Efficient multi-response adaptive sampling algorithm for construction of variable- fidelity aerodynamic tables

Maxim TYAN,Jae-Woo LEE

Department of Aerospace Information Engineering,Konkuk University,Seoul 05029,Korea

KEYWORDS Adaptive sampling;Aerodynamic database;Multi-response system;Surrogate modeling;Variable fidelity modeling

Abstract Adaptive sampling is an iterative process for the construction of a global approximation model.Most of engineering analysis tools computes multiple parameters in a single run.This research proposes a novel multi-response adaptive sampling algorithm for simultaneous construction of multiple surrogate models in a time-efficient and accurate manner.The new algorithm uses the Jackknife cross-validation variance and a minimum distance metric to construct a sampling criterion function.A weighted sum of the function is used to consider the characteristics of multiple surrogate models.The proposed algorithm demonstrates good performance on total 22 numerical problems in comparison with three existing adaptive sampling algorithms.The numerical problems include several two-dimensional and six-dimensional functions which are combined into singleresponse and multi-response systems.Application of the proposed algorithm for construction of aerodynamic tables for 2D airfoil is demonstrated.Scaling-based variable- fidelity modeling is implemented to enhance the accuracy of surrogate modeling.The algorithm succeeds in constructing a system of three highly nonlinear aerodynamic response surfaces within a reasonable amount of time while preserving high accuracy of approximation.

1.Introduction

In recent years,the use of surrogate models in aerospace and other engineering communities has grown significantly.The high demand for accuracy of the analysis methods used in engineering design or simulation necessitates the use of computationally expensive High-Fidelity(HF)analysis or a real experimental setup.Even though computational power is growing exponentially according to Moore's law,there is still a limitation of using HF analysis directly to solve engineering problems when many function calls are needed.Examples of such problems are design optimization,1-3generation of aerodynamic look-up tables for flight simulation,4and helicopter rotor blade design.5Surrogate modeling has its roots in structural design and optimization and has been applied to solve a variety of structural design problems.6It aims to reduce the number of simulations using efficient numerical interpolation,extrapolation,and design space exploration strategies.Surrogate modeling is a general process of constructing a computationally cheap mathematical model through either approximation or the interpolation of data spread over a certain domain.There exist a variety of surrogate models that include response surface methods,7splines,Bezier surfaces,Gaussian processes(Kriging),8Radial Basis Function(RBF)networks,9support vector machines,and others.10Introduction of automatic differentiation methods such as adjoint method11forced development of surrogate modeling algorithms that are able to efficiently utilize gradient information.Gradient enhanced Kriging method12-14uses values of the approximated function as well as its gradients to improve the approximation accuracy.The method requires significantly lower amount of sample points to construct an accurate model.

While surrogate models are responsible for the accuracy of approximation,design space exploration takes care of the distribution of sample points to construct a surrogate.Design of Experiments(DoE)is a discipline that studies the effects of parameter variation on a function response.In short,DoE is often used to generate the set of sample points distributed within a specific domain.There are many different algorithms capable of generating samples.Full Factorial Design(FFD)7is considered as the basic type of DoE;it covers the whole design space with a uniformly spaced grid.FFD provides a good distribution of samples at the cost of the large computational power required to evaluate a function and construct a surrogate model.FFD is rarely used for approximation of functions with more than three variables.Latin Hypercube Sampling(LHS)has recently become very popular for the design of computer experiments.15-17The number of samples is independent of the number of design variables.This property is very important when the work is related to costly analysis methods and grants additional flexibility in selection of the initial design.

