An Effective Approach for Uncertain Aerodynamic Analysis of Airfoils via the Polynomial Chaos Expansion

,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 model, whereas the expansion coefficient is expressed as the multivariate stochastic integral of the input random vector. In this regard, the statistical regression in conjunction with a small number of representative samples is employed to determine the expansion coefficient. ,en, a combination of the PCE surrogate model with brutal-force Monte Carlo simulation allows to determine numerical results for the uncertain aerodynamic analysis. Potential applications of this approach are first illustrated by the uncertainty analysis of the Helmholtz equation with spatially varied wave-number random field, and its effectiveness is further examined by the uncertain aerodynamic analysis of the NACA 63-215 airfoil. Results for the small regression error and a close agreement between simulated and benchmark results have confirmed numerical accuracy and efficiency of this approach. It, therefore, has a potential to deal with computationally demanding aerodynamical models for the uncertainty analysis.


Introduction
With the fast development of computer and simulation techniques, numerical methods for the aerodynamic analysis of airfoils have been received considerable attentions during past decades [1,2]. Various simulation codes based on the finite difference, the finite element, and the finite volume were extensively investigated [3,4]. In this regard, parameters of the aerodynamic model, i.e., the geometry dimensions and the boundary conditions are usually assumed as deterministic [5][6][7].
However, the uncertainty is universally existed in aerodynamic models of an airfoil [8][9][10]. Statistical characteristics of the average wind velocity and the turbulence intensity are necessarily described by using the probability distribution [11]. In this regard, Ernst et al. [12] considered geometric uncertainties of rotor blades that arise from manufacturing tolerances and operational wears, and the input uncertainty indeed leads to a significant scatter of the lift and drag coefficients. Zhu and Qiu [13] investigated the possibility of utilizing perturbation approaches for uncertain aerodynamic loads modelling, and their combined effects on the response interval prediction of lift coefficients.
Rather than utilizing the first-order or the Neumann series to approximate the aerodynamic response function, the Monte Carlo simulation can deal with the uncertain aerodynamic analysis in a much more straightforward manner [14]. With a large number of samples to represent the input uncertainty, the question of dealing with the uncertain aerodynamic analysis is degenerated as a series of deterministic model evaluations [15]. is routine can be applied for the moment analysis, the prediction of response distribution, and the global sensitivity analysis [16,17]. erefore, combined with the parallel computing technology, results determined by the Monte Carlo simulation method is assumed to provide benchmarks for uncertainty aerodynamic analysis of airfoils in this paper.
However, the convergence rate of the Monte Carlo simulation (MCS) method is fairly slow, i.e., 1/ ����� N MCS for a total of N MCS samples used in this simulation [18]. is implies that a large number of random samples are necessary for an accurate estimation result. If each round of the deterministic aerodynamic analysis is relatively time consuming, e.g., about three minutes, the overall uncertain aerodynamic analysis with 10 4 samples would be about 20.83 days. is motivates an efficient approximation method for the uncertainty aerodynamic analysis in this paper. Besides the brutal-force random sampling method, several improved versions of the MCS based on low-discrepancy points [19], the Quasi Monte Carlo method [20], and the active-learning approach [21,22] were alternatively developed. It can improve numerical efficiency of the random simulation method to some extents. However, this paper specially focuses on the polynomial chaos expansion method, which is able to provide a global approximation result of the mechanistic model for the uncertain aerodynamic analysis of airfoils. e ideal of the polynomial chaos expansion was initially introduced by Wiener [23] to investigate the Gaussian turbulence models. Li and Ghanem [24] further investigated the utility of the chaos expansion approach for the stochastic finite element analysis of various solid mechanics models [25]. Combined with the Askey scheme, Xiu and Karniadakis [26] extended this expansion approach to deal with continuous random variables. To achieve this, the basis functions for input random variables are first constructed by using the chaotic rule of marginal orthogonal polynomials [27].
is allows to represent a multivariate model with the generalized Fourier series expansion [28]. Once the PCE model for an airfoil model is determined, the Gaussian quadrature approach can be used to deal with multivariate integrals on the expansion coefficient [29][30][31]. is results in the Galerkin projection [32], the stochastic collocation [33][34][35], and the statistical regression methods [36] in the literature.
In general, numerical algorithms for this PCE model are divided as the intrusive and nonintrusive approaches. e first category requires to continuously revise the deterministic aerodynamic model, which is time consuming and difficult for complex simulation models [37]. In the contrast, the nonintrusive approach treats the deterministic aerodynamic analysis as a black-box function and represents random variables as a limited number of deterministic values. After substituting for representative values of a random variable, statistical characteristics of the uncertain aerodynamic response can be obtained based on a simple postprocessing procedure in terms of some deterministic analyses.
Since a large number of aerodynamic analyses are embedded in the brutal-force Monte Carlo simulation, this generally results in computationally demanding cost.
e PCE approach was combined with the Gaussian quadrature scheme and the dimensional reduction method [38]. However, the standard Gaussian method might result in the problem of the curse of dimensionality [39], which motives the nonintrusive and the statistical regression method to determine expansion coefficients in this paper.
With low-discrepancy samples generated based on the Sobol' or the Latin Hypercube scheme, the estimation of PCE coefficients can be realized by the generalized statistical regression procedure as shown in numerical examples. Even though sparse regression algorithms based on the orthogonal matching pursuit [40], the backward-and-forward selection, and the least-angle regression [41] can be alternatively realized, the paper specially focuses on the utility of the full PCE model and the standard regression approach due to its robustness.
To summarize, an objective of this paper is to present an effective approach for uncertainty aerodynamic analysis of airfoil models via the polynomial chaos expansion. A statistical regression method in conjunction with the lowdiscrepancy samples is employed to determine the PCE coefficient. To demonstrate potential applications of this method, examples in the literature are presented to examine its numerical performance. e rest of the manuscript is organized as follows. Section 2 briefly summarizes the method of the polynomial chaos expansion for the uncertain aerodynamic analysis of an airfoil, and a statistical regression approach is presented in Section 2.3 to determine the expansion coefficient. Section 3 illustrates an application of this method for uncertain analysis of the Helmholtz equation with a spatially varying wave-number model. Results for uncertain aerodynamic analysis of the NACA 63-215 airfoil are discussed in Section 4, and conclusions are summarized in Section 5.

