An Effective Approach for Uncertain Aerodynamic Analysis ...downloads.hindawi.com/journals/mpe/2020/7417835.pdf · ResearchArticle An Effective Approach for Uncertain Aerodynamic

Research ArticleAn Effective Approach for Uncertain Aerodynamic Analysis ofAirfoils via the Polynomial Chaos Expansion

Xufang Zhang 1 and Jiafei Sun2

1School of Mechanical Engineering amp Automation Northeastern University Shenyang Liaoning 110819 China2BYD Auto Industry Co Ltd Shenzhen Guangdong 518118 China

Correspondence should be addressed to Xufang Zhang zhangxfmailneueducn

Received 19 September 2019 Revised 11 November 2019 Accepted 17 December 2019 Published 26 February 2020

Academic Editor Luca Chiapponi

Copyright copy 2020 Xufang Zhang and Jiafei Sun is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper presents an effective approach for uncertain aerodynamic analysis of airfoils via the polynomial chaos expansion(PCE) To achieve this the multivariate polynomial is first setup to represent random factors within the aerodynamic modelwhereas the expansion coefficient is expressed as the multivariate stochastic integral of the input random vector In this regard thestatistical regression in conjunction with a small number of representative samples is employed to determine the expansioncoefficient en a combination of the PCE surrogate model with brutal-force Monte Carlo simulation allows to determinenumerical results for the uncertain aerodynamic analysis Potential applications of this approach are first illustrated by theuncertainty analysis of the Helmholtz equation with spatially varied wave-number random field and its effectiveness is furtherexamined by the uncertain aerodynamic analysis of the NACA 63-215 airfoil Results for the small regression error and a closeagreement between simulated and benchmark results have confirmed numerical accuracy and efficiency of this approach Ittherefore has a potential to deal with computationally demanding aerodynamical models for the uncertainty analysis

1 Introduction

With the fast development of computer and simulationtechniques numerical methods for the aerodynamic analysisof airfoils have been received considerable attentions duringpast decades [1 2] Various simulation codes based on thefinite difference the finite element and the finite volumewere extensively investigated [3 4] In this regard param-eters of the aerodynamic model ie the geometry dimen-sions and the boundary conditions are usually assumed asdeterministic [5ndash7]

However the uncertainty is universally existed inaerodynamic models of an airfoil [8ndash10] Statistical char-acteristics of the average wind velocity and the turbulenceintensity are necessarily described by using the probabilitydistribution [11] In this regard Ernst et al [12] consideredgeometric uncertainties of rotor blades that arise frommanufacturing tolerances and operational wears and theinput uncertainty indeed leads to a significant scatter of thelift and drag coefficients Zhu and Qiu [13] investigated the

possibility of utilizing perturbation approaches for uncertainaerodynamic loads modelling and their combined effects onthe response interval prediction of lift coefficients

Rather than utilizing the first-order or the Neumannseries to approximate the aerodynamic response functionthe Monte Carlo simulation can deal with the uncertainaerodynamic analysis in a much more straightforwardmanner [14] With a large number of samples to representthe input uncertainty the question of dealing with theuncertain aerodynamic analysis is degenerated as a series ofdeterministic model evaluations [15] is routine can beapplied for the moment analysis the prediction of responsedistribution and the global sensitivity analysis [16 17]erefore combined with the parallel computing technol-ogy results determined by the Monte Carlo simulationmethod is assumed to provide benchmarks for uncertaintyaerodynamic analysis of airfoils in this paper

However the convergence rate of the Monte Carlosimulation (MCS) method is fairly slow ie 1

NMCS

1113968for a

total of NMCS samples used in this simulation [18] is

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7417835 13 pageshttpsdoiorg10115520207417835

implies that a large number of random samples are necessaryfor an accurate estimation result If each round of the de-terministic aerodynamic analysis is relatively time con-suming eg about three minutes the overall uncertainaerodynamic analysis with 104 samples would be about 2083days is motivates an efficient approximation method forthe uncertainty aerodynamic analysis in this paper

Besides the brutal-force random sampling methodseveral improved versions of the MCS based on low-dis-crepancy points [19] the Quasi Monte Carlo method [20]and the active-learning approach [21 22] were alternativelydeveloped It can improve numerical efficiency of the ran-dom simulation method to some extents However thispaper specially focuses on the polynomial chaos expansionmethod which is able to provide a global approximationresult of the mechanistic model for the uncertain aerody-namic analysis of airfoils

e ideal of the polynomial chaos expansion wasinitially introduced by Wiener [23] to investigate theGaussian turbulence models Li and Ghanem [24] furtherinvestigated the utility of the chaos expansion approachfor the stochastic finite element analysis of various solidmechanics models [25] Combined with the Askeyscheme Xiu and Karniadakis [26] extended this expansionapproach to deal with continuous random variables Toachieve this the basis functions for input random vari-ables are first constructed by using the chaotic rule ofmarginal orthogonal polynomials [27] is allows torepresent a multivariate model with the generalizedFourier series expansion [28] Once the PCE model for anairfoil model is determined the Gaussian quadratureapproach can be used to deal with multivariate integralson the expansion coefficient [29ndash31] is results in theGalerkin projection [32] the stochastic collocation[33ndash35] and the statistical regression methods [36] in theliterature

In general numerical algorithms for this PCE model aredivided as the intrusive and nonintrusive approaches efirst category requires to continuously revise the deter-ministic aerodynamic model which is time consuming anddifficult for complex simulation models [37] In the contrastthe nonintrusive approach treats the deterministic aerody-namic analysis as a black-box function and representsrandom variables as a limited number of deterministicvalues After substituting for representative values of arandom variable statistical characteristics of the uncertainaerodynamic response can be obtained based on a simplepostprocessing procedure in terms of some deterministicanalyses

Since a large number of aerodynamic analyses areembedded in the brutal-force Monte Carlo simulationthis generally results in computationally demanding coste PCE approach was combined with the Gaussianquadrature scheme and the dimensional reductionmethod [38] However the standard Gaussian methodmight result in the problem of the curse of dimensionality[39] which motives the nonintrusive and the statisticalregression method to determine expansion coefficients inthis paper

With low-discrepancy samples generated based on theSobolrsquo or the Latin Hypercube scheme the estimation ofPCE coefficients can be realized by the generalized statisticalregression procedure as shown in numerical examples Eventhough sparse regression algorithms based on the orthog-onal matching pursuit [40] the backward-and-forward se-lection and the least-angle regression [41] can bealternatively realized the paper specially focuses on theutility of the full PCE model and the standard regressionapproach due to its robustness

To summarize an objective of this paper is to present aneffective approach for uncertainty aerodynamic analysis ofairfoil models via the polynomial chaos expansion A sta-tistical regression method in conjunction with the low-discrepancy samples is employed to determine the PCEcoefficient To demonstrate potential applications of thismethod examples in the literature are presented to examineits numerical performance

e rest of the manuscript is organized as followsSection 2 briefly summarizes the method of the polynomialchaos expansion for the uncertain aerodynamic analysis ofan airfoil and a statistical regression approach is presentedin Section 23 to determine the expansion coefficient Section3 illustrates an application of this method for uncertainanalysis of the Helmholtz equation with a spatially varyingwave-number model Results for uncertain aerodynamicanalysis of the NACA 63-215 airfoil are discussed in Section4 and conclusions are summarized in Section 5

2 Uncertainty Analysis via the PolynomialExpansion Method

e aerodynamic response of an airfoil (eg the pressureand the velocity field and the lift or the drag coefficient)would become stochastic if input random variables areconsidered in the model function η(X u v w) Hereinu v andw represent three physical dimensions of theaerodynamic model η(middot) whereas the random vector X

[X1 middot middot middot Xd]T consists of all uncertain factors Examples ofthe random variable include the average wind speed the airdensity and the viscosity parameters as shown in numericalexamples To account for this input uncertainty a regres-sion-based polynomial chaos expansionmethod is presentedas follows

21 AReview of the Polynomial Chaos Expansion To developthe polynomial chaos expansion model for the aerodynamicanalysis we first define an index vector α [α1 middot middot middot αd]T

with each integer αi isin [0 p] en a set of chaos polyno-mials defined by utilizing the highest order-parameter p andd-dimensional random vector X are expressed as