The DoE methods discussed above do not use the information about the approximated function.These methods generate samples in order to maximize/minimize some specific metric,like minimum distance,determinants,or eigenvalues of corresponding information matrices.7Adaptive sampling methods use DoE for initial sampling(pre-sampling)and then iteratively update the surrogate model in the region of interest.18The Sampling Criterion Functions(SCFs)usually cover uniform sampling,the shape of an approximated function,and other factors.Jin et al.proposed the use of leave-one-out cross-validation error scaled by the distance to the nearest point.19Mackman and Allen used a function gradient and the Laplacian as a cost function for the placement of another sample.20Gaussian process based frameworks are also popular.Lee et al.used the variance of a Kriging model as an indicator of poor approximation.16Their approach for the construction of 3D flight dynamics tables shows good approximation of CFD-based aerodynamic functions.Da Ronch et al.also implemented a Kriging-based variance characteristic to cope with the use of multiple data sources for the construction of aerodynamic tables.4

The majority of engineering analysis tools are able to calculate multiple parameters in a single run.For example,once a CFD calculation has been completed,an engineer is able to extract the information about force and moment coefficients,pressure and velocity distributions, flow paths,and other parameters.Conventional adaptive sampling methods do not consider the multi-response nature of engineering tools.In that case,surrogate models can be constructed based on the SCF of a single function.There exist a few researches aimed at the refinement of multiple models at the same time.Liu et al.21extended the cross-validation Voronoi sampling method by including the weighted sum of the metric.Aute22treated the multi-response sampling problem as a multi-objective optimization and performed refinement of a surrogate based on the computed Pareto front.

During the past decade,a family of methods called Variable-Fidelity Modeling(VFM)has become widespread to improve the accuracy of approximation by introducing a secondary,Low-Fidelity(LF)analysis.23-25While surrogate modeling is a pure mathematical process,VFM uses a surrogate model to correct an LF function to approximate the HF one.LF analysis is a computationally cheap analysis method based on simplified physical principles with a greater number of assumptions and limitations compared to HF analysis.Combined use of the LF function with a limited number of HF sample points generally produces more accurate approximation than a surrogate model constructed with the same number of HF samples.The majority of VFM algorithms are based on conventional surrogate modeling techniques from Taylor series approximation26,27to RBF networks15,28and Kriging.16,24Surrogate models describe correlation between HF and LF functions.VFM can describe the difference between function responses.15In this case,algorithms can be grouped by type of scaling function.There exist additive,29multiplicative30and several types of hybrid12,24scaling functions.Space mapping method introduced by Bandler et al.31corrects input variables rather than function response.LF model is stretched and shifted to align the contours of a HF function.The approach proposed by Koziel and Leifsson32aims to find values B and q such that fHF（x） ≈fLF（Bx+q）.The authors used this method for airfoil shape optimization combining CFD analysis with different grid resolution.Kriging surrogate modeling method provides a good approximation in coupe with flexibility of the algorithm,and thus many variants of the algorithm exist,10including versions for VFM.Method called co-Kriging33uses cheap and expensive sets of data to construct surrogate model.The expensive model is approximated as the cheap one multiplied by a constant scaling factor plus a Gaussian process10.Han and G?rtz34proposed the modification of a universal Kriging for variable- fidelity use.A developed hierarchical Kriging replaces low-order polynomial regression of a universal Kriging with LF kriging model as the model trend of the main kriging approximation.It is important to note that not only the final output of a function can be approximated.Koziel and Leifsson35constructed a surrogate model of a pressure distribution instead of parameters of interest like lift or drag coefficient.Correction is applied to correct LF pressure distribution with the HF.The method shows its efficiency,but the major drawback is the requirement to adjust the method for each individual problem.VFM can be divided into two classes by application area:optimization and full domain approximation.The first class is used for design optimization problems.15,27,28The algorithms ensure that the points of the function's minima are properly located,and thus prediction of the general shape of the global design space and accurate approximation of the optimum point neighborhood are important properties.The second class requires accurate approximation of the whole domain.16Such algorithms can be used for the generation of look-up tables.The requirement for global approximation accuracy is strict in this case.

