dynamic stall flutter analysis with uncertainties using multi-element probabilistic collocation

Dynamic stall flutter analysis with uncertainties

using Multi-Element Probabilistic Collocation

G.J.A. Loeven,∗§ S. Sarkar,†§ J.A.S. Witteveen,§ H. Bijl‡§

Delft University of Technology, The Netherlands

In this paper the effect of uncertainty present in external torque on the bifurcation be-

havior of a dynamic stall flutter problem is analyzed using the Multi-Element Probabilistic

Collocation method. A Multi-Element Probabilistic Collocation approach is followed since

the bifurcation in the response causes oscillations for the ordinary Probabilistic Collocation

method. Two elements are used, with one element covering the damped response and the

other the period-one oscillations. This is only possible if the elements are separated near

the bifurcation point, of which the location is in general not known in advance. A search

algorithm is used to find a coarse estimate of the bifurcation point to divide the domain

into two elements. The stochastic bifurcation plot and the probability distribution of the

bifurcation point show a large variation in the output due to the uncertain external torque.

I. Introduction

Uncertainty quantification becomes more important in engineering since the accuracy of deterministicnumerical simulations has significantly improved. More information is now required about how variations,which are inherently present in reality, affect the numerical simulation. Especially in fluid-structure inter-action problems the dynamic coupling between the fluid and the structure can make the system sensitive tosmall variations in the input parameters. Deterministic computations can already be computationally inten-sive, but due to the increase in computer resources and the development of efficient stochastic algorithms itis feasible to include uncertainties in the computations nowadays.

Dynamic stall flutter is a fluid-structure interaction, which is of great importance in structural analysisof aircraft wings and wind turbine blades. Dynamic stall increases the loadings on the structure significantlyand the flutter influences the fatigue life of the structure. A lot of effort is, therefore, put in the prediction ofthe bifurcation behavior of the system. Since the dynamical systems that describe dynamic stall flutter areknown to be sensitive to variations in the parameters, uncertainty quantification has already been extensivelyapplied.10, 11, 16–18

The output of the dynamic stall flutter model is the pitch angle, whereas the output of interest is abifurcation diagram. A deterministic bifurcation diagram shows the minimum and maximum pitch angleof the asymptotic state of the system for a range of free stream velocities.20 For uncertainty quantificationthe Galerkin Polynomial Chaos method provides the stochastic properties of the pitch angle from whichthe stochastic bifurcation diagram has to be computed. The Probabilistic Collocation method1, 12 obtainsthe stochastic properties of the minimum and maximum pitch angle and bifurcation point directly. Sincethe collocation approach is applied to the functional of the stochastic solution, it is more efficient andaccurate than the Galerkin approach. The response of the system, however, does not comply with thesmoothness conditions to assure convergence of the Probabilistic Collocation method. The response of thepitch angles contains discontinuous derivatives, leading to an oscillatory solution due to the global polynomialapproximation. Multi-Element approaches are more robust for stochastic solutions involving a bifurcated

∗Ph. D. researcher, email: [email protected], member AIAA†Post-doctoral researcher‡Full Professor, member AIAA.§Department of Aerospace Engineering, P.O. Box 5058, 2600 GB Delft, The Netherlands

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American Institute of Aeronautics and Astronautics

48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br> 15th23 - 26 April 2007, Honolulu, Hawaii

AIAA 2007-1964

Copyright © 2007 by G.J.A. Loeven, S. Sarkar, J.A.S. Witteveen and H. Bijl. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

response. Various approaches are developed, like Wiener-Haar expansions8, 9, 17 or Galerkin PolynomialChaos expansions.24

In this paper the Multi-Element Probabilistic Collocation method is shown, which is a multi elementformulation in stochastic space based on the Probabilistic Collocation method.1, 12 It is shown that dampedresponse and period-one oscillation approximations match with Monte Carlo results using one element.However, when the system has a bifurcated response one element results in an oscillating solution. Since theProbabilistic Collocation method works well for the damped part and the period-one oscillations part, thedomain is divided into two elements. This means that the elements meet at the bifurcation point, of whichthe location in general is not known in advance. A search sample algorithm is used to get an estimate for thebifurcation point. The Multi-Element Probabilistic Collocation method is shown to result in approximationsthat correspond with Monte Carlo simulation and has a significant increase in efficiency.

