A novel uncertainty quantification routine in the genre of adaptive sparse grid stochastic collocation (SC) has been proposed in this study to investigate the propagation of parametric uncertainties in a stall flutter aeroelastic system. In a hypercube stochastic domain, presence of strong nonlinearities can give way to steep solution gradients that can adversely affect the convergence of nonadaptive sparse grid collocation schemes. A new adaptive scheme is proposed here that allows for accelerated convergence by clustering more discretization points in regimes characterized by steep fronts, using hat-like basis functions with nonequidistant nodes. The proposed technique has been applied on a nonlinear stall flutter aeroelastic system to quantify the propagation of multiparametric uncertainty from both structural and aerodynamic parameters. Their relative importance on the stochastic response is presented through a sensitivity analysis.

References

References
1.
Lee
,
B. H. K.
,
Price
,
S. J.
, and
Yong
,
Y. S.
,
1999
, “
Nonlinear Aeroelastic Analysis of Airfoils: Bifurcation and Chaos
,”
Prog. Aerosp. Sci.
,
35
(
3
), pp.
205
334
.
2.
Li
,
D.
,
Guo
,
S.
, and
Xiang
,
J.
,
2012
, “
Study of the Conditions That Cause Chaotic Motion in a Two-Dimensional Airfoil With Structural Nonlinearities in Subsonic Flow
,”
J. Fluids Struct.
,
33
, pp.
109
126
.
3.
Ilango
,
S. J. J.
, and
Sarkar
,
S.
,
2017
, “
An Efficient Stochastic Framework to Propagate the Effect of the Random Solid-Pore Geometry of Porous Media on the Pore-Scale Flow
,”
Comput. Methods Appl. Mech. Eng.
,
315
, pp.
73
99
.
4.
Ghanem
,
R.
, and
Spanos
,
P. D.
,
1991
,
Stochastic Finite Elements: A Spectral Approach
, Vol.
41
,
Springer
, Berlin.
5.
Xiu
,
D.
,
2010
,
Numerical Methods for Stochastic Computations, a Spectral Method Approach
,
Princeton University Press
,
Princeton, NJ
.
6.
Ghanem
,
R.
,
1999
, “
Ingredients for a General Purpose Stochastic Finite Elements Implementation
,”
Comput. Methods Appl. Mech. Eng.
,
168
(
1–4
), pp.
19
34
.
7.
Lin
,
G.
,
Su
,
C.-H.
, and
Karniadakis
,
G. E.
,
2006
, “
Predicting Shock Dynamics in the Presence of Uncertainties
,”
J. Comput. Phys.
,
217
(
1
), pp.
260
276
.
8.
Witteveen
,
J. A. S.
,
Sarkar
,
S.
, and
Bijl
,
H.
,
2007
, “
Modeling Physical Uncertainties in Dynamic Stall Induced Fluid–Structure Interaction of Turbine Blades Using Arbitrary Polynomial Chaos
,”
Comput. Struct.
,
85
(
11–14
), pp.
866
878
.
9.
Babuška
,
I.
,
Nobile
,
F.
, and
Tempone
,
R.
,
2010
, “
A Stochastic Collocation Method for Elliptic Partial Differential Equations With Random Input Data
,”
SIAM Rev.
,
52
(
2
), pp.
317
355
.
10.
Loeven
,
A.
,
Witteveen
,
J. A. S.
, and
Bijl
,
H.
,
2007
, “
Probabilistic Collocation: An Efficient Non-Intrusive Approach for Arbitrarily Distributed Parametric Uncertainties
,”
AIAA
Paper No. 2007-317.
11.
Garcke
,
J.
,
Griebel
,
M.
, and
Thess
,
M.
,
2001
, “
Data Mining With Sparse Grids
,”
Computing
,
67
(
3
), pp.
225
253
.
12.
Bungartz
,
H.-J.
, and
Griebel
,
M.
,
2004
, “
Sparse Grids
,”
Acta Numer.
,
13
, pp.
147
269
.
13.
Gerstner
,
T.
, and
Griebel
,
M.
,
1998
, “
Numerical Integration Using Sparse Grids
,”
Numer. Algorithms
,
18
(
3/4
), pp.
209
232
.
14.
Ganapathysubramanian
,
B.
, and
Zabaras
,
N.
,
2007
, “
Sparse Grid Collocation Schemes for Stochastic Natural Convection Problems
,”
J. Comput. Phys.
,
225
(
1
), pp.
