A Data-based Approach for Fast Airfoil Analysis and Optimization

Transcription

1 A Data-based Approach for Fast Airfoil Analysis and Optimization Jichao Li *, Mohamed Amine Bouhlel, and Joaquim R. R. A. Martins University of Michigan, Ann Arbor, MI, 48109, USA Airfoils are of great importance in aerodynamic design, and there is a need for tools that can evaluate and optimize their performance. Existing tools are usually either accurate or efficient, but not both. This paper presents a tool that can analyze airfoils in both subsonic and transonic regimes in about one hundredth of a second, and optimize airfoil shapes in a few seconds. We use camber and thickness mode shapes derived from over one thousand existing airfoils to parameterize the airfoil shape, which reduces the number of design variables. More than one hundred thousand RANS evaluations associated with different airfoils and flow conditions (M, ) are selected in a well-defined way to obtain aerodynamic coefficients (, C d and C m ) and their gradients. The gradients for the RANS evaluations are computed using an adjoint method. This data is used to train a surrogate model that combines gradient-enhanced kriging, partial least squares (GE-KPLS), and mixture of experts. Two global models composed of dozens of GE-KPLS models are constructed in the subsonic and transonic regimes. These surrogate models provide fast aerodynamic analysis and gradient computation, which are coupled with a gradient-based optimizer to perform rapid airfoil shape design optimization. In a validating test of 2741 airfoils and 989 airfoils in the subsonic and transonic regimes, the mean differences of the surrogate models from RANS evaluations in the drag coefficient are 9 and 0.8 counts, respectively. We also validate the airfoil optimization for various cases by comparing with drag minimization based on direct RANS analysis. The largest differences in C d are 4 counts for subsonic cases, and 2.5 counts for transonic cases. This approach opens the door for interactive airfoil analysis and design using any modern computer or mobile device. I. Introduction Airfoil design plays a fundamental role in aircraft design. Components, such as wings, horizontal and vertical stabilizers in fixed-wing aircraft, and rotor blades in helicopters are designed using airfoil-shaped cross sections. A wide variety of airfoils have been developed for different applications. Airfoils used to be designed exclusively based on wind tunnel experiments, which are reliable but fairly expensive and time-consuming. In the last several decades, airfoils are increasingly designed using computational aerodynamics, sometimes with the aid of numerical optimization to automatically find the shape. There are several kinds of numerical models that can be used for airfoil aerodynamic analysis, including potential flow theory, Euler equations, and Reynolds-averaged Navier Stokes (RANS) equations. Aerodynamic analysis by potential flow theory can be performed rapidly with panel methods, but these do not consider viscous or compressibility effects. To obtain higher accuracy, potential flow theory can be coupled with a boundary layer model to model the viscosity, as done in the XFOIL [1] software, which is widely used in the analysis of subsonic airfoils. However, higher fidelity models, such as RANS, are required in the transonic regime. Despite the rapid increase in computational power, aerodynamic analysis using these methods is still time-consuming, especially when multiple design iterations are required. Airfoil design optimization requires a choice of shape parameterization to control the geometry. Generally, a parameterization method that requires few parameters and provides a high accuracy in representing airfoil geometries is attractive. Masters et al. [2] compared seven different airfoil parametrization schemes by considering the geometric shape recovery of over 2000 airfoils. In their comparison, they included class-shape transformations [3], Hicks Henne bump functions [4], singular-value decomposition (airfoil modes) [5, 6, 7, 8], B-splines, radial basis function domain elements, Bézier surfaces, and the parameterized sections method. They concluded that the airfoil modes method was the most efficient approach in them. Based on this study, we adopt a parametrization approach based on the identification of airfoil modes shapes, since a fewer number of parameters is of great help to the construction of a more accurate surrogate model. *Visiting Scholar, Department of Aerospace Engineering Post-Doctoral Fellow, Department of Aerospace Engineering Professor, Department of Aerospace Engineering, AIAA Associate Fellow 1 of 25