The Multi-Element Probabilistic Collocation method is a general applicable method. The use of twoelements with search samples is developed for special cases when the response is known to have two domainswhich both are smooth, like damped solutions and period-one oscillations of the dynamic stall flutter problem.This analysis uses one uncertain parameter. An extension of the this framework to multiple uncertainparameters and responses with more than two smooth domains is required for applicability to more generalproblems.

Here, the dynamic stall flutter is modeled using a two dimensional symmetric pitching airfoil attached toa torsional spring. The aerodynamic loading is computed using an engineering dynamic stall model given byOnera.15, 21 This stall model approximates the loads well and is less computationally expensive than a fullviscous flow solver.2 Price and Keleris19 showed that the choice of external torque has a strong effect on theresponse of the system. The stochastic bifurcation plot is computed using the Multi-Element ProbabilisticCollocation method assuming a lognormally distributed external torque. It shows the mean of the minimumand maximum pitch angles with 99.8% uncertainty bars and the probability distribution of the bifurcationpoint. The uncertain external torque results in large variations of the output.

This paper is organized as follows. First the Multi-Element Probabilistic Collocation method is explainedin section II. In section III the dynamic stall flutter model is given, after which the uncertainty quantificationis performed in section IV. Section V provides the conclusions.

II. The Multi-Element Probabilistic Collocation Method

The Probabilistic Collocation method1, 12 is based on the idea of a chaos transformation, used in thePolynomial Chaos methods7, 22, 24, 26 combined with the non-intrusive approach of the collocation methodof Mathelin and Hussaini.13, 14 The Probabilistic Collocation method converges exponentially for arbitraryprobability distributions, as explained later in this section. In the Probabilistic Collocation method a poly-nomial chaos expansion is constructed based on Lagrange polynomials. With the resulting polynomial chaosexpansion it is possible to describe the input distribution with two collocation points only (just as for thePolynomial Chaos method, where two polynomial coefficients describe the input distribution). Secondly,Gauss quadrature weighted by the probability density function of the uncertain parameter is used to com-pute the Galerkin projection and the integration of the approximation of the distribution function. Asordinary Gauss quadrature is able to integrate polynomials exactly, the properly weighted Gauss quadratureintegrates polynomial chaoses exactly. By using Gauss quadrature a decoupled set of equations and a higherorder approximation of the mean and variance are obtained. Since the bifurcation of the response of thedynamical system results in oscillating solutions of the ordinary Probabilistic Collocation method, here aMulti-Element approach is followed.

Multi-Element Probabilistic Collocation expansion

The solution, in this case the pitch angle α, and each variable depending on the uncertain input parameteris expanded as follows:

α(τ, ω) =

Ne∑

i=1

Np∑

j=1

αij(τ)hij (ξi(ω)) , (1)

where the solution α(τ, ω), a function of the dimensionless time τ and the random event ω ∈ Ω, is writtenas a summation over the number of elements Ne and the number of collocation points in each element

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Np. The complete probability space is given by (Ω,F , P ), with Ω the set of outcomes, F ⊂ 2Ω the σ-algebra of events and P : F → [0, 1] a probability measure. Furthermore, αij(τ) is the solution α(τ, ω) atthe collocation point ωij , which is the collocation point j in element i; hij is the Lagrange interpolatingpolynomial chaos corresponding to the collocation points ωij ; ξi is the basis random variable in element i.The Lagrange interpolation results in polynomial chaoses instead of ordinary polynomials. The Lagrangeinterpolating polynomial is a function in terms of the random variable ξ(ω), which is chosen on the standarddomains [−1, 1], [0,∞) or (−∞,∞), such that the uncertain input parameter is a linear transformation ofξ(ω). In the Multi-Element formulation, the basis ξ(ω) is split into a basis random variable in each element

ξ(ω) =∑Ne

i=1 ξi(ω). The Lagrange interpolating polynomial chaos is the polynomial chaos hij (ξi(ω)) of orderNp − 1 that passes through the Np collocation points in element i. It is given by:

hij (ξi(ω)) =

Np∏

k=1k 6=j

ξi(ω) − ξi(ωik)

ξi(ωij) − ξi(ωik), i = 1, . . . , N

e, (2)

with hij (ξi(ωik)) = δjk . The collocation points are chosen such that they correspond to the Gauss quadraturepoints used to integrate the function α(τ, ω) in the ω domain. For convenience of notation the argumentω is omitted from here on. The solution has to be integrated in order to obtain for instance the mean orvariance. To find the suitable Gauss quadrature points and weights the procedure below is followed.

