The Effect of Multiple Additional Sampling with Multi-Fidelity, Multi-Objective Efficient Global Optimization Applied to an Airfoil Design

Abstract

The multiple additional sampling point method has become popular for use in Efficient Global Optimization (EGO) to obtain aerodynamically shaped designs in recent years. It is a challenging task to study the influence of adding multi-sampling points, especially when multi-objective and multi-fidelity requirements are applied in the EGO process, because its factors have not been revealed yet in the research. In this study, the addition of two (multi-) sampling points (2-MAs) and four (multi-) sampling points (4-MAs) in each iteration are used to study the proposed techniques and compare them against results obtained from a single additional sampling point (1-SA); this is the approach that is conventionally used for updating the hybrid surrogate model. The multi-fidelity multi-objective method is included in EGO. The performance of the system, the computational convergence rate, and the model accuracy of the hybrid surrogate are the main elements for comparison. Each technique is verified by mathematical test functions and is applied to the airfoil design. Class Shape Function Transformation is used to create the airfoil shapes. The design objectives are to minimize drag and to maximize lift at designated conditions for a Reynolds number of one million. Computational Fluid Dynamics is used for ensuring high fidelity, whereas the panel method is employed when ensuring low fidelity. The Kriging method and the Radial Basis Function were utilized to construct high-fidelity and low-fidelity functions, respectively. The Genetic Algorithm was employed to maximize the Expected Hypervolume Improvement. Similar results were observed from the proposed techniques with a slight reduction in drag and a significant rise in lift compared to the initial design. Among the different techniques, the 4-MAs were found to converge at the greatest rate, with the best accuracy. Moreover, all multiple additional sampling point techniques are shown to improve the model accuracy of the hybrid surrogate and increase the diversity of the data compared to the single additional point technique. Hence, the addition of four sampling points can enhance the overall performance of multi-fidelity, multi-objective EGO and can be utilized in highly sophisticated aerodynamic design problems.

1. Introduction

Due to the high computation time and costs of aerodynamic analysis, Efficient Global Optimization [1] (EGO) is one of the most popular approaches for aerodynamic shape optimization design. The strategy of EGO [2] is to combine a fitting response surface with a small sampling size and an arbitrary optimization method. The EGO process consists of the design of experiment (DOE), an evaluation, a surrogate model, and data improvement. The crucial process of EGO is data improvement or infilled sampling criteria, because a surrogate model is constructed with a small sampling size which might have an effect on the accuracy of a surrogate model. The addition of sampling points after the optimization process is essential for decreasing the inaccuracy of a surrogate model. The data improvement process or infilled sampling criteria are used to find the point in the area with a high probability of improvement, and that point is provided to update the surrogate model. Originally, the addition of sampling points requires the presence of only one sampling point in each iteration. Single-fidelity, single-objective, single additional sampling EGO (SF-SO-SA) was developed to handle a large number of geometric constraints using the Kreisselmeier–Steinhauser method [3]. This study focused on improving a surrogate model for a design with a large number of geometric constraints. However, using only one additional sampling point in each iteration did not fully improve the EGO entire process. Multi-fidelity, single-objective, single additional sampling EGO (MF-SO-SA) was developed for application in various aerodynamic-shape-optimization designs. The helicopter blade [4], supersonic wing [5], and rectangular wing [6] were successfully designed using MF-SO-SA EGO. Unquestionably, aerodynamic evaluation has a high computation cost and time, in the same way as Computational Fluid Dynamics (CFD). Studies have performed multi-fidelity methods for aerodynamics analysis to make hybrid surrogate models more precise, and to assist in ensuring that the computation time for the process is faster. Nevertheless, using only one additional sampling point in each iteration caused a failure of the opportunity to utilize other devices that were available when gathering the yield of a new sampling point. Additionally, MF-SO-SA EGO was developed using the fidelity of three or more variables [7] or using data mining techniques with the covariance matrix adaptation evolution strategy (CMA-ES) [8]. Adding one additional sampling point makes the algorithm stable, but it trades this with the slower convergence of the process. Single-fidelity, multi-objective, single additional sampling EGO (SF-MO-SA) is often used for real-world design problems, because multiple purposes are required in practical applications. The addition of a new sampling point for updating a surrogate model with multiple objectives is rather more difficult than with a single objective, because the objectives have increased dimensions [9]. Several researchers have improved on the infilled sampling criteria for multiple objectives; examples include statistical improvement criteria for multi-objective design [10], multi-objective expected improvement for aerodynamic design [11], differentiable expected hypervolume improvement for multi-objective design [12], and multi-objective particle swarm assisted by expected hyper-volume improvement [13]. These research studies tried to develop methods for the acquisition of data and employed proposed methods such as the expected hyper-volume improvement (EHVI) method, the dynamic expected hyper-volume improvement (DEHVI) method, the q-expected hyper-volume improvement (q-EHVI) method, etc. However, multi-objective problems are susceptible to divergence in computation. Therefore, one additional sampling point after the sub-optimization process was still used, because it was added carefully into the system and made the computation process stable. Moreover, SF-MO-SA EGO was applied to low-boom supersonic transport platforms and succeeded in obtaining optimal shapes [14]. However, this research was evaluated by CFD, and it took a long time, especially when analyzing the flow in supersonic regions, so one additional sampling point was also used. Therefore, this research suffered from computational cost and time. Multi-fidelity, multi-objective, single additional sampling EGO (MF-MO-SA) was used to design airfoils with two and three objectives [15]. This work had to handle multiple objectives, so multi-fidelity was provided to construct the hybrid surrogate model to reduce computation time. However, it was insufficient because of using one additional sampling point in each iteration. Hence, the addition of one additional sampling point is still a major problem for the computation time in an EGO process.

