Research Article Smolyak-Grid-Based Flutter Analysis with...

Research ArticleSmolyak-Grid-Based Flutter Analysis withthe Stochastic Aerodynamic Uncertainty

Yuting Dai and Chao Yang

School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

Correspondence should be addressed to Yuting Dai; [email protected]

Received 6 January 2014; Accepted 14 February 2014; Published 23 March 2014

Academic Editor: Guanghui Wen

Copyright © 2014 Y. Dai and C. Yang. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to estimate the stochastic aerodynamic parametric uncertainty on aeroelastic stability is studied in this current work. Theaerodynamic uncertainty is more complicated than the structural one, and it takes more significant effect on the flutter boundary.First, the nominal unsteady aerodynamic influence coefficients were calculated with the doublet lattice method. Based on thisnominal model, the stochastic uncertainty model for unsteady aerodynamic pressure coefficients was constructed with physicalmeaning. Afterwards, the methodology for flutter uncertainty quantification due to aerodynamic perturbation was developed,based on the nonintrusive polynomial chaos expansion theory. In order to enhance the computational efficiency, the integrationalgorithm, namely, Smolyak sparse grids, was employed to calculate the coefficients of the stochastic polynomial basis. Finally,the flutter uncertainty analysis methodology was applied to an aircraft’s wing model. The influence of uncertainty with uniformdistribution for aerodynamic pressure coefficients on flutter boundary was quantified. The numerical results indicate that, theinfluence of unsteady aerodynamic pressure due to the motion of coupling modes takes significant effect on flutter boundary. Itis validated that the flutter uncertainty analysis based on Smolyak sparse grids integration is efficient and accurate for quantifyinginput uncertainty with high dimensions.

1. Introduction

Flutter is an aeroelastic instability phenomenon, whichinvolves the interaction of elastic structures, aerodynamicforce, and inertial force.Thoughwith high fidelity, the definitemodels to describe aeroelastic behavior of real aircrafts andmissiles are so complicated that uncertainties or nonlin-earities are inevitable. Pettit summarized the results andadvances about uncertainty quantification in aeroelasticity[1] extensively. This academic paper is focused on the studyof aeroelastic stability, response, and design problems withuncertainties. Most of the research topics can be classifiedinto two aspects, either probabilistic aeroelasticity or non-probabilistic one. For the nonprobabilistic aeroelasticity, itis of no need to know the probabilistic distributions ofthe uncertainties for input physical parameters and outputaeroelastic response [2]. In addition, the extreme range ofaeroelastic response is paid attention to, namely, “worst case”or “robust” [3]. Sometimes, the robust flutter boundary maybe overconservative as there is “bad experimental data.”

This is why aeroelastic analysis with probabilistic uncertaintycomes into ourmind. For probabilistic aeroelasticity, both theinput and output uncertainties are characterized as randomvariables. This is originated from the areas of structurereliability analysis and system engineering. The Monte CarloSimulation (MCS) [4, 5], the Polynomial Chaos Expansion(PCE) [6], and the Stochastic Collocation [7] methods arethe well-known theories in probabilistic uncertainty quan-tification. However, whether they are applicable in flutteruncertainty quantification is not clear. Generally, among thethreemethods,MCS is regarded as the straightforward choicefor uncertainty quantification of probabilistic aeroelasticity.The influence of uncertainty for structural parameters onflutter boundary was analyzed by Pitt et al. [8]. In particular,he strongly recommended that uncertainty margin shouldbe introduced into aircraft certification process. When weare applying computational fluid dynamics to calculate theunsteady aerodynamics, MCS may increase the computa-tional time rapidly [1, 9]. Hence, the potential of PCEmethodwas tremendous in the field of computational aeroelasticity

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014, Article ID 174927, 8 pageshttp://dx.doi.org/10.1155/2014/174927