2 There are two broad categories of optimization methods: gradient-free [9, 10] and gradient-based [11, 12, 13] methods. Gradient-free methods usually need to evaluate thousands of different shapes when performing airfoil optimization, and thus the overall computational cost is prohibitive when using RANS. A common way to handle this issue is to use gradient-free methods together with surrogate models [14, 15, 16] that are constructed in advance and enriched during the optimization procedure. This type of method work well in small-scale cases, but problems of dimension disaster may be produced when they are used in high-dimensional problems. Moreover, these surrogate models are generally designed for the optimization of a specific airfoil and are trained in a perturbation subspace around an initial airfoil under a given objective and some constraints. They are not generic, and basically they are not suitable for the design of other airfoils. So a generic and accurate surrogate for fast aerodynamic analysis and optimization is of great interest. On the other hand, gradient-based optimization methods require fewer iterations to converge to optimal airfoils. Using the adjoint method [17, 18, 19], the computational cost to obtaining the gradient information is dramatically reduced compared to the finite-difference or complex-step methods [20]. Usually, a gradient-based airfoil optimization with an efficient adjoint solver converges to the optimal solution using dozens to hundreds of aerodynamic analyses. While RANS-based airfoil analysis can be done in minutes using 16 processors in a high-performance parallel computer and shape optimization can be performed within one or two hours, a much faster analysis would be desirable for a more interactive design process that can be done with much more modest hardware. To address the need of fast interactive airfoil analysis and design optimization, we adopt a data-based approach that provides accurate results for airfoils in both subsonic and transonic flow regimes. The main difference of this work from previous surrogate-based airfoil optimization literature is as follows: (i) the considered design space is much larger than what literature suggests, (ii) high accuracy of the surrogate model is required throughout the whole design space rather than in a neighborhood of an initial airfoil or near the optimum of a given objective function. We perform a singular-value decomposition (SVD) modal extraction method to derive mode shapes from existing airfoils to control the shape of airfoils in each flow regime. The modes for the subsonic regime is based on over 1100 existing airfoils taken from the University of Illinois Urbana Champaign (UIUC) airfoil database a, while the modes for the transonic regime is based on the NASA SC(2) airfoils. Unlike approaches that use full-airfoil modes [5, 6, 7, 8, 2], we compute separate camber and thickness modes to parameterize the airfoils, because they are more suitable for optimization problems with thickness constraints. The design space contains all the existing airfoils that are used for the decomposition, and we use a strategy to exclude abnormal airfoils that do not have a positive contribution to the surrogate model. Then, we create two databases containing more than one hundred thousand RANS samples with different shapes and flow conditions generated using an adaptive approach with an enriching Latin hypercube sampling (LHS) method to fill the subsonic and transonic design spaces. We use these samples to train a recently developed surrogate model, which combines gradient-enhanced kriging, partial least squares (GE-KPLS), and mixture of experts. The gradients for the RANS evaluations are computed using an adjoint method, which contributes towards the overall efficiency and accuracy of the surrogate model construction. Two global models composed of dozens of GE-KPLS models are constructed for the subsonic and transonic flow regimes. These surrogate models provide fast aerodynamic analysis and gradient computation. In order to perform rapid airfoil shape design optimization, they are coupled with a gradient-based optimizer, pyopt [21], which is a Python interface to efficient gradient-based optimizers like SNOPT (Sparse Nonlinear OPTimizer) [22]. We demonstrate airfoil design optimization for a wide range of airfoils under kinds of constraints can be done almost in real time without any call of the CFD solver. The rest of this paper is organized as follows: First, we introduce the derivation of airfoil mode shapes in Section II. Then, we describe how we define the geometry design space in Section III, and how the samples in our CFD databases are selected and evaluated in Section IV. We then introduce the data-driven approach including the surrogate model and the mixture of experts method in Section V. Finally, we verify this approach by comparing to CFD analysis in Section VI and to CFD-based optimization in Section VII. We end with a summary of the conclusions in Section VIII. II. Airfoils Shape Parametrization We have found that working in a design space defined by mode shapes as opposed to using control points works better when creating surrogate models. One of the reasons is that fewer parameters are required by this method to meet a certain geometry accuracy in airfoil shape representations [2]. In addition, the information from the airfoil mode shapes of existing airfoils provides a way to exclude abnormal airfoils that would never be useful in practice, like for example, airfoils with excessive oscillations in their shape. a database.html 2 of 25

3 We use SVD on the airfoil databases to find the modes that parameterize the airfoil shape. To cover the largest number of airfoils, we first investigate airfoils in the UIUC airfoil database and obtain about 1100 airfoils after some selection and smoothing operations. Then, we derive mode shapes for these airfoils to control airfoil shapes in the subsonic regime. In addition, we derive specific modes from a series of supercritical airfoils to control the airfoil in the transonic regime. In this section we start by discussing how to obtain the mode shapes, including full-airfoil, camber, and thickness modes from a given airfoil database (Section 1), and then we explain how the UIUC airfoil raw data is processed before the mode derivation (Section 2). 1. Mode Shape Derivation We can derive airfoil mode shapes through SVD using an airfoil database of m airfoils that share the same x coordinates. These modes can be found using a deformation formula, but due to the large geometry domain involved in this paper, the constructive formula is adopted. Assuming that airfoils are represented by n surface points (x i, y i ), i = 1,..., n, all the y coordinates values of each airfoil in the database are assembled in the snapshot matrix as Performing SVD on A, we obtain y 11 y y m1 y 12 y y m2 A =.. (1.1)..... y 1n y 2n... y mn A = UΣV T. Each column in U is one full-airfoil mode, and these modes are sorted by their importance. Not all the modes are required, so we use Φ to represent the first n f full-airfoil modes that we want to consider. For a new airfoil, the Y coordinates values can be expressed by modes Φ and their coefficients x f. Y airfoil = Φx f (1.2) Figure 1 shows the first 20 full-airfoil modes that are derived from the 1100 UIUC airfoils after the preprocessing operations described in Section 2. We can see that some modes, such as modes 3, 8, and 10, represent mostly camber, while others include camber and thickness information. However, we cannot attribute any design meaning in general to these modes. #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20 Figure 1. First 20 full-airfoil mode shapes derived from 1100 UIUC airfoils. 3 of 25