The current research proposes a novel approach for multiresponse adaptive sampling based on combining Jackknife variance prediction,the maximin distance metric,and the weighted sum of a single surrogate's SCF.The case study presents the construction of three aerodynamic tables for Clark-Y airfoil using the proposed algorithm and scaling-based VFM and its comparison with single- fidelity aerodynamic analysis and Kriging-based multi-response sampling.Potential flow and Reynolds-Averaged Navier-Stokes(RANS)CFD solvers are used as the LF and HF functions respectively.

2.Methodology

This section describes the proposed approach for multiresponse adaptive sampling including details of the algorithm and the numerical demonstration.A comparison of the algorithm with the modified single-response method and pure space- filling approaches is also presented.

Fig.1 shows the process of the proposed multi-response adaptive sampling algorithm.All the methods used in this process are general,that is,they can be applied to any type of surrogate modeling technique which provides an interpolation option.In this research,the Gaussian process interpolation of the open-source scikit-learn library is used.36The library is the implementation of Kriging algorithm by Rasmussen and Williams.37

The process starts with the generation of initial samples.The typical method used for construction of the initial surrogate model is LHS,FFD,or a combination of the two.The number of samples at the initial stage may be relatively small.The main purpose of initial sampling is to generate the points in the whole domain in order to capture the global behavior of the approximated function.In this research,the use of a combination of optimal LHS and two-level FFD(corner points)is proposed.The optimal LHS provides a uniform distribution of the samples within the design space.An important property of LHS is that the number of samples is independent of the number of variables.By using the optimal LHS,it is possible to preserve flexibility in the selection of initial samples while preserving the space- filling requirement.Two-level FFD adds samples at all corners of a domain.These samples are added to prevent extrapolation at the corners.Since the goal of adaptive sampling is an accurate approximation of the whole domain,extrapolation is highly undesirable.An example of an initial DoE with fifteen LHS and four FFD points is shown in Fig.2.

All the variables and function values are normalized to the range［-1，1］.There are two main reasons for normalization.First,a priori knowledge of the range reduces the probability of obtaining a degenerated surrogate model.38Second,normalization provides a common accuracy and convergence metric for all functions in a system.Normalization of function responses is performed using the minimum and maximum values of the function for the initial samples.The normalization bounds are evaluated once after the initial sampling stage and do not change during iteration.

Fig.2 Example of initial sampling and evaluation grid.

Fig.1 Jackknife-maximin distance adaptive sampling algorithm.

In the next stage,the variance of the surrogate model is calculated.Jackknife is a statistical Cross-Validation(CV)based method.39,40Jackknife is a more advanced version of the conventional leave-one-out CV technique.Unlike conventional CV methods,Jackknife predicts the variance of a model as a continuous function(at any point of a domain).It also predicts the mean and the variance of the surrogate model σ.These properties make the method more convenient for use as an SCF for adding new samples.

SCF is an artificial function that represents the metric for refinement of a surrogate model.New samples are added to the location at which the value of the SCF is maximized.SCF is a highly nonlinear and multimodal function.Gradient based optimization algorithms fail to find the maximum.41Surrogate model is a computationally cheap method.It is possible to perform thousands of calls within several seconds.The most robust and efficient way to find the location of its maximum is through a grid search.16The values of the SCF are evaluated on a fine evaluation gridXevalwith resolution from 50 to 200 points.Fig.2 shows the example of evaluation grid with resolution of 25 points.

The convergence criteria of the algorithm are the maximum number of samples and the mean Jackknife variance.The mean is calculated on an evaluation grid as

where neis the number of evaluation points.If the convergence criterion is not met,the SCF C（x）is calculated.The SCF must take into account the simultaneous behavior of all approximated functions.It is proposed that a weight factor be used to account for the mean and maximum variances of each function in a multi-response system.

The weight factor gives more preference to a function that has higher mean and maximum variance,and thus it forces both global and local refinement.In order to avoid clustering of the samples,a maximin distance(maximizing the minimum distance)metric is also implemented.42d（x）is the minimum distance from x to the closest sample point.