Uncertainty Analysis via the Polynomial
Expansion Method e aerodynamic response of an airfoil (e.g., the pressure and the velocity field and the lift or the drag coefficient) would become stochastic, if input random variables are considered in the model function η(X; u, v, w). Herein, u, v, and w represent three physical dimensions of the aerodynamic model η(·), whereas the random vector X � [X 1 , · · · , X d ] T consists of all uncertain factors. Examples of the random variable include the average wind speed, the air density, and the viscosity parameters, as shown in numerical examples. To account for this input uncertainty, a regression-based polynomial chaos expansion method is presented as follows.

A Review of the Polynomial Chaos Expansion.
To develop the polynomial chaos expansion model for the aerodynamic analysis, we first define an index vector α � [α 1 , · · · , α d ] T with each integer α i ∈ [0, p]. en, a set of chaos polynomials defined by utilizing the highest order-parameter p and d-dimensional random vector X are expressed as Herein, X k is kth input random variable, whereas φ α k (X k ) denotes α k th-order orthogonal polynomials that is used to represent the random variable X k .

2
Mathematical Problems in Engineering Specially, the order of a d-variate polynomial is defined as p � d k�1 α k , and the number of elements within the polynomial set ϕ i (X) Besides, arbitrary two elements within the polynomial set ϕ i (X), i � 0, . . . , N − 1 are orthogonally defined. is is 〈ϕ i (x), ϕ j (x)〉 � δ ij (as i, j � · · · , n), where δ ij denotes the Kronecker delta symbol. erefore, the chaotic polynomial set ϕ i (X), i � 0, 1, . . . , n − 1 constitute a complete set to represent a d-dimensional real-valued space when n ⟶ +∞, and the structural model response can be spectrally represented as Herein, a i (u, v, w) denotes deterministic expansion coefficients: Herein, f X (x) denotes the joint probability density function the input random vector X.
Since the PCE model contains infinite terms in the definition, the approximation for an aerodynamic response can be numerically realized by truncating the expression starting an nth term: which is used to approximate the true mechanistic model in numerical simulations. e normalization constant of Jacobi polynomials: Specially, the orthogonal chaos polynomial depends largely on the probability distribution of input random variables. Table 1 summarizes the relation between the probability distribution of random variables and the orthogonal polynomial used to develop the PCE model. Once the spacial-dependent coefficients in equation (4) are numerically determined, uncertain response of the aerodynamic model η(X; u, v, w) can be theoretically represented by the truncated PCE model. erefore, numerical approaches for the calculation of the PCE coefficient are presented as follows.

