Uncertainty Quantification of an Aeroelastic System using Polynomial Chaos Expansion

S. Venkatesh1, Ajit Desai2 and Sunetra Sarkar3 1,2Masters Student, Department of Aerospace Engineering, IIT Madras, Chennai 600036

3Assistant Professor, Department of Aerospace Engineering, IIT Madras, Chennai 600036

sunetra.sarkar@gmail.com

System parametric uncertainties in modern wind turbine rotors with highly flexible

blades can alter the stability characteristics significantly and hence it is an important

design criterion. A nonlinear aeroelastic system with parametric uncertainties has been

considered. The analysis has been put in a stochastic framework and the propagation of

system uncertainties have been quantified using a spectral uncertainty quantification

tool called Polynomial Chaos Expansion. A projection based non-intrusive Polynomial

Chaos approach with Gauss-Hermite quadrature rule is applied, as it is more efficient

than its Galerkin based counterpart with increasing order of chaos expansions. Effect of

system randomness on the bifurcation behavior and flutter has been significant and the

results pertaining to it has been presented.

Nomenclature = Natural frequency ratio in plunge and pitch

= Non-dimensional distance from airfoil mid-chord to elastic axis

= Airfoil semi-chord length

xα = Non-dimensional distance from elastic axis to centre of mass

rα = Radius of gyration about elastic axis

CL = Aerodynamic lift coefficient

CM = Aerodynamic moment coefficient

Kα = Torsional spring stiffness

Kh = Plunge spring stiffness

U = Non-dimensional velocity

I. Introduction The Prediction of onset of flutter is of great importance in the field of aeroelasticity due to diverging growth

of structural oscillations after the flutter point. Nonlinear parameters present in the system can stabilize the

diverging growth of flutter oscillations to a limit cycle oscillation (LCO). However, sustained LCO can lead to

fatigue failure of rotating structures such as wind turbine rotors. Hence, it is an important design concern in

aeroelastic analysis. Moreover, with system uncertainty, flutter point can vary (1), hence, increases the risk.

Therefore, interest is growing in understanding how these system uncertainties in structural and aerodynamic

parameters and initial conditions propagate through the system. Uncertainty quantification in a stochastic

framework with stochastic inputs has traditionally been analyzed with Monte Carlo simulations (MCS). To

apply this procedure one should use the distribution of the input parameters to generate a large number of

realizations of the response. Probability density function (PDF), probability of failure (POF) and other required

statistics are then approximated from these response realizations. However, it is computationally expensive,

especially for complex problems, which need large number of simulations. Hence, it is used only as last

resource when all other methods fail. Therefore, there is a need to develop alternate approaches, which are

computationally cheaper than direct MCS procedure. Stochastic spectral methods based on generalized

polynomial chaos expansion (gPCE) are more effective approach, pioneered by Ghanem and Spanos (3; 4),

proposed first in the structural mechanics finite elements area. It is a spectral representation of the uncertainty in

terms of orthogonal polynomials. The stochastic input is represented spectrally by employing orthogonal

polynomial functionals from the Askey scheme as basis in the random space. The original homogeneous PCE

was based on Hermite polynomials from the Askey family (6). It can give optimal exponential convergence for

Gaussian inputs (4). The non-intrusive PCE approach and its implementation to a nonlinear aeroelastic model

with parametric uncertainty modeled in more than one system parameter are discussed in detail in the

subsequent sections.

II. Nonlinear Aeroelastic Model

A schematic plot of the two degree-of-freedom pitch-plunge aeroelastic system and also the notations used in

the analysis is shown in Fig.1.

FIG.1 The schematic of a symmetric airfoil with pitch and plunge degree of freedom

The aeroelastic equations of motion for the linear system have been derived by Fung (2). For nonlinear restoring

forces such as with cubic springs in both pitch and plunge, the mathematical formulation is given by Lee (8) as

given above Eq. 1. The above equations are shown in the nondimensional form. The nondimensional parameters

are given below. The plunge deflection is considered positive in the downward direction and the pitch angle.

Among the nondimensional quantities, ε=h\b = nondimensional displacement of the elastic axis point; τ = vt/b =

nondimensional time. In the structural part, are the damping ratios in plunge and pitch respectively.