2 Discrete Dynamics in Nature and Society

due to its high efficiency. In this method, the uncertaintiesof inputs and outputs of aeroelastic system are projected intoa stochastic space. Pettit et al. successfully applied the PCEmethod to the analysis of aeroelastic limit cycle oscillations[10, 11]. The stochastic basis construction of the PCE methodfor harmonic oscillations is an outstanding contribution inhis research. The advantage and drawback of MCS and PCEwere compared thoroughly by Badcock et al. [12]. However,the present research of stochastic uncertainty quantificationin aeroelasticity is mainly determining the effect of thestructural uncertainty on aeroelastic stability. However, fewof published works are about the uncertainty modelingand quantification of unsteady aerodynamics. Due to theimportance of unsteady aerodynamics on flutter boundary,the effect of stochastic aerodynamic uncertainty on fluttervelocity will be studied and quantified in the present work.First, the stochastic uncertainty model of aerodynamic pres-sure in the frequency domain was constructed. Combinedwith the widely used pk method, the nonintrusive PCEmethod is developed to analyze the influence of aerodynamicuncertainty on flutter boundary. Since the dimension ofaerodynamic uncertainty is moderately large, the compu-tational time of full tensor-product integration increasesexponentially. Hence, the modified numerical integrationmethod, namely, Smolyak sparse grids numerical integrationalgorithm [13], is developed to calculate the formulationof uncertain flutter boundary. Finally, a numerical exampleof aircraft’s wing model is applied to validate the flutteruncertainty quantification framework.

2. Nominal Flutter Solution: pk Method

When there is no external force, the generalized aeroelasticequation of motion without uncertainty can be written as

M0q̈ + C

0q̇ + K

0q − 𝑞

0(𝜌, 𝑉)Q

0(Ma, 𝑘) q = 0. (1)

The above aeroelastic equation is represented under thesystemof themodal basis coordinates. q represents themodalcoordinate vector. M

0, C0, and K

0are the generalized mass,

damping, and stiffness, respectively. Q0is the generalized

unsteady aerodynamic influence coefficients. It is related tothe Mach number Ma and the reduced frequency 𝑘. 𝑞

0=

(1/2)𝜌𝑉

2 represents the dynamic pressure and 𝑉 is the free-stream velocity.

Equation (1) is a characteristic equation essentially. How-ever, it is not so easy for the flutter solution since Q

0is

related to the discrete reduced frequency but not a continuousknown formulation. How to calculate it accurately in thefrequency domain is important to flutter analysis. The pkalgorithm is widely used in flutter solution for engineeringapplications, to solve the above characteristic equation [14].For the algorithm, it is assumed that

q (𝑡) = q𝑒𝑝𝑡, (2)

where 𝑝 = 𝜔(𝛾 + 𝑖) is the complex eigenvalue. As we know,at the flutter velocity condition, the damping is zero andthe aircraft is critically undergoing a harmonic oscillating.

Hence, the unsteady aerodynamic force is also assumed to beoscillating with flutter frequency. Then, the modified flutterequation can be written as

[M0𝑝

2+ C0𝑝 − (

𝑞

0Im (Q

0) 𝑏

𝑘𝑉

)𝑝

+K0− 𝑞

0Re (Q

0) ] q = 0,

(3)

where the reduced frequency is 𝑘 = 𝜔𝑏/𝑉 = (𝑏/𝑉) Im(𝑝) and𝑏 is the half of chord length.

The characteristic equation of (3) can be rewritten in astandard form:

𝑝[

q𝑝q] = A

0[

q𝑝q] , (4)

where

A0=

[

[

0 I

M−10

(𝑞

0Re (Q

0) − K0) M−10

(

𝑞

0Im (Q

0) 𝑏

𝑘𝑉

− C0)

]

]

.

(5)

The procedure of flutter solution with pk algorithm isas follows, given the flight velocity 𝑉and an initial value of𝑘, for each mode, solving the characteristic equation of (4)to obtain an eigenvalue 𝑝. Comparing its imaginary part of𝑝 with 𝑘, iterating the above equation until the discrepancybetween the imaginary part of 𝑝 and 𝑘 is small enough.Then, the frequency and damping are obtained at this givenflight velocity 𝑉. The same procedure is repeated for severalvelocities. The flutter velocity 𝑉

𝑓and frequency 𝜔

𝑓will be

reached with the zero damp at a certain velocity.