4 Another approach to representing an airfoil is to use the camber and thickness lines. Using the same airfoil database, the camber and thickness line of each airfoil can be represented by n h points (x i, y ci ) and (x i, y ti ), i = 1,..., n h. As in the full-airfoil mode derivation, all the y coordinates values in the camber and thickness line of each airfoil can be assembled in snapshot matrices A c and A t, respectively: y c11 y c21... y cm1 y t11 y t21... y tm1 y c12 y c22... y cm2 y t12 y t22... y tm2 A c =....., A t = y c1nh y c2nh... y cmnh y t1nh y t2nh... y tmnh Thus, the modes of camber and thickness lines, U c and U t, can be obtained by SVD as follows A c = U c Σ c V c T, A t = U t Σ t V t T. The first 10 camber and thickness modes are shown in Figure 2. #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 a) First 10 camber modes #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 b) First 10 thickness modes Figure 2. Camber and thickness modes derived from 1100 UIUC airfoils. When representing the y coordinates values of a new airfoil by the first n c camber modes Φ c and the first n t thickness modes Φ t, we have [ ] [ ] [ ] Yupper Φc Φ Y air f oil = = t xc = Φ Y lower Φ c Φ t x ct x ct. (1.3) t The rank of Φ ct is not necessarily equal to n c + n t. However, after many tests, we find that every camber and thickness mode cannot be represented by Φ t and Φ c, respectively. That is to say, if n f = n c + n t, rank(φ) = rank(φ ct ). This means that the orders of the geometry spaces spanned by the full-airfoil modes and camber-thickness modes are the same if the same number of modes are used. Nevertheless, the ability of these modes to represent airfoils might be not the same. From the mathematical point of view, full-airfoil modes would be the most accurate in representing the snapshot airfoils. To study the difference in airfoil representation, we define the error, G error = m Y (i) ori Y (i) mode 1 m Y (i) ori, (1.4) 1 i=1 4 of 25

5 which we use to evaluate the ability of the parameterization to represent the m UIUC airfoils in the database. In the above equation, Y (i) mode can be obtained from Eqs. (1.2) and (1.3), where x(i) f = Φ T Y (i) ori, and x(i) c = Φ T c C (i) ori, x (i) t = Φ T t T (i) ori. C(i) ori and T (i) ori are the y coordinates values of the camber and thickness lines in the ith airfoil, respectively Full-airfoil modes Camber-thickness modes G error The number of modes Figure 3. Accuracy of full-airfoil modes and camber-thickness modes in representing 1100 UIUC airfoils. The error (1.4) reduces as the number of modes is increased, as shown in Fig. 3. For a certain number of modes, there are several different combinations of n c and n t. We find the highest accuracy is achieved when: { nc = n t, if n c + n t is even (1.5) n c = n t 1, otherwise In Fig. 3, the combination of Eq. (1.5) is used for each total number of modes used in the camber-thickness approach. Both types of modes are equally accurate, especially for 10 modes or more. Thickness constraints Bounds of modes The second mode UIUC airfoils Test airfoils The first mode Figure 4. Optimization bounds of full-airfoil modes satisfying given thickness constraints 5 of 25

6 0.8 Test airfoils Bounds of modes The first camber mode Thickness constraints UIUC airfoils The first thickness mode Figure 5. Optimization bounds of camber-thickness modes satisfying given thickness constraints From the designer point of view, separating camber and thickness separately is more intuitive and practical. For example, in airfoil optimization, it is common to enforce thickness constraints. In full-airfoil modes method, each mode contains both camber and thickness information, so there is no simple way to bound thicknesses directly. For camber-thickness modes, however, there is a direct relationship to thickness and camber, and it is much easier to enforce bounds for a certain problem. As an example, consider a two-mode case shown in Figs. 4 and 5. The thickness does not increase along each mode for full-airfoil method in Fig. 4, and a practical design domain would have large bounds, as shown by the orange box. For camber-thickness modes, however, we can clearly see in Fig. 5 a more direct relationship between the modes and the corresponding thickness and camber characteristics. This is both more meaningful and helpful because smaller design space bounds simplify the optimization and the surrogate model construction. 2. Airfoil Data Preprocessing The UIUC airfoil database contains more than 1500 subsonic airfoils that are suitable for a variety applications. The raw UIUC airfoil data is not provided in a uniform format and not directly usable for our purposes, so this data requires some preprocessing. The preprocessing is composed of smoothing the shape when needed, rotating airfoils to make their angles of attack be zero, and changing the trailing edge so that all airfoils have a sharp trailing edge. Suppose we have n h discrete points (x 1, y 0 1 ),..., (x n h, y 0 n h ) on the upper and lower surfaces of a given airfoil. We use Laplacian smoothing by modifying the Y values of Eq. (2.1) as follows y k+1 i = ρ (x i ) (y k i 1 + yk i+1 ) + ( ρ (x i))y k i, 1 < i < n h, (2.1) where ρ (x i ) is a weighted function and its value increases (to a maximum of 5) with the rise of the curvature at point (x i, y i ). The modification is terminated when max y k 1 i n i y 0 i > ɛ. The treatment of the trailing edge is the same as h presented by Masters et al. [2], where all airfoils are modified to have a sharp trailing edge. Airfoils designed for the transonic regime (supercritical airfoils), have different characteristics than those designed for the subsonic regime, so it makes sense to derive different modes for these two regimes. After eliminating some unsmooth airfoils even after processing, about 1100 airfoils are left, which we use to derive the mode shapes for subsonic airfoils because most of them are designed for subsonic usage. The first ten camber and thickness modes that we derived are shown in Figure 2. 6 of 25