The mass ratio µ is defined as m/(πρv∞). denotes coefficients of cubic spring in pitch and plunge

respectively. For incompressible, inviscid flow, Fung (2) gives the expressions for unsteady lift and pitching

moment coefficients, CL and CM. Using Wagner function and introducing the following new variables w1, w2, w3

and w4 the original integro-differential equations for aeroelastic system given by Eq. (1) are reformulated into

set of first order autonomous differential equations. For details refer Lee et al. (5).

III. Polynomial chaos expansion

Polynomial Chaos expansion is a spectral uncertainty quantification tool, which is based on the homogeneous

chaos theory of Wiener (6). In its original form, it employs Hermite polynomials as basis from generalized

Askey scheme and Gaussian random variables. In the classical Galerkin-PCE approach (also called intrusive

approach), the polynomial chaos expansion of the system response is substituted into the governing equation

and a Galerkin error minimization in the probability space is followed. This results in a set of coupled equations

in terms of the polynomial chaos coefficients. The resulting system is deterministic, and can be solved using

standard techniques to get chaos coefficients. After solving these sets of coefficient equations, they are

substituted back to get the system response. As per the Cameron-Martin theorem (3), a random process X (t, θ)

(as function of random event θ) that is second order stationary can be written as,

where, denotes the Hermite polynomial of order n in terms of n-dimensional

independent standard gaussian random variables with zero mean and unit variance.

The above equation is the discrete version of the original wiener polynomial chaos expansion, and the

continuous integrals are replaced by summations. For notational convenience Eq.3 can be rewritten as:

chaos expansions uses Hermite polynomials. The form of Hermite polynomials basis for a second order

expansion over two random dimensions are,

where are the one-dimensional Hermite polynomials as given below

One can also use the most appropriate orthogonal polynomials from the generalized Askey scheme for some

standard non-Gaussian input uncertainty distributions such as gamma and beta (8). With this approach, Xiu and

Karniadakis (8) achieved exponential convergence for number of different input distributions. For any arbitrary

input distribution, a Gram-Schmidt orthogonalization can be employed to generate the orthogonal family of

polynomials (7).

Any stochastic process governed by Gaussian random variables can always be normalized

as a standard Gaussian one) can be approximated by the following truncated series:

Note that the infinite upper limit of Eq.4 is replaced by p, called the order of expansion. For n number of

random variables and polynomial order np , p is given by the following (8).

As the Galerkin approach is more time consuming, the non-intrusive approach has been developed in which

a deterministic solver is used repeatedly at certain collocation points as in MCS. We call this step as pseudo-

Monte Carlo simulations. To calculate the response PCE coefficients, we employ the spectral projection

approach, which projects the response against each basis function using inner products and employs the

polynomial orthogonality to extract each coefficient.

The denominator in above equation can be shown to satisfy for non-normalized Hermite

polynomials (13). So for the evaluation of the projection we have the following

Where the weighing function is the Gaussian probability density function. A Gauss-Hermite

quadrature will be suitable for evaluating the above integral as the domain is and the weight function

is Gaussian PDF. Above Eq. 10 is re-written as

Where the couples represent the one-dimensional Gauss-Hermite quadrature points and weights.

IV. Results and Discussions

The main focus of the present study is quantifying the effect of parametric uncertainties on the bifurcation

behavior and the flutter boundary of the nonlinear aeroelastic system. The system parameters that are assumed

to be uncertain are different combinations of the following parameters The rest of the parameters

are deterministic and taken from (5). In all the cases, we have restricted our domain to (-4 to 4) as 99.99% of

realizations fall in this region. In all the cases, the U at which PDFs are drawn is so chosen that it lie close to

first bifurcation point. RK4 variable step method is employed for the time integration.

A. Results for as a Random parameter

We have assumed to be a Gaussian random variable with mean = 0.2 and standard deviation = 0.02. All other

parameters are deterministic and considered to be same as earlier. A 15th order PCE gives good results for which

16-point quadrature is minimum, but we have used 20-quadrature points, which is well above the minimum to

get the final response. The realisations of PCE matches well with that of Monte-Carlo at the beginning. As a

result, the PDF plots by MCS and PCE agree at time = 3000 (Fig. 4). In Fig. (3), we see that the velocity at

which flutter occurs itself is a random variable. The PDF and CDF corresponding to it has been shown in Fig. 5.

From Fig. 5, we can determine the probability of flutter occuring below deterministic linear flutter speed.

