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Uncertainties in fluid-structure interaction simulations

Hester Bijl

Aukje de Boer, Alex Loeven, Jeroen Witteveen, Sander van Zuijlen

Faculty of Aerospace Engineering

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Some Fluid-Structure Interactions

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Flexible wing motion simulation

Flow: CFD

Structure: FEM

Damped flutter computation for the AGARD 445.6 wing

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Result of simulation

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Helios encountering turbulence ..

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Transonic flow over NACA0012 airfoil with uncertain Mach numberMach number M on the surface?

Uncertainty:

• Min

• Lognormal• Mean = 0.8• CV = 1%

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Large effect uncertainty due to sensitive shock wave location

ASFE

Original global polynomial

Robust approximation Adaptive Stochastic Finite Elements

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Polynomial Chaos uncertainty quantification framework selected• Probabilistic description uncertainty • Global polynomial approximation response• Weighted by input probability density• More efficient than Monte Carlo simulation• No relation with “chaos”

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Polynomial Chaos expansion

Polynomial expansion in probability space in terms of random variables and deterministic coefficients:

u(x,t,ω) = Σ ui(x,t)Pi(a(ω))i=0

p

u(x,t,ω) uncertain variableω in probability space Ωui(x,t) deterministic coefficientPi(a) polynomial

a(ω) uncertain input parameterp polynomial chaos order

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Robust uncertainty quantification needed

Singularities encountered in practice:

• Shock waves in supersonic flow

• Bifurcation phenomena in

fluid-structure interaction

Singularities are of interest:

• Highly sensitive to input uncertainty

• Oscillatory or unphysical predictions

shock

NACA0012 at M=0.8

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Adaptive Stochastic Finite Elements approach for more robustness• Multi-element approach:

Piecewise polynomial approximation response

• Quadrature approximation in the elements:Non-intrusive approach based on deterministic solver

• Adaptively refining elements:Capturing singularities effectively

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Adaptive Stochastic Finite Element formulation

Probability space subdivided in elements For example for stochastic moment μk’:

μk’ = ∫ x(ω)kdω = ∑ ∫ x(ω)kdω

Quadrature approximation in elements:

μk’ ≈ ∑ ∑ cjxi,jk

i=1

i=1 j=1

Ω Ωi

NΩ

NΩ NsNΩ # stochastic elementsNs # samples in elementcj quadrature coefficients

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Based on Newton-Cotes quadrature in simplex elements

• Newton-Cotes quadrature:midpoint rule, trapezoid rule, Simpson’s rule, …

• Simplex elements:line element, triangle, tetrahedron, …

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Lower number of deterministic solves

Due to location of the Newton-Cotes quadrature points:

• Samples used in approximating response in multiple elements

• Samples reused in successive refinement steps

Example: refinement quadratic element with 3 uncertain parametersStandard 54 deterministic solvesNewton-Cotes <5 deterministic solves

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Adaptive refinement elements captures singularities

Refinement measure: • Curvature response surface weighted by

probability density• Largest absolute eigenvalue of the Hessian in

element

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Monotonicity and optima of the samples preserved

Polynomial approximation with maximum in element:

• Element subdivided in subelements • Piecewise linear approximation of the response• Without additional solves

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Numerical results

1. One-dimensional piston problem

2. Pitching airfoil stall flutter

3. Transonic flow over NACA0012 airfoil

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1. One-dimensional piston problem

Mass flow m at sensor location?

Uncertainties:

• upiston

• ppre

• Lognormal• Mean = 1• CV = 10%

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ASFE Global polynomial

Oscillatory and unphysical predictions in global polynomial approximationDiscontinuity in response due to shock wave

uncertain upiston

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Discontinuity captured by adaptive grid refinement

2 elemen

ts

10 element

s

Monotone approximation of discontinuity

uncertain upiston and ppre

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Mass flow highly sensitive to input uncertainty

50 element

s

100 elements

Input coefficient of variation: 10%Output coefficient of variation: 184%

uncertain upiston and ppre

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2. Pitching airfoil stall flutter

Pitch angle ?

Uncertainty:

• Fext

• Lognormal• Mean = 0.002• CV = 10%

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Discontinuous derivative due to bifurcation behavior

Accurately resolved by Adaptive Stochastic Finite Elements

ASFE

Global polynomial

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Conclusion

Adaptive Stochastic Finite Element method allows

robust uncertainty quantification, ex.

- bifurcation in FSI

- shock wave in supersonic flow

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Thank you