3. Stochastic UncertaintyModel for Aerodynamics

In order to solve the flutter problem in (1), the determinationof the unsteady aerodynamic force due to elastic motion in adiscrete frequency domain is essential to us. The linear dou-blet lattice method (DLM) is well accepted by the aerospaceindustry and widely employed in aeroelastic applications tocalculate the unsteady aerodynamics, since it can handle con-figurationswithout extensive time formost subsonic aircrafts.It relates motions of the dynamic down wash to the pressurecoefficients for each aerodynamic panel, based on unsteadylinear small-disturbance equations for fluid dynamics. Threedifferent calculation architectures can describe the unsteadyaerodynamic model in DLM scheme. Those are the modelsfor aerodynamic influence coefficients (AIC matrix), theaerodynamic pressure coefficients (C

𝑝matrix), and the gen-

eralized aerodynamic influence coefficients (GAIC matrix)indicated in (1). In fact, all the three aerodynamic descriptionsshould be transformed to the GAIC form. However, we caninvestigate in more details with the first two descriptions.Nowadays, only the C

𝑝matrix can be measured in the

wind tunnel test. Hence, the nominal aerodynamics togetherwith its uncertainty model should be constructed in the C

𝑝

Discrete Dynamics in Nature and Society 3

architecture. Based on this consideration, the generalizedaerodynamic force (denoted as GAIC) in (1) is rewritten as

Q0(Ma, 𝑘) = Φ𝑇

𝑝SC𝑝, (6)

where Φ𝑝is the modal matrix of the pressure nodes on

the aerodynamic planar elements. S = diag(𝑆1, . . . , 𝑆

𝑛𝐴)

represents the diagonal matrix composed of area for eachplanar element.𝑛

𝐴is the total number of devised aerody-

namic planar elements. C𝑝(Ma, 𝑖𝑘) ∈ R𝑛𝐴×𝑁𝑀 represents

the dynamic pressure coefficients on aerodynamic elementsdue to harmonic modal motion, at given reduced frequency𝑘. 𝑁𝑀

is the number of normal modes for the structuraldynamics.

3.1. Uncertainty Model for Aerodynamic Pressure Coefficients.Considering the numerical error of the unsteady pressurecoefficients C

𝑝(Ma, 𝑖𝑘), (6) can be rewritten as

C𝑝= C𝑝0

+Wcpl × Δcp ×Wcpr, (7)

where theWcpl andWcpr represent the left and right weight-ing matrixes for pressure coefficient uncertainty, respectively.They scale the pressure perturbation such that the stochasticuncertain variables 𝜉

𝑖in Δcp are subject to certain distri-

butions in [−1, 1]. For simplicity, all the uncertainties areindependent and identically and uniformly distributed.

Usually, the weighting matrixes ofWcpl andWcpr shouldbe verified by experiments or accurate data with high fidelitymodel. Afterwards, the generalized aerodynamic force withuncertainty is reformulated as follows:

Q0(Ma, 𝑘) = Φ𝑇

𝑝S (C𝑝0

+Wcpl × Δcp ×Wcpr) . (8)

If the matrix of Δcp is a full nonzero matrix, there will be𝑛

𝐴×𝑁

𝑀stochastic uncertainties in the input parameters. Too

many uncertain variables will increase the complexity of theproblem. In order to reduce the number of uncertain inputvariables, we assume that, for every normal vibration mode,the relative pressure coefficients on different aerodynamicpanels perturb in a uniform manner. By this assumption, theperturbed pressure coefficientΔ𝐶

𝑝(𝑗, 𝑖) on each aerodynamic

panel 𝑗 for every mode 𝑖 can be represented as

Δ𝐶

𝑝(𝑗, 𝑖) = 𝐶

𝑝0(𝑗, 𝑖) × 𝑤

𝑗,𝑖× 𝜉

𝑖. (9)

That is to say, after the scaling of 𝑤, 𝜉cp𝑖 = 1 meansthat the perturbed pressure coefficients on different patches 𝑗and 𝑘 will reach their largest values 𝐶

𝑝0(𝑗, 𝑖)𝑤

𝑗,𝑖, 𝐶𝑝0(𝑘, 𝑖)𝑤

𝑘,𝑖

simultaneously. Under this hypothesis, the number of inde-pendent aerodynamic uncertainties, denoted by 𝑛

𝛿, equals

𝑁

𝑀. Comparing (8) with (9), it is clear thatWcpr = I

𝑁𝑀×𝑁𝑀.

