerkin (difficult to implement, non-trivial
parallelisation, the cheapest from all, dim. depen.)

7. Outline
Overview
Modelling of free stream turbulence
Numerics
Uncertainties in geometry
Numerics
Low-rank approximation of the solution
Numerics

8. Modelling of uncertainties in free stream turbulence
α
v
v
u
u’
α’
v1
2
Random vectors v1 and v2 model turbulence in the atmosphere.

9. Truncated Polynomial Chaos Expansion
We represent CL in a Hermitian basis Hβ, β ∈ J .
CL(θ) =
β∈J
Hβ(θ)CLβ, (1)
where θ a vector of random Gaussian variables, J is a multiindex set
and β = (β1, ..., βj , ...) ∈ J a multiindex.
CLβ =
1
β! Θ
Hβ(θ)CL(θ) P(dθ). (2)
CLβ ≈
1
β!
n
i=1
Hβ(θi )CL(θi )wi , (3)
where weights wi and points θi are defined from sparse
Gauss-Hermite integration rule.

10. Uniform and Gaussian distributions of α and Ma
The following experiments are done for:
Profiles: RAE-2822, Wilcox-k-w turbulence model
Turbulence intensity I = 0.001
mean st. deviation σ variance σ2
α 2.790 0.1 1.0e-2
Ma 0.734 0.005 2.5e-5
Table: Mean values and standard deviations

11. Sparse Gauss-Hermite Quadratures
Figure: Sparse Gauss-Hermite grids for the perturbed angle of attack α
′
and
the Mach number Ma
′
, n = {13, 29, 137}.

12. Sparse Gauss-Hermite grid, 5 points
[min, max] mean variance st. dev. σ σ/mean
α [2.69, 2.89] 2.79 0.01 0.1 0.036
Ma [0.729, 0.739] 0.734 0.00003 0.005 0.007
CL [0.837, 0.872] 0.853 0.00032 0.018 0.021
CD [0.0178, 0.0233] 0.0206 0.00001 0.0031 0.151

13. Sparse Gauss-Hermite grid, 13 points.
[min, max] mean variance st. dev. σ σ/mean
α [2.62, 2.96] 2.79 0.01 0.1 0.036
Ma [0.725, 0.743] 0.734 0.00002 0.005 0.007
CL [0.823, 0.884] 0.853 0.0003 0.0174 0.02
CD [0.0161, 0.0254] 0.0206 0.00001 0.003 0.146

14. Sparse Gauss-Hermite grid, 29 points.
[min, max] mean variance st. dev. σ σ/mean
α [2.56, 3.024] 2.79 0.01 0.1 0.036
Ma [0.722, 0.746] 0.734 0.00002 0.005 0.007
CL [0.812, 0.893] 0.852 0.0003 0.018 0.021
CD [0.0148, 0.0271] 0.0206 0.00001 0.0031 0.151

15. Table: Statistic obtained from 1500 MC simulations, α and Ma have Gaussian
distributions.
[min, max] mean variance st. dev. σ σ/mean
α [2.5, 3.12] 2.789 0.0095 0.0973 0.035
Ma [0.718, 0.75] 0.734 0.00002 0.0049 0.007
CL [0.805, 0.905] 0.8525 0.00030 0.0172 0.02
CD [0.0127, 0.0301] 0.0206 0.00001 0.0030 0.146

16. 0.75 0.8 0.85 0.9 0.95
0
5
10
15
20
25
Lift: Comparison of densities
0.005 0.01 0.015 0.02 0.025 0.03 0.035
0
50
100
150
Drag: Comparison of densities
0.75 0.8 0.85 0.9 0.95
0
0.2
0.4
0.6
0.8
1
Lift: Comparison of distributions
0.005 0.01 0.015 0.02 0.025 0.03 0.035
0
0.2
0.4
0.6
0.8
1
Drag: Comparison of distributions
sgh13
sgh29
MC

17. Table: Statistic obtained from 3800 MC simulations, α and Ma have uniform
distribution.
[min, max] mean variance st. dev. σ σ/mean
α [2.69, 2.89] 2.787 0.0034 0.058 0.021
Ma [0.729, 0.739] 0.734 0.00001 0.003 0.004
CL [0.831, 0.8728] 0.853 0.0001 0.0104 0.012
CD [0.0164, 0.0247] 0.0205 0.00000 0.0018 0.088

18. Figure: [min, max] intervals in each point of RAE2822 airfoil for the cp and cf.
The data are obtained from 645 solutions, computed in n = 645 nodes of
sparse Gauss-hermite grid.