The addition of multiple sampling points is explored instead of one additional sampling point. Single-fidelity, multi-objective, multiple additional sampling EGO (SF-MO-MA) was verified by test function [16] and applied to alloy wheel design [17] and aerodynamic shape design in hypersonic flow [18]. These research studies used single fidelity, which caused the computational convergence to be slow. However, using multiple additional sampling points to update a surrogate model can assist in attaining a faster computational convergence rate, and a surrogate model became more accurate because multiple additional sampling points increased the diversity of data when a surrogate model was updated. Multi-fidelity, single-objective, multiple additional sampling EGO (MF-SO-MA) verified its algorithm by a test function and was applied to an axial flow compressor shape design [19]. The effect of adding multiple additional sampling points into MF-SO-MA EGO was studied in [20]. This research accomplished the use of multiple additional sampling points with multi-fidelity, which can enhance the overall performance of MF-SO-MA EGO, save computation time, and attain an optimal shape for the compressor. In addition, the results from both research studies revealed that it was possible to use four additional sampling points in each iteration and maintain the model accuracy of the hybrid surrogate. However, it was verified for only a single objective. Using the full EGO process, the multi-fidelity, multi-objective, multiple additional sampling (MF-MO-MA) was applied for airfoil design [21] and a micro-aerial vehicle fuselage design [22]. This research was successful in obtaining the optimal shape for its applications, increasing the computational convergence rate and maintaining the model accuracy of the hybrid surrogate with multiple purposes for a practical design. But this research was performed with only two additional sampling points in each iteration for the multi-fidelity application with and multiple objectives. Therefore, the influence of adding multiple additional sampling points with multi-fidelity, multi-objective calculations should be considered, especially when adding more than two sampling points in each iteration. Although multiple additional sampling points can accelerate the computation time of the EGO process, many factors that are caused by multiple additional sampling have not been revealed yet in other research.

Hence, the influence of adding multiple additional sampling points with a multi-fidelity, multi-objective EGO approach was proposed for study in this work. The additions of two (multi-) sampling points (2-MAs) and four (multi-) sampling points (4-MAs) in each iteration were applied to study the main proposed techniques and compare them against the results obtained from the single additional sampling point (1-SA) that is conventionally used for updating the hybrid surrogate model. The proposed techniques are verified by test functions and applied to airfoil design for practical applications in aerodynamic shape optimization design. The performance of the system, such as the optimal point or pareto front shape or stability of convergence, the computational convergence rate, and the model accuracy of the hybrid surrogate are the main elements of this work for studying the influence of adding multiple additional sampling points with the multi-fidelity, multi-objective EGO approach.