In this case, the left and right weighting matrixes are asfollows:

Wcpl = C𝑝0

× w, Wcpr = I𝑁𝑀×𝑁𝑀

. (10)

With this uncertainty model, it is easy to understandthe physical meaning of aerodynamic perturbation manner.

Moreover, its uncertainty distribution may be measured bythe wind tunnel test. The advantage of introducing thehypothesis is that the uncertainty size is much less than theaerodynamic panels in most general cases. This handling canreduce much time in the numerical calculation. However, thedrawback is that the pressure coefficients on all patches areperturbed in the same manner.

3.2. Analysis of Flutter Uncertainty. Since parametric uncer-tainty exists in the aerodynamic model, the flutter boundarysolved by (1) will also be a random variable. Similar to thenominal harmonic assumption for motions at the fluttervelocity, the random response q(𝜉(𝜃), 𝑡) can also be writtenin an exponential form as follows:

q (𝜉 (𝜃) , 𝑡) = 𝑒

𝑝𝑡

𝑁𝑝

∑

𝑗=1

q𝑗𝑃

𝑗(𝜉 (𝜃)) . (11)

Comparing (2) with (11), the modal motion q(𝜉(𝜃)) in thefrequency domain is written as a series of random polyno-mials, where 𝑃

𝑗(𝜉(𝜃)) is the 𝑗th polynomial basis and 𝜉(𝜃) =

[𝜉

1(𝜃), . . . , 𝜉

𝑁𝑀(𝜃)] is the random input vector. q

𝑗represents

the modal vector of the jth polynomials in the frequencydomain.

Substituting both of the input uncertainties of aerody-namics and output random response of motion into (1), theuncertain flutter equation can be rewritten as

[M0𝑝

2+ C0𝑝 − (

𝑞

0Im (Q

0) 𝑏

𝑘𝑉

)𝑝 + K0

− (

𝑞

0Im (Φ

𝑇

𝑝SWcpl × Δcp ×Wcpr) 𝑏

𝑘𝑉

)𝑝

− 𝑞

0Re (Φ𝑇

𝑝SWcpl × Δ𝑐𝑝 ×Wcpr)

−𝑞

0Re (Q

0) ]P𝑠(𝜉 (𝜃)) q

𝑠= 0,

(12)

where q𝑠

= [q𝑇0, . . . , q𝑇

𝑖, . . . , q𝑇

𝑁𝑝]

𝑇, 𝑖 = 1, 2, . . . , 𝑁

𝑝.

P𝑠= [𝑃

0(𝜉)I𝑁𝑀

, . . . 𝑃

𝑖(𝜉)I𝑁𝑀

, . . . , 𝑃

𝑁𝑝(𝜉)I𝑁𝑀

]. 𝑁𝑝is the total

number of items for stochastic polynomials. With total-orderexpansion strategy for multidimensional uncertainties,𝑁

𝑝is

written as

𝑁

𝑝+ 1 =

(𝑁

𝑀+ 𝑃)!

𝑁

𝑀!𝑃!

, (13)

where 𝑃 is the polynomial order bound for the one-dimensional uncertainty. Comparing (12) with the nominalflutter equation of pk method, the uncertain characteristicequation can be rearranged as

𝑝[

q𝑠

𝑝q𝑠

] = A𝑠[

q𝑠

𝑝q𝑠

] , (14)


where

A𝑠=

[

[

[

0 I

M−10

(𝑞

0Re (Q

0) − K0− 𝑞

0Re (Φ𝑇

𝑝SWcpl × Δcp ×Wcpr)) M−1

0(

𝑞

0Im (Q

0) 𝑏

𝑘𝑉

− (

𝑞

0Im (Φ

𝑇

𝑝SWcpl × Δcp ×Wcpr) 𝑏

𝑘𝑉

) − 𝐶

0)

]

]

]

.

