Generation of surrogate-based aerodynamic model of an UCAV configuration using an adaptive co-Kriging method

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Generation of surrogate-based aerodynamic model of an UCAV configuration using an adaptive co-Kriging method Halil Kaya ∗,1 , Hakan Tiftikçi 2 , Ümit Kutluay 3 , Evren Sakarya 4 Turkish Aerospace Industries Inc., Ankara, 06980, Turkey

a r t i c l e

i n f o

Article history: Received 31 January 2019 Received in revised form 25 April 2019 Accepted 27 April 2019 Available online xxxx

a b s t r a c t This work presents a multi-fidelity aerodynamic modeling approach for the aerodynamic model generation of the NATO AVT – 251 Task Group’s UCAV configuration named MULDICON. The aerodynamic data were generated with CFD simulations. A large number of cheap but less accurate, hence lower fidelity, Euler simulations are coupled with a smaller number of expensive but more accurate, hence higher fidelity, RANS simulations. Although the current study is limited to one-dimensional data, findings of the study suggest that the demonstrated method has the potential of being useful in obtaining highresolution multi-dimensional aerodynamic models for modern aircraft with reasonable effort. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction The NATO AVT – 251 Task Group aimed re-design an effective and agile UCAV configuration (named MULDICON) [1]. It is known that, novel aircraft configurations, like MULDICON (Fig. 1), are flown by the on-board automatic flight control systems, which rely on the state of the art flight control algorithms. To assess the stability and control characteristics of these aircraft and design flight control algorithms, a high-resolution and adequate fidelity aerodynamic model that covers the entire flight envelope is essential. That is, to test the control laws in a simulation environment, the aerodynamic model should include the effect of each and every control effector and their interactions for the entire flight envelope. Computational fluid dynamics (CFD) methods are extremely useful during the early design phases of an aircraft as they allow the engineers to understand the principal phenomena about the newly designed aircraft. Numerical RANS methods are a valid highly accurate tool in the vicinity of the design point. Even for off design conditions there are examples where RANS methods are well validated within NATO community [1], [2], [3]. However, CFD simulations have to be supported by experimental data since the accuracy and reliability for off design conditions is diminished [4].

*

Corresponding author. E-mail addresses: [email protected] (H. Kaya), [email protected] (H. Tiftikçi), [email protected] (Ü. Kutluay), [email protected] (E. Sakarya). 1 Aerodynamics Specialist Engineer, UAV Group. 2 Guidance, Navigation and Control Senior Specialist Engineer, UAV Group. 3 Experimental Aerodynamics Chief, Department of Engineering. 4 Structural Dynamics & Aeroelasticity Specialist Engineer, Aircraft Group. https://doi.org/10.1016/j.ast.2019.105511 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

Fig. 1. MULDICON.

For early stability and control assessment and control law design, a comprehensive database is needed which can not be realized nowadays by use of highly accurate CFD methods due to limited time and resources. Therefore, Multi-Fidelity and ROM methods are needed. These conditions should ideally include effect of each independent variable and their interactions. The independent variables are the flow conditions (Mach number, angle of attack and angle of side slip) and deflections of control surfaces (including high lift devices). Such databases are traditionally built by the “one factor at a time” – OFAT approach, in a “Brute-force” manner [5]. Thus, all independent variables but one is kept constant at a time and the effect of each factor is collected sequentially within the data [6], [7]. OFAT approach, neglects the effects of combined changes of independent variables (and their interactions), with the assumptions of linear aerodynamics, superposition, quasi-steady

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Nomenclature C CD CFD CL CM D dc e h hf M mf OFAT RANS

Covariance matrix Drag force coefficient Computational Fluid Dynamics Lift force coefficient Pitching moment coefficient Difference Drag count (10 −4 CD value) error Basis of regression function High-fidelity Mach number Multi-fidelity One Factor at A Time Reynolds Averaged Navier–Stokes