ϕi(X) i 0 n minus 11113864 1113865 ≔ cup|α|lep

1113945

d

k1φαk

Xk( 1113857 (1)

Herein Xk is kth input random variable whereasφαk

(Xk) denotes αk th-order orthogonal polynomials that isused to represent the random variable Xk

2 Mathematical Problems in Engineering

Specially the order of a d-variate polynomial is definedas p 1113936

dk1αk and the number of elements within the

polynomial set ϕi(X)1113864 1113865nminus 1i0 is [24]

n

d + p

p

⎛⎝ ⎞⎠ (d + p)

dp (2)

Besides arbitrary two elements within the polynomialset ϕi(X) i 0 N minus 11113864 1113865 are orthogonally definedis islangϕi(x) ϕj(x)rang δij (as i j middot middot middot n) where δij denotes theKronecker delta symbol erefore the chaotic polynomialset ϕi(X) i 0 1 n minus 11113864 1113865 constitute a complete set torepresent a d-dimensional real-valued space whenn⟶ +infin and the structural model response can bespectrally represented as

η(X u v w) 1113944infin

i0ai(u v w)ϕi(X) (3)

Herein ai(u v w) denotes deterministic expansioncoefficients

ai(u v w) 1113946Xϕi(x)η(X u v w)fX(x)dx i 0 n minus 1

(4)

Herein fX(x) denotes the joint probability densityfunction the input random vector X

Since the PCE model contains infinite terms in thedefinition the approximation for an aerodynamic responsecan be numerically realized by truncating the expressionstarting an nth term

1113954η(X u v w) 1113944nminus 1


which is used to approximate the true mechanistic model innumerical simulations

e normalization constant of Jacobi polynomialsCJacobi (2a+b+12i + a + b + 1) (Γ(i + a + 1)Γ(i + b + 1)Γ(i + a + b + 1)i)

Specially the orthogonal chaos polynomial dependslargely on the probability distribution of input randomvariables Table 1 summarizes the relation between theprobability distribution of random variables and the or-thogonal polynomial used to develop the PCE model Oncethe spacial-dependent coefficients in equation (4) are nu-merically determined uncertain response of the aerody-namic model η(X u v w) can be theoretically representedby the truncated PCE model erefore numerical ap-proaches for the calculation of the PCE coefficient arepresented as follows

22 5e Calculation of PCE Coefficients e key issue torealize the PCE method for the uncertainty quantification isto determine expansion coefficients ak(u v w)

(k 0 1 middot middot middot n minus 1) in equation (5) which are generallymodelled by using the following optimization problem

Find 1113954a0(u v w) 1113954a1(u v w) middot middot middot 1113954anminus 1(u v w)1113864 1113865

Minimize η x(k)( 1113857 minus 1113936Nminus 1i0 1113954ai(u v w)ϕi x(k)( 1113857

q

⎧⎨

⎩

(6)

Herein x(k) represents the kth realization of the inputrandom vector X whereas middotq denotes a q-norm operatorwith respect to totally n0 residual terms of η(middot) as referring tothe PCE approximation result 1113954η(middot)

If the parameter q 2 and n0 n the probabilisticoptimization problem directly leads to an n-order algebraicsystem in term of the PCE coefficients

1113898 η(X u v w) minus 1113944nminus 1

k0ak(u v w)lang ϕk(X)⎡⎣ ⎤⎦ϕi(X)1113899 0

i 0 n minus 1

(7)

which can be re-expressed as

langη(X u v w)ϕi(X)rang 1113944nminus 1

k0ak(u v w)langϕk(X) ϕi(X)rang

i 0 n minus 1

(8)

Considering that langϕk(X) ϕi(X)rang δik asi k 0 n minus 1 this finally derives totally n d-dimensionalintegrals as expressed in equation (4)

e brutal-forceMCSmethod and its optimized versionshave been used in the literature to deal with the high-di-mensional integration Results for theMCS based estimationof the expansion coefficient are

1113954ai(u v w) asymp1

NMCS1113944

NMCS

j1η x(j)

u v w1113872 1113873ϕi x(j)1113872 1113873

i 0 n minus 1

(9)

where the symbol x(i) represents the ith realization of theinput random vector X and NMCS represents the totalnumber of samples used in the Monte Carlo simulation

Note that numerical realization of such large number ofhigh-dimensional integrals is a computationally intensivetask given that the deterministic aerodynamic responseη(X u v w) is possibly spatially modelled based on a finiteelement scheme In this regard the brutal-force MCSmethod is replaced with the statistical regression method asfollows

23 5e Statistical Regression Method Denote thatζ x(1) middot middot middot x(m)1113864 1113865 consists of m realizations of the inputrandom vector X generated based on a quasi-simulationmethod in conjunction with a low-discrepancy sequence(such as the Sobolrsquo Halton or Hammersley) Correspondingnumerical realizations of the chaos polynomial set ϕi(x)1113864 1113865

nminus 1i0

would be

Mathematical Problems in Engineering 3

ξ

ϕ0 x(1)( 1113857 ϕ1 x(1)( 1113857 middot middot middot ϕnminus 1 x(1)( 1113857


⋮ ⋮ ⋱ ⋮

ϕ0 x(m)( 1113857 ϕ1 x(m)( 1113857 middot middot middot ϕnminus 1 x(m)( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(10)

en a regression model can be obtained as1113954Y(u v w) Ξ1113954a(u v w) + ϵ(u v w) (11)

is is an n-dimensional vector 1113954a(u v w)

[1113954a0(u v w) 1113954a1(u v w) 1113954anminus 1(u v w)]T which repre-sents the regression coefficients that are attached to ex-planatory variables Ξ [ϕ0(X) ϕ1(X) ϕnminus 1(X)]whereas the symbol ϵ(u v w) denotes residual errors

ϵ(u v w) y(u v w) minus ξ1113954a(u v w) (12)

which consists of residual errors for all spatial and time reali-zations of the model Following the theory of the multivariateregression the unknown regression coefficients 1113954a in equation(11) can be expressed in terms of the training data set ξ andresponse sample y as

1113954a(u v w) ξTξ1113872 1113873minus 1ξTy(u v w) (13)

Once the training matrix ξ is defined in equation (10)andthe corresponding aerodynamic response samples y areavailable the minimization of the Euclidian-norm-based re-sidual errors ϵ(i) (with i 1 middot middot middot m) allows finally deriving theregression-based surrogate model noted in equation (11)which will be useful to mimic the true but computationallyintensive model η(middot) in the afterwards uncertainty simulations

24 Uncertainty Analysis Based on the PCE Surrogate ModelOnce a surrogate model 1113954η(X u v w) is obtained eitheranalytically or numerically the mean value and the varianceof the aerodynamic response can be approximated as

ıCD[η(X u vw)] asymp ıCD 1113954a0(u vw) + 1113944nminus 1

i11113954ai(u vw)ϕi(X)⎡⎣ ⎤⎦

1113954a0(u v w)

Var[η(X u v w)] asymp Var 1113954a0(u v w) + 1113936nminus 1

i11113954ai(u v w)ϕk(X)1113890 1113891

1113936nminus 1

i11113954a2

i (u v w)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

Besides the Sobolrsquo index used for the variance-basedglobal sensitivity analysis can be easily expressed bymeans of

the PCE coefficients [42] To implement an index set ofrepresenting the ith variance component Vi uniquely con-tributed by the random variable Xi is first defined as

β1113864 1113865i arg α cup|α|lep

αi ne 01113858 1113859cap 1113858αj 01113859d

j1jnei1113882 11138831113896 1113897 (15)

Similarly a q-dimensional index set q i1 middot middot middot iq1113966 1113967 thatmeasures the joint variance effect due to multiple inputrandom variables Xi1

middot middot middot Xiq1113882 1113883 acting together is presented as

β1113864 1113865q arg α cup|α|lep

αk ne 01113858 1113859forallk isin q1113864 1113865cap αk 01113858 1113859forallk notin q1113864 11138651113896 1113897

(16)

Based on abovementioned notations results for themainand the joint Sobolrsquo sensitivity indices for the variance-basedglobal sensitivity analysis can be readily expressed as

Si(u v w) 1113936αisin β i

a2α(u v w)