19. Figure: Intervals [mean − σ, mean + σ], σ standard deviation, in each point of
RAE2822 airfoil for the pressure, density, cp and cf. Build for 645 points of
sparse Gauss-Hermite grid.

20. α(θ1, θ2), Ma(θ1, θ2), where θ1, θ2 have Gaussian distributions
Table: Statistics obtained on sparse Gauss-Hermite grid with 137 points.
[min, max] mean variance st. dev. σ σ/mean
α [2.04, 3.56] 2.8 0.041 0.2 0.071
Ma [0.72, 0.74] 0.73 0.00001 0.0026 0.0036
CL [0.7047, 0.967] 0.85 0.0014 0.0373 0.044
CD [0.0113, 0.0313] 0.01871 0.00001 0.00305 0.163

21. Table: Comparison of results obtained by a sparse Gauss-Hermite grid (n
grid points) with 17000 MC simulations.
n 137 381 645 MC,
17000
σCL
CL
0.044 0.042 0.042 0.0145
σCD
CD
0.163 0.159 0.16 0.1589
|CL−CL0|
CL
7.6e-4 1.3e-3 1.6e-3 4.2e-4
|CD−CD0|
CD
1.66e-2 1.46e-2 1.4e-2 2.1e-2

22. Outline
Overview
Modelling of free stream turbulence
Numerics
Uncertainties in geometry
Numerics
Low-rank approximation of the solution
Numerics

23. Uncertainties in geometry
Random boundary perturbations:
∂Dε(ω) = {x + εκ(x, ω)n(x) : x ∈ ∂D}.
where κ(x, ω) is a random field.
How to generate geometry with uncertainties ?
Algorithm:
1. Assume cov. function cov(x, y) for random field κ(x, ω) given
2. Compute Cij := cov(xi, xj ) for all grid points (in a sparse format!)
3. Solve eigenproblem Cφi = λi φi
4. Then κ(x, ω) ≈
m
i=1
√
λi φi ξi (ω), where ξi (ω) are uncorrelated
random variables.
Sparse approximation of dense matrix C is done in [Khoromskij,
Litvinenko, Matthies, 2009]

24. 69 RAE-2822 airfoils with uncertainties
Covariance function - Gaussian.

25. Uncertainties in geometry
[min, max] mean variance
, σ2
st. dev.
σ
σ/mean
CL [0.828, 0.863] 0.8552 0.00002 0.0049 0.0058
CD [0.017, 0.022] 0.0183 0.00000 0.00012 0.0065
PCE of order 1 with 3 random variables and sparse Gauss-Hermite
grid wite 25 points were used.

26. Outline
Overview
Modelling of free stream turbulence
Numerics
Uncertainties in geometry
Numerics
Low-rank approximation of the solution
Numerics

27. Low-rank approximation of the solution
U VΣ
T=M
U
VΣ∼
∼ ∼ T
=M
∼
Figure: Reduced SVD, only k biggest singular values are taken.

28. Decay of eigenvalues
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−20
−15
−10
−5
0
5
log, #eigenvalues
log,values
pressure
density
cp
cf
Figure: Decay (in log-scales) of 100 largest eigenvalues of the combined
matrix constructed from 645 solutions (pressure, density, cf, cp) on the
surface of RAE-2822 airfoil.

29. Low-rank approximation of the solution matrix
M = [density; pressure; cp; cf] ∈ R2048×645
(4)
M in dense matrix format requires 10.6 MB.
rank k M − ˜Mk 2/ M 2 memory, kB
1 0.82 22
2 0.21 43
5 0.4 108
10 5e-3 215
20 5e-4 431
50 1.2e-5 1080

30. Literature
1. A.Litvinenko, H. G. Matthies, Sparse Data Representation of
Random Fields, PAMM, 2009.
2. B.N. Khoromskij, A.Litvinenko, H. G. Matthies, Application of
hierarchical matrices for computing the Karhunen-Lo`eve
expansion, Springer, Computing, 84:49-67, 2009.
3. B.N. Khoromskij, A.Litvinenko, Data Sparse Computation of the
Karhunen-Lo`eve Expansion, AIP Conference Proceedings,
1048-1, pp. 311-314, 2008.
4. H. G. Matthies, Uncertainty Quantification with Stochastic Finite
Elements, Encyclopedia of Computational Mechanics, Wiley,
2007.

31. Acknowledgement
Elmar Zander
A Malab/Octave toolbox for stochastic Galerkin methods
(KLE, PCE, sparse grids, tensors, many examples etc)
http://ezander.github.com/sglib/