flow and no interdependence of states and controls [6]. Obviously, from a non-linear flight simulation point of view, neither of these assumptions is desirable. Also, generating thousands of data points in OFAT manner is not cost and time effective. Thus, approaches other than OFAT are strongly supported among stability and control engineers. There are many studies available in the literature about the utilization of modern design of experiments methods to aerodynamic data and subsequent surrogate model generation, mostly based on wind tunnel tests, [7], [8], [9]. Also in recent years, interesting approaches have been made to make the most out of CFD, such as the application of system identification methods to “simulated flight tests” within the CFD domain [10] and using multi-fidelity data fusion approach in aerodynamic model generation for flight simulation purposes [5], [11], [12], [13]. It is anticipated that, for the flight mechanical analysis of UCAV configuration MULDICON of the NATO AVT 251 Task Group [1], a high resolution aerodynamic model will be required. The multifidelity data fusion approach may provide the required model with relatively small effort, in comparison to the OFAT approach. To assess the feasibility of multi-fidelity data fusion approach, this study was initiated with a single independent variable, i.e. angle of attack. The selected data fusion approach is co-Kriging method. An object oriented design and analysis of computer experiments (ooDACE)Matlab kriging toolbox, [14], [15], [16], was used for surrogate modeling of the data. The main idea behind co-Kriging approach is to get the trend of the model from “cheap data” and exact values from “expensive data”. However, for the co-Kriging method to yield reasonable models, the data sources need to be similar at least in trends, thus they need to be correlated at some level. In the present study, Euler simulations were employed to get “cheap data”, and RANS simulations were employed to get “expensive data”. So, it is expected that Euler simulations will provide the trends of the aerodynamics coefficients and RANS simulations will provide the exact values. Furthermore, the main features of the vortex dominated flow topology (vortex interaction, vortex breakdown etc.) of blended wing UCAV geometries can be captured by Euler simulations [17], [18], [19]. Reference [20], presents a computational study that investigated low-speed aerodynamic and flowfield characteristics of a representative blended-wing UCAV configuration. In the context of the study, Euler and RANS simulations utilizing different turbulence models conducted, and the results were compared with the results of wind tunnel experiments. When the results of the CFD analyses are considered, it is seen that Euler simulations were able to capture the main features such as nonlinear behavior for lift coefficient and polynomial behavior of drag

RMSD P Xc Xe Yc Ye

α β

ρ λ

σ  ˆ

Root Mean Square Deviation Smoothness parameter of correlation function Cheap sample locations Expensive sample locations Cheap data Expensive data Angle of attack Coefficients of regression function Scaling factor Weighting factor Standard deviation Correlation matrix Estimation

coefficient with respect to angle of attack. Furthermore, Euler was also able to capture the occurrence of two sharp pitch breaks, yet failed to predict the correct angle of attack values for these pitch breaks (it is assumed that correct value is provided by the experimental data). Thus, the references show that, Euler solutions for an UCAV can be used as the “cheap data” for co-Kriging approach, since they can predict the trends in the aerodynamic characteristics. Hence, a similar trend can be expected for MULDICON, and co-Kriging approach that utilizes a large number of Euler simulation results and a smaller number of RANS simulations can yield an accurate model for the aerodynamic coefficients. Any potential flow methods is not preferred to get cheap data, because vortex interactions that cause sudden rise of pitching moment can not be predicted by use of any potential flow methods, but by Euler methods. One issue of importance is the determination of the number and the locations of expensive data that are required to get a model with an acceptable level of accuracy. To solve this issue, a robust sample-point refinement strategy as defined in Reference [21] is employed in this paper. The strategy assumes that the model constructed based on multi-fidelity data is always more accurate than the model constructed based on high-fidelity data. If a very large number of high-fidelity simulations are available, both models would converge to the same result. Therefore, by adding new sample points at the locations with maximum difference between multi-fidelity and high-fidelity models, the difference is decreased and this procedure is repeated until the average difference is below a defined threshold value. The details and the success of the methodology are considered in the next sections. 2. Numerical simulations 2.1. Geometry and computational grid The present study was performed on the AVT-251 Task Group MULDICON geometry. Pointwise’s meshing software was utilized to generate both the inviscid and the viscous grid required to perform Euler and RANS simulations. Both grids were generated on the half geometry. The mean aerodynamic chord length is 6 m and the refence area of half geometry is 38.9 m2 . The surfaces of the wings were discretized using ≈240 K triangular elements. For the viscous grid, the height of the first viscous layer was 3e-6 m ensuring y+ < 1 over the airfoil surface. There were 35 prismatic layers generated with a growth rate of 1.25. The viscous grid contains≈25 million hybrid volume elements (Fig. 2). The inviscid grid contains≈5 million tetrahedral elements.

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Fig. 2. The viscous computational grid.