1113936nminus 1k1a

2k(u v w)

i 1 2 d

Sq(u v w) 1113936αisin β q

a2α(u v w)

1113936nminus 1k1a

2k(u v w)

1le i1 iq le d

(17)

Note that the parameter aα(u v w) denotes the PCEcoefficient attached to a d-dimensional polynomial ϕα(X)erefore results for Sobolrsquo sensitivity index are purelyexpressed in terms of the PCE coefficients

To summarize the proposed approach for uncertainaerodynamic analysis of an airfoil includes (1) the developmentof the chaos polynomial set ϕi(X)1113864 1113865

nminus 1i0 in equations (1) and (2)

the calculation of PCE coefficients by using the statistical re-gression method in equations (3) and (13) the sensitivityanalysis and the response distribution estimation in Section 24erefore in conjunction with a small number of low-dis-crepancy simulations and a simple postprocessing procedurethe uncertainty analysis of an airfoil model can be effectivelyrealized via the polynomial chaos expansion method

3 Uncertainty Analysis of theHelmholtz Equation

is section illustrates an application of the polynomialchaos expansion approach for the uncertainty analysis of thetwo-dimensional Helmholtz equation which is generallydefined as

Table 1 Distribution of random variables and orthogonal polynomials [26]

Polynomial Weight function Domain Orthogonality DistributionHermit exp(minus x22) (minus infin +infin) langψiψrangj

2π

radiciδij Normal

Legendre 1 [minus 1 1] langψiψjrang (22i + 1)δij UniformJacobi (1 minus x)a(1 + x)b [minus 1 1] langψiψjrang CJacobiδij Beta

Laguerre exp(minus x) [0 +infin) langψiψjrang (Γ(i + 1)i)δij Exponential

General Laguerre Γ(x a + 1 1) [0 +infin) langψiψjrang i + a

i1113888 1113889δij Gamma


minusy2(u v)

zu2 +y2(u v)

zv21113888 1113889 + ω(u v)y(u v) g0(u v)

Ω isin [a b] times[c d]

y(u v)|Γ1 g1(u v)

zy(uv)

zn

11138681113868111386811138681113868 Γ2 g2(u v)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

where Γ1 represents the Dirichlet boundary condition and Γ2denotes the Neumann boundary condition For uncertaintyanalysis the wave-number function ω(u v θ) is molded byusing the Gaussian random field is implies the responsequantity of this equation is spatially stochastic asu v isin [a b] times [c d]

31 5e KarhunenndashLoeve Expansion for Spatial VariabilitiesAccording to the theory of a random field simulation thespatially varying two-dimensional random propertyω(u v θ) can be effectively represented by means of theKarhunenndashLoeve (K-L) expansion [43] and its variants [44]

H(u v θ) μ(u v) + 1113944infin

i1

λi

1113969

Xi(θ)ϕi(u v) (19)

Herein μ(u v) denotes the mean value of the randomfield whereas the statistically Gaussian random variablesXi(θ) in this example can be obtained by

Xi(θ) 1λi

1113968 1113946Ω

1113946Ω

[H(u v θ) minus μ(u v θ)]ψi(u v)du dv

(20)

where λi and ϕi(u v) denote eigenvalues and eigenfunctionsfor the covariance function Cov(u v) respectively whichare solutions of the following integral eigenvalue problem(IEVP)

1113946Ω

1113946ΩCov(u v)ψi uprime vprime( 1113857duprimedvprime λiψi(u v) (21)

Once a numerical solution of the integral integration isavailable numerical realization of the random field based onan m-term K-L expansion is given as

1113957H(u v θ) μ(u v) + 1113944M

i1

λi

1113969

ψi(u v)Xi(θ) (22)

In numerical simulations the following exponentialmodel is assumed for the spatially varying wave-numberfunction

Cov(u v) σ20exp minusu minus uprime

11138681113868111386811138681113868111386811138681113868

δuminus

v minus vprime1113868111386811138681113868

1113868111386811138681113868

δv1113888 1113889 as u v isin Ω

(23)

Figure 1 presents simulation results for random field ofthe wave-number function ω(u v θ) based on 104 randomsamples Herein parameters in the covariance function aregiven as σ0 π δu 08 and δv 1 With the truncationparameter M 5 the error for the variance simulationresult in Figure 1(c) is determined as 415 which impliesthe simulation variance is relatively lower than that of thebenchmark variance π

32 Numerical Results for the Uncertainty Analysis Withnumerical simulation results for random field of the wave-number function the procedure summarized in Section24 is used for uncertainty analysis of the Helmholtzequation e multivariate Hermit polynomials are firstdetermined to represent the Gaussian random variablesproduced by the K-L expansion and the PCE model forstochastic response of the Helmholtz equation is deter-mined as

1113954η(X u v) 1113944nminus 1

k0ak(u v)ϕk(X) (24)

where the random vector is defined as X [X1 X5]T

and Xi is the standard Normal random variable determinedby the K-L expansion in equation (20)

Assume that the highest order of the chaotic poly-nomial is p 6 is implies the total number of poly-nomial terms in equation (24) will be 462 If the Gaussianquadrature method used to evaluate the PCE coefficientsit involves 7776(65) functional evaluations in totalHerein each random variable is represented by sixGaussianndashHermit nodes To reduce the computationalcost for multivariate cases the statisticalregression method in Section 23 with 500 samples werealternatively used and the corresponding error term isdetermined to evaluate numerical accuracy of this PCEmodel

ε2(u v) 1113936

NMCSi1 η x(i) u v( 1113857 minus 1113954η x(i) u v( 11138571113960 1113961

2

1113936

NMCS

i1η x(i)

u v1113872 1113873 minus Y(u v)1113960 11139612 (25)

where η(middot) and 1113954η(middot) represent the simulated and the predictedmodel responses respectively whereas Y(u v) denotes themean value of the model response ie

Y(u v) (1NMCS)1113936NMCSi1 η(x(i) u v)erefore an indicator

R2 is defined as R2(u v) 1 minus ε2(u v) in the followingsimulations to quantify numerical accuracy of the PCE model


minus04 minus02 0 02 04u


minus04 minus02 0 02 04minus04

minus02

0

02

04

u

ν


35π

4π

45π

5π

55π

6π

65π

(a)

ν

uminus04 minus02 0 02 04

minus04

minus02

0

02

04

4985π

5π

5015π

(b)

ν

minus04

minus02

0

02

04


094π

095π

096π

097π

098π

099π

(c)

Figure 1 Results for Monte Carlo simulation of the two-dimensional random field in terms of the wave-number function (mean-valuefunction μ(u v) 5π and the sample size is 104) (a) Four realizations for random field of the wave-number function ω(u v θ) (b) Meanvalue (c) Standard deviation

minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(a)

minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(b)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(c)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(d)

Figure 2 Mean value and standard deviation for stochastic response of the Helmholtz equation (a) Mean value brutal-force simulation (b)Mean value the PCE method (c) Standard deviation brutal-force simulation (d) Standard deviation the PCE method


Figure 2 depicts results for themean value and the standarddeviation for stochastic responses of the Helmholtz equationWith the spatially varying random property for the wave-number function ω(u v θ) results determined by the brutal-force MCS with 104 samples are provided for numericalverifications e proposed regression algorithm with 500 low-discrepancy training samples is able to determine accuratesimulation results for the PCEmodelemaximal fitting errorresult is determined as maxuvisinΩ ε2(u v)1113864 1113865 1355 times 10minus 3esmall regression error has verified numerical accuracy of thisapproach in developing the PCEmodel for uncertainty analysisof the Helmholtz equation

Given the PCE model for the system response Figure 3further presents simulation results of Y(X u v) for somelocations within the simulation domain Ω [minus 04 04] times

[minus 04 04] Note that 104 samples were used to determine thebenchmark result A close agreement of the simulation and the

predicted results has confirmed the effectiveness of this PCEmodel for uncertain analysis of the Helmholtz equation withspatially varying wave-number property

4 UncertainAerodynamicAnalysisof theNACA63-215 Airfoil

is section considers uncertain aerodynamic analysis of theNACA 63-215 airfoil via the polynomial chaos expansionapproach To achieve this the deterministic aerodynamicsimulation is first realized based on a finite element model inconjunction with the SpalartndashAllmaras (S-A) turbulencemodel In this simulation the angle of attack was consideredfrom minus 4∘ to 18∘ with an incremental step 05∘