e Calculation of PCE Coefficients.
e key issue to realize the PCE method for the uncertainty quantification is to determine expansion coefficients a k (u, v, w) (k � 0, 1, · · · , n − 1) in equation (5), which are generally modelled by using the following optimization problem: Herein, x (k) represents the kth realization of the input random vector X, whereas ‖·‖ q denotes a q-norm operator with respect to totally n 0 residual terms of η(·) as referring to the PCE approximation result η(·).
If the parameter q � 2 and n 0 � n, the probabilistic optimization problem directly leads to an n-order algebraic system in term of the PCE coefficients: which can be re-expressed as (4). e brutal-force MCS method and its optimized versions have been used in the literature to deal with the high-dimensional integration. Results for the MCS based estimation of the expansion coefficient are

this finally derives totally n d-dimensional integrals as expressed in equation
where the symbol x (i) represents the ith realization of the input random vector X and N MCS represents the total number of samples used in the Monte Carlo simulation.
Note that numerical realization of such large number of high-dimensional integrals is a computationally intensive task, given that the deterministic aerodynamic response η(X; u, v, w) is possibly spatially modelled based on a finite element scheme. In this regard, the brutal-force MCS method is replaced with the statistical regression method as follows.

2.3.
e Statistical Regression Method. Denote that ζ � x (1) , · · · , x (m) consists of m realizations of the input random vector X generated based on a quasi-simulation method in conjunction with a low-discrepancy sequence (such as the Sobol', Halton, or Hammersley). Corresponding numerical realizations of the chaos polynomial set ϕ i (x) en, a regression model can be obtained as which consists of residual errors for all spatial and time realizations of the model. Following the theory of the multivariate regression, the unknown regression coefficients a in equation (11) can be expressed in terms of the training data set ξ and response sample y as Once the training matrix ξ is defined in equation (10)and the corresponding aerodynamic response samples y are available, the minimization of the Euclidian-norm-based residual errors ϵ (i) (with i � 1, · · · , m) allows finally deriving the regression-based surrogate model noted in equation (11), which will be useful to mimic the true but computationally intensive model η(·) in the afterwards uncertainty simulations.

Uncertainty Analysis Based on the PCE Surrogate Model.
Once a surrogate model η(X; u, v, w) is obtained either analytically or numerically, the mean value and the variance of the aerodynamic response can be approximated as Besides, the Sobol' index used for the variance-based global sensitivity analysis can be easily expressed by means of the PCE coefficients [42]. To implement, an index set of representing the ith variance component V i uniquely contributed by the random variable X i is first defined as Similarly, a q-dimensional index set q � i 1 , · · · , i q that measures the joint variance effect due to multiple input random variables X i 1 , · · · , X i q acting together is presented as Based on abovementioned notations, results for the main and the joint Sobol' sensitivity indices for the variance-based global sensitivity analysis can be readily expressed as Note that the parameter a α (u, v, w) denotes the PCE coefficient attached to a d-dimensional polynomial ϕ α (X). erefore, results for Sobol' sensitivity index are purely expressed in terms of the PCE coefficients.
To summarize, the proposed approach for uncertain aerodynamic analysis of an airfoil includes (1) the development of the chaos polynomial set ϕ i (X) (1) and (2) the calculation of PCE coefficients by using the statistical regression method in equations (3) and (13) the sensitivity analysis and the response distribution estimation in Section 2.4. erefore, in conjunction with a small number of low-discrepancy simulations and a simple postprocessing procedure the uncertainty analysis of an airfoil model can be effectively realized via the polynomial chaos expansion method.

Uncertainty Analysis of the Helmholtz Equation
is section illustrates an application of the polynomial chaos expansion approach for the uncertainty analysis of the two-dimensional Helmholtz equation, which is generally defined as Table 1: Distribution of random variables and orthogonal polynomials [26].

Polynomial
Weight function Domain Orthogonality Distribution Hermit where Γ 1 represents the Dirichlet boundary condition and Γ 2 denotes the Neumann boundary condition. For uncertainty analysis, the wave-number function ω(u, v; θ) is molded by using the Gaussian random field. is implies the response quantity of this equation is spatially stochastic as

e Karhunen-Loève Expansion for Spatial Variabilities.
According to the theory of a random field simulation, the spatially varying two-dimensional random property ω(u, v; θ) can be effectively represented by means of the Karhunen-Loève (K-L) expansion [43] and its variants [44]: Herein, μ(u, v) denotes the mean value of the random field, whereas the statistically Gaussian random variables X i (θ) in this example can be obtained by where λ i and ϕ i (u, v) denote eigenvalues and eigenfunctions for the covariance function Cov(u, v), respectively, which are solutions of the following integral eigenvalue problem (IEVP): Once a numerical solution of the integral integration is available, numerical realization of the random field based on an m-term K-L expansion is given as In numerical simulations, the following exponential model is assumed for the spatially varying wave-number function: Figure 1 presents simulation results for random field of the wave-number function ω(u, v; θ) based on 10 4 random samples. Herein, parameters in the covariance function are given as σ 0 � π, δ u � 0.8, and δ v � 1. With the truncation parameter M � 5, the error for the variance simulation result in Figure 1(c) is determined as 4.15%, which implies the simulation variance is relatively lower than that of the benchmark variance π.