In each element the collocation points are computed based on the weighting function in the elementw(ξi) = fξi

(ξi). The weighting function fξi(ξi) is set equal to the global weighting function fξ(ξ) inside

the element i and is set to zero outside the element. Figure 1 shows a lognormal distribution, which isapproximated using 4 elements of equal probability. The left figures show how the cumulative distributionfunction and the probability density function are separated into 4 elements. The right figures show theresulting weighting functions of each element, which are used to compute the collocation points and weightsinside the elements.

Computing Gauss-Chaos quadrature points with corresponding weights

A powerful method to compute Gaussian quadrature rules is by means of the Golub-Welsch algorithm.6

This algorithm requires the recurrence coefficients5 of polynomials which are orthogonal with respect to theweighting function w(ξ) of the integration. Exponential convergence for arbitrary probability distributionsis obtained when the polynomials are orthogonal with respect to the probability density function of ξ, sow(ξ) = fξ(ξ). The recurrence coefficients are computed using the discretized Stieltjes procedure,4 which isa stable method for arbitrary distribution functions.

First the recurrence coefficients are computed. Orthogonal polynomials with respect to the weightingfunction of the element are constructed in each element using the recurrence relation

Ψi0(ξi) = 0, Ψi1(ξi) = 1,

Ψi,j+1(ξi) = (ξi − αij)Ψij(ξi) − βijΨi,j−1, j = 2, 3, . . . , Np, i = 1, . . . , Ne. (3)

Now αij and βij are the recurrence coefficients determined by the weighting function wi(ξi) and Ψij(ξi)Np

j=1

is a set of (monic) orthogonal polynomials with Ψij(ξi) = ξji +O(ξj−1

i ), j = 1, 2, . . . , Np in element i. Ganderand Karp4 showed that discretizing the weighting function leads to a stable algorithm. Therefore, the dis-cretized Stieltjes procedure is used to obtain the recurrence coefficients. From the recurrence coefficients αij

and βij , i = 1, . . . , Np, j = 1, . . . , Ne, the collocation points ξij and corresponding weights wij are computedusing the Golub-Welsch algorithm.6 Now the collocation points ξij in the ξ-domain are known. They aremapped to the ω-domain using the distribution function of ξ. The collocation points ωij are then found by

ωij = Fξ(ξij), i = 1, . . . , Ne, j = 1, . . . , Np. (4)

For a uniformly distributed parameter and ξ(ω) = U(−1, 1) the Probabilistic Collocation method results inthe collocation method of Mathelin and Hussaini.13

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Figure 1. The cumulative distribution function and probability density function (left) of a lognormal basis withµ = 10 and σ2 = 25. On the right the weighting functions in each of the four elements.

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Application to a general stall flutter model

In this section the application of the Multi-Element Probabilistic Collocation method to a general modelis shown. It is demonstrated how the method is used when the parameter of interest is a functional of thesolution, as is the case for the dynamic stall flutter model. Expansion (1) is substituted into the dynamicstall flutter model, represented here by the operator L, which depends on an uncertain input parametera(ω):

L (a (ω)) α(τ, ω) = S(τ). (5)

A Galerkin projection on each basis hik (ξi(ω)) is applied:

⟨

L (a (ω))

Ne∑

i=1

Np∑

j=1

αij(τ)hij , hik

⟩

= 〈S, hik〉 , k = 1, . . . , Np, i = 1, . . . , Ne. (6)

This projection is approximated using Gaussian quadrature, with collocation points and correspondingweights based on the input distribution. The result is a deterministic system of equations which is fullydecoupled

L (a (ωik)) αik(τ) = S(τ), k = 1, . . . , Np, i = 1, . . . , Ne. (7)