Together with results provided by the wind tunnel test[45] and XFoil computational package [46] the applicabilityof the simulation model is first verified Note that the

minus08 minus07 minus06 minus05 minus04 minus03 minus020

1

2

3

4

5

6

Uncertain response Y (X u v)

PDF

minus01

PCEMCS

(a)

0

1

2

3

4

PDF


PCEMCS

minus02 0 02 06 0804

(b)

065 07 075 080

5

10

15

20

25

30

35

PDF


PCEMCS

(c)

Uncertain response Y (X u v)minus08 minus07 minus06 minus05 minus04 minus03 minus020

1

2

3

4

5

6

PDF

minus01

PCEMCS

(d)

Figure 3 Verifications for output probability distribution of the Helmholtz equation (a) Position A (minus 0268 0297) (b) Position B (minus 02970297) (c) Position C (minus 0268 0268) (d) Position D (minus 0297 0268)


following simulation parameters are considered the windvelocity V 16ms the air density ρ 1225 kgm3 and theviscosity parameter 17894 times 10minus 5 kgmmiddots

Figure 4 presents the simulation results for the liftingand the drag coefficients of this airfoil It is observed thatresults for the lift coefficient provided by the FEM with theS-A turbulence model are closely agreed with the resultsprovide by the XFoil and the experimental data Comparedwith the wind tunnel test data the drag coefficient resultpredicted by the S-A is more accurate than that of the XFoilerefore the uncertain aerodynamic quantification of theNACA 63-215 airfoil is implemented based on the S-Amodel for the pressure and the velocity response fields Oncea numerical simulation model of the airfoil is available thecorresponding uncertainty analysis can be realized via thepolynomial chaos expansion approach Note that themaximal lift-drag ratio is determined as AOAlowast 95∘ in thisdeterministic simulation and the PCE approach is furtherapplied to provide estimation results for the probabilitydistribution of the drag the lift and the life-to-drag ratio atthis AOA value as follows

41 Results for the Uncertainty Analysis To implement theuncertain aerodynamic analysis the probabilistic charac-teristics of input random factors are listed in Table 2 isincludes the inflow velocity Vinfin the air density ρ and theviscosity parameter ν e PCE algorithm is realized for theuncertain analysis and the brutal-force Monte Carlo sim-ulation with 104 samples is assumed to provide thebenchmark result for the numerical verification

At first the PCE models for the coefficient of pressureCp(X uprime vprime) and the pressure response of the simulationfield are separately developed

1113954Cp(X u v) 1113944dminus 1

k0aCpk(u v)ϕk(X)

1113954P(X u v) 1113944dminus 1

k0aPk(u v)ϕk(X)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

in which the chaotic polynomial terms ϕk(X) are expressedas the tensor product of the Hermit polynomials to representnormal random variables in Table 2 With 500 samplesgenerated based on the LSH approach the statistical re-gression method in Section 23 is used to determine the PCEcoefficient

In this PCE model the polynomial order parameter isgiven as p 3 is determines totally 20 terms for nu-merical representation of the uncertain aerodynamic re-sponse Following the moment results based on the PCEcoefficients in equation (14) the first-two orders of momentsfor the aerodynamic responses are presented in Figure 5 Aclose agreement between the simulation and the predictedmoment results has confirmed the high accuracy of this PCEmethod for the pressure coefficient Cp of this airfoil

Results for the mean value and the standard deviationof this pressure response field are determined as shown

Angle of attack

ndash05

0

05

1

15Li

fting

coef

ficie

nt (C

l)

S-A modelXfoil resultExperimental data

ndash5 0 5 10 15 20

(a)


0

005

01

015

02

Dra

g co

effic

ient

(Cd)

Angle of attackndash5 0 5 10 15 20

(b)

Figure 4 Results for aerodynamic characteristic of the NACA 63-215 airfoil determined by various turbulence models (a) Coefficients oflift (b) Coefficients of drag

Table 2 Random variables for uncertainty aerodynamic analysis ofthe NACA 63-215 airfoil

Random variables Symbol Distribution Mean COVWind velocity (ms) Vinfin Gaussian 20 005Air density (kgm3) ρ Gaussian 1225 005Viscosity parameter(cP) ν Gaussian 17894 times 10minus 2 005


0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)

Figure 5 Numerical results for the mean and the standard deviation of Cp at the airfoil surface (a) e mean value (b) e standarddeviation

u

ν

minus05 0 05 1 15minus06

minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)

Figure 6 Results for the mean value and the standard deviation of the pressure response field (a) Mean value the Monte Carlo simulation(b) Mean value the PCE method (c) Standard deviation the MCS method (d) Standard deviation the PCE method


u

ν

(minus00492 minus00485)

(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D

Figure 7 Reference positions for uncertainty analysis of the response pressure field

40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)

minus120 minus100 minus80 minus60 minus400

001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)

Figure 8 Results for probability distribution of the pressure response at four positions (a) Position A (minus 00492 minus 00485) (b) Position B(06394 01428) (c) Position C (11756 minus 00708) (d) Position D (05307 minus 03557)


in Figure 6 Note that results for all simulation pointswithin the investigated domain were determined eregression parameters ak(u v) are spatially dependentCombined with benchmark result provided based on 104samples the proposed approach is fairly efficient fornumerical estimation of the moment of the responserandom filed

In this surrogate model deterministic simulations werenumerically realized based on a personal computer with theCPU of Interl(R)Core(TM)i7 minus 3770340GHz and the 4GBphysical memories In addition four-thread parallel com-puting technique was implemented to accelerate the sim-ulation process Each round of the deterministic simulationapproximately needs 3 seconds erefore with 500 samplesto develop the PCE surrogate model the total simulationtime is around three minutes in total However the brutal-force Monte Carlo simulation with 104 samples requires 95hours is has demonstrated the high efficiency of thisproposed approach for uncertain aerodynamic analysis ofthe NACA 63-215 airfoil

Numerical verification of the PCE approach is furtherextended to the probability distribution for pressure re-sponses at four locations as shown in Figure 7 is includesthe point A(minus 00492 minus 00485) the point B(06394 01428)the point C(11756 minus 00708) and the pointD(05307 minus 03557) Results for the probability distribution ofthe pressure are summarized in Figure 8 Note that results forthe accuracy measure in equation (25) are generally less than60 times 10minus 8 Together with accuracy simulation results forprobability distribution of the lift and drag coefficients inFigure 9 for the case of AOAlowast 95∘ it has confirmed theeffectiveness of utilizing this polynomial chaos expansion forstochastic aerodynamic analysis of airfoils

42 Results for the Global Sensitivity Analysis e sectionfurther determines results for the global sensitivity anal-ysis of the NACA 63-215 airfoil with random variables interms of the inflow velocity Vinfin the air density ρ and thedynamic viscosity To achieve this surrogate models forCl Cd and Cld are separately determined In this regard

the regression error are determined as 314 times 10minus 6326 times 10minus 7 and 228 times 10minus 7 for Cl Cd and Cl d respec-tively e small regression errors (le10minus 5) for all PCEmodels have confirmed the high accuracy of this proposedapproach

With the polynomial order parameter p 3 the chaoticHermit polynomials are used to develop the PCE surrogatemodel for aerodynamic response of the NACA 63-215airfoil e statistical regression method with 500 Sobolrsquosamples was used to determine the expansion coefficientBesides the brutal-force Monte Carlo simulation with 104samples was used to provide benchmark results for thisglobal sensitivity analysis

Based on the PCE coefficients determined with thestatistical regression method Table 3 summarizes resultsfor the global sensitivity analysis of this uncertain aero-dynamic model with random variables in terms of the Vinfinthe ρ and the dynamic viscosity parameter ] It is ob-served that these three random factors have almostcontributed equally for total aerodynamic variance of ClCd and Cl d whereas the parameter of the wind velocityVinfin has been ranked as the principle factor given thelargest value of the global sensitivity result is is fol-lowed by the air density ρ and the dynamic viscosityparameter ] Specially results for the total sensitivityindex STi

are slightly larger than those of the primarysensitivity index Si given that the total variance includesjoint variance contributions of Xi and Xj together Tosummarize the proposed PCE surrogate model canprovide reliable estimation results for the global sensi-tivity analysis of the NACA 63-215 airfoil

PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)

Figure 9 Results for probability distribution of (a) the lift coefficient (b) the drag coefficient and (c) the ratio of lift-to-drag

Table 3 Results for sensitivity analysis of the airfoil model

Sensitivity Cl Cd Cld

S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions

e paper presents an effective approach for uncertainaerodynamic analysis of airfoils via the polynomial chaosexpansion To archive this the spatially varied randomproperties of a partial differential equation are first resentedas the normal random variables based on the K-L expansionmethod en a set of basis functions are determined byusing the chaotic orthogonal polynomials is allows todeveloping a statistical regression model for the uncertainaerodynamic analysis via the polynomial chaos expansionapproach In numerical simulations the Helmholtz equationwith a spatially varied wave-number function is consideredWith the truncated K-L expansion scheme to represent theinput random field the polynomial chaos expansion methodis able to provide reliable estimation results for the statisticalmoment and the probability distribution of the model re-sponse is verification is further extended to the uncertainaerodynamic analysis of the NACA 63-215 airfoil which ismodelled by using the normal random variables e de-terministic aerodynamic simulation is realized by using thefinite element method in conjunction with the Spa-lartndashAllmaras turbulence model With a small number ofdeterministic model evaluations results for the responsepressure and velocity fields of this airfoil are determinedCompared to benchmark results provided by the brutal-force Monte Carlo simulation it has confirmed the highaccuracy of this method for the global sensitivity analysis ofaerodynamic models e close agreement between thesimulation and the predicted results for the lift the drag andthe lift-to-drag ratio has confirmed the effectiveness of thisapproach for uncertain aerodynamic analysis of airfoils

Data Availability

e simulation data within this submission are availablebased on the request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors would like to appreciate the National NaturalScience Foundation of China (Grant numbers 51775095 and51605083) and the Fundamental Research Funds for CentralUniversities (N180703018 and N170308028) for financiallysupporting the research

References

[1] A Chehouri R Younes A Ilinca and J Perron ldquoReview ofperformance optimization techniques applied to wind tur-binesrdquo Applied Energy vol 142 pp 361ndash388 2015

[2] R Barrett and A Ning ldquoComparison of airfoil pre-computational analysis methods for optimization of windturbine bladesrdquo IEEE Transactions on Sustainable Energyvol 7 no 3 pp 1081ndash1088 2016

[3] P J Schubel and R J Crossley ldquoWind turbine blade designrdquoEnergies vol 5 no 9 pp 3425ndash3449 2012

[4] B Li H Ma X Yu J Zeng X Guo and B Wen ldquoNonlinearvibration and dynamic stability analysis of rotor-blade systemwith nonlinear supportsrdquo Archive of Applied Mechanicsvol 89 no 7 pp 1375ndash1402 2019

[5] A F P Ribeiro A M Awruch and H M Gomes ldquoAn airfoiloptimization technique for wind turbinesrdquo Applied Mathe-matical Modelling vol 36 no 10 pp 4898ndash4907 2012

[6] Y Liu J Han Z Xue Y Zhang and Q Yang ldquoStructuralvibrations and acoustic radiation of bladendashshaftingndashshellcoupled systemrdquo Journal of Sound and Vibration vol 463Article ID 114961 2019

[7] S Yin D Yu Z Ma and B Xia ldquoA unified model approachfor probability response analysis of structure-acoustic systemwith random and epistemic uncertaintiesrdquo Mechanical Sys-tems and Signal Processing vol 111 pp 509ndash528 2018

[8] H S Toft L Svenningsen W Moser J D Soslashrensen andM L oslashgersen ldquoAssessment of wind turbine structuralintegrity using response surface methodologyrdquo EngineeringStructures vol 106 pp 471ndash483 2016

[9] H Ma J Zeng R Feng X Pang Q Wang and B WenldquoReview on dynamics of cracked gear systemsrdquo EngineeringFailure Analysis vol 55 pp 224ndash245 2015

[10] L Wang Y Liu and Y Liu ldquoAn inverse method for dis-tributed dynamic load identification of structures with in-terval uncertaintiesrdquo Advances in Engineering Softwarevol 131 pp 77ndash89 2019

[11] Standard International Electrotechnical Commission WindTurbines-Part 1 Design Requirements Standard InternationalElectrotechnical Commission Geneva Switzerland 2005

[12] B Ernst H Schmitt and J R Seume ldquoEffect of geometricuncertainties on the aerodynamic characteristic of offshorewind turbine bladesrdquo Journal of Physics Conference Seriesvol 555 Article ID 012033 2014

[13] J Zhu and Z Qiu ldquoInterval analysis for uncertain aerody-namic loads with uncertain-but-bounded parametersrdquoJournal of Fluids and Structures vol 81 pp 418ndash436 2018

[14] J D Soslashrensen and H S Toft ldquoProbabilistic design of windturbinesrdquo Energies vol 3 pp 241ndash257 2010

[15] S Wang Q Li and G J Savage ldquoReliability-based robustdesign optimization of structures considering uncertainty indesign variablesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 280940 8 pages 2015

[16] L Wang C Xiong J Hu X Wang and Z Qiu ldquoSequentialmultidisciplinary design optimization and reliability analysisunder interval uncertaintyrdquo Aerospace Science and Technol-ogy vol 80 pp 508ndash519 2018

[17] P Wang S Wang X Zhang et al ldquoRational construction ofCoOCoF2 coating on burnt-pot inspired 2D CNs as thebattery-like electrode for supercapacitorsrdquo Journal of Alloysand Compounds vol 819 Article ID 153374 2020

[18] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 463ndash474 2004

[19] H Dai and W Wang ldquoApplication of low-discrepancysampling method in structural reliability analysisrdquo StructuralSafety vol 31 no 1 pp 55ndash64 2009

[20] H Zhang H Dai M Beer and W Wang ldquoStructural reli-ability analysis on the basis of small samples an interval quasi-Monte Carlo methodrdquo Mechanical Systems and Signal Pro-cessing vol 37 no 1-2 pp 137ndash151 2013


[21] X Zhang L Wang and J D Soslashrensen ldquoA novel active-learning function toward adaptive Kriging surrogate modelsfor structural reliability analysisrdquo Reliability Engineering ampSystem Safety vol 185 pp 440ndash454 2019

[22] X Zhang L Wang and J D Soslashrensen ldquoAKOIS an adaptiveKriging oriented importance sampling method for structuralsystem reliability analysisrdquo Structural Safety vol 82 ArticleID 101876 2020

[23] N Wiener ldquoe homogeneous chaosrdquo American Journal ofMathematics vol 60 no 4 pp 897ndash936 1938

[24] R Li and R Ghanem ldquoAdaptive polynomial chaos expansionsapplied to statistics of extremes in nonlinear random vibra-tionrdquo Probabilistic Engineering Mechanics vol 13 no 2pp 125ndash136 1998

[25] Y Liu Q Meng X Yan S Zhao and J Han ldquoResearch on thesolution method for thermal contact conductance betweencircular-arc contact surfaces based on fractal theoryrdquo Inter-national Journal of Heat and Mass Transfer vol 145 ArticleID 118740 2019

[26] D Xiu and G E Karniadakis ldquoe wienerndashaskey polynomialchaos for stochastic differential equationsrdquo SIAM Journal onScientific Computing vol 24 no 2 pp 619ndash644 2002

[27] D Xiu and G Em Karniadakis ldquoModeling uncertainty insteady state diffusion problems via generalized polynomialchaosrdquo Computer Methods in Applied Mechanics and Engi-neering vol 191 no 43 pp 4927ndash4948 2002

[28] S Yin D Yu Z Luo and B Xia ldquoUnified polynomial ex-pansion for interval and random response analysis of un-certain structure-acoustic system with arbitrary probabilitydistributionrdquo Computer Methods in Applied Mechanics andEngineering vol 336 pp 260ndash285 2018

[29] H Lu G Shen and Z Zhu ldquoAn approach for reliability-basedsensitivity analysis based on saddlepoint approximationrdquoProceedings of the Institution of Mechanical Engineers Part OJournal of Risk and Reliability vol 231 no 1 pp 3ndash10 2017