Deterministic linear flutter speed = 6.285, now from the CDF plot, we see that for a speed of 6.1 the chance of

flutter occuring is 10%.

Fig. 2 Random :A typical realization time Fig. 3 Random :Stochastic Bifurcation plot

history with 15th order PCE at U = 6.8

Fig. 4 Random :Comparison of the PDF by Fig. 5 Random : PDF and CDF plots for

PCE and MCS at U=6.8 with time = 3000 U about first bifurcation point

B. Results for as a Random parameter

We have assumed to be Gaussian distributed with a mean of 0.25 and standard deviation of 0.025. From the

bifurcation plot (Fig. 9), we see that bifurcation is not only a function of U but also a function of itself.

Fig. 2 Random :A typical realization time Fig. 3 Random :Stochastic Bifurcation plot

history with 20th order PCE at U = 7.77

3000 4000 5000 6000 7000 80000.3

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for mean value of RV xfor min. value of RV xfor max. value of RV x

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LCO amplitudes at mean values of RVsLCO amplitudes at maxiumu values of RVsLCO amplitudes at minimum values of RVsUnstable branch

Similar to case A, we have drawn PDF and CDF for U as shown in Fig. (11). In the Fig. (8), we see that

realizations of MCS and PCE match well at beginning. Hence the PDFs in Fig. (10) exactly match. We have

used 20th order of PCE for which minimum 21 quadrature points are required but we have used 31 quadrature

points for computation, which is above our requirement.

Fig. 10 Random :Comparison of the PDF by Fig. 11 Random :PDF and CDF plots for U

PCE and MCS at U=7.77 with time = 2000 about first bifurcation point.

C. Results for and as a Random parameters

Fig. 14 Random and :A typical realization time Fig. 15 Random and :Stochastic plot

history with 28th order of PCE at U=8.5 of Bifurcation

Fig. 16 Random and :CDF for U Fig. 17 Random and :PDF for U

about the first Bifurcation point. about the first Bifurcation point.

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.50

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PCEMCS

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minimum value of RVmean value of RVmaximum value of RV

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Fig. 18 Random and :Comparison PDF Fig. 19 Random and :Comparison of the PDF

by MCS and PCE at time =2000 and U=8.5 by MCS and PCE at time =4000 and U=8.5

Both and are assumed to be independent Gaussian random variables with values as specified earlier.

From Fig. (14), we see that the time history of realization by MCS and PCE does not match because degeneracy

in this case sets in very early and that there is no region where the PCE solution and MCS solution matches. As

a result, the PDFs obtained by PCE and MCS in Fig. (18) and Fig. (19) does not match. Fig. (16) and Fig. (17)

gives the CDF and PDF for random variable U about the first bifurcation point respectively. Fig. (15), Fig. (16)

and Fig. (17) are all obtained by using MCS. For this case we have used 28th order PCE with 31 Quadrature

points.

References

[1] A. Desai and S. Sarkar, “Analysis of a nonlinear aeroelastic system with parametric uncertainties using

polynomial chaos expansion,” Mathematical Problems in Engineering, vol. 379472, p.21, 2010

[2] C. Fung, Y., An Introduction to the Theory of Aeroelasticity. John Wiley & Sons, Inc., New York, 1955.

[3] R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements: a Spectral Approach. Newblock Springer-

Verlag, New York, 1991.

[4] R. Ghanem and D. Spanos, “Polynomial chaos in stochastic finite elements” Journal of Applied Mechanics,

vol. 57, pp. 197-202, 1990.

[5] B. H. K. Lee, Y. B. Jiang and Y. S. Wong, “Flutter of an airfoil with cubic nonlinearity restoring force,”

AIAA Journal, vol. 1725, pp. 237-257, 1998.

[6] N. Weiner, “The homogeneous chaos,” American Journal of mathematics, vol. 60, pp. 897-936, 1938.

[7] J. A. S. Witteveen, S. Sarkar and H. Bijl, “Modeling physical uncertainties in dynamic stall induced fluid-

structure interaction of turbine blades using arbitrary polynomial chaos,” Computers and Structures, vol. 85,

pp. 866-878, 2007.

[8] D. Xiu and G. E. Karniadakis, “The Wiener-Askey polynomial chaos for stochastic differential equations,”

SIAM Journal on Scientific Computing, vol. 24, no. 2, pp. 619-644, 2002

1 0.5 0 0.5 10

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