The SCF then takes the form

where nsis the total number of evaluation points,and nfis the number of response functions.An example of an SCF for a single function f（x）=x·sinxand its components is shown in Fig.3.The Jackknife variance provides a good metric to indicate whether refinement is needed near the sample point,while the minimum distance restricts the clustering of samples.

Fig.3 Proposed sampling criterion function for f（x）=x·sinx.

A new sample point is added to the location of the SCF's maximum.Iteration continues until one of the convergence criteria is met.

It is important to track the accuracy of a constructed surrogate model.Ideally,a separate set of data is used to calculate the error.In many cases,the use of an additional data set is a luxury.Jackknife is a procedure proposed by Quenouille43to calculate the bias of an estimator using the“l(fā)eave-one-out”CV technique.The concept of CV,which is popular nowadays,proposes the iterative construction of a surrogate model excluding one of the data sets and the calculation of the differences between the model constructed with and without the test set.There are different ways of dividing the data into training and testing data.44The K-fold strategy randomizes the samples,divides them into k-equal parts,and then uses one set for testing and the other for training.The leave-p-out strategy also randomizes samples and then uses（ns-p）samples for training and p samples for testing.The most accurate and computationally expensive variant of the leave-p-out strategy is leave-one-out,where p=1.A surrogate model must be constructed（ns+1）times.In this research,the use of the leave-one-out strategy for calculating the Jackknife variance is proposed.However,if the computational load becomes too high,p can be increased.

In order to calculate variance of the surrogate model,the concept of pseudo-value is introduced.The pseudo-value of the Jackknife^fj（x）at a given point x is the difference between the surrogate model value constructed with all the samples and that constructed without the jth set.

The variance of these pseudo-values is the estimate of the surrogate model's variance.

Then the 95% confidence interval is

An example of the Jackknife confidence interval is shown in Fig.4.

3.Numerical demonstration

3.1.Two-dimensional problem

Four two-dimensional numerical functions were used to demonstrate the efficiency of the proposed algorithm.Four single-response problems and ten multi-response systems are demonstrated.

Six-hump camel back function(Function 1):

Styblinski-Tang function(Function 2):

where d is the number of dimensions.In this research,d=2.

Currin exponential function(Function 3):

Branin function(Function 4):

The coefficients of the Branin function area=1，b=5.14/π2，c=5/π，r=6，s=10，and t=1/（8π）.

Fig.5 shows all the benchmark functions.The functions have different behavior.Functions 1 and 2 are multimodal and have highly nonlinear shapes.Function 3 is flat in most of the domain with an abrupt descent near x1=0.

Fig.4 Jackknife confidence interval of f（x）=x·sinx.

The proposed algorithm was tested for single functions and 10 multi-response systems using different combinations of the four functions.Ten initial samples were generated using LHS and additional four FFD samples were generated on the corners.Four different types of algorithms for multi-response adaptive sampling were used in benchmarking.LHS with optimal spacing is the pure non-adaptive space- filling DoE method.A full set of LHS points is generated using JMP software.45A Kriging-variance based adaptive sampling method is modified for use in multi-response systems.A single-response Kriging-based sampling4,16inserts new point where the Kriging variance is maximized.In the modified version,the largest variance among all the surrogate models is identified.The third method is maximin distance design,42where a new sample is inserted at the most distant location from other samples.The fourth method is the proposed Jackknife-maximin distance sampling.The SCF is calculated on a 100×100 evaluation grid.The algorithm terminates when σ-becomes less than 10-3.The second convergence criterion of the maximum number of samples is disabled in the numerical demonstration.

