1. Uncertainties in fluid-structure interaction simulations
Enorme advances voor mijn groep. Dit project mijn eerste proposal
als verse UD bij LR, 1e AIO: Sander.
Hierna nog veel proposals en bijbehorende AIOs en Postdocs.
Nu HL, met groep van 7 AIOs en 2 Postdocs. Werken aan instat. Stromingen,
FSI, numerieke methoden en toepassingen.
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 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 Ns
NΩ # stochastic elements
Ns # samples in element
cj quadrature coefficients
i=1
j=1

13. Based on Newton-Cotes quadrature in simplex elements
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