2.2. Flow solver Numerical simulations were performed using ANSYS Fluent Release 16.1. The steady-state simulations were conducted using cell-centered, implicit, pressure-based formulation. For the pressure-velocity coupling a Semi-Implicit Method for PressureLinked Equations (SIMPLE) has been applied. A second order upwind scheme was used for solving advection terms. For RANS simulations, diffusion terms were computed through a second order central difference scheme. The closure of the viscous RANS equations was achieved by using the Spalart-Allmaras one-equation turbulence model. The computational cost of cheap simulations was compared with the expensive simulations on a run time basis. The cheap simulations, on the average, took about 1/50 times shorter than the expensive simulations. 3. Data fusion and sample point refinement strategy Fig. 3. A one variable co-kriging example, [23].

It is a well-known fact that high fidelity aerodynamics analyses require high computational power and time. Hence, it may be infeasible to generate an aerodynamic model required for flight mechanical analysis based on large number of high fidelity analyses. Here, the aim is to generate a high accuracy aerodynamic model using minimum number of high fidelity simulations coupled with large number of low fidelity simulations. So, a methodology to fuse multi-fidelity data, and a strategy to achieve high level of accuracy with small number of high fidelity simulations are needed (Fig. 3). The methodology that is utilized to fusion multi-fidelity data in the study is co-Kriging. The theoretical formulation and different presentations of co-Kriging method are well documented in the literature, [12], [13], [14], [22] and [23]. The ooDACE toolbox, which is utilized in the study, employs the auto-regressive co-Kriging model of Kennedy et al. [22]. Actually, co-Kriging is a form that correlates multiple sets of data. Here, only, two sets of data, high fidelity and low fidelity (called also expensive and cheap) are considered. The formulation begins by concatenating the sample locations and the value at the locations to give the combined set of sample locations and sample values,

⎛

(1 )

Xc

⎜ . ⎜ ..   ⎜ ⎜ (nc ) Xc ⎜ X = ⎜ c(1) X= Xe ⎜ Xe ⎜ ⎜ .. ⎝ . (ne )

Xe

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

⎛

(1 )

Yc (Xc )

⎜ .. ⎜ . ⎜   ⎜ (n ) Yc ⎜ Y (X c ) = ⎜ c c(1) Y= Ye ⎜ Ye (Xe ) ⎜ ⎜ .. ⎝ . (n ) Ye (Xe e )

where X is the sample locations, Y is the values at the given locations, and n is the number of sample points. Moreover, the subscripts e and c stand for expensive and cheap, respectively.

 The auto-regressive model assumes that cov {Y e x(i ) , Y c (x) |





Y c x(i ) } = 0, ∀x(i ) = x. That means if the value of expensive function at a point is available, then the cheap function value at that point is totally ignored and the expensive function value is considered correct. The auto-regressive model approximates expensive function as the cheap function multiplied by a scaling factor plus a Gaussian process which describes the residual between expensive function and scaled cheap function, i.e. yˆ (x) = ρ y c (x) + y d (x). Thus, the method can be considered as building two Kriging models in sequence. Firstly, a model based on cheap data is constructed. After that, a second kriging model is built on the residuals of the expensive data and the scaled cheap data with a factor ρ . The value of scaling factor is calculated as part of maximum likelihood estimation of the second model. A general (employing universal Kriging) co-Kriging model is formulated as,



⎞

yˆ (x) = h (x)T βˆ + c T C−1 Y − Hβˆ

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

where C is the data covariance matrix is thus:



(1)

C=

σc2 c (Xc , Xc ) σc2 ρ c (Xc , Xe ) 2 2 2 σc ρ c (Xe , Xc ) σc ρ c (Xc , Xc ) + σd2 d (Xe , Xe )

(2)

 (3)





Since the model assumes that cov {Y e x(i ) ,

(i ) auto-regressive  i) ( Y c (x) | Y c x } = 0, ∀x = x, the value σc2 ρ c (Xe , Xc ) = 0. cT is the correlations of the point x,

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c (x)T =



σc2 ρ c (x, Xc ) σc2 ρ 2 d (x, Xe ) + σd2 d (x, Xe )



(4)

Moreover, correlation function assumed in the form,

⎛

ψ (i ) = exp ⎝−

k 

⎞

θ j | x j ) − x j |p j ⎠ (i

(5)

j =1

The universal Kriging treats the regression function as a multivariate polynomial, namely,

f ( x) =

r 

h i (x) βi

(6)

i =1

where h i (x) are i = 1, . . . , r are basis in the regression function and β = (β1 , . . . , βr ) denotes the coefficients of the regression function. We utilize the same basis with different coefficients for Kriging models. Thus, h (x)T = (ρ h (x)T , h (x)T ). The regression part is encoded in the model matrix H:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ H=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

T

h x1c



.. .

n T

1 T

h xc c

ρ h xe

.. .