(15)

We can clearly see that matrix A𝑠contains the stochastic

uncertain variables 𝜉(𝜃), different from the nominal A0in

(5). If the stochastic parametric uncertainties are given byrandom sampling, we can also utilize the pk algorithm inSection 1 to calculate the flutter boundary one sample byone sample. It is straightforward but time consuming. Inthis case, the individual flutter velocity is associated with thestochastic parameter’s uncertainty, and the distribution offlutter velocities can be obtained by estimating the statisticmeasurement of all the samples. Regardless of the calculationtime of the sampling-based method, it is appropriate andstraightforward for flutter uncertainty quantification. How-ever, for large engineering problems, the computational timemay be too long to employ. Except for the standard MCSmethod, similar to the stochastic form for motion responseq(𝜉(𝜃), 𝑡), the flutter velocity is also written as a series ofstochastic polynomials; that is,

𝑉

0𝜉=

𝑁𝑝

∑

𝑗=1

𝑉

0𝑗𝑃

𝑗(𝜉 (𝜃)) , (16)

where 𝑉0𝑗is the coefficient of stochastic polynomial basis.

This type of polynomials should be selected accordingto the probability distribution of the input uncertainties.Note that when the stochastic uncertainty satisfies a uniformdistribution located in the range of [−1, 1], the Legendrepolynomials are orthogonal within the probability densityfunction. The formulation of one-dimensional Legendrepolynomials is written as

𝑃

0= 1, 𝑃

1= 𝜉, 𝑃

𝑚+1=

2𝑚 + 1

𝑚 + 1

𝜉𝑃

𝑚−

𝑚

𝑚 + 1

𝑃

𝑚−1.

(17)

The orthogonality property for one-dimensional Legen-dre polynomials is written as

∫

1

−1

𝑃

𝑚𝑃

𝑗

1

2

𝑑𝜉 = 𝐿

𝑗𝛿

𝑚𝑗,

(18)

where𝐿𝑗is 𝑗th constant for orthogonality and 𝛿

𝑚𝑗is theDirac

constant. It is obvious that 𝐿

𝑗is 1 at 𝑚 = 𝑗 and is 0 for

other 𝑚-𝑗 pairs. Multiplication by the 𝑗th polynomial basisin two sides of (16) and integration result in determining𝑉

0𝑗

as follows:

𝑉

0𝑗=

∫

1

−1𝑉

0𝜉𝑃

𝑗𝑑𝜉

𝐿𝑒

𝑗

.

(19)

After the coefficients of polynomial basis 𝑉0𝑗are calcu-

lated, 𝑉0𝑗

is substituted into (16). Hence, the relationship offlutter velocity and aerodynamic uncertainty can explicitlybe represented. Consequently, the influence of aerodynamicparametric uncertainty to aeroelastic stability can be quan-tified, which helps engineers make reasonable decision ofaircraft design.

3.3. Numerical Integration of Smolyak Sparse Grids. In orderto solve the flutter equation of (14) with uncertainty, thecoefficients of polynomial basis in (16) should be estimatedby (19) firstly. In this section, we will develop an algorithmto calculate the numerical integration in the numeratorefficiently.

For discrete function, we have several methods fornumerical integrals with high dimensions. Those includethe sampling-based MCS method, the point collocationmethod, and the full tensor-product numerical integrationone. MCS is to sample the uncertainty within the densityof the weighting function and to calculate the expectationof the response-basis product. It is different from the abovedirect MCS for response. TheMCS here is only employed fordetermining the coefficients of polynomials not to quantifythe flutter uncertainty directly.The point collocationmethod,known as stochastic response surfaces method, solves a leastsquare response equation to best match the response values.It is more feasible than the MCS method since its number ofcollocation points is much less [15]. The extension of tensor-product for one-dimensional quadrature is a direct approachto estimate the multidimensional integrals in (19) for severaluncertainties.