Numerical Results for the Uncertainty Analysis.
With numerical simulation results for random field of the wavenumber function, the procedure summarized in Section 2.4 is used for uncertainty analysis of the Helmholtz equation.
e multivariate Hermit polynomials are first determined to represent the Gaussian random variables produced by the K-L expansion, and the PCE model for stochastic response of the Helmholtz equation is determined as where the random vector is defined as X � [X 1 , . . . , X 5 ] T and X i is the standard Normal random variable determined by the K-L expansion in equation (20). Assume that the highest order of the chaotic polynomial is p � 6. is implies the total number of polynomial terms in equation (24) will be 462. If the Gaussian quadrature method used to evaluate the PCE coefficients, it involves 7,776(� 6 5 ) functional evaluations in total. Herein, each random variable is represented by six Gaussian-Hermit nodes. To reduce the computational cost for multivariate cases, the statistical regression method in Section 2.3 with 500 samples were alternatively used, and the corresponding error term is determined to evaluate numerical accuracy of this PCE model: where η(·) and η(·) represent the simulated and the predicted model responses, respectively, whereas Y(u, v) denotes the mean value of the model response, i.e., . erefore, an indicator R 2 is defined as R 2 (u, v) � 1 − ε 2 (u, v) in the following simulations to quantify numerical accuracy of the PCE model.

Uncertain Aerodynamic Analysis of the NACA 63-215 Airfoil
is section considers uncertain aerodynamic analysis of the NACA 63-215 airfoil via the polynomial chaos expansion approach. To achieve this, the deterministic aerodynamic simulation is first realized based on a finite element model in conjunction with the Spalart-Allmaras (S-A) turbulence model. In this simulation, the angle of attack was considered from − 4 ∘ to 18 ∘ with an incremental step 0.5 ∘ .
Together with results provided by the wind tunnel test [45] and XFoil computational package [46], the applicability of the simulation model is first verified. Note that the following simulation parameters are considered: the wind velocity V � 16 m/s, the air density ρ � 1.225 kg/m 3 , and the viscosity parameter 1.7894 × 10 − 5 kg/m·s. Figure 4 presents the simulation results for the lifting and the drag coefficients of this airfoil. It is observed that results for the lift coefficient provided by the FEM with the S-A turbulence model are closely agreed with the results provide by the XFoil and the experimental data. Compared with the wind tunnel test data, the drag coefficient result predicted by the S-A is more accurate than that of the XFoil. erefore, the uncertain aerodynamic quantification of the NACA 63-215 airfoil is implemented based on the S-A model for the pressure and the velocity response fields. Once a numerical simulation model of the airfoil is available, the corresponding uncertainty analysis can be realized via the polynomial chaos expansion approach. Note that the maximal lift-drag ratio is determined as AOA * � 9.5 ∘ in this deterministic simulation, and the PCE approach is further applied to provide estimation results for the probability distribution of the drag, the lift, and the life-to-drag ratio at this AOA value as follows.