Table 1 shows the results of surrogate model construction for a single function and multiple functions.Results of numerical demonstration for single functions are discussed first.The optimal LHS shows the worst performance with an average of 80.25 samples to achieve the required mean variance.The generation of optimal LHS samples requires extensive integer optimization.46The typical selection is Monte Carlo and simulated annealing algorithms.47Some computational effort is needed to guarantee the optimal spacing for a large number of samples.Thus,there are two main reasons why the optimal LHS has the worst performance:unused information about function shape and possible poor convergence of the integer optimization algorithm when number of samples is large.The maximin distance method is an adaptive method.refinement of the existing optimal LHS samples may guarantee true space filling.The performance of the method is better than that of optimal LHS with an average of 74 samples for the construction of a single response surrogate model.The two methods that use different types of variances show the best performance among the four.The Kriging-variance based sampling uses an average of 66.75 sample points and the proposed Jackknife-maximin distance method uses 63.Jackknife cannot directly locate the point of maximum prediction error.Instead,it indicates whether there is a region of a surrogate that is highly dependent on a single sample.

The results of multi-response systems show a trend similar to that of a single function approximation.The Jackknifeminimax distance and Kriging-variance based methods show the best performance,with the proposed method having a slight advantage.It is also observed that the number of samples required for multi-response systems is equal to the number of samples for a single function with optimal LHS and maximin distance methods.The Kriging variance algorithm was modified for multi-response use and the proposed Jackknifemaximin distance was developed for that purpose.

3.2.Multi-dimensional problem

This section demonstrates the benchmark of the proposed three multi-dimensional numerical functions.Three sixdimensional numerical functions are selected.Total six numer-ical problems were solved for single and multi-response systems.The functions used are

Fig.5 2D benchmark functions.

Table 1 Results of numerical demonstration for a two-dimensional problem.

Rosenbrock function(Function 5):

evaluated at xi∈ ［-2，2］，i=1，2，...，6.

Dixon-Price function(Function 6):

evaluated at xi∈ ［-10，10］，i=1，2，...，6.Zhou function(Function 7):

where φ（x）=（2π）n/2exp（-0.5||x||2）, evaluated at xi∈ ［-0，1］，i=1，2，...，6.

25 initial samples were generated using LHS.The highdimensional approximations are widely used in design optimization and rarely for construction of a full database due to extreme computational load as discussed previously.The termination condition for this numerical example is set to σ to be less than 10-2,and thus the general shape of approximation can be captured accurately.

Table 2 shows the results of surrogate model construction for single-response and multi-response functions.The hierarchy of the algorithm performance results is similar to that of a two-dimensional problem.Optimal LHS shows the worst performance on a six-dimensional benchmark problem,while two algorithms that employ the variance prediction perform best.

All the algorithms show similar performance of singleresponse and multi-response systems.It is clear that the two methods that do not use the information about the function shape have the worst performance.The new method performs better due to combined usage of space filling,Jackknife variance,and multi-response weights.

4.Generation of aerodynamic table for Clark-Y airfoil

The generation of aerodynamic tables is an essential problem for many fields of aerospace engineering.Two-dimensional airfoil tables are primarily used for analysis using Blade Element Theory(BET)and its modifications.BET is widely used for rotary wing analysis,such as helicopter blades,propellers,and wind turbines.X-Plane and YaSIM flight dynamic models also use airfoil tables and BET for simplified 3D wing aerodynamic analysis.Advanced HF simulators use more sophisticated full aerodynamic tables48that describe the effects of static,dynamic,and control behavior of an aircraft.A typical practice is to model each of six force and moment coefficients using a look-up table build-up method.The total coefficient is decomposed into several 1D,2D or 3D tables.It is then estimated as a sum of these tables.For example,Pamadi et al.49constructed the total force coefficient for X-34 vehicle using a basic 2D table for angle of attack and Mach number and several incremental 1D tables for the effect control surface deflections.Da Ronch et al.4constructed the basic force coefficient using the 3D look-up table complemented by several 2D and 3D tables of force coefficient increments due to angular rates,and control deflection.Flight simulation software rarely uses aerodynamic look-up tables with number of dimensions(variables)more than 348-50due to complexity in generation of high-dimensional tables and implementation of them for flight dynamics models.In this research,Clark-Y airfoil table generation is used to demonstrate the developed approach for multiresponse adaptive sampling.Comparison of the new algorithm with the existing Kriging-variance based algorithm is performed for construction of aerodynamic tables with implementation of single- fidelity and variable- fidelity analysis tools.Detailed convergence of the Jackknife-maximin distance algorithm with variable- fidelity analysis is also discussed.Clark-Y is a classical airfoil that is often used for propeller sections.The lift,drag,and moment coefficients at a range of angles of attack and Mach numbers compose the three aerodynamic tables in C81 format.The majority of aerodynamic analysis tools compute all three coefficients in a single run.Thus,a multi-response adaptive sampling algorithm can be implemented.