7 The modes for the transonic airfoils are based on 21 supercritical airfoils that NASA developed in their SC(2) series. Their thicknesses vary from 2 to 18 percent and the design lift coefficients vary from 0 to. The first five camber and thickness modes of these supercritical airfoils are shown in Figure 6. #1 #2 #3 #4 #5 a) The first 5 camber modes #1 #2 #3 #4 #5 b) The first 5 thickness modes Figure 6. Supercritical camber and thickness modes derived from the NASA SC(2) series airfoils. Considering manufacturing issues, airfoils are always associated with a blunt trailing edge in practice. However, the modes derived in the above approach can only lead to airfoils with a sharp trailing edge. One could use another variable to control the thickness of the trailing edge, but it would lead to one more design variable and make it more difficult to model the aerodynamics by surrogates. Another way to address this issue is to fix all airfoils with the same thickness on the trailing edge, but it might be not reasonable to make two airfoils with different thickness have the same thickness at the trailing edge. After investigating the SC(2) airfoils, we find that there is a relationship between the thickness of the trailing edge and the coefficient of the first thickness mode, from which we can determine the thickness of the airfoil. As shown in Figure 7, and based on the information in these 21 SC(2) airfoils, we propose a linear regression equation to determine the thickness of the trailing edge by the coefficient of the first thickness mode. That is, T T E = K T E C T1, where T T E is the thickness of the trailing edge, C T1 is the coefficient of the first thickness mode and K T E = 06. In this paper, we use this equation to determine the trailing edge thickness for all airfoils in both regimes. This approach makes airfoils have a reasonable trailing edge thickness without introducing an additional design variable Regressed K TE = T TE SC(2) airfoils C T1 Figure 7. The relationship between the first thickness mode and the thickness of the trailing edge III. Design Space Definition We now explain how we defined the airfoil design space using the modes that we computed in the previous section. There are two main questions: How many modes to use, and what the bounds should be on the mode amplitudes. 1. Number of Mode Shapes There is always a trade-off when selecting the number of geometry parameters in aerodynamic surrogate models. On the one hand, increasing the number of geometry parameters would decrease the missing information G error in representing airfoils, but on the other, if more geometry parameters are used, it would become more difficult for 7 of 25

8 surrogate models to capture the aerodynamic feature and, thus, the surrogate modes would be less accurate. In the transonic regime, due to complex flow phenomena such as multiple shock waves, it is difficult for surrogate models to capture the aerodynamic profiles. In Fig. 8, we show a test in the transonic regime with conventional modes where the surrogate model error becomes larger when predicting airfoils with increasingly strong double shock waves. C P Airfoil 20 Error of C D (count) Surrogate model θ (from p 1 to p 2 ) Figure 8. The performance of surrogate model in transonic regime (M = 0.73) along a line between two airfoils(p 1 and p 2 ) in the design space. The surrogate model is trained with 1000 LHS samples with GE-KPLS method. Three camber and four thickness modes are used in this test and the flow conditions are fixed to M = 0.73, =, and Re = In this paper, we define a design space for the subsonic regime and another for the transonic. Airfoils in the subsonic regime are controlled by modes derived from 1100 UIUC airfoils, which we call conventional modes, while in the transonic regime, airfoils are governed by supercritical modes derived from SC(2) airfoils. We set a geometry missing error of G error = % in both spaces, which guarantees an accurate geometry recovery. To satisfy this condition, seven camber modes with seven thickness modes and four camber modes with four thickness modes are required in the subsonic and transonic regime, respectively. 2. Mode Shape Bounds Our aim is to capture existing airfoils as many as possible, but this does not mean that we deal with all kinds of geometries. If the bounds of airfoil modes are too large, it is quite risky to involve abnormal airfoils in our database. These abnormal airfoils are usually meaningless, and they are useless in practice. These bounds, due to abnormal airfoils, would make the aerodynamic function profile more complex and, thus, make the construction of an accurate surrogate model more challenging. We determine the bounds of each mode by calculating the camber and thickness mode coefficients of each airfoil from the existing database. We set these as C ( j) R m, for j = 1,..., n c (for camber) and T ( j) R m, j = 1,..., n t (for thickness), where m, n c, and n t are, respectively, the numbers of airfoils, camber modes, and thickness modes. Then, the bounds of the j th camber and thickness mode are Cbound ( j) lower = min j) C( 1 i m i Tbound ( j) lower = min T ( j) 1 i m i,. (2.1) Cbound upper ( j) j) = maxc( Tbound upper ( j) = max 1 i m i T ( j) 1 i m i However, bounds determined by Eq. (2.1) are generally too large when kinds of airfoils are taken into account. One reason is that the distribution feature among parameters is neglected. The variation ranges of higher-order modes are generally constrained by lower-order modes which determine the outline of the airfoil shape. The first camber and thickness modes provide a rough representation of the airfoil shapes. For existing airfoils in UIUC database, two airfoils tend to have similar shapes if they are close in the first camber and thickness modes (as shown in Figure 9). Thus, we interpolate surfaces with respect to the first camber and thickness modes to capture the distributions of other 8 of 25