As an extension, the multidimensional integrals by fulltensor-product in (19) is written as

𝑉

0𝑗=

∑

𝑚𝑖1

𝑗1=1⋅ ⋅ ⋅ ∑

𝑚𝑖𝑛

𝑗𝑛=1𝑉

0𝜉𝑃

𝑗(𝜉

𝑖1

𝑗1, . . . , 𝜉

𝑖𝑛

𝑗𝑛) (𝑑

𝑖1

𝑗1⊗ ⋅ ⋅ ⋅ ⊗ 𝑑

𝑖𝑛

𝑗𝑛)

𝐿𝑒

𝑗

,

(20)

where 𝑑

𝑖𝑛

𝑗𝑛is the integration weight coefficient, 𝜉𝑖𝑛

𝑗𝑛is the 𝑗

𝑛th

integration point for the 𝑖𝑛th uncertainty, and𝑚

𝑖𝑛is the total

number of points for 𝑖

𝑛th uncertainty. From (20), it can be

seen that the inner product needs multivariate integration ofthe uncertainties, and the total number for tensor-productfunction evaluations is 𝑚

𝑖1× 𝑚

𝑖2⋅ ⋅ ⋅ × 𝑚

𝑖𝑛. It is feasible for

integrals when the number of uncertainties and the numberof collocation point in each direction are small. Usually, wecan limit the collocation point for each uncertainty to get anacceptable accuracy. However, the uncertainty size is beyondour manipulation. It depends on the specific problem. It is


easy to understand that the number of evaluation pointsgrows exponentially with the size of uncertainties. Hence,the approximations based on tensor-product grids may sufferfrom the curse of dimensionality, which limits its applica-tion to complicated engineering problems. Therefore, thenumerical integration of Smolyak sparse grids is developedto solve “dimension catastrophe.” It is originated from thefull tensor-product, but it reduces the number of integralpoints, while preserving a high accuracy [13]. Motivated bythis advantage, the effective Smolyak sparse grid integrationmethod is developed alongwith flutter analysis algorithm, forhigh dimensional uncertainty quantification.

The Smolyak sparse grid integration is modified accord-ing to the full tensor-product. Note that a given level 𝑙

𝑤is

introduced for different integration points. It is independentof the uncertainty dimension. By the transformation, thenumerical integration formula of Smolyak sparse grids canbe written as [16]

∫

1

−1

𝑉

0𝜉𝑃

𝑗𝑑𝜉 = ∑

𝑙𝑤+1≤|𝑖|≤𝑙𝑤+𝑁𝑀

(−1)

𝑙𝑤+𝑁𝑀−|𝑖|(

𝑁

𝑀− 1

𝑙

𝑤+ 𝑁

𝑀− |𝑖|

)

× 𝑉

0𝜉𝑃

𝑗(𝜉

𝑖1

𝑗1, . . . , 𝜉

𝑖𝑛

𝑗𝑛) (𝑑

𝑖1

𝑗1⊗ ⋅ ⋅ ⋅ ⊗ 𝑑

𝑖𝑛

𝑗𝑛) ,

(21)

where |𝑖| = 𝑖

1+ 𝑖

2+ ⋅ ⋅ ⋅ 𝑖

𝑛is the summation of all the

uncertainty indexes. 𝑉0𝜉

is evaluated with flutter solution ateach integration point. The significant difference of Smolyaksparse grid and full tensor-product is the choice of integrationpoints. Here, the common nonlinear Clenshaw-Curtis inte-gration formula is used for point number at each direction.The number of integration points in one dimension is givenas follows:

𝑚

𝑖= {

1, 𝑖 = 1,

2

𝑖−1+ 1, 𝑖 > 1,

(22)

where 𝑖 is the index.When the collocation point of integrationis the roots of polynomial basis of order 𝑝, the accuracy ofnumerical integration is 2𝑝.

From the comparison of (20) and (21), it can be seen thatthe Smolyak sparse grids integration only accounts for thecollocation point whose index is between 𝑙

𝑤+ 1 and 𝑙

𝑤+𝑁

𝑀.

It can significantly reduce the number of integration pointsto improve the computational efficiency.