The mean and variance of the solution are found by

µα =

Ne∑

i=1

Np∑

j=1

αij(τ)wij , (8)

σ2α =

Ne∑

i=1

Np∑

j=1

(αij(τ))2wij −

Ne∑

i=1

Np∑

j=1

αij(τ)wij

2

, (9)

where wij are the weights corresponding to the collocation points ωij . These relations are derived fromthe definition of the mean and variance. The solution of the dynamic stall flutter problem is given as abifurcation diagram, so the minimum and maximum pitch angles of the asymptotic state of the system arerequired. For the minimum it is given by the following expansion

αmin(τ, ω) =

Ne∑

i=1

Np∑

j=1

αmin,ij(τ)hij (ξi(ω)) , (10)

with

αmin,ij = min (αij(τ)) for the time interval τ = [τa, τb], (11)

with the time interval [τa, τb] sufficiently large to contain at least one period of oscillation and is outsidethe transient part of the response. For the maximum a similar expression is used. Reconstruction of thestochastic properties of the minimum and maximum pitch angles (and thus the bifurcation diagram) isefficient and accurate due to the collocational approach. αmin and αmax are functionals of α(τ, ω). For theGalerkin Polynomial Chaos method the functionals have to be applied to the reconstructed solution α(τ, ω),which leads to problems with the long time integration.23 Another drawback of the Galerkin PolynomialChaos method to this dynamic stall flutter model is the presence of the nonlinear step function ∆Cm. Inthis paper the influence of one uncertain parameter is investigated. If multiple uncertain parameters arepresent, the collocation points are found using tensor products of one dimensional points or using a sparsegrid approach.27

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A stochastic computation is now performed using the following steps:

Determine uncertain parameters

specify the probability density functions

Compute collocation points and weights

compute parameter values for each collocation point

Run deterministic solver for each collocation point

post process the deterministic results

Compute the probability distribution, mean and variance of the solution

III. The dynamic stall flutter model

To investigate the dynamic stall flutter, the aircraft wing or wind turbine blade is modeled as a twodimensional rigid airfoil. A NACA0012 airfoil is attached to a torsional spring and is subjected to theaerodynamic moment obtained from the Onera dynamic stall model. This is standard practice in engineeringaeroelastic problems.15, 19, 21 The structural model is shown in figure 2. The equation of motion for the singledegree-of-freedom pitching airfoil is given below:3

Iαα + Iαω2αα + Knl1α

3 = M(t) + Mext1. (12)

Here, Iα is the wing mass moment of inertia; α is the pitch angle, ωα the natural frequency of the pitchelastic mode, Knl1 a nonlinear stiffness term accounting for concentrated structural nonlinearities in thetorsional direction, M(t) is the time dependent aerodynamic moment and Mext1 a constant external torqueapplied to the airfoil. First the equations of motion are made non-dimensional to be able to investigatethe effect of system parameters. The non-dimensional equation of motion of the single degree-of-freedompitching oscillation is given by:

α′′ + α1

U2+ Knlα

3 =1Cm

πµr2α

+ Mext, (13)

where ′′ denotes the second derivative with respect to τ = tV /b the non-dimensional time, with V the freestream velocity, Knl is the non-dimensional form of Knl1, rα = Iα/mb2 the radius of gyration, µ = m/πρb2

the mass ratio, U = V /bωα the dimensionless free stream velocity, Cm the aerodynamic moment coefficientdepending on τ obtained from the dynamic stall model and Mext is the non-dimensional form of Mext1. Theexternal torque Mext is used to give the model a structural equilibrium angle.19

The aerodynamic model is the Onera dynamic stall model governed by the following set of differentialequations

Cm = smα′ + kvmα′′ + Cm1 + Cm2 (14)

C ′m1 + λmCm1 = λm (aomα + σmα′) + αm (aomα′ + σmα′′) (15)

C ′′m2 + 2dωC ′

m2 + w2(

1 + d2)

Cm2 = −w2(

1 + d2)

(∆Cm|α + e∆C ′m|α) , (16)

where Cm is the aerodynamic moment. The aerodynamic moment is split into two parts: the inviscidcirculatory part Cm1 and the viscous part Cm2. The other coefficients sm, kvm, λm, αm, aom, σm, d, w and eare empirically determined coefficients by parameter identification techniques using experimental data. Thevalues for the coefficients for the NACA0012 airfoil have been obtained from Dunn and Dugundji.2 Thecubic structural stiffness Knl is set to zero. The function ∆Cm is a step function which activates a nonlinearfunction of α above the static stall angle of 12 and is linear below the static stall angle.