The scaling-based VFM approach15is implemented to combine an HF ANSYS Fluent RANS CFD solver with lowfidelity Javafoil analysis.VFM uses scaling functions to correct(scale)the LF function to match its values with the HF function.The approximation of a HF function is performed as

Table 2 Results of numerical demonstration for a multi-dimensional problem.

Here,γ（x）is the additive scaling function.The additive scaling function is the error between the HF and LF functions.

Values of the scaling function are evaluated at sample points.A surrogate model γ～（x）of the scaling function is then constructed.Approximation of the HF function is then shown in Eq.(15).This approach generally produces more accurate and physically meaningful approximation than direct approximation of the HF function.10,15

Fig.6 shows an example of a lift coefficient VFM based on the additive scaling model and comparison to ordinary Kriging approximation constructed with only four samples.The aerodynamic coefficients in the figure are estimated under Mach number of 0.1 and sea level atmosphere conditions.Kriging is a pure mathematical approximation,while VFM follows the general shape of an LF function and matches its values with the HF samples.The difference can be clearly observed at α ∈ ［5°，20°］,where the VFM curve follows the HF curve almost exactly with only two samples in that region.

Initial samples are generated on a domain bounded by a Mach number of 0.1-0.8 and angles of attack of-20°to 20°.Ten optimal LHS and four FFD initial samples are generated as discussed in the methodology section.The HF analysis is represented by an ANSYS Fluent RANS CFD solver,evaluated on a 301×101 C-type structural mesh with the Spalart-Allmaras turbulence model.A single function evaluation takes approximately 30 min on a desktop PC with i7-4770 CPU and 32 GB of RAM(see Fig.7).

Javafoil software is used for LF aerodynamic analysis.Javafoil is a relatively simple program that uses potential flow with boundary layer analysis to predict the aerodynamic characteristics of an airfoil.The software uses Karman and Tsien subsonic compressibility correction to account for Mach number variation.The LF analysis is valid under subsonic flight conditions and has large prediction errors under transonic flow.

Fig.8 shows aerodynamic coefficients for Clark-Y airfoil at low subsonic speeds of Mach number 0.1,and transonic speed of Mach number 0.8.LF prediction stays relatively accurate at low speeds for lift and drag curves.LF also preserves its shape at higher speeds,while the HF analysis captures transonic effects and the curves become different.It is also noted by several authors that pitch moment coefficient database is more complicated to construct and requires separate treatment.4,51

Fig.6 Example of scaling-based VFM.

Fig.7 High- fidelity aerodynamic analysis.

After calculating the scaling factors for the lift,drag,and moment coefficients,the initial VFMis constructed.Then the iterative refinement process starts with the procedure explained in the methodology section until the convergence criterion of σ-≤10-3is achieved.

Airfoil aerodynamic table is constructed 4 times with HF and VF functions,and Jackknife-maximin and Kriging based multi-response sampling techniques.

Fig.9 shows the convergence history of all four cases.14 initial samples are generated first.The proposed Jackknifemaximin algorithm converges in 87 iterations in case of VFM and in 101 iterations with pure HF function.Multiresponse sampling for Kriging case converges in 117 and 120 iterations.It can be noted that the new algorithm's error varies significantly in first 50 iterations,which indicates radical changes of the surrogate's shape.Opposed to that,Kriging based algorithm behaves more like pure space- filling algorithm.