2. Efficient Global Optimization

Efficient Global Optimization is defined by Jonas Mockus [1] as an optimization method based on the minimization of the expected deviation from the extremum of a stochastic function. The EGO process consists of four main steps. Firstly, the input of such a problem is generated through DOE. Secondly, the output is collected by the evaluated function. Thirdly, the input and output data are compared to find a correlation and construct a surrogate model. The EGO strategy sees a surrogate model as a random function and adds a kernel function that conforms to the relationship between the input and output data. Lastly, the data improvement step is conducted to maximize the acquisition function that uses an arbitrary optimization method to determine what the next query point should be for the infilled sampling point. The original EGO is performed only for SA-SF-SO, as shown in Figure 1a, while MA-MF-MO is shown in Figure 1b.

2.1. Multi-Fidelity Requirements for the Hybrid Surrogate Model

The concept of multi-fidelity for the hybrid surrogate model is depicted in Figure 2 [2]. The input, x, is correlated to the output, f(x). This relationship is an expensive function known as a real function (solid black line). Then, the four initial points (red circles) are evaluated through high-fidelity evaluation and are constructed to become a high-fidelity function or Single-fidelity surrogate model (dashed black line). Due to the small sample size, a Single-fidelity surrogate model may not be able to capture the landscape of the real function. If this problem needs to be minimized, a Single-fidelity surrogate model cannot provide the global optimum solution. Hence, a multi-fidelity surrogate model is addressed. The 10 initial points (blue squares) are evaluated through low-fidelity evaluation and are constructed to become a low-fidelity function (dotted blue line). Due to the large sample size, a low-fidelity function can emulate the landscape of the real function better than a Single-fidelity surrogate model can. However, the exactness of the low-fidelity function has been found to be too poor when compared to a real function. The integration of the high-fidelity function and the low-fidelity function is achieved by maintaining the values of the high-fidelity data and developing a trend for the high-fidelity function by imitating the low-fidelity function to become a multi-fidelity function or a hybrid surrogate model (long-dashed red line). The trend of a multi-fidelity function is quite similar to that of a real function.

For the construction of the high-fidelity function, the Kriging model [23] is often used in soil science and geology for generating an estimated surface from a scattered set of points with the measured value at the $ith$ location. It is impossible to gather a large volume of data because of geographical features. Hence, the volume of data is limited according to a small sampling point for the high-fidelity function. The Kriging model calculates a global model at each point of the measured value; to keep an accurate black-box function available, the Kriging model also estimates the local deviation to ensure high accuracy in the surrogate model, because the global model is constant. Additionally, the Kriging model is time-consuming to build because of the maximization problem of the likelihood function estimation (MLE). The Kriging model represents a high-fidelity function as the value-estimated response, $y={\left\{y^{(1)},y^{(2)},\dots ,y^{(n)}\right\}}^T$, at the unknown design point, $x={\left\{x^{(1)},x^{(2)},\dots ,x^{(n)}\right\}}^T$ as:

$${\widehat{y}}_{Kriging}(x)=\mu +\epsilon (x)$$

(1)

where ${\widehat{y}}_{Kriging}(x)$ represents a high-fidelity function, $\mu$ is expressed as the constant global model, and $\epsilon (x)$ represents the local deviation from the global model at each point of a set of sampling points. The local deviation is expressed as follows:

$$\epsilon (x)=r{(x)}^T{R}^{-1}(f-1\mu )$$

(2)

where $r(x)$ is a vector that is expressed in terms of $x$ and is assigned as the sampling point, $R$ is a matrix that denotes the relationship between each of the sampling points, $f$ is a vector that obtains the response value of each sampling point and $1\mu$ is a mean of the random field in which $1$ is an $n x 1$ column vector of ones. The correlation between $y(x^i)$ and $y(x^j)$ is related to the distance between the corresponding points using the basis function expressions, $x^i$ and $x^j$, as follows:

$$Corr\left[y(x^i),y(x^j)\right]=\mathit{exp}\left(-{\displaystyle \sum _{k=1}^n{\theta }_k\left|x_k^{(i)}-x_k^{(j)}\right|}{p}_k\right)$$

(3)

The local deviation at each sampling point is expressed through stochastic processes. The input of x and the response of y are calculated and then interpolated using a Gaussian random function as the correlation function for the estimation of the trend of a surrogate model.