[30] H Lu Z Zhu and Y Zhang ldquoA hybrid approach for reli-ability-based robust design optimization of structural systemswith dependent failure modesrdquo Engineering Optimizationvol 52 no 3 pp 384ndash404 2020

[31] Q Zhao X Chen Z D Ma and Y Lin ldquoReliability-basedtopology optimization using stochastic response surfacemethod with sparse grid designrdquo Mathematical Problems inEngineering vol 2015 Article ID 487686 13 pages 2015

[32] A Doostan R G Ghanem and J Red-Horse ldquoStochasticmodel reduction for chaos representationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 196no 37ndash40 pp 3951ndash3966 2007

[33] I Babuska F Nobile and R Tempone ldquoA stochastic collo-cation method for elliptic partial differential equations withrandom input datardquo SIAM Journal on Numerical Analysisvol 45 no 3 pp 1005ndash1034 2007

[34] L Wang C Xiong X Wang G Liu and Q Shi ldquoSequentialoptimization and fuzzy reliability analysis for multidisci-plinary systemsrdquo Structural and Multidisciplinary Optimi-zation vol 60 no 3 pp 1079ndash1095 2019

[35] D Poljak S Sesnic M Cvetkovic et al ldquoStochastic collo-cation applications in computational electromagneticsrdquoMathematical Problems in Engineering vol 2018 Article ID1917439 13 pages 2018

[36] G Blatman and B Sudret ldquoAn adaptive algorithm to build upsparse polynomial chaos expansions for stochastic finite el-ement analysisrdquo Probabilistic Engineering Mechanics vol 25no 2 pp 183ndash197 2010

[37] A Desai and S Sarkar ldquoAnalysis of a nonlinear aeroelasticsystem with parametric uncertainties using polynomial chaosexpansionrdquoMathematical Problems in Engineering vol 2010Article ID 379472 21 pages 2010

[38] X Wei H Chang B Feng and Z Liu ldquoSensitivity analysisbased on polynomial chaos expansions and its application inship uncertainty-based design optimizationrdquo MathematicalProblems in Engineering vol 2019 Article ID 749852619 pages 2019

[39] B Sudret Uncertainty propagation and sensitivity analysis inmechanical models ndash Contributions to structural reliability andstochastic spectral methods Habilitation a diriger desrecherches Universite Blaise Pascal Clermont-FerrandFrance 2007

[40] T Zhang ldquoSparse recovery with orthogonal matching pursuitunder RIPrdquo IEEE Transactions on Information5eory vol 57no 9 pp 6215ndash6221 2011

[41] G Blatman and B Sudret ldquoAdaptive sparse polynomial chaosexpansion based on least angle regressionrdquo Journal ofComputational Physics vol 230 no 6 pp 2345ndash2367 2011

[42] T Ishigami and T Homma ldquoAn importance quantificationtechnique in uncertainty analysis for computer modelsrdquo inProceedings of the IEEE First International Symposium onUncertainty Modeling and Analysis pp 398ndash403 Los Ala-mitos CA USA 1990

[43] R Ghanem and P Spanos Stochastic Finite Elements ASpectral Approach Springer-Verlag New York NY USA1991

[44] X Zhang Q Liu and H Huang ldquoNumerical simulation ofrandom fields with a high-order polynomial based Ritz-Galerkin approachrdquo Probabilistic Engineering Mechanicsvol 55 pp 17ndash27 2019

[45] P Fuglsang I Antoniou N Soslashrensen and H AagaardMadsenValidation of aWind Tunnel Testing Facility for BladeSurface Pressure Measurements International Energy AgencyParis France 1998

[46] M Drela XFOIL An Analysis and Design System for LowReynolds Number Airfoils Low Reynolds Number Aerody-namics T J Mueller Ed Springer Berlin Heidelberg BerlinGermany 1989


implies that a large number of random samples are necessaryfor an accurate estimation result If each round of the de-terministic aerodynamic analysis is relatively time con-suming eg about three minutes the overall uncertainaerodynamic analysis with 104 samples would be about 2083days is motivates an efficient approximation method forthe uncertainty aerodynamic analysis in this paper

Besides the brutal-force random sampling methodseveral improved versions of the MCS based on low-dis-crepancy points [19] the Quasi Monte Carlo method [20]and the active-learning approach [21 22] were alternativelydeveloped It can improve numerical efficiency of the ran-dom simulation method to some extents However thispaper specially focuses on the polynomial chaos expansionmethod which is able to provide a global approximationresult of the mechanistic model for the uncertain aerody-namic analysis of airfoils

e ideal of the polynomial chaos expansion wasinitially introduced by Wiener [23] to investigate theGaussian turbulence models Li and Ghanem [24] furtherinvestigated the utility of the chaos expansion approachfor the stochastic finite element analysis of various solidmechanics models [25] Combined with the Askeyscheme Xiu and Karniadakis [26] extended this expansionapproach to deal with continuous random variables Toachieve this the basis functions for input random vari-ables are first constructed by using the chaotic rule ofmarginal orthogonal polynomials [27] is allows torepresent a multivariate model with the generalizedFourier series expansion [28] Once the PCE model for anairfoil model is determined the Gaussian quadratureapproach can be used to deal with multivariate integralson the expansion coefficient [29ndash31] is results in theGalerkin projection [32] the stochastic collocation[33ndash35] and the statistical regression methods [36] in theliterature

In general numerical algorithms for this PCE model aredivided as the intrusive and nonintrusive approaches efirst category requires to continuously revise the deter-ministic aerodynamic model which is time consuming anddifficult for complex simulation models [37] In the contrastthe nonintrusive approach treats the deterministic aerody-namic analysis as a black-box function and representsrandom variables as a limited number of deterministicvalues After substituting for representative values of arandom variable statistical characteristics of the uncertainaerodynamic response can be obtained based on a simplepostprocessing procedure in terms of some deterministicanalyses

Since a large number of aerodynamic analyses areembedded in the brutal-force Monte Carlo simulationthis generally results in computationally demanding coste PCE approach was combined with the Gaussianquadrature scheme and the dimensional reductionmethod [38] However the standard Gaussian methodmight result in the problem of the curse of dimensionality[39] which motives the nonintrusive and the statisticalregression method to determine expansion coefficients inthis paper

With low-discrepancy samples generated based on theSobolrsquo or the Latin Hypercube scheme the estimation ofPCE coefficients can be realized by the generalized statisticalregression procedure as shown in numerical examples Eventhough sparse regression algorithms based on the orthog-onal matching pursuit [40] the backward-and-forward se-lection and the least-angle regression [41] can bealternatively realized the paper specially focuses on theutility of the full PCE model and the standard regressionapproach due to its robustness

To summarize an objective of this paper is to present aneffective approach for uncertainty aerodynamic analysis ofairfoil models via the polynomial chaos expansion A sta-tistical regression method in conjunction with the low-discrepancy samples is employed to determine the PCEcoefficient To demonstrate potential applications of thismethod examples in the literature are presented to examineits numerical performance

e rest of the manuscript is organized as followsSection 2 briefly summarizes the method of the polynomialchaos expansion for the uncertain aerodynamic analysis ofan airfoil and a statistical regression approach is presentedin Section 23 to determine the expansion coefficient Section3 illustrates an application of this method for uncertainanalysis of the Helmholtz equation with a spatially varyingwave-number model Results for uncertain aerodynamicanalysis of the NACA 63-215 airfoil are discussed in Section4 and conclusions are summarized in Section 5

2 Uncertainty Analysis via the PolynomialExpansion Method

e aerodynamic response of an airfoil (eg the pressureand the velocity field and the lift or the drag coefficient)would become stochastic if input random variables areconsidered in the model function η(X u v w) Hereinu v andw represent three physical dimensions of theaerodynamic model η(middot) whereas the random vector X

[X1 middot middot middot Xd]T consists of all uncertain factors Examples ofthe random variable include the average wind speed the airdensity and the viscosity parameters as shown in numericalexamples To account for this input uncertainty a regres-sion-based polynomial chaos expansionmethod is presentedas follows

21 AReview of the Polynomial Chaos Expansion To developthe polynomial chaos expansion model for the aerodynamicanalysis we first define an index vector α [α1 middot middot middot αd]T

with each integer αi isin [0 p] en a set of chaos polyno-mials defined by utilizing the highest order-parameter p andd-dimensional random vector X are expressed as