ρ h xne e

⎞

0

T

.. . 0

T

h x1e



.. .

n

h xe e

T

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(7)

The coefficients of the regression functions are determined by Generalized Least Squares,

βˆ = (βˆ c , βˆ d )T = (HT C−1 H)−1 HT C−1 Y

(8)

Since cheap data is considered to be independent of expensive data, one can find maximum likelihood estimation for the parameters of kriging model of cheap data by maximizing,

−

nc 2

−

ln



σc2 −

1 2

2σc2

(9)

Maximum likelihood estimation for the parameters of the Kriging model, which describes the residual between expensive data and scaled cheap data, and the scaling factor, ρ , can be found by maximizing,

−

ne 2

−

ln



σd2 −

1 2

ln |d (Xe , Xe )|

(yd − βˆ d 1)T d (Xe , Xe )−1 (yd − βˆ d 1) 2σd2

 

(10)

where yd = y e − ρ y c , and if y c is not available at Xe , one may use the values of yˆ c (Xe ). In the literature, depending on the form of the regression function, Kriging has been prefixed with different names. A popular version is ordinary Kriging. It assumes a constant but unknown regression function, whereas a more general version, that is universal Kriging, assumes a general polynomial trend. ooDACE toolbox allows users to select the type of Kriging models. Hence, the successes of both kriging versions were considered. When employing universal Kriging, a second order regression function was utilized as global trend. Moreover, smoothness parameter of the correlation function was fixed at 2 and to maximize ln-likelihood, QuasiNewton Method was employed.

 

i i e i =  yˆ mf − yˆ hf  , i = 1, 2, . . . , n

(11)

i i where nt is the number of test points, yˆ mf and yˆ hf are the predicted value by the model based on multi-fidelity data and the model based high-fidelity data at the ithtest point, respectively. Since all error values at test points are obtained using constructed model, the computational cost is negligible. The initial data points are selected at extremums of angle of attack, i.e. boundaries of the problem and also mid-point of the domain. For refinement of the models based on high-fidelity and multi-fidelity data, the location with the maximum error is selected. The method is repeated such that the average error is below a certain threshold. In the present study, there will be three models to be constructed describing lift, drag and pitching moment coefficients. Therefore, an error definition considering total error of the models was made as follows,

e = λ1 e 1 + λ2 e 2 + λ3 e 3

(12)

Where subscripts 1, 2, 3 represent lift, drag and moment coefficient models, respectively and λ is the weighting factor. Equal weighting is applied, so all the weighting value, λ, is fixed at 1/3. Moreover, the average error in percentage is calculated as follows,

e=

ln |c (Xc , Xc )|

(Yc − βˆ c 1)T c (Xc , Xc )−1 (Yc − βˆ c 1)

To be able to achieve high level of accuracy with a small number of high-fidelity simulations, it is needed to use an efficient and robust sample-point refinement strategy. So, in the present study the methodology that is presented in the reference [21] was implemented. The method assumes that the model constructed based on multi-fidelity data is always more accurate than the model constructed based on high-fidelity data. Only when a very large number of high-fidelity simulations are available, both models will converge to the same result. Hence, the error between the model based on multi-fidelity data and the model based on high-fidelity data can be measured as follows,

nt 1 

nt

i i i ˆi ˆi (λ1 e1i / yˆ mf 1 + λ2 e 2 / ymf 2 + λ3 e 3 / ymf 3 ) × 100

(13)

i =1

4. Results and discussion The success of the approach is tested by modeling the aerodynamic coefficients of the AVT-251 Task Group MULDICON geometry, as a function of one independent variable that is the angle of attack. The aerodynamic coefficients were calculated running Euler and RANS simulations. The Euler simulation results are considered as cheap data, hence low-fidelity and RANS simulation results are considered as expensive data, hence high-fidelity. Moreover, a set of RANS simulations were conducted to validate the constructed model. Angle of attack was in the range from −10◦ to 20◦ . The Mach number was fixed at 0.2. The Reynolds number at mean aerodynamic chord (6 m) was 28 × 106 for the RANS simulations. RANS simulations, which were utilized to validate the model, and Euler simulations were conducted at one degree intervals (31 samples), whereas only 3 initial samples were employed for RANS simulations. Then, co-Kriging approach was utilized to model lift, drag and pitching moment coefficient values. As depicted in Fig. 4, the Euler simulations were able to capture the main features of the aerodynamic coefficients characteristics with respect to angle of attack. Thus, co-Kriging method is expected to generate a model with an acceptable level of accuracy. When Fig. 4 (a) and (b) are considered, there is a good agreement between co-Kriging predictions and validation data in lift and drag coefficients without necessity of refinement. Moreover, universal Kriging models of lift and drag coefficients based on

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Fig. 4. Comparisons of aerodynamic coefficient predictions, left ordinary kriging, right universal kriging.