9 modes in existing airfoils, and then the upper and lower bounds can be defined by adding or subtracting a margin relative to these surfaces. The first camber mode The first thickness mode Figure 9. The distribution of over 1100 UIUC airfoils with respect to the first camber and thickness mode. Nevertheless, if we directly use the information in existing airfoils to interpolate a surface and shift it up and down to define the upper and lower bound surfaces, we would get rather non-smooth bounds, because these surfaces are kind of over-fitted due to kinds of noise. Also, we find that some parts of an upper bound are less than zero and others are greater than zero in a lower bound. Thus, in this paper, a margin strategy with smoothing operations is adopted to define the bounds on the mode shape amplitudes. For the first camber and thickness modes, bounds can be obtained by Eq. (2.1), i.e., lower and upper bounds are the minimum/maximum values of existing airfoils, respectively. While for the other modes, the upper and lower bounds are first defined by interpolated surfaces with respect to the first camber and thickness modes and then modified by a Laplacian smoothing method. For a specific mode except the first camber and thickness modes, we apply four steps to obtain their bounds: 1. Interpolate a surface function S using the data of existing airfoils and generate a discrete grid G i j (N i N j ) in the first camber-thickness plane and compute the surface function value S i j of the grid. 2. Generate the upper and lower bounds coordinates S upper i j and S i lower j = min { }, S i j. and S lower i j based on S i j, where S upper i j = max {, S i j } 3. Smooth the coordinates by the Laplacian smoothing method. For the upper bound, make sure S upper i j > max { }, S i j and S upper i j < max { }, S i j +margin, while for the lower bound, S lower i j < min { }, S i j and S lower i j > min { }, S i j margin. margin is defined to be 10% of the variance range of this mode in existing airfoils database; i.e., margin = 0.1 ( max { } { }) S i j min S i j. 4. Interpolate the upper and lower bound surfaces with cubic interpolation method. Inputs are G i j and outputs are and S i lower j for the upper and lower bounds, respectively. S upper i j In Figure 10, we compare the samples generated with different bounds of the modes. This test is conducted with the 14 conventional modes in the subsonic regime. In both kinds of definitions, the bounds of the first camber and thickness modes are both determined by Eq. (2.1). We first generate 150 normalized sampling seeds in a 14-D space by the LHS method. Based on the bounds defined in each approach, these sampling seeds are transferred to corresponding ranges. Then we use the transferred sampling coefficients to generate airfoils and plot them in Figure 10 based on their values of first camber and thickness modes. It can be noticed that there are some airfoils with strange shapes where 9 of 25

10 the bounds are defined by Eq. (2.1), while the sampling airfoils are much more reasonable where the margin strategy is used to determine the bounds. Bounds defined by Eq. (7) Bounds defined by smoothing margin strategy The first camber mode The first thickness mode Figure 10. Sampling tests with different bounds. IV. Airfoil Database Sampling Given that aerodynamic analysis and optimization will be done without any CFD calls, many airfoils filling the design space need to be sampled and evaluated in advance. We have constructed two aerodynamic databases for subsonic and transonic regimes. In the subsonic database, airfoils are controlled by 14 conventional modes (7 camber and 7 thickness), [ 2.0, 6.0], and M [0.3, ]. The air conditions are set as the standard atmosphere at the altitude of meters, so based on the variation of M, Re changes in the range of [ , ]. While for the transonic regime, 8 supercritical modes (4 camber and 4 thickness) are used to control airfoil shapes. [ 1.5, 4.5], M [5, 0.85], and Re [ , ]. Figure 11. CFD meshes of some airfoils generated by pyhyp. We adopt the LHS method to generate samples in the design space. Since the bounds of other modes are mutative with respect to the first camber and thickness modes, we first generate a normalized coefficient from to for each of these modes, and based on the bounds calculated by the values of first camber and thickness modes, these 10 of 25

11 normalized coefficients are transferred between the lower and upper bounds. We also adopt an adaptive enriching LHS strategy to add more samples based on the existing LHS samples. Before we conduct sampling, we divide the design space into sub-domains evenly with respect to M,, the first camber mode, and the first thickness mode. Then we generate the same number of sampling airfoils in each sub-domain, and meanwhile, the same number of validating airfoils are generated in each sub-domain by the enriched LHS algorithm. After running CFD simulations of these sampling and validating airfoils, validation tests are conducted in each sub-domain. If the error on these validating airfoils is not acceptable in one sub-domain, these validating airfoils would be added in the sampling airfoils and some other validating airfoils would be generated in this sub-domain by the enriched LHS method. Table 1. Description of two aerodynamic databases Subsonic Database Transonic Database Variable Number Shape Varibale Flow Variable 14 2 (7 camber + 7 thickness) (M+) 8 2 (4 camber + 4 thickness) (M+) Sample Number We use an in-house code pyhyp to automatically generate CFD meshes of the sampling and validating airfoils. Figure 11 shows the meshes of some airfoils generated by pyhyp. The RANS solver ADflow is used to do CFD simulations with these meshes. The Spalart Allmaras turbulence model [23] is adopted in these RANS simulations. The coefficients of lift, drag C d, and pitching moment C m (at the 1/4 chord point) are obtained after the CFD simulation. Meanwhile, the adjoint solver in ADflow is called to compute the gradient with respect to all modes, M, and because the gradient information is required for the construction of the surrogate model, and more details are given in Section V. Finally, we obtain two databases for subsonic and transonic regime, respectively. Table 1 lists the details of two databases. In total, we have sampled more than airfoils in two design spaces. V. Surrogate Model 1. GE-KPLS Gradient-Enhanced Kriging with Partial-Least Squares Surrogate models, also known as response surfaces, provide a low computational cost alternative to time-consuming aerodynamic models. It consists of approximate functions over a design space defined by input variables with a limited number of high-fidelity function evaluations. One of the most popular surrogate models is the kriging model [24, 25, 26, 27]. Kriging has been widely used in airfoil design [9, 28, 29]. In addition, kriging can be extended to utilize gradient information, called gradient-enhanced kriging (GEK), that improves the accuracy of the model in particular when the number of input variables is high. However, GEK is not computationally efficient when the number of inputs, the number of sampling points, or both, are high. This is mainly due to the size of the corresponding correlation matrix that increases proportionally with both the number of inputs and the number of sampling points. In addition, the correlation matrix becomes ill-conditioned when the sampling points are close to each other. Another difficulty is to estimate the hyper-parameters of the model which involves solving a multimodal optimization problem maximizing the likelihood function whose number of variables is proportional to the problem dimension. To address these issues, we use a new developed approach proposed by Bouhlel and Martins [30] called gradientenhanced kriging with partial-least squares (GE-KPLS). This method exploits the gradient information with a slight increase of the size of the correlation matrix and reduces the number of hyper-parameters. The key idea of GE-KPLS consists in generating a set of approximating points around each sampling points using the first order Taylor approximation method. We then apply the PLS method several times, each time on a different number of sampling points with the associated approximating points. Each PLS provides a set of coefficients that gives the contribution of each variable nearby the associated sampling point to the output. Next, we compute the average of all PLS coefficients with respect to each variable to get the global influence to the output. Denoting these coefficients by w (l) av = [ w (l) av1,..., ] w(l) avd for l = 1,..., h, where d is the dimension of the problem and h is the number of principal components of the PLS, and using the mathematical property that the tensor product of several kernels is a kernel, we finally build the GE-KPLS kernel h k 1:h (x, x ( ) = k l (F l (x), F l x ) ), x, x B, (1.1) l=1 11 of 25

