Uncertainties in fluid-structure interaction simulations
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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