4. Numerical Example

An aircraft’s wing model is selected to validate the flutteruncertainty analysis. Its structure and finite element modelare shown in Figure 1. The finite element model is used tocalculate the model’s modal frequencies and modal shapes.Its aerodynamic model is based on the subsonic DLM.Whenthe aerodynamic uncertainty is not considered, the fluttervelocity of the wing model is 36.8m/s and the frequency is5.71Hz by the nominal pk method. The velocity-frequencyplot shows that the coupling of the wing’s second bendingmode and the first torsion mode results in occurrence offlutter phenomenon.

Table 1: Flutter uncertainty by the PCE method.

One-dimensionalpolynomial order 𝑃

Mean valuefor flutter

velocity: m/s

Standard errorfor flutter

velocity: m/s

Velocityrange: m/s

1 36.5 0.7835 [35.0, 38.1]2 36.5 0.7840 [35.1, 38.2]3 36.5 0.7877 [34.9, 38.2]4 36.5 0.7987 [34.8, 38.3]5 36.5 0.8137 [34.5, 38.6]

Afterwards, the stochastic uncertainty for the aerody-namic pressure is considered to quantify flutter uncertainty.Here, the uncertainty level 𝑤

𝑖of 0.1 is given for the aero-

dynamic pressure coefficients. It means that the calculationerror of the aerodynamic force due to themodalmotionsmaybe up to 10% of their nominal values. The four parametricuncertainties are described as random variables with inde-pendently uniform distribution.

First, theMCS is employed to simulate the distribution offlutter velocities [17] numerically. It is regarded as a criterionand a basic method for comparison. 10000 random samplesare simulated on aThinkpad T430 laptop with computationaltime of 4814 seconds. The simulation error for MCS is 1%,approximated with the sampling number [16]. According tothe statistic data of these input and outputs samples, themeanvalue of flutter velocity is 36.5m/s with its standard errorof 0.7905m/s. In this specific random simulation, the fluttervelocity scatters from 35.1m/s to 38.3m/s.

Our objective is to enhance the efficiency of flutteruncertainty analysis. Hence, the PCE approach along withdifferent integration algorithms was applied for comparison.A straightforward PCE approach is to utilize the MCS simu-lation results. Naturally, the stochastic aerodynamic samplesand flutter velocities obtained by the MCS above wereemployed to estimate the coefficients of polynomial basis in(16). Therefore, the flutter uncertainty can be representedexplicitly by the stochastic polynomials, whose stochasticdata is shown in Table 1, compared with different orders ofLegendre polynomials.

From Table 1, we can conclude that the stochastic poly-nomials with only one order are accurate enough to depictthe flutter uncertainty. And the coefficients for polynomialsof high order are not so accurate since the velocity rangein the third column of Table 1 became larger with the orderof polynomials increasing. It infers that the sampling-basedPCE method does not always converge with high orderof polynomials. With one-order stochastic polynomials, thestatistical distribution for flutter velocity is shown in Figure 2.From the comparison of the results by MCS and PCE, thetwo probability density functions agree well with each other.Consequently, in the following, the polynomials of one orderare employed for more detailed investigation.

By means of the PCE method with coefficients estimatedby the MCS, we can write the formulation for flutter uncer-tainty of velocity explicitly as follows:

𝑉

𝑓= 36.5 + 0.17𝜉

1− 0.06𝜉

2− 1.36𝜉

3− 0.02𝜉

4. (23)


(a) (b)

Figure 1: Structure and its finite element model for a wing.

35 35.5 36 36.5 37 37.5 38 38.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

pdf

Direct MCSPCE based on MCS

V (m/s)

Figure 2:The probability density for flutter uncertainty byMCS andPCE.

00.5

10

0.51

35

36

37

38

39

Flut

ter v

eloci

ty

−1−1

−0.5−0.5

Variable1Variable

3

Figure 3: The statistic distribution of flutter velocity.