12 where B is a hypercube expressed by the product between the intervals of each direction space, k l : B B R is an isotropic stationary kernel, and F l is a linear map given by F l :B In this paper, we use the Gaussian kernel given by k(x, x ) = σ 2 h l=1 d i=1 x B [ w (l) av1 x 1,..., w (l) avd x ] (1.2) d. [ ( ) exp θ l w (l) av i x i w (l) av i x 2 ] i, θ l [0, + ], x, x B. (1.3) The GE-KPLS method reduces the number of hyper-parameters to be estimated from d to h, where h << d, thus drastically decreasing the time to construct the model. As previously explained, we locally apply the PLS technique with respect to each sampling point, which provides the influence of each input variable around that point. The idea here is to add only n m approximating points (n m [1, d]) around each sampling point. To this end, we consider only the n m highest coefficients given by the first principal component, which usually contains the most useful information. Using this construction, we improve the accuracy of the model with respect to relevant directions and increase the size of the correlation matrix to only n(n m +1) n(n m +1) compared to n(d + 1) n(d + 1) with classical approaches, where n m << d. We use a 16-D subsonic problem (14 mode shapes, M, and ) to validate the advantage of GE-KPLS over Kriging and GE-Kriging methods. To reduce the design of experiments choice s influence on the results, we test each model for 20 times with different LHS sampling airfoils. The results shown in Figure 12 are the average of 20 tests in predicting 1000 validating airfoils, and demonstrate the advantage of GE-KPLS in accuracy and computation cost over Kriging and GE-Kriging, respectively Kriging Relative Error (%) GE-Kriging GE-KPLS Time (s) Figure 12. Comparison of 3 models in a subsonic case with 16 parameters. M [, ] and [, 2.0]. 2. GE-KPLS Coupled with Mixture of Experts GE-KPLS have an obvious advantage in computation cost when the number of sampling points is large. However, it is still difficult for one single GE-KPLS model to handle dozens of thousands of sampling points in a 10-D or 16-D problem. The heavy ill-condition issue would lead to crashes in training the model. To solve this issue, we adopt a mixture of exports strategy. Mixture of experts (ME) [31] is a combining method, which has great potential to improve performance of surrogate models. ME is established based on the divide-and-conquer principle in which the problem space is divided between a few experts. The original ME model was introduced in [32], and a cluster-based ME model was proposed in [33]. Typically, the general ME formulation can be expressed: K ŷ (x 0 ) = π k (x 0 ) ŷ k (x 0 ) k=1 with π k [, ] and K π k =. k=1 (2.1) 12 of 25