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Fig. 5. Comparisons of aerodynamic coefficient predictions after refinement, left ordinary kriging, right universal kriging.

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RANS simulations are also in a good agreement with validation data. Since the behavior of lift and drag coefficient fits a second order polynomial, it is an expected result. The effect of implementing universal kriging can be clearly seen in Fig. 4 (b). Root mean square deviation of the universal kriging model from the validation data is 42 drag counts, whereas root mean square deviation of the ordinary kriging model is 398 drag counts. However, coKriging model of the drag coefficient is still in a good agreement between the validation data, because co-Kriging model predicts the trend using Euler simulation results and the exact values are obtained from RANS simulation results. Fig. 4 (c) shows the comparisons of pitching moment coefficient predictions. The behavior of pitching moment coefficient with respect to angle of attack is highly non-linear and co-Kriging method fails to accurately predict the pitching moment coefficient without refinement. Since the estimated value of scaling factor ρ is too large, the trend of Euler results is exaggerated by the co-Kriging models. The initial co-Kriging models were refined by adding new expensive samples adaptively at the test location with the maximum error. The process of adding samples continued until the average error defined in eq. (13) is below 5%. In case of the co-Kriging approach with universal Kriging, the error in lift coefficient was reduced to below the threshold value by just the addition of a single sample, where as 2 additional samples were enough for the error in drag coefficient to fall below the threshold. However, to achieve an average accuracy below the threshold value of 5%, it was required to add 4 expensive samples. This was due to the highly non-linear behavior of the pitching moment coefficient. In contrast, the co-Kriging model that employs ordinary Kriging did not converge to an average error below 5% by the addition of 4 more expensive samples (Fig. 5). It is required one more expensive sampling point to achieve the defined average error threshold in lift coefficient, and nine more expensive sampling points to achieve the defined average error threshold in drag coefficient. Ordinary co-Kriging required 14 more samples to achieve an average error below 5% (due to the non-linearity in pitching moment coefficient). Due to variable behavior of aerodynamic coefficient with respect to angle of attack, behavior of aerodynamic coefficients can be modeled better with a Kriging model that utilizes a global polynomial trend instead of a constant value. So, it is an expected situation that the difference between universal co-Kriging model approximations and the high-fidelity universal Kriging approximations are lower than the difference between ordinary co-Kriging model approximations and high-fidelity ordinary Kriging approximations. One crucial parameter is the achieved RMSD value. Addition of 4 new samples, yields RMSD values (from the validation data) of 0.006, 0.0017, and 0.0008 for lift coefficient, drag coefficient and pitching moment coefficient, respectively, in the case of co-Kriging model that uses universal Kriging (Fig. 5). Whereas, the RMSD values are 2 to 7 times larger (0.01, 0.0047, and 0.0056, respectively) in the case of co-Kriging model that uses ordinary Kriging (Fig. 5). Hence, it is concluded that universal co-Kriging model can achieve a desired level of accuracy using smaller number of expensive samples, when there is a varying trend. Generally, the behaviors of the aerodynamic coefficients with respect to other possible variables, such as angle of side slip, elevator deflection, aileron deflection etc., are non-linear. So, to generate an aerodynamic model, choice of the universal Kriging is expected to be superior to ordinary Kriging. 5. Conclusion The NATO AVT 251 Task Group focused on the tools and methods that can be utilized during the preliminary design of an aircraft. With this study, the application of multi-fidelity modelling