The dimensionless free stream velocity U is used as bifurcation parameter for the analysis of this system.The external torque Mext is set such that the system has an equilibrium pitch angle of 4. The systemis perturbed by an instantaneous pitch angle of 10. After a transient part the system is in equilibrium.The deterministic bifurcation plot of the pitch angle for a range of U ∈ [12, 22] is shown in figure 3. The

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Figure 3. Deterministic bifurcation plot with the bifurcation point at U = 15.765.

bifurcation diagram shows the minimum and maximum pitch angles of the time interval τ = 750−800. Thistime interval covers at least one period of oscillation and is outside the transient part of the response. Thebifurcation plot shows that the supercritical Hopf bifurcation is located at Ubif,det = 15.765. To the left ofthe bifurcation point, U ∈ [12, 15.765], are the damped solutions, where the minimum and maximum pitchangles are equal. To the right, U ∈ [15.765, 22], the solutions are period-one oscillations, represented by adifferent value of the minimum and maximum pitch angle.

Since the choice of the external torque Mext has a strong effect on the bifurcation diagram it has tobe chosen carefully. Price and Keleris19 used values of the order of 10−4. Here the mean µMext

is set to8.3 · 10−4, which corresponds to a structural equilibrium angle of 4. A lognormal distribution is assumedwith a coefficient of variation of CVMext

= 15%.

IV. Uncertainty quantification for the dynamic stall flutter problem

This section treats the uncertainty quantification of the dynamic stall flutter problem and leads to astochastic bifurcation plot and the stochastic properties of the bifurcation point. The external torque Mext

is assumed to be uncertain with mean µMext= 8.3 ·10−4, which corresponds to a structural equilibrium angle

of 4. The first three sections show the possible responses the system can have, i.e. all damped solutions,all period-one oscillations and a bifurcated response. The Multi-Element Probabilistic Collocation is usedwith one element for the first two response types and with two elements when the system has a bifurcatedresponse. To check the response type first two samples are taken at ω = 0.001 and ω = 0.999. If bothsolutions are of the same type, damped or period-one oscillations, no bifurcation is present in the 99.8%domain. Then a single element is used for the propagation of the uncertainty. If one sample shows a dampedsolution and the other a period-one oscillation two elements are used for the uncertainty propagation. Itis shown that a single element Probabilistic Collocation approach results in an oscillatory solution for abifurcated response and how two elements are used to solve this problem. The two elements have to beseparated at the bifurcation point, otherwise the elements that contains the bifurcation point will producea oscillatory solution. Therefore, several search samples are used to estimate the bifurcation point, which isthen used to separate the domain into two elements.

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A. Damped solutions, U = 12

The non-dimensional velocity U is set to 12, such that all responses are damped. For this range the equationsare linear, since the pitch angle remains below the static stall angle of 12. A third order approximationsis used, which means four deterministic solves. The mean of the pitch angle is µα = 6.95 with a varianceof σ2

α = 3.31 · 10−4, this results in a coefficient of variation of the angle of attack is 15%. Due to the lineardependence the coefficient of variation remains the same. Figure 4(a) shows that the cumulative distributionfunction computed by the Probabilistic Collocation method corresponds to 100 Monte Carlo samples. Theprobability density function is shown in figure 4(b).

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Figure 4. Distribution functions of the pitch angle α, (a) the cumulative distribution function Fα(α) and (b)the probability density function fα(α), resulting from a lognormally distributed external torque Mext withµMext

= 8.3 · 10−4 and CVMext= 15% for U = 12.