13 In Eq. (2.1), K is the total number of local experts and ŷ k (k = 1,..., K) is the k th local surrogate model. π k (x 0 ) is the mixing proportion, which depends on the location of x 0 in the design space. The output of ME is a weighted combination of K local models. The construction of a cluster-based ME can be divided into three steps: 1) Divide the samples into K clusters by an unsupervised clustering algorithm, and train local surrogate models with the samples in each cluster. 2) Use a supervised learning algorithm to obtain cluster posterior probability, which can be used to evaluate the mixing proportions. 3) Provide the weighted combination output using Eq. (2.1). In this paper, we adopt K-means algorithm to do the unsupervised clustering of the samples. The clustering inputs contain the first camber and thickness modes because these two modes are enough for the rough airfoil shape. To make the samples in the same cluster have similar aerodynamic features, the clustering inputs also contain the output function, C d or C m. Different clusters would be generated for different aerodynamic function to predict. For each aerodyanmic function, a 3-D K-means clustering is involved in dividing the samples. The aerodynamic coefficients are normalized before the clustering process to make them have a similar range with the two modes. The mixing proportion is typically derived based on the cluster posterior probability, i.e., the probability that the k th cluster is active at x 0. Similar with the approach introduced by Liem et al. [34], we use the regularized Gaussian classifier as the supervised algorithm to provide the cluster posterior probability. The regularized Gaussian classifier are trained by the samples labeled by the clustering results. It only takes the first two modes to train the classification and to predict unknown points. We use the modified softmax function proposed to obtain the mixing proportion[34]. The mixing proportion of the k th cluster at point x 0 is expressed in Eq. (2.2) and ω = 5.0. π k (x 0 ) = exp (ωa k (x 0 )) Kj=1 exp ( ), (2.2) ωa j (x 0 ) where a j = ln [ p ( x z j = 1 ) p ( z j = 1 )]. (2.3) Figure 13 shows the mixing proportion of 6 clusters in the subsonic design space. The clusters are trained by 5000 samples with M [, ] and [, 2.0]. Due to the clustering strategy described previously, different aerodynamic coefficients are associated to different clusters. This fiugre shows the clustering results for and C d. The first camber mode The first camber mode The first thickness mode The first thickness mode a) 6 clusters for b) 6 clusters for C d Figure 13. Mixing proportion π k plots. Different colors refer to different clusters/experts. The flow condition parameters, M and, have a distinct influence on the aerodynamic coefficients. Two samples of big difference in M or should not be divided into the same cluster even if they are close in geometry. In practice, we first adopt a hard split of the flow condition domain to make it into even parts, and then use K-means clustering in each part. The numbers of flow condition parts in the subsonic and transonic regimes are 9 and 12, respectively. The best number of clusters in each part is determined based on the performance of the ME model in predicting validation airfoils in the part. VI. Aerodynamic Analysis In this section, the surrogate model is validated by aerodynamic analysis of airfoils. To demonstrate the robustness, three different methods are used: 13 of 25

14 1. Analysis of random airfoils. These airfoils are generated by an enriching LHS sampling method to ensure they are not included in the training database. This test gives us a statistical insight of the surrogate model s performance. 2. Analysis of given airfoils. These airfoils are generated along lines across the design space. These lines are selected between two airfoils of distinct camber or thickness characteristics. This test demonstrates the accuracy of the surrogate model with the variation of airfoils camber and thickness. Meanwhile, it is also a good way to show which airfoils are in the design space. 3. Aerodynamic coefficients curves of some airfoils. The surrogate model is used to plot, C d, and C m curves along of some airfoils. This test is to show the performance of the surrogate model with the increase of. 1. Aerodynamic Analysis of Randomly Generated Airfoils 2745 and 1017 airfoils are generated by an enriching LHS method in the 14-D conventional airfoil design space and 8-D supercritical airfoil design space, respectively. For conventional airfoils, the flow conditions are M = 5 and = 2.5, while for supercritical airfoils, M = 0.73 and =. These airfoils are first evaluated by ADflow. Some of these CFD simulations do not converge finally since these airfoils are randomly generated and might not be suitable for the given Mach number and angle of attack. The numbers of conventional and supercritical validating airfoils that converge in ADflow are 2741 and 989, respectively. Then these converged airfoils are evaluated by the corresponding surrogate model. In the following, we use Eq. (1.1) to compute the relative error of the surrogate model. Relative Error = V CFD V 2 V CFD 2 100%, (1.1) where V CFD and V are the output vectors of ADflow and the surrogate model. Error (count) Relative Error 0.145% C d 55% C m 0.160% Figure 14. Validation of the subsonic surrogate model by aerodynamic analysis of 2741 randomly generated airfoils in the conventional design space Figure 14 illustrates the error of the subsonic surrogate model comparing with ADflow in evaluating, C d, and C m of 2741 conventional airfoils. On the left side of the figure, the absolute error in count (one count equals a,, or C m of 001) is shown by a scatter plot and a corresponding box plot. For, about 86.6% airfoils are predicted with an error less than 10 counts. For C d and C m, the percentages of validating airfoils with an error less than 1 count are about 99.6% and 63.8%, respectively. There are a few airfoils with relatively large error. For example, one airfoil is with a C d error about 10 counts. About six airfoils are given with a error more than 50 counts. Since the absolute value of, C d and C m are different, it is not fair to use absolute error to compare their performance. On the right side of the figure, we provide the total relative errot for the three coefficients, which are all less than 0.3%. We also find that it is more difficult to predict C d than and C m in the subsonic regime. 14 of 25

