Post on 09-Apr-2017
Stochastic methods for simulatinguncertainties in free stream turbulence and in
the geometry
Project MUNA: Final Workshop,Alexander Litvinenko,
Institut fur Wissenschaftliches Rechnen,TU Braunschweig0531-391-3008,
litvinen@tu-bs.de
March 22, 2010
Outline
Overview
Modelling of free stream turbulenceNumerics
Uncertainties in geometryNumerics
Low-rank approximation of the solutionNumerics
Outline
Overview
Modelling of free stream turbulenceNumerics
Uncertainties in geometryNumerics
Low-rank approximation of the solutionNumerics
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.
Our Aims
1. Sparse representation of the input data (random fields)
2. The whole computation process must be sparse and done in areasonable time
3. Changes in the deterministic solver so small as possible (use asa black-box)
4. A sparse format for the solution
Stochastical Methods overview
1. Monte Carlo Simulations (easy to implement, parallelisable,expensive, dim. indepen.).
2. Stoch. collocation methods with global or local polynomials (easyto implement, parallelisable, cheaper than MC, dim. depen.).
3. Stochastic Galerkin (difficult to implement, non-trivialparallelisation, the cheapest from all, dim. depen.)
Outline
Overview
Modelling of free stream turbulenceNumerics
Uncertainties in geometryNumerics
Low-rank approximation of the solutionNumerics
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.
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 setand β = (β1, ..., βj , ...) ∈ J a multiindex.
CLβ =1β!
∫Θ
Hβ(θ)CL(θ) P(dθ). (2)
CLβ ≈ 1β!
n∑i=1
Hβ(θi)CL(θi)wi , (3)
where weights wi and points θi are defined from sparseGauss-Hermite integration rule.
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-2Ma 0.734 0.005 2.5e-5
Table: Mean values and standard deviations
Sparse Gauss-Hermite Quadratures
Figure: Sparse Gauss-Hermite grids for the perturbed angle of attack α′
andthe Mach number Ma
′
, n = {13, 29, 137}.
Sparse Gauss-Hermite grid, 5 points
[min, max] mean variance st. dev. σ σ/meanα [2.69, 2.89] 2.79 0.01 0.1 0.036Ma [0.729, 0.739] 0.734 0.00003 0.005 0.007CL [0.837, 0.872] 0.853 0.00032 0.018 0.021CD [0.0178, 0.0233] 0.0206 0.00001 0.0031 0.151
Sparse Gauss-Hermite grid, 13 points.
[min, max] mean variance st. dev. σ σ/meanα [2.62, 2.96] 2.79 0.01 0.1 0.036Ma [0.725, 0.743] 0.734 0.00002 0.005 0.007CL [0.823, 0.884] 0.853 0.0003 0.0174 0.02CD [0.0161, 0.0254] 0.0206 0.00001 0.003 0.146
Sparse Gauss-Hermite grid, 29 points.
[min, max] mean variance st. dev. σ σ/meanα [2.56, 3.024] 2.79 0.01 0.1 0.036Ma [0.722, 0.746] 0.734 0.00002 0.005 0.007CL [0.812, 0.893] 0.852 0.0003 0.018 0.021CD [0.0148, 0.0271] 0.0206 0.00001 0.0031 0.151
Table: Statistic obtained from 1500 MC simulations, α and Ma have Gaussiandistributions.
[min, max] mean variance st. dev. σ σ/meanα [2.5, 3.12] 2.789 0.0095 0.0973 0.035Ma [0.718, 0.75] 0.734 0.00002 0.0049 0.007CL [0.805, 0.905] 0.8525 0.00030 0.0172 0.02CD [0.0127, 0.0301] 0.0206 0.00001 0.0030 0.146
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
Table: Statistic obtained from 3800 MC simulations, α and Ma have uniformdistribution.
[min, max] mean variance st. dev. σ σ/meanα [2.69, 2.89] 2.787 0.0034 0.058 0.021Ma [0.729, 0.739] 0.734 0.00001 0.003 0.004CL [0.831, 0.8728] 0.853 0.0001 0.0104 0.012CD [0.0164, 0.0247] 0.0205 0.00000 0.0018 0.088
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 ofsparse Gauss-hermite grid.
Figure: Intervals [mean− σ, mean + σ], σ standard deviation, in each point ofRAE2822 airfoil for the pressure, density, cp and cf. Build for 645 points ofsparse Gauss-Hermite grid.
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/meanα [2.04, 3.56] 2.8 0.041 0.2 0.071Ma [0.72, 0.74] 0.73 0.00001 0.0026 0.0036CL [0.7047, 0.967] 0.85 0.0014 0.0373 0.044CD [0.0113, 0.0313] 0.01871 0.00001 0.00305 0.163
Table: Comparison of results obtained by a sparse Gauss-Hermite grid (ngrid points) with 17000 MC simulations.
n 137 381 645 MC,17000
σCL
CL0.044 0.042 0.042 0.0145
σCD
CD0.163 0.159 0.16 0.1589
|CL−CL0|
CL7.6e-4 1.3e-3 1.6e-3 4.2e-4
|CD−CD0|
CD1.66e-2 1.46e-2 1.4e-2 2.1e-2
Outline
Overview
Modelling of free stream turbulenceNumerics
Uncertainties in geometryNumerics
Low-rank approximation of the solutionNumerics
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 = λiφi
4. Then κ(x , ω) ≈ ∑mi=1
√λiφiξi(ω), where ξi (ω) are uncorrelated
random variables.
Sparse approximation of dense matrix C is done in [Khoromskij,Litvinenko, Matthies, 2009]
Uncertainties in geometry
[min, max] mean variance, σ2
st. dev.σ
σ/mean
CL [0.828, 0.863] 0.8552 0.00002 0.0049 0.0058CD [0.017, 0.022] 0.0183 0.00000 0.00012 0.0065
PCE of order 1 with 3 random variables and sparse Gauss-Hermitegrid wite 25 points were used.
Outline
Overview
Modelling of free stream turbulenceNumerics
Uncertainties in geometryNumerics
Low-rank approximation of the solutionNumerics
Low-rank approximation of the solution
U VΣ T=M
UVΣ∼
∼ ∼ T
=M∼
Figure: Reduced SVD, only k biggest singular values are taken.
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
, va
lue
s
pressuredensitycpcf
Figure: Decay (in log-scales) of 100 largest eigenvalues of the combinedmatrix constructed from 645 solutions (pressure, density, cf, cp) on thesurface of RAE-2822 airfoil.
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, kB1 0.82 222 0.21 435 0.4 10810 5e-3 21520 5e-4 43150 1.2e-5 1080
Literature
1. A.Litvinenko, H. G. Matthies, Sparse Data Representation ofRandom Fields, PAMM, 2009.
2. B.N. Khoromskij, A.Litvinenko, H. G. Matthies, Application ofhierarchical matrices for computing the Karhunen-Loeveexpansion, Springer, Computing, 84:49-67, 2009.
3. B.N. Khoromskij, A.Litvinenko, Data Sparse Computation of theKarhunen-Loeve Expansion, AIP Conference Proceedings,1048-1, pp. 311-314, 2008.
4. H. G. Matthies, Uncertainty Quantification with Stochastic FiniteElements, Encyclopedia of Computational Mechanics, Wiley,2007.