ϕi(X) i 0 n minus 11113864 1113865 ≔ cup|α|lep

1113945

d

k1φαk

Xk( 1113857 (1)

Herein Xk is kth input random variable whereasφαk

(Xk) denotes αk th-order orthogonal polynomials that isused to represent the random variable Xk





n

d + p

p

⎛⎝ ⎞⎠ (d + p)

dp (2)






(4)



1113954η(X u v w) 1113944nminus 1









q

⎧⎨

⎩

(6)





i 0 n minus 1

(7)




i 0 n minus 1

(8)




NMCS1113944

NMCS

j1η x(j)

u v w1113872 1113873ϕi x(j)1113872 1113873

i 0 n minus 1

(9)




nminus 1i0

would be


ξ



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(10)










i11113954ai(u vw)ϕi(X)⎡⎣ ⎤⎦

1113954a0(u v w)


i11113954ai(u v w)ϕk(X)1113890 1113891

1113936nminus 1

i11113954a2

i (u v w)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



β1113864 1113865i arg α cup|α|lep

αi ne 01113858 1113859cap 1113858αj 01113859d

j1jnei1113882 11138831113896 1113897 (15)



β1113864 1113865q arg α cup|α|lep


(16)



a2α(u v w)

1113936nminus 1k1a

2k(u v w)

i 1 2 d


a2α(u v w)

1113936nminus 1k1a

2k(u v w)

1le i1 iq le d

(17)









2π

radiciδij Normal




i1113888 1113889δij Gamma


minusy2(u v)

zu2 +y2(u v)

zv21113888 1113889 + ω(u v)y(u v) g0(u v)


y(u v)|Γ1 g1(u v)

zy(uv)

zn

11138681113868111386811138681113868 Γ2 g2(u v)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




i1

λi

1113969

Xi(θ)ϕi(u v) (19)


Xi(θ) 1λi

1113968 1113946Ω

1113946Ω


(20)


1113946Ω



1113957H(u v θ) μ(u v) + 1113944M

i1

λi

1113969

ψi(u v)Xi(θ) (22)



11138681113868111386811138681113868111386811138681113868

δuminus

v minus vprime1113868111386811138681113868

1113868111386811138681113868

δv1113888 1113889 as u v isin Ω

(23)



1113954η(X u v) 1113944nminus 1





ε2(u v) 1113936


2

1113936

NMCS

i1η x(i)

u v1113872 1113873 minus Y(u v)1113960 11139612 (25)








minus02

0

02

04

u

ν


35π

4π

45π

5π

55π

6π

65π

(a)

ν


minus04

minus02

0

02

04

4985π

5π

5015π

(b)

ν

minus04

minus02

0

02

04


094π

095π

096π

097π

098π

099π

(c)


minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(a)

minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(b)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(c)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(d)











1

2

3

4

5

6


PDF

minus01

PCEMCS

(a)

0

1

2

3

4

PDF


PCEMCS

minus02 0 02 06 0804

(b)

065 07 075 080

5

10

15

20

25

30

35

PDF


PCEMCS

(c)


1

2

3

4

5

6

PDF

minus01

PCEMCS

(d)







1113954Cp(X u v) 1113944dminus 1

k0aCpk(u v)ϕk(X)

1113954P(X u v) 1113944dminus 1

k0aPk(u v)ϕk(X)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Angle of attack

ndash05

0

05

1

15Li

fting

coef

ficie

nt (C

l)


ndash5 0 5 10 15 20

(a)


0

005

01

015

02

Dra

g co

effic

ient

(Cd)


(b)





0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)


u

ν


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)



u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References




















































n

d + p

p

⎛⎝ ⎞⎠ (d + p)

dp (2)






(4)



1113954η(X u v w) 1113944nminus 1









q

⎧⎨

⎩

(6)





i 0 n minus 1

(7)




i 0 n minus 1

(8)




NMCS1113944

NMCS

j1η x(j)

u v w1113872 1113873ϕi x(j)1113872 1113873

i 0 n minus 1

(9)




nminus 1i0

would be


ξ



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(10)










i11113954ai(u vw)ϕi(X)⎡⎣ ⎤⎦

1113954a0(u v w)


i11113954ai(u v w)ϕk(X)1113890 1113891

1113936nminus 1

i11113954a2

i (u v w)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



β1113864 1113865i arg α cup|α|lep

αi ne 01113858 1113859cap 1113858αj 01113859d

j1jnei1113882 11138831113896 1113897 (15)



β1113864 1113865q arg α cup|α|lep


(16)



a2α(u v w)

1113936nminus 1k1a

2k(u v w)

i 1 2 d


a2α(u v w)

1113936nminus 1k1a

2k(u v w)

1le i1 iq le d

(17)









2π

radiciδij Normal




i1113888 1113889δij Gamma


minusy2(u v)

zu2 +y2(u v)

zv21113888 1113889 + ω(u v)y(u v) g0(u v)


y(u v)|Γ1 g1(u v)

zy(uv)

zn

11138681113868111386811138681113868 Γ2 g2(u v)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




i1

λi

1113969

Xi(θ)ϕi(u v) (19)


Xi(θ) 1λi

1113968 1113946Ω

1113946Ω


(20)


1113946Ω



1113957H(u v θ) μ(u v) + 1113944M

i1

λi

1113969

ψi(u v)Xi(θ) (22)



11138681113868111386811138681113868111386811138681113868

δuminus

v minus vprime1113868111386811138681113868

1113868111386811138681113868

δv1113888 1113889 as u v isin Ω

(23)



1113954η(X u v) 1113944nminus 1





ε2(u v) 1113936


2

1113936

NMCS

i1η x(i)

u v1113872 1113873 minus Y(u v)1113960 11139612 (25)








minus02

0

02

04

u

ν


35π

4π

45π

5π

55π

6π

65π

(a)

ν


minus04

minus02

0

02

04

4985π

5π

5015π

(b)

ν

minus04

minus02

0

02

04


094π

095π

096π

097π

098π

099π

(c)


minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(a)

minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(b)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(c)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(d)











1

2

3

4

5

6


PDF

minus01

PCEMCS

(a)

0

1

2

3

4

PDF


PCEMCS

minus02 0 02 06 0804

(b)

065 07 075 080

5

10

15

20

25

30

35

PDF


PCEMCS

(c)


1

2

3

4

5

6

PDF

minus01

PCEMCS

(d)







1113954Cp(X u v) 1113944dminus 1

k0aCpk(u v)ϕk(X)

1113954P(X u v) 1113944dminus 1

k0aPk(u v)ϕk(X)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Angle of attack

ndash05

0

05

1

15Li

fting

coef

ficie

nt (C

l)


ndash5 0 5 10 15 20

(a)


0

005

01

015

02

Dra

g co

effic

ient

(Cd)


(b)





0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)


u

ν


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)



u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References

















































ξ



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

mtimesn

(10)










i11113954ai(u vw)ϕi(X)⎡⎣ ⎤⎦

1113954a0(u v w)


i11113954ai(u v w)ϕk(X)1113890 1113891

1113936nminus 1

i11113954a2

i (u v w)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



β1113864 1113865i arg α cup|α|lep

αi ne 01113858 1113859cap 1113858αj 01113859d

j1jnei1113882 11138831113896 1113897 (15)



β1113864 1113865q arg α cup|α|lep


(16)



a2α(u v w)

1113936nminus 1k1a

2k(u v w)

i 1 2 d


a2α(u v w)

1113936nminus 1k1a

2k(u v w)

1le i1 iq le d

(17)









2π

radiciδij Normal




i1113888 1113889δij Gamma


minusy2(u v)

zu2 +y2(u v)

zv21113888 1113889 + ω(u v)y(u v) g0(u v)


y(u v)|Γ1 g1(u v)

zy(uv)

zn

11138681113868111386811138681113868 Γ2 g2(u v)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




i1

λi

1113969

Xi(θ)ϕi(u v) (19)


Xi(θ) 1λi

1113968 1113946Ω

1113946Ω


(20)


1113946Ω



1113957H(u v θ) μ(u v) + 1113944M

i1

λi

1113969

ψi(u v)Xi(θ) (22)