For construction of a low-fidelity function, the Radial Basis Function [24] is an interpolation function that approximates any smooth, continuous function that takes input and output based on the distance between the input value projected in space from a central fixed point. The RBF is one of the artificial neural networks (ANNs) which has only one hidden layer. The RBF is used as an activation function in the hidden layer. There is an input layer that obtains several neurons which represent design variables and the output layer, as the response value, has a weighted sum of outputs from the hidden layer to form the network output. The RBF can solve problems with large datasets that have complicated nonlinear distributions and only one hidden layer. Since it is simple to use, the RBF is an appropriate function that is used for surrogate modelling of low-fidelity data. The RBF represents a low-fidelity function as the value of interpolating response, $y={\left\{y^{(1)},y^{(2)},\dots ,y^{(n)}\right\}}^T$, at the unknown design point, $x={\left\{x^{(1)},x^{(2)},\dots ,x^{(n)}\right\}}^T$, as follows:

$${\widehat{y}}_{RBF}(x)={\displaystyle \sum _{i=1}^N{w}_i\cdot \mathit{exp}\left(-\psi {\left|x_i-c^i\right|}^2\right)}$$

(4)

where ${\widehat{y}}_{RBF}(x)$ represents a low-fidelity function, N is the number of inputs of the design variable of the low-fidelity data, $w_i$ is denoted as the weighted coefficient of the ith design variable, $\psi$ is the $c^i$ vector containing the values of basis function, and $c^i$ is the center of the basis function. $\psi$ is evaluated at the Euclidean distance between the prediction site $x_i$ and the center, $c^i$.

For the construction of a multi-fidelity function or the hybrid surrogate model, the combination of the high-fidelity and low-fidelity functions is activated, which means that ${\widehat{y}}_{Kriging}(x)$ is integrated into ${\widehat{y}}_{RBF}(x)$. The low-fidelity function from ${\widehat{y}}_{RBF}(x)$ has the contribution of the high-fidelity function from ${\widehat{y}}_{Kriging}(x)$. The mean value of the Kriging model is modified following the trend of the low-fidelity function, but still uses the value from the high-fidelity data. The hybrid surrogate model is expressed as follows:

$$\widehat{y}(x)=\mu +f_{RBF}(x)+r{(x)}^T{R}^{-1}(f-1\mu -1f_{RBF}(x))$$

(5)

where $\widehat{y}(x)$ represents the hybrid surrogate model from each design point, $x={\left\{x^{(1)},x^{(2)},\dots ,x^{(n)}\right\}}^T$, and response, $y={\left\{y^{(1)},y^{(2)},\dots ,y^{(n)}\right\}}^T$. $f_{RBF}(x)$ is estimated from the low-fidelity data generated from low-fidelity evaluations.

The hybrid surrogate model is updated by an additional sampling point to enhance the accuracy and diversity of the data. Hence, the coefficient of determination $\left(R^2\right)$ is needed to verify how well the hybrid surrogate model predicts a response. The $R^2$ is calculated from Equation (6)

$$R^2=1-{\displaystyle \frac{{{\displaystyle \sum _{i=1}^n\left(y_{true,i}-{\widehat{y}}_{predicted}\right)}}^2}{{{\displaystyle \sum _{i=1}^n\left(y_{true,i}-{\overline{y}}_{average}\right)}}^2}}$$

(6)

where ${\widehat{y}}_{true,i}$ is a real function value evaluated at the $ith$ sampling point, ${\widehat{y}}_{predicted}$ is the prediction from the hybrid surrogate model, ${\overline{y}}_{average}$ is an average of the real function value, and $n$ is the number of sampling points.

2.2. Multi-Objective, Multiple Additional Sampling for EGO Process

The hybrid surrogate model is a combination of a high-fidelity function and a low-fidelity function that represents only a single objective. Thus, DOE, evaluation and surrogate model procedures are performed for all objectives. For an n-dimensional objective, the hybrid surrogate model would be $\widehat{y}(x)={\left[{\widehat{y}}_{obj,1}(x),{\widehat{y}}_{obj,2}(x),\dots ,{\widehat{y}}_{obj,n}(x)\right]}^T$. After the hybrid surrogate model is completed, data improvement will be performed. The data improvement works in such ways to insert all objective functions into the integration function, and then it is optimized by an arbitrary optimization method, searching for an optimal point; that point will be the next candidate point for updating the hybrid surrogate model to increase the accuracy of the hybrid surrogate model. Additionally, it helps increase the data diversity to prevent a local optimum trap. This procedure is also called the infilled sampling criteria procedure or the additional sampling point technique. The expected improvement (EI) indicator is used for infilled sampling criteria by maximizing an EI value to search for the next sampling point candidate, as follows:

$$EI(x)={\displaystyle {\int }_{-\infty }^{y_{\min }}(y_{\min }-\widehat{y}(x))\varphi (}\widehat{y}(x))dy$$

