Uncertainties in fluid-structure interaction simulations

1. Uncertainties in fluid-structure interaction simulations Hester Bijl Aukje de Boer, Alex Loeven, Jeroen Witteveen, Sander van Zuijlen Faculty of Aerospace Engineering

2. Some Fluid-Structure Interactions

3. Flexible wing motion simulation Flow: CFD Damped flutter computation for the AGARD 445.6 wing Structure: FEM

4. Result of simulation

5. Helios encountering turbulence ..

6. Transonic flow over NACA0012 airfoil with uncertain Mach number Mach number M on the surface? Uncertainty: • Min • Lognormal • Mean = 0.8 • CV = 1%

7. Large effect uncertainty due to sensitive shock wave location Robust approximation Adaptive Stochastic Finite Elements ASFE Original global polynomial

8. 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”

9. 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)) u(x,t,ω) uncertain variable ω in probability space Ω ui(x,t) deterministic coefficient Pi(a) polynomial a(ω) uncertain input parameter p polynomial chaos order p i=0

10. 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

11. 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

12. 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 NΩ i=1 Ω Ωi NΩ # stochastic elements Ns # samples in element cj quadrature coefficients NΩ Ns i=1 j=1

13. Based on Newton-Cotes quadrature in simplex elements • Newton-Cotes quadrature: midpoint rule, trapezoid rule, Simpson’s rule, … • Simplex elements: line element, triangle, tetrahedron, …

14. 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 parameters Standard 54 deterministic solves Newton-Cotes <5 deterministic solves

15. Adaptive refinement elements captures singularities Refinement measure: • Curvature response surface weighted by probability density • Largest absolute eigenvalue of the Hessian in element

16. 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

17. Numerical results • One-dimensional piston problem • Pitching airfoil stall flutter • Transonic flow over NACA0012 airfoil

18. 1. One-dimensional piston problem Mass flow m at sensor location? Uncertainties: • upiston • ppre • Lognormal • Mean = 1 • CV = 10%

19. Oscillatory and unphysical predictions in global polynomial approximation Discontinuity in response due to shock wave ASFE Global polynomial uncertain upiston

20. Discontinuity captured by adaptive grid refinement Monotone approximation of discontinuity 2 elements 10 elements uncertain upiston and ppre

21. Mass flow highly sensitive to input uncertainty Input coefficient of variation: 10% Output coefficient of variation: 184% 50 elements 100 elements uncertain upiston and ppre

22. 2. Pitching airfoil stall flutter Pitch angle ? Uncertainty: • Fext • Lognormal • Mean = 0.002 • CV = 10%

23. Discontinuous derivative due to bifurcation behavior Accurately resolved by Adaptive Stochastic Finite Elements ASFE Global polynomial

24. Conclusion Adaptive Stochastic Finite Element method allows robust uncertainty quantification, ex. - bifurcation in FSI - shock wave in supersonic flow

25. Thank you