11138681113868111386811138681113868111386811138681113868

δuminus

v minus vprime1113868111386811138681113868

1113868111386811138681113868

δv1113888 1113889 as u v isin Ω

(23)



1113954η(X u v) 1113944nminus 1





ε2(u v) 1113936


2

1113936

NMCS

i1η x(i)

u v1113872 1113873 minus Y(u v)1113960 11139612 (25)








minus02

0

02

04

u

ν


35π

4π

45π

5π

55π

6π

65π

(a)

ν


minus04

minus02

0

02

04

4985π

5π

5015π

(b)

ν

minus04

minus02

0

02

04


094π

095π

096π

097π

098π

099π

(c)


minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(a)

minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(b)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(c)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(d)











1

2

3

4

5

6


PDF

minus01

PCEMCS

(a)

0

1

2

3

4

PDF


PCEMCS

minus02 0 02 06 0804

(b)

065 07 075 080

5

10

15

20

25

30

35

PDF


PCEMCS

(c)


1

2

3

4

5

6

PDF

minus01

PCEMCS

(d)







1113954Cp(X u v) 1113944dminus 1

k0aCpk(u v)ϕk(X)

1113954P(X u v) 1113944dminus 1

k0aPk(u v)ϕk(X)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Angle of attack

ndash05

0

05

1

15Li

fting

coef

ficie

nt (C

l)


ndash5 0 5 10 15 20

(a)


0

005

01

015

02

Dra

g co

effic

ient

(Cd)


(b)





0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)


u

ν


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)



u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References

















































minusy2(u v)

zu2 +y2(u v)

zv21113888 1113889 + ω(u v)y(u v) g0(u v)


y(u v)|Γ1 g1(u v)

zy(uv)

zn

11138681113868111386811138681113868 Γ2 g2(u v)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




i1

λi

1113969

Xi(θ)ϕi(u v) (19)


Xi(θ) 1λi

1113968 1113946Ω

1113946Ω


(20)


1113946Ω



1113957H(u v θ) μ(u v) + 1113944M

i1

λi

1113969

ψi(u v)Xi(θ) (22)



11138681113868111386811138681113868111386811138681113868

δuminus

v minus vprime1113868111386811138681113868

1113868111386811138681113868

δv1113888 1113889 as u v isin Ω

(23)



1113954η(X u v) 1113944nminus 1





ε2(u v) 1113936


2

1113936

NMCS

i1η x(i)

u v1113872 1113873 minus Y(u v)1113960 11139612 (25)








minus02

0

02

04

u

ν


35π

4π

45π

5π

55π

6π

65π

(a)

ν


minus04

minus02

0

02

04

4985π

5π

5015π

(b)

ν

minus04

minus02

0

02

04


094π

095π

096π

097π

098π

099π

(c)


minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(a)

minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(b)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(c)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(d)











1

2

3

4

5

6


PDF

minus01

PCEMCS

(a)

0

1

2

3

4

PDF


PCEMCS

minus02 0 02 06 0804

(b)

065 07 075 080

5

10

15

20

25

30

35

PDF


PCEMCS

(c)


1

2

3

4

5

6

PDF

minus01

PCEMCS

(d)







1113954Cp(X u v) 1113944dminus 1

k0aCpk(u v)ϕk(X)

1113954P(X u v) 1113944dminus 1

k0aPk(u v)ϕk(X)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Angle of attack

ndash05

0

05

1

15Li

fting

coef

ficie

nt (C

l)


ndash5 0 5 10 15 20

(a)


0

005

01

015

02

Dra

g co

effic

ient

(Cd)


(b)





0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)


u

ν


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)



u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References




















































minus02

0

02

04

u

ν


35π

4π

45π

5π

55π

6π

65π

(a)

ν


minus04

minus02

0

02

04

4985π

5π

5015π

(b)

ν

minus04

minus02

0

02

04


094π

095π

096π

097π

098π

099π

(c)


minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(a)

minus040

04

minus04

0

04minus1

0

1

uv

Mea

n va

lue

(b)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(c)

minus040

04

minus04

0

040

01

02

uv

Stan

dard

dev

iatio

n

(d)











1

2

3

4

5

6


PDF

minus01

PCEMCS

(a)

0

1

2

3

4

PDF


PCEMCS

minus02 0 02 06 0804

(b)

065 07 075 080

5

10

15

20

25

30

35

PDF


PCEMCS

(c)


1

2

3

4

5

6

PDF

minus01

PCEMCS

(d)







1113954Cp(X u v) 1113944dminus 1

k0aCpk(u v)ϕk(X)

1113954P(X u v) 1113944dminus 1

k0aPk(u v)ϕk(X)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Angle of attack

ndash05

0

05

1

15Li

fting

coef

ficie

nt (C

l)


ndash5 0 5 10 15 20

(a)


0

005

01

015

02

Dra

g co

effic

ient

(Cd)


(b)





0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)


u

ν


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)



u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References

























































1

2

3

4

5

6


PDF

minus01

PCEMCS

(a)

0

1

2

3

4

PDF


PCEMCS

minus02 0 02 06 0804

(b)

065 07 075 080

5

10

15

20

25

30

35

PDF


PCEMCS

(c)


1

2

3

4

5

6

PDF

minus01

PCEMCS

(d)







1113954Cp(X u v) 1113944dminus 1

k0aCpk(u v)ϕk(X)

1113954P(X u v) 1113944dminus 1

k0aPk(u v)ϕk(X)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Angle of attack

ndash05

0

05

1

15Li

fting

coef

ficie

nt (C

l)


ndash5 0 5 10 15 20

(a)


0

005

01

015

02

Dra

g co

effic

ient

(Cd)


(b)





0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)


u

ν


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)



u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References





















































1113954Cp(X u v) 1113944dminus 1

k0aCpk(u v)ϕk(X)

1113954P(X u v) 1113944dminus 1

k0aPk(u v)ϕk(X)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)




Angle of attack

ndash05

0

05

1

15Li

fting

coef

ficie

nt (C

l)


ndash5 0 5 10 15 20

(a)


0

005

01

015

02

Dra

g co

effic

ient

(Cd)


(b)





0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)


u

ν


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)



u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References

















































0 02 04 06 08 1minus5

minus4

minus3

minus2

minus1

0

1

2

Position

C p

PCEMCS

(a)

C p

PCEMCS

0 02 04 06 08 10

0002

0004

0006

0008

001

Position

(b)


u

ν


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

(a)

uminus05 0 05 1 15

minus500

minus400

minus300

minus200

minus100

0

100

200

ν

minus06

minus04

minus02

0

02

04

06

(b)

ν

uminus05 0 05 1 15

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(c)

uminus05 0 05 1 15

ν

minus06

minus04

minus02

0

02

04

06

0

20

40

60

80

(d)



u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References

















































u

ν


(06394 01428)

(11756 minus00708)

(05307 minus03557)


minus04

minus02

0

02

04

06

minus500

minus400

minus300

minus200

minus100

0

100

200

A

B

C

D


40 60 80 100 120 1400

001

002

003

004

Pressure (Pa)

PDF

PCEMCS

(a)


001

002

003

004

005

Pressure (Pa)

PDF

PCEMCS

(b)

10 15 20 25 30 350

005

01

015

02

Pressure (Pa)

PDF

PCEMCS

(c)

20 25 30 35 40 45 50 550

002

004

006

008

01

Pressure (Pa)

PDF

PCEMCS

(d)











PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References

























































PCEMCS

1040

50

100

150

200

250Pr

obab

ility

den

sity

func

tion

1025 103 1035 1045

(a)

PCEMCS

002150

500

1000

1500

2000

00195 002 00205 0021 0022

(b)

PCEMCS

460

01

02

03

04

05

06

48 50 52 54

(c)




S1 03901 04221 03942ST1

03929 04324 04032S2 03454 03237 03457ST2

03536 03339 03504S3 02645 02543 02601ST3

02851 02946 02735


5 Conclusions


Data Availability




Acknowledgments


References

















































5 Conclusions


Data Availability




Acknowledgments


References












































































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An Effective Approach for Uncertain Aerodynamic Analysis ...downloads.hindawi.com/journals/mpe/2020/7417835.pdf · ResearchArticle An Effective Approach for Uncertain Aerodynamic