(7)

where $EI(x)$ is an expected improvement value for each design point, $x$. The $y_{\min }$ is the minimum global solution for the minimization problem (note that, for the maximization problem, one must insert a minus (−) sign in front of the integrate symbol). The $\varphi$ denotes the standard normal distribution function and $\widehat{y}(x)$ is the predicted value from the surrogate model. The integration over the difference of $y_{\min }$ and $\widehat{y}(x)$ from $-\infty$ to $y_{\min }$ means to find the maximum uncertainty area of confidence interval of the EI value.

This work proposes multi-objective optimization; then, EHVI is used as an acquisition function to search for the next sampling point candidate at the pareto front. The EHVI transforms a multi-objective function into a single uncertainty-aware metric and EHVI is the function of hypervolume improvement (HVI) integrated with the uncertainty area of the additional sampling point. The HVI is computed from the improvement of the hypervolume of the additional sampling point and the non-dominated solution, as shown in Figure 3a, and is compared to the general non-dominated solution of the multi-objective optimization problem, as shown in Figure 3b. EHVI is expressed as follows:

$$\begin{array}{l}EHVI\left[f_1(x),f_2(x),\dots ,f_N(x)\right]=\\ {\displaystyle {\int }_{-\infty }^{f_{ref,1}}{\displaystyle {\int }_{-\infty }^{f_{ref,2}}\dots {\displaystyle {\int }_{-\infty }^{f_{ref,N}}HVI}}\left[f_1(x),f_2(x),\dots ,f_N(x)\right]}\times {\varphi }_1(F_1){\varphi }_2(F_2)\dots {\varphi }_N(F_N)d{F}_{1}d{F}_2\dots d{F}_N\end{array}$$

(8)

where $f_i$ is the objective function or the hybrid surrogate model, given the mean and variance $\left[{\widehat{f}}_i(x),{{\widehat{s}}_i}^2(x)\right]$. ${\varphi }_i(F_i)$ is the probability density function of the normal distributed function and $f_{ref,i}$ is the reference value that is used for calculating the hypervolume value.

The result of maximizing EHVI is the additional sampling point. However, it provides only a single additional sampling point in each iteration. Thus, to accelerate the computation time of the EGO process, multiple additional sampling points or parallel infilled sampling points are addressed. The concept of the multiple additional sampling point technique involves the prediction of the next sampling point candidate, exploiting the prior hybrid surrogate model. For an illustration of the concept of the multiple additional sampling point technique, Figure 1b is presented. The data improvement procedure performs and provides the next sampling point candidate to update the hybrid surrogate model. The next sampling point candidate has not been evaluated yet, but an estimate of the response value from the previous hybrid surrogate model of this additional sampling point is evaluated. Thus, the response value or yield point of the next sampling point candidate is estimated from the hybrid surrogate model, not from the evaluation tools. The next sampling point candidate and the response value are found to update the hybrid surrogate model. The EGO process continues to iteratively calculate until the optimal solution converges. The number of N-additional sampling points that is needed is $N-1$ from the predicted hybrid surrogate model. Illustrations of the single additional sampling point and multiple additional sampling point techniques are shown in Figure 4a,b, respectively [15].

The acquisition function used in this work is the EHVI function, that integrates multi-objective functions into a one-objective function (Equation (8)). Hence, an arbitrary optimization method for a single objective is available. In addition, the hybrid surrogate model is treated as a black-box function that cannot be arranged into a closed form. Evolutionary optimization is popular for solving unknown closed form functions. The Genetic Algorithm is one among the metaheuristic optimization methods for solving a single-objective optimization problem and is inspired by natural selection. Design variables are treated as chromosomes and responses are treated as objective functions. The GA [25] process commences by initializing the population, i.e., a group of chromosomes. The initial population comprises parents, which are then the first generation. After that, the population is evaluated through an objective function; if the process can search optimal chromosomes, then the process is terminated. If not, then a generic operation is invoked to evaluate previous chromosomes for procreating the next generation, called the children. The genetic operation consists of selection, crossover, and mutation. The selection step involves choosing some parts of the chromosome of the parent that have sufficient strength for dealing with the objective function. The crossover step involves assembling the parent’s chromosomes to form the chromosomes of the children for the next generation. The mutation step involves modifying individual parent’s chromosomes to maintain genetic diversity to form the children’s chromosomes. The GA procedure is iterated until it receives a strong child chromosome for evaluating the objective function. The schematics process of GA and the genetic operations are presented in Figure 5.