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approaches to aerodynamic model generation is demonstrated on MULDICON. Multi-fidelity surrogate modeling was found to be very useful in reducing the total computational time required for obtaining a high quality aerodynamic model. Adaptive sampling approach allowed reducing the number of RANS simulations (expensive data) at a minimum, thus helped further reducing the overall cost. Although, the current study demonstrated the application on a single dimension, i.e. angle of attack, the utilized approach can easily be extended to multi-dimensional aerodynamic models. As a future work, a multi-dimensional study is being planned to obtain a high resolution model for stability and control analysis of MULDICON. Declaration of competing interest No competing interest. References [1] R.M. Cummings, C. Liersch, A. Schütte, Multi-disciplinary design and performance assessment of effective, agile NATO air vehicles, in: AIAA Aviation and Aeronautics Forum and Exposition, 2018. [2] A. Schuette, D. Hummel, S.M. Hitzel, Numerical and experimental analyses of the vortical flow around the SACCON configuration, in: 28th AIAA Applied Aerodynamics Conference, Fluid Dynamics and Co-Located Conferences, 2010. [3] A.J. Lofthouse, M. Ghoreyshi, A. Jirasek, R.M. Cummings, Static and dynamic simulations of a generic UCAV geometry using the kestrel flow solver, in: 32nd AIAA Applied Aerodynamics Conference, AIAA AVIATION Forum, 2018. [4] A. Jirásek, R.M. Cummings, A. Schütte, K. Huber, Extended assessment of stability and control prediction methods for NATO air vehicles: summary, J. Aircr. 55 (2018) 623–637, https://doi.org/10.2514/1.C033818. [5] M. Zhang, M. Tomac, C. Wang, A. Rizzi, Variable fidelity methods and surrogate modelling of critical loads on X-31 aircraft, in: 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2013. [6] E.A. Morelli, R. DeLoach, Wind tunnel database development using modern experiment design and multivariate orthogonal functions, in: 41st Aerospace Sciences Meeting and Exhibit, 2003. [7] R. DeLoach, Tactical defenses against systematic variation in wind tunnel testing, in: 40th Aerospace Sciences Meeting and Exhibit, 2002. [8] J.A. Grauer, E.A. Morelli, Generic global aerodynamic model for aircraft, J. Aircr. 52 (2015) 13–20, https://doi.org/10.2514/1.C032888. [9] C.C. de Visser, J.A. Mulder, Global aerodynamic modeling with multivariate splines, in: AIAA Modeling and Simulation Technologies Conference and Exhibit, 2008. [10] J.P. Dean, S.A. Morton, D.R. McDaniel, S. Görtz, Efficient high resolution modeling of fighter aircraft with stores for stability and control clearance, AIAA (2007), https://doi.org/10.2514/6.2007-1652. [11] M. Ghoreyshi, K.J. Badcock, M.A. Woodgate, Integration of multi-fidelity methods for generating an aerodynamic model for flight simulation, in: 46th AIAA Aerospace Sciences Meeting and Exhibit, 2008. [12] Y. Kuya, K. Takeda, X. Zhang, A.I.J. Forrester, Multifidelity surrogate modeling of experimental and computational aerodynamic data sets, AIAA 49 (2011) 289–298, https://doi.org/10.2514/1.J050384. [13] M.P. Rumpfkeil, P. Beran, Construction of multi-fidelity surrogate models for aerodynamic databases, in: 9th International Conference on Computational Fluid Dynamics (ICCFFD9), 2016. [14] I. Couckuyt, T. Dhaene, P. Demeester, ooDACE Toolbox: a flexible object – oriented kriging implementation, J. Mach. Learn. Res. 15 (2014) 3183–3186. [15] I. Couckuyt, F. Declercq, T. Dhaene, H. Rogier, L. Knockaert, Surrogate-based infill optimization applied to electromagnetic problems, Int. J. RF Microw. Comput.-Aided Eng. (2010) 492–501, https://doi.org/10.1002/mmce.20455. [16] I. Couckuyt, K. Crombecq, D. Gorissen, T. Dhaene, Automated response surface model generation with sequential design, in: 1st International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering (CSC), 2009. [17] P.J. O’Neil, R.M. Barnett, C.M. Louie, Numerical simulation of leading edge vortex breakdown using an Euler code, J. Aircr. 29 (1992) 301–307, https:// doi.org/10.2514/3.46161. [18] K. Fujii, S. Gavali, T.L. Holst, Evaluation of Navier-Stokes and Euler solutions for leading-edge separation vortices, Int. J. Numer. Methods Fluids 8 (1988) 1319–1329, https://doi.org/10.1002/fld.1650081014. [19] S. Görtz, Unsteady Euler and detached-eddy simulations of vortex breakdown over a full-span delta wing, in: European Congress on Computational Methods in Applied Sciences and Engineering, 2004. [20] K. Petterson, CFD analysis of the low-speed aerodynamic characteristics of a UCAV, in: 44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meeting, AIAA, 2006.

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