15 Error (count) Relative Error 04% C d 0.826% C m 0.966% Figure 15. Validation of the transonic surrogate model by aerodynamic analysis of 989 randomly generated airfoils in the supercritical design space Figure 15 shows the results in evaluating 989 supercritical airfoils in the transonic regime. The error here becomes bigger comparing with that in the subsonic test. This is due to the stronger nonlinearity in the transonic regime. For C d, the biggest error in the test is about 23 counts, while there are still more than 77% airfoils whose error is less than 1 count. For, there are 14 airfoils with an error over 100 counts, and more than 50% are associated with an error less than 10 counts. For C m, the percentage of airfoils with an error less than 10 counts is about 80.3%. The total relative error of, C d, and C m has increased by a factor of, respectively, 1.78, 2.23, and 5.03 comparing with the subsonic regime, while for the average error, the values have increased by a factor of 3.2, 8.9, and 6.7. Thus, it can be seen that surrogate models for C d and C m suffer more from the non-linearity in the transonic regime. Nevertheless, the performance of the transonic surrogate models remains acceptable. 2. Aerodynamic Analysis of Given Airfoils In this test, four airfoils of different camber or thickness characteristics are selected in both design spaces. These four airfoils are a) p 1 airfoil: large thickness, large camber; b) p 2 airfoil: small thickness, small camber; c) p 3 airfoil: large thickness, small camber; d) p 4 airfoil: small thickness, large camber. They are close to the 4 corners of the T 1 -C 1 plane (T 1 and C 1 refer to the first thickness and camber modes, respectively). Airfoils: Airfoils: ADflow ADflow C d (count) C d (count) C m C m θ (from p 1 to p 2) θ (from p 3 to p 4) a) Line 1: from p 1 tot p 2 b) Line 2: from p 3 tot p 4 Figure 16. Aerodynamic analysis of given airfoils along two lines through the conventional airfoil design space. Sampling airfoils along two lines from p 1 to p 2 and from p 3 to p 4 are analyzed by both ADflow and the surrogate 15 of 25

16 model (noted as in the figure). The modes coefficients of airfoils along these lines are linearly interpolated by the corresponding coefficients on the vertexes, so along these lines, the airfoil from one vertex to the other can be controlled by a factor θ [, ]. Along the line from p 1 to p 2, the camber and thickness both decreases with θ increasing. While along the other line, the camber and thickness increases and decreases, respectively. Some airfoils along these lines are demonstrated on the top of each sub-figure in Figs. 16 and 17. For conventional airfoils, M = 5 and =. The analysis results are shown in Figure 16. For and C m, the real coefficient values are shown, while C d is shown in count. The surrogate model has captured those function curves rather accurately. For C d, the error of most airfoils is within 0.1 count, and the biggest one is about count. The average error of and C m is 2 counts and count, respectively, while the largest ones are 11.2 counts and 1.5 counts. These distributions in error agree well with previous tests with random airfoils. Airfoils: Airfoils: ADflow ADflow C d (count) C d (count) C m 0.15 C m θ (from p 1 to p 2) θ (from p 3 to p 4) a) Line 1: from p 1 tot p 2 b) Line 2: from p 3 tot p 4 Figure 17. Aerodynamic analysis of given airfoils along two lines through the supercritical airfoil design space. Figure 17 shows a similar test in transonic regime with M = 0.73 and =. Airfoils are selected in the supercritical airfoil design space. The average values of error for, C d, and C m are 7.6,, and 2.8 counts, while the largest errors are 42.3, 1.2, and 8.9 counts. These error distribution also agrees well with that shown in Figure 15. From Figure 16, we also find that the coefficients of lift and nose-down pitching moment turn larger with the increase of the first camber mode which generally means more camber in the airfoil, while the thickness does not have an obvious influence on these two coefficients in the subsonic regime. Less camber or thickness generally results in a smaller C d. From Figure 17, we find a similar trend except for the variation of with θ [0, ] in Figure 17a. In this range, gets smaller with the increase of camber, because the thickness also increases and the shock waves are getting stronger. Meanwhile, there is an increasingly obvious flow separation on the upper side of the trailing edge. The flow in this range is rather complex. Nevertheless, we can still find that the output of the surrogate model agrees well with the one obtained by ADflow. 3. Aerodynamic Coefficients Curves Along The surrogate model is used to plot the aerodynamic curves of some existing airfoils with respect to. In the subsonic regime, three famous airfoils NACA4412, ClarkY, and NLF1015 are selected. The Mach number is 5 in this validation test. While for the transonic regime, three supercritical airfoils SC(2)0404, SC(2)0606, and SC(2)0710 are selected. The Mach number for the transonic validation is set to be To be noticed, since we have modified the existing airfoils to make them smoother and with an zero angle of attack before the airfoil mode derivation, airfoils used here are the modified ones subject to those smoothing and rotation modifications. 16 of 25

17 In Figs. 18 and 19, we find that for both subsonic and transonic regimes, the accuracy of the surrogate model keeps in consistence with increasing from the lower bound to the upper bound. The average error for, C d, and C m is, respectively, 33.9, 0.3, and 9.9 for the three airfoils with different angles of attack in the subsonic regime. While the average error in predicting the three supercritical airfoils is 40.1, 4.5, and 14.9 counts for, C d, and C m. The largest error in predicting C d (2.3 counts) in the subsonic regime occurs in NLF1015 when is around 3.5. This is due to the conjunction issue of different local experts in ME. As described in the previous section, different from the K-means clustering for geometry parameters, we adopt a hard-cut strategy in splitting and M domain. For the subsonic regime, we have divided the range of into 3 parts in the ME model, that is, [ 2, 7], [7, 3, 33], and [3.33, 6]. Similar situations also appear in the transonic regime in predicting and C m of SC(2)0710 when is around 2.5 and, respectively. In the transonic regime, we have divided the domain into [ 1.5, ], [, 2.5], and [2.5, 4.5], and the largest errors also occur in some of the conjunction parts. Despite of these conjunction issues in ME, the surrogate model is accurate enough ADflow C d (count) C m a) NACA ADflow C d (count) C m b) ClarkY ADflow C d (count) C m c) NLF1015 Figure 18. Aerodynamic coefficients curves obtained by ADflow and the surrogate model in the subsonic regime (M = 5) 17 of 25