652
685
.
15.
Babuska
,
I.
,
Tempone
,
R.
, and
Zouraris
,
G.
,
2004
, “
Galerkin Finite Elements Approximation of Stochastic Finite Elements
,”
SIAM J. Numer. Anal.
,
42
(
2
), pp.
800
825
.
16.
Wan
,
X.
, and
Karniadakis
,
G. E.
,
2005
, “
An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations
,”
J. Comput. Phys.
,
209
(
2
), pp.
617
642
.
17.
Foo
,
J.
,
Wan
,
X.
, and
Karniadakis
,
G. E.
,
2008
, “
The Multi-Element Probabilistic Collocation Method (ME-PCM): Error Analysis and Applications
,”
J. Comput. Phys.
,
227
(
22
), pp.
9572
9595
.
18.
Chassaing
,
J.-C.
,
Lucor
,
D.
, and
Trégon
,
J.
,
2012
, “
Stochastic Nonlinear Aeroelastic Analysis of a Supersonic Lifting Surface Using an Adaptive Spectral Method
,”
J. Sound Vib.
,
331
(
2
), pp.
394
411
.
19.
Ma
,
X.
, and
Zabaras
,
N.
,
2009
, “
An Adaptive Hierarchical Sparse Grid Collocation Algorithm for the Solution of Stochastic Differential Equations
,”
J. Comput. Phys.
,
228
(
8
), pp.
3084
3113
.
20.
Bungartz
,
H.-J.
, and
Dirnstorfer
,
S.
,
2003
, “
Multivariate Quadrature on Adaptive Sparse Grids
,”
Computing
,
71
(
1
), pp.
89
114
.
21.
Witteveen
,
J. A. S.
, and
Iaccarino
,
G.
,
2012
, “
Simplex Stochastic Collocation With Random Sampling and Extrapolation for Nonhypercube Probability Spaces
,”
SIAM J. Sci. Comput.
,
34
(
2
), pp.
A814
A838
.
22.
Witteveen
,
J. A. S.
, and
Iaccarino
,
G.
,
2013
, “
Simplex Stochastic Collocation With ENO-Type Stencil Selection for Robust Uncertainty Quantification
,”
J. Comput. Phys.
,
239
, pp.
1
21
.
23.
Witteveen
,
J. A. S.
, and
Iaccarino
,
G.
,
2013
, “
Subcell Resolution in Simplex Stochastic Collocation for Spatial Discontinuities
,”
J. Comput. Phys.
,
251
, pp.
17
52
.
24.
Nobile
,
F.
,
Tempone
,
R.
, and
Webster
,
C.
,
2008
, “
A Sparse Grid Collocation Method for Elliptic Partial Differential Equations With Random Input Data
,”
SIAM J. Numer. Anal.
,
46
(5), pp.
2309
2345
.
25.
Devathi
,
H.
, and
Sarkar
,
S.
,
2016
, “
Study of a Stall Induced Dynamical System Under Gust Using the Probability Density Evolution Technique
,”
Comput. Struct.
,
162
, pp.
38
47
.
26.
Fung
,
Y. C.
,
1969
,
An Introduction to the Theory of Aeroelasticity
,
Dover Publications
,
Mineola, NY
.
27.
Leishman
,
J.
, and
Beddoes
,
T.
,
1989
, “
A Semi-Empirical Model for Dynamic Stall
,”
J. Am. Helicopter Soc.
,
34
(
3
), pp.
3
17
.
28.
Sobol
,
I. M.
,
2001
, “
Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates
,”
Math. Comput. Simul.
,
55
(
1–3
), pp.
271
280
.
29.
Saltelli
,
A.
,
Annoni
,
P.
,
Azzini
,
I.
,
Campolongo
,
F.
,
Ratto
,
M.
, and
Tarantola
,
S.
,
2010
, “
Design and Estimator for the Total Sensitivity Index. Variance Based Sensitivity Analysis of Model Output
,”
Comput. Phys. Commun.
,
181
(
2
), pp.
259
270
.
30.
Fragiskatos
,
G.
,
1999
, “Nonlinear Response and Instabilities of a Two-Degree-of-Freedom Airfoil Oscillating in Dynamic Stall,”
Master's thesis
, McGill University, Montreal, QC, Canada.
You do not currently have access to this content.