B. Period-one oscillations, U = 22

The next case is a non-dimensional velocity of U = 22. All realizations result in a period-one oscillation. Thesystem is in the nonlinear regime, since pitch angles above 12 occur, which activate the ∆Cm function. Sinceno bifurcation is present in the response, a single element is sufficient for the Probabilistic Collocation method.The system is nonlinear now, which requires a higher order approximation. A third order approximationis used, so 4 deterministic solves are required. Figures 5(a) and (b) show the distribution functions ofthe minimum and maximum pitch angles during the oscillation. The mean of the minimum pitch angle isµα,min = 6.82 with a variance of σ2

α = 7.81·10−4, this results in a coefficient of variation of the minimum pitchangle is 23.5%. For the maximum pitch angle the mean is µα,max = 14.27 with a variance of σ2

α = 1.49·10−4,so a coefficient of variation of 4.9% for the maximum angle of attack is obtained. Since the input uncertaintyhas a coefficient of variation of 15%, it is increased for the minimum and decreased for the maximum pitchangle. The cumulative distribution function, shown in figure 5(a), matches with the Monte Carlo results.Figure 5(b) shows the probability density function of the minimum and maximum pitch angle.

C. Damped solutions and period-one oscillations, U = 16

The third case, U = 16, has a bifurcated response. The results of a single element computation are shownin figure 6. Due to the presence of the bifurcation the global polynomial approximation of the ProbabilisticCollocation method oscillates. For the maximum pitch angle the oscillations are not that severe, since thegraph is still smooth. For the minimum a strong discontinuous derivative results in a non-smooth solution,which yields an oscillating approximation. Both cumulative distribution functions, figure 6(b), do not matchwith the Monte Carlo results.

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Figure 5. Distribution functions of the pitch angle α, (a) the cumulative distribution function Fα(α) and (b)the probability density function fα(α), resulting from a lognormally distributed external torque Mext withµMext

= 8.3 · 10−4 and CVMext= 15% for U = 22.

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Figure 6. The approximated response (a) of the pitch angle using the Probabilistic Collocation method andthe resulting cumulative distribution functions (b), resulting from a lognormally distributed external torqueMext with µMext

= 8.3 · 10−4 and CVMext= 15% for U = 16.

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D. Multi Element Probabilistic Collocation with search samples

To compute the cases with a bifurcated response, the number of elements is increased. To keep the algorithmas efficient as possible the least amount of elements is used. Two elements suffice for this case, since it wasshown that the damped response and the period-one oscillations are approximated separately. One elementis used for the approximation of the damped solutions and one for the period-one oscillations. This meansthat the two elements have to be separated by the bifurcation point. Since the location of the bifurcationpoint is in general not known in advance a search algorithm is used to find a coarse estimate of the bifurcationpoint.

Search samples

First two samples at ω = 0.001 and 0.999 were used to determine if the bifurcation is present. If there isa bifurcation, then a search algorithm is started. The search algorithm subsequentially halves the part ofthe parameter domain that contains the bifurcation. The third sample will in the middle of the parameterdomain and so forth. The search is stopped when the period-one oscillation with the smallest amplitudeis below a threshold, here set to 0.2 degrees pitch angle. Finally, one additional sample is added for morerobustness and accuracy of estimating the bifurcation point. This extra sample is located between thethird sample, in the middle of the domain and the boundary sample which has the same type of response.Figure 7(a) shows the search samples used for the case with U = 16. The search samples are numbered toshow how the search is performed. The final sample, number 6, here is chosen halfway between sample 3and 2, since they result both in a period-one oscillation.

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(b) Two Element Collocation points

Figure 7. Used samples and collocation points for the uncertainty propagation when the bifurcation is presentsomewhere inside the response domain. The figures show (a) the search samples in the response domain and(b) the collocation points of element I() and II() based on the estimated bifurcation point(∗).

Bifurcated response with Multi-Element Probabilistic Collocation

Figure 8 shows the new results for the distribution functions for a non-dimensional velocity U of 16. Again a3rd order approximation is obtained using two elements both of order 3. For this case a total of 14 samples isused, 6 search samples and 4 samples per element for a 3rd order approximation. So the increase in robustnessof this approach is worth the effort, since 14 samples are sufficient to coincide with the Monte Carlo samples.The mean of the minimum pitch angle is µα,min = 10.62 with a variation of σ2

α,min = 3.38 · 10−4, and for

the maximum µα,max = 11.82 and σ2α,max = 4.14 · 10−4. Both have a coefficient of variation of 9.9%, so the

input uncertainty is damped in the output.

