Uncertainty Quantification in numerical Aerodynamics

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Dr. A. Litvinenkohttp://sri-uq.kaust.edu.sa/

September 29, 2013

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Introduction to uncertainties

Uncertain Input:Variables (α, Ma), geometry of airfoil

Uncertain solution:

1. statistical moments of (v , p, ρ), exceedance probabilitiesP(ρ > ρ∗)

2. probability density functions of CL, CD , position of shock.

Our aims:

1. Sparse representation of the input data (random fields)

2. Computing process in a reasonable time

3. Use the deterministic solver as a black box

4. A sparse format for the solution

5. Postprocessing in the sparse formatCenter for UncertaintyQuantification

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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-trivialparallelisation, the cheapest from all, dim. depen.)

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Example 1:

500 MC realisations of cp in dependence on αi and Mai

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Example 2:

500 MC realisations of cf in dependence on αi and Mai

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Example 3:

5% and 95% quantiles for cp from 500 MC realisations.

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Example 4:

5% and 95% quantiles for cf from 500 MC realisations.

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Quantification of uncertainties of the solution

Compute difference with the deterministic solution (α = 2.74,Ma = 0.73)

(a) det.turb. viscos. ∈ (2e − 5, 3.6e −4)

(b) det. density ∈ (0.45, 1.25)

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Comparison of determ. and uncert. solutions

(c) det. pressure ∈ (0.5, 1.4) (d) det.kin.energy ∈ (2e− 3, 3.8e− 2)

(e) det. velocity. in x ∈ (0, 1.4) (f) det. velocity in z ∈ (−0.2, 0.7)

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Conclusion 1:

I It is important to model and to quantify uncertainties.

I One can see significant influence of uncertainties on thesolution.

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Part II. Compressing of data

I Compressing of N snapshots

[q(x ,θ1), ..., q(x ,θN)] ≈ ABT .

influence on the mean value and variance of q(x ,θ) ?

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Approximate mean of the pressure

(a) (k=5) (b) (k=10)

(c) (k=30) (d) (k=50)

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Approximate variance of the pressure

(a) (k=5) (b) (k=10)

(c) (k=30) (d) (k=50)

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Case 1, density

(a) |ρ̄PCE137 − ρ̄MC |F = 0.5534relative error: 0.00222

(b) |ρ̄PCE281 − ρ̄MC |F = 0.5670relative error: 0.00228

(c) |var(ρPCE137) − var(ρMC )|F =0.2503

(d) |var(ρPCE281) − var(ρMC )|F =0.2502

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Case 1, Pressure

(a) |p̄PCE137 − p̄MC |F = 0.7114relative error: 0.00282

(b) |p̄PCE281 − p̄MC |F = 0.7240relative error: 0.00287

(c) |var(pPCE137) − var(pMC )|F =0.4714

(d) |var(pPCE281) − var(pMC )|F =0.4713

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Case 9, density

(a) |ρ̄PCE29 − ρ̄MC |F = 7.0045relative error: 0.01431

(b) |ρ̄PCE201 − ρ̄MC |F = 7.2286relative error: 0.01477

(c) |var(ρPCE29) − var(ρMC )|F =0.7885

(d) |var(ρPCE201) − var(ρMC )|F =0.6205

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Case 9, Pressure

(a) |p̄PCE29 − p̄MC |F = 9.0311relative error: 0.01820

(b) |p̄PCE201 − p̄MC |F = 9.3225relative error: 0.01878

(c) |var(pPCE29) − var(pMC )|F =1.3030

(d) |var(pPCE201) − var(pMC )|F =1.0294

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Results are published

total 390 pages, our results on pp. 265-281

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