3. Mathematics Test Functions

The proposed techniques of MA-MF-MOEGO should be verified by mathematical test functions before launching them for an airfoil design to ensure that the proposed techniques are judicious. The mathematical test functions included two functions, and each function had two objectives with low- and high-fidelity functions. The first test function and the second test function were expressed as follows.

The low-fidelity and high-fidelity functions of first and second objectives for the first test function are denoted by $f_{L,1}$, $f_{L,2}$, $f_{H,1}$, and $f_{H,2}$.

$$\mathit{min}:\hspace{1em}{f}_{H,1}=x_1+x_2+x_3+\mathit{sin}(x_1)+\mathit{sin}(x_2)+\mathit{sin}(x_3)$$

(9)

$$\mathit{min}:\hspace{1em}{f}_{H,2}={(x_1-1)}^2+{(x_2-1)}^2+{(x_3-1)}^2$$

(10)

$$f_{L,1}=x_1+x_2+x_3+0.1x_1+\mathit{sin}(x_1)+0.1\mathit{sin}(x_2)+0.1\mathit{sin}(x_3)$$

(11)

$$f_{L,2}={(x_1-2)}^2+{(x_2-2)}^2+{(x_3-2)}^2$$

(12)

The design range of the first test function was $x_1,x_2,x_3 \in [-5,5]$. The low-fidelity and high-fidelity functions of the first and second objective for the second test function were denoted by ${\widehat{f}}_{L,1}$, ${\widehat{f}}_{L,2}$, ${\widehat{f}}_{H,1}$, and ${\widehat{f}}_{H,2}$.

$$\mathit{min}:\hspace{1em}{\widehat{f}}_{H,1}=x_1^2+x_2^2+x_3^2$$

(13)

$$\mathit{min}:\hspace{1em}{\widehat{f}}_{H,2}=-{(x_1-1)}^2-{(x_2-2)}^2-{(x_3-3)}^2$$

(14)

$${\widehat{f}}_{L,1}=0.8x_1^2+0.8x_2^2+0.8x_3^2$$

(15)

$${\widehat{f}}_{L,2}=-{(x_1-1)}^2-{(x_2-2)}^2-{(x_3-3)}^2-{(x_1-x_2)}^2-{(x_2-x_3)}^2$$

(16)

The design range of the second test function was $x_1,x_2,x_3 \in [-10,10]$. A total of 20 initial sampling points, used as high-fidelity data, and 100 initial sampling points, used as low-fidelity data, were generated by LHS. The Kriging method and RBF function were used to construct the high-fidelity function and the low-fidelity function, respectively. The GA deployed an optimization process to find the next sampling point candidate. There were 20 additional sampling points, so the total number of sampling points was 40.

3.1. Result and Discussion of First Test Function

The entire set of sampling points, including the initial and the additional sampling points of the first test function, are plotted as a pareto graph. Figure 6a, Figure 6b and Figure 6c show the initial sampling points development to the optimal sampling points of the 1-SA, 2-MAs, and 4-MAs techniques, respectively. The proposed approach can be applied to the first test function and can search for the pareto optimal solution. All additional sampling points are better than the initial sampling points. Therefore, the first test function was suitable for the proposed approach. The shapes of the pareto solutions for all the techniques are quite similar, as shown in Figure 6d. This indicates that the 2-MAs and 4-MAs techniques can be applied for the test function instead of the 1-SA technique. Additionally, the 2-MAs technique has the widest pareto solution shape. Thus, the 2-MAs approach creates an alternative pareto solution which is better than that of the 1-SA technique.

The computational convergence rates of $f_1$ and $f_2$ for the first test function are displayed in Figure 7 and Figure 8, respectively. For the first objective of the first test function, the 1-SA and 4-MAs techniques terminate simultaneously and reach the first objective target at the 23rd point. This indicates that the 4-MAs technique converges faster than the 1-SA technique does, because the 23rd point of the 4-MAs is still in the first iteration, but the 23rd point of the 1-SA is in the third iteration. Although the 2-MAs technique converges at the 38th point in the ninth iteration, the solution is better than the other techniques. For the second objective of the first test function, all techniques converge at the 24th point in the fourth, second, and first iterations for the 1-SA, 2-MA, and 4-MAs techniques, respectively. Both Figure 7 and Figure 8 indicate that the proposed techniques converge faster than the conventional technique. Thus, the proposed techniques have been qualified for application as test functions instead of the 1-SA technique.