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Figure 8. The distribution functions of the minimum and maximum angle of attack α, (a) the cumulative distri-bution function Fα(α) and (b) the probability density function fα(α), resulting from a lognormally distributedexternal torque Mext with µMext

= 8.3 · 10−4 and CVMext= 15% for U = 16.

E. Bifurcation plot with uncertainty bars

The previous section shows that the Multi-Element Probabilistic Collocation method can produce goodresults for all types of output. Figure 9 shows the bifurcation plot with uncertainty bars arising from anuncertain external torque Mext. A total of 120 deterministic solves are used for the complete computation,including the search samples. The uncertainty bars show the intervals which contain 99.8% of all possiblesolutions, obtained from the distribution function of the minimum and maximum pitch angles. It can be seenthat apart from the region around the bifurcation point the mean is close to the deterministic computation.The uncertainty bars, however, indicate that the solution is sensitive to variations in the external torque.

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Figure 9. Stochastic bifurcation diagram with uncertain Mext. Uncertainty bars show the interval that contains99.8% of all possible values. The deterministic bifurcation diagram is shown by the solid black line.

F. Stochastic properties of the bifurcation point

For engineers the bifurcation point and the influence of uncertainties on the bifurcation point is of greatimportance, since flutter can lead to failure of the structure. Due to the uncertain external torque thebifurcation point becomes a random variable as well. The mean is Ubif,mean = 15.87, with a variance ofUbif,var = 1.37 this results in a coefficient of variation of 7.4%. Figure 10 shows the cumulative distribution

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function and the probability density function of the bifurcation point, with deterministic value indicatedby the dashed line. It shows that there is a probability of 47% that the bifurcation happens before thedeterministically predicted value for an input uncertainty of 15%.

PSfrag replacements

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bif)

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12 13 14 15 16 17 18 19 20 210

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Ubif

fU

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Figure 10. Cumulative distribution function (a) and the probability density function (b) of the bifurcationpoint with uncertain external torque Mext with a coefficient of variation of CVext = 15%. The deterministicvalue of the bifurcation point Ubif,det = 15.765 is indicated by the dashed line (- -).

V. Conclusions

The Multi-Element Probabilistic Collocation method is applied to uncertainty quantification for thedynamic stall flutter analysis of a pitching airfoil. The stochastic bifurcation diagram requires the stochasticproperties of the minimum and maximum pitch angle of the asymptotic state of the system. The minimumand maximum pitch angle are a functional of the pitch angle response resulting from the dynamic stall fluttermodel. Due to the collocational approach functionals of the solution are efficiently and accurately obtained.A standard high order multi element approach would result in an oscillatory approximation in the elementwhere the bifurcation is present. Therefore, two elements are used with the separation of the elements at anestimation of the bifurcation point. Several search samples are used to estimate the bifurcation point. Usingthe two element approach increases the robustness of the method, it shows no oscillations in the solution.

The stochastic bifurcation plot shows that with an uncertain external torque large uncertainty bars arepresent. The mean of the minimum and maximum pitch angle are shown, which shows no clear bifurcationpoint. The mean shows a bifurcation region around the deterministic bifurcation point. Away from thebifurcation point the mean coincides with the deterministic values. The uncertainty bars are, however,relatively large in the entire domain. From the distribution functions of the bifurcation point it can beconcluded that a 15% input uncertainty results to a 7.4% output uncertainty for the bifurcation pointprediction, with 47% probability that the value is lower than the deterministic prediction. These results showthat uncertainty quantification provides valuable information about the bifurcation behavior of a dynamicalsystem and should become standard practice in engineering.

VI. Acknowledgments

The presented work is supported by the NODESIM-CFD project (Non-Deterministic Simulation forCFD based design methodologies); a collaborative project funded by the European Commission, ResearchDirectorate-General in the 6th Framework Programme, under contract AST5-CT-2006-030959.

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dynamic stall flutter analysis with uncertainties using multi-element probabilistic collocation

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