From the above equation, it is clear that the aerodynamicuncertainty due to the third modal motion, that is, thetorsional mode, has significant effect on the flutter boundary.While the aerodynamic uncertainties of the second and forthmodes have little effect on flutter boundary. This propertycan also be seen from Figures 3 and 4. In Figure 3, theflutter velocity is nearly a surface, which decreases with the

-0.5 0 0.5 135

35.5

36

36.5

37

37.5

38

38.5

Velo

city

(m/s

)

−1

Variable3

Figure 4: The statistic distribution of flutter velocity with theuncertainty of 3rd aerodynamic pressure coefficient.

aerodynamic pressure due to the torsional mode. Therefore,in order to get a high accuracy of flutter boundary, we shouldimprove the accuracy of aerodynamics of key modes, such asthe torsional mode in this example.

The above flutter uncertainty quantification is essentiallybased on random sampling, either by the standard MCSapproach or by the PCEmethod. For this simple wing model,we still need 4814 seconds to complete flutter uncertaintyquantification, let alone a complicated full aircraft model.

In this section, the numerical integration algorithminstead of random sampling is employed to calculate thecoefficients of the flutter polynomial basis in (16). First,the full tensor-product integral is conducted with 6-orderpolynomials. In order to achieve a high accuracy of thecoefficient, the integration points are located at the rootsof the Legendre polynomials. In this case, from (20), therewill be 64 flutter evaluations for the full tensor-productmethod. 1296 eigenvalue solutions were obtained with thecomputational time of 481 seconds. After the coefficients ofpolynomials were calculated, the flutter uncertainty can bewritten as

𝑉

𝑓= 36.5 + 0.19𝜉

1− 0.06𝜉

2− 1.36𝜉

3− 0.03𝜉

4. (24)

From the comparison of (23) and (24), it can be seen thatthe coefficients with the two approaches agree well with eachother. In addition, the computational time (481 s) reducesmuch, comparedwith the results byMCS-based PCEmethod(4841 s). The results indicated that the PCE based on full


tensor-production is more advantageous than the one basedon MCS sampling.

In order to further reduce the computational timefor numerical integration, the Smolyak sparse grids wereemployed to reduce the collocation points. In this wing’sexample, the grid level 𝑙

𝑤= 2was selected. According to (21),

83 collocation points of flutter solution should be calculated.From the roots of Legendre polynomials and Clenshaw-Curtis formula, we first determine the integration pointand weighting coefficients of Smolyak grids. Then, at eachintegration point, the flutter equation will be solved by thepk algorithm, to calculate the individual flutter velocity. Fromthe PCE analysis above, it is known that the polynomials ofone order are enough for a good accuracy. Consequently,the coefficients of polynomials in Smolyak grid method arecalculated, and the uncertainty of flutter velocity is written as

𝑉

𝑓= 36.5 + 0.19𝜉

1− 0.06𝜉

2− 1.36𝜉

3− 0.03𝜉

4. (25)

The flutter uncertainty formula by Smolyak sparse girdis completely the same as the one by the full tensor-productapproach. Notably, the computational time of Smolyak sparsegrid is only 35 seconds. It significantly reduces the computa-tional time while preserving the accuracy.

5. Conclusions

The stochastic uncertainty model for unsteady aerodynamicpressure coefficients is constructed in this paper. Basedon the uncertainty model, the polynomial chaos expansionmethod based on Smolyak sparse grid is developed in theanalysis of flutter uncertainty.The flutter uncertainty formuladue to aerodynamic uncertainty can be obtained with highefficiency. By the comparison of accuracy and efficiency ofthree different approaches, it indicates that the PCE withSmolyak sparse grids is advantageous for flutter uncertaintyquantification. By the validation of a wing’s example, thefollowing is concluded.

(1) It is feasible to construct uncertainty model foraerodynamic pressure coefficients which has physicalmeaning. The aerodynamic accuracy of key modeis vital for flutter boundary. Its calculation accuracyshould be paid much attention to.

(2) The PCE method together with Smolyak sparse gridsis applicable for aeroelastic stability analysis and espe-cially suitable with high dimensional uncertainties.It can improve the efficiency of uncertain aeroelasticanalysis, which is helpful for flutter uncertainty quan-tification of full aircrafts.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (nos. 11302011 and 11172025) and the

Research Fund for the Doctoral Program of Higher Educa-tion of China (no. 20131102120051).

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