Results for the Uncertainty Analysis.
To implement the uncertain aerodynamic analysis, the probabilistic characteristics of input random factors are listed in Table 2. is includes the inflow velocity V ∞ , the air density ρ, and the viscosity parameter ν. e PCE algorithm is realized for the uncertain analysis, and the brutal-force Monte Carlo simulation with 10 4 samples is assumed to provide the benchmark result for the numerical verification.
At first, the PCE models for the coefficient of pressure Cp(X; u ′ , v ′ ) and the pressure response of the simulation field are separately developed: in which the chaotic polynomial terms ϕ k (X) are expressed as the tensor product of the Hermit polynomials to represent normal random variables in Table 2. With 500 samples generated based on the LSH approach, the statistical regression method in Section 2.3 is used to determine the PCE coefficient. In this PCE model, the polynomial order parameter is given as p � 3.
is determines totally 20 terms for numerical representation of the uncertain aerodynamic response. Following the moment results based on the PCE coefficients in equation (14), the first-two orders of moments for the aerodynamic responses are presented in Figure 5. A close agreement between the simulation and the predicted moment results has confirmed the high accuracy of this PCE method for the pressure coefficient C p of this airfoil.
Results for the mean value and the standard deviation of this pressure response field are determined, as shown     in Figure 6. Note that results for all simulation points within the investigated domain were determined. e regression parameters a k (u, v) are spatially dependent. Combined with benchmark result provided based on 10 4 samples, the proposed approach is fairly efficient for numerical estimation of the moment of the response random filed.
In this surrogate model, deterministic simulations were numerically realized based on a personal computer with the CPU of Interl(R)Core(TM)i7 − 37703.40 GHz and the 4 GB physical memories. In addition, four-thread parallel computing technique was implemented to accelerate the simulation process. Each round of the deterministic simulation approximately needs 3 seconds. erefore, with 500 samples to develop the PCE surrogate model, the total simulation time is around three minutes in total. However, the brutalforce Monte Carlo simulation with 10 4 samples requires 9.5 hours.
is has demonstrated the high efficiency of this proposed approach for uncertain aerodynamic analysis of the NACA 63-215 airfoil.
Numerical verification of the PCE approach is further extended to the probability distribution for pressure responses at four locations, as shown in Figure 7 Figure 8. Note that results for the accuracy measure in equation (25) are generally less than 6.0 × 10 − 8 . Together with accuracy simulation results for probability distribution of the lift and drag coefficients in Figure 9 for the case of AOA * � 9.5 ∘ , it has confirmed the effectiveness of utilizing this polynomial chaos expansion for stochastic aerodynamic analysis of airfoils.

Results for the Global Sensitivity Analysis.
e section further determines results for the global sensitivity analysis of the NACA 63-215 airfoil with random variables in terms of the inflow velocity V ∞ , the air density ρ, and the dynamic viscosity. To achieve this, surrogate models for C l , C d , and C ld are separately determined. In this regard, the regression error are determined as 3.14 × 10 − 6 , 3.26 × 10 − 7 , and 2.28 × 10 − 7 for C l , C d , and C l d , respectively. e small regression errors ( ≤10 − 5 ) for all PCE models have confirmed the high accuracy of this proposed approach.
With the polynomial order parameter p � 3, the chaotic Hermit polynomials are used to develop the PCE surrogate model for aerodynamic response of the NACA 63-215 airfoil.
e statistical regression method with 500 Sobol' samples was used to determine the expansion coefficient. Besides, the brutal-force Monte Carlo simulation with 10 4 samples was used to provide benchmark results for this global sensitivity analysis.
Based on the PCE coefficients determined with the statistical regression method, Table 3 summarizes results for the global sensitivity analysis of this uncertain aerodynamic model with random variables in terms of the V ∞ , the ρ, and the dynamic viscosity parameter ]. It is observed that these three random factors have almost contributed equally for total aerodynamic variance of C l , C d , and C l d , whereas the parameter of the wind velocity V ∞ has been ranked as the principle factor given the largest value of the global sensitivity result.
is is followed by the air density ρ and the dynamic viscosity parameter ]. Specially, results for the total sensitivity index S T i are slightly larger than those of the primary sensitivity index S i , given that the total variance includes joint variance contributions of X i and X j together. To summarize, the proposed PCE surrogate model can provide reliable estimation results for the global sensitivity analysis of the NACA 63-215 airfoil.

Conclusions
e paper presents an effective approach for uncertain aerodynamic analysis of airfoils via the polynomial chaos expansion. To archive this, the spatially varied random properties of a partial differential equation are first resented as the normal random variables based on the K-L expansion method. en, a set of basis functions are determined by using the chaotic orthogonal polynomials.
is allows to developing a statistical regression model for the uncertain aerodynamic analysis via the polynomial chaos expansion approach. In numerical simulations, the Helmholtz equation with a spatially varied wave-number function is considered. With the truncated K-L expansion scheme to represent the input random field, the polynomial chaos expansion method is able to provide reliable estimation results for the statistical moment and the probability distribution of the model response. is verification is further extended to the uncertain aerodynamic analysis of the NACA 63-215 airfoil, which is modelled by using the normal random variables. e deterministic aerodynamic simulation is realized by using the finite element method in conjunction with the Spalart-Allmaras turbulence model. With a small number of deterministic model evaluations, results for the response pressure and velocity fields of this airfoil are determined. Compared to benchmark results provided by the brutalforce Monte Carlo simulation, it has confirmed the high accuracy of this method for the global sensitivity analysis of aerodynamic models. e close agreement between the simulation and the predicted results for the lift, the drag, and the lift-to-drag ratio has confirmed the effectiveness of this approach for uncertain aerodynamic analysis of airfoils.

Data Availability
e simulation data within this submission are available based on the request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.