Computational cost of all cross-validation based algorithms is a hot discussion topic.Fig.10 shows time required by the proposed algorithm.Jackknife-variance computation time is calculated as total time for a single iteration of the algorithm excluding the CFD running time.Curve on a secondary axis shows the time required to construct single surrogate model with given number of samples.

Fig.8 Prediction of aerodynamic coefficients using HF and LF analysis.

Both curves show almost linear dependency of computational time number of sample points.The algorithm requires 56 s for the final iteration and 2618 s(43.6 min)in total.Kriging based algorithms require only single construction of a surrogate,and computational time for adaptive sampling algorithm is negligibly small.Airfoil table generation case study shows additional 30 evaluations of CFD analysis performed by Kriging based algorithm (87 versus117 iterations).In this particular case,the proposed Jackknifeminimax distance algorithm is computationally superior when HF analysis requires more than 87 s.Typical HF analysis requires much longer time.The computational cost may become an issue for a very large number of sample points and design variables.However,the cross-validation based methods fi t well with problems related to construction of aerodynamic tables as discussed earlier in this section.

Fig.9 Convergence history of airfoil aerodynamic table construction.

Fig.10 Computational time of Jackknife evaluation.

The calculation terminated after 87 iterations with a total of 101 HF function evaluations.Fig.11 shows the variation of weight factors through the calculation.The figure shows that the lift and moment weight factors dominate.Fig.12 shows the highly nonlinear lift and moment surfaces,while drag is less complicated to approximate.From here,it can be concluded that SCFs of the lift and moment functions have a higher influence on selection of a new sample.

Fig.11 Variation of weight factors.

Fig.13 Convergence of variances for airfoil table construction.

Fig.12 Airfoil coefficient surfaces and sample points.

Fig.13 shows the oscillatory nature ofˉσ until iteration 50.The algorithm performs global refinement at that stage by locating important nodes that may change the behavior of the model.In later iterations,the variance constantly decreases;at this stage there is no point that may change the global variance of the model,and thus space- filling process is running.

Fig.12 shows the results of multi-response adaptive sampling for the Clark-Y airfoil.The constructed surrogate model almost exactly matches the HF function.The differences are barely noticeable visually.More samples can be observed near the edges of the domain.That is a known issue of all crossvalidation based algorithms,when error is overestimated near the edges.In the proposed algorithm,the product of Jackknife variance and minimum distance is used,and thus oversaturation is less possible.On the other hand,fewer samples are placed on a plateau in the middle of the domain for all three surfaces.The algorithm converged successfully with the real mean variances of 4.04×10-4，2.17×10-3，4.29×10-3for the lift,drag,and moment coefficients respectively.

5.Conclusions

An efficient algorithm for the construction of multi-response surrogate models is proposed.The algorithm uses the Jackknife variance and minimum distance metrics to create the sampling criterion function.Weight factors calculated using the mean and maximum variances of each surrogate model are implemented to calculate the multi-response SCF.The 11 combinations of four 2D functions,and 7 combinations of three 6D functions were successfully solved using the proposed algorithm,optimal LHS,the multi-response version of Kriging-variance based adaptive sampling,and the pure space- filling maximin distance algorithm.The proposed algorithm outperforms the other three algorithms in most of the numerical case studies.

The construction of airfoil aerodynamic tables is performed to demonstrate the use of the proposed algorithm for realworld problems.Lift,drag,and moment coefficient surfaces were created simultaneously at a range of Mach numbers and angles of attack.The potential flow solver Javafoil and the ANSYS Fluent RANS CFD solver are used as the HF and LF functions,respectively.Scaling-based VFM is applied to improve the quality of approximation and to reduce the computational load.All three surrogates are created using 101 samples with mean variances of 4.04×10-4，2.17×10-3，and4.29×10-3for the lift, drag,and moment,respectively.

Acknowledgments

This research was supported by the Konkuk University Brain Pool 2018 and the National Research Foundation of Korea(NRF)[Grant NRF-2018R1D1A1B07046779]funded by the Korean government(MISP).