The cross-validation of the hybrid surrogate model of the 1-SA, 2-MAs, and 4-MAs techniques for the first test functions are compared against the exact test function, as shown in Figure 9a,b, Figure 10a,b and Figure 11a,b, for $f_1$ and $f_2$, respectively.

The $R^2$ values of $f_1$ for the 1-SA, 2-MAs, and 4-MAs techniques are equal to 0.9999, 0.9995, and 0.9999, respectively. Figure 9a, Figure 10a and Figure 11a have $R^2$ values of $f_1$ close to 1, which means that the hybrid surrogate model for $f_1$ is very well predicted; this is because the low-fidelity function is created by 200 samples, assisting the high-fidelity function in being more accurate; 20 additional sampling points is sufficient for all techniques. Additionally, the proposed techniques are available for the first test function because the model accuracy of the hybrid surrogate is the same as that for the 1-SA technique. The $R^2$ value of $f_2$ for the 1-SA, 2-MAs, and 4-MAs techniques are equal to 0.9743, 0.9722, and 0.9717, respectively. Figure 9b, Figure 10b and Figure 11b indicate that the highest precise predictions of $f_2$ were 1-SA, 2-MAs, and 4-MAs, consecutively. However, the 1-SA technique has an $R^2$ value greater than those of the other techniques, but they are close to each other. This indicates that the proposed techniques can still be applied to the test function.

3.2. Results and Discussion of Second Test Function

The entire set of sampling points, including the initial and additional sampling points of the second test function, are plotted as pareto graphs. Figure 12a, Figure 12b and Figure 12c show the initial sampling point development to the optimal sampling points of the 1-SA, 2-MAs, and 4-MAs techniques, respectively. The proposed approach accomplishes the goal of finding the optimal solution for the second test function. The 2-MAs technique has the widest pareto solution line shape. This indicates that the proposed techniques found a better optimal solution than that which was obtained conventionally for only one sampling point. Additionally, the shapes of the pareto solution lines for all the techniques are the same; this might influence one’s choice of using the proposed techniques instead of the 1-SA technique.

The computational convergence rates of $f_1$ and $f_2$ for the second test function are displayed in Figure 13 and Figure 14, respectively. The second test function is a good example of an instance in which the proposed techniques are not better than the 1-SA technique. The 1-SA technique has converged at the 23rd point in the third iteration, while the 2-MAs and 4-MAs techniques converged at the 32nd point in the sixth iteration and at the 38th point in the fifth iteration for the first objective. In this case, the 1-SA technique terminates faster than the other techniques and the optimal solution of the 1-SA technique is the best. This indicates that sophisticated functions or unstable problems should also be deployed by the 1-SA technique. However, the proposed techniques can still be applied to the second test function because the optimal values of all techniques make a small difference, while the proposed techniques converge slower than the 1-SA technique. The 1-SA, 2-MA, and 4-MAs techniques have converged at the 32nd point in the twelfth iteration, at the 40th point in the tenth iteration, and at the 32nd point in the third iteration for the second objective, respectively. The 1-SA technique completed the computation faster than the other techniques because 1-SA converges first. However, the 4-MAs can find the best solution for the second objective. Thus, the proposed techniques are still essential for solving the second test function, while the proposed techniques terminate slowly.

The cross-validation of the hybrid surrogate models of the 1-SA, 2-MAs, and 4-MAs techniques for the second test function are compared against the exact test function as shown in Figure 15a,b, Figure 16a,b and Figure 17a,b, for $f_1$ and $f_2$, respectively.

The $R^2$ values for $f_1$ with the 1-SA, 2-MAs, and 4-MAs techniques are equal to 0.9977, 0.9976, and 0.9976, as shown in Figure 15a, Figure 16a and Figure 17a, respectively. The $R^2$ values for $f_2$ with the 1-SA, 2-MAs, and 4-MAs techniques are equal to 0.9561, 0.9467, and 0.9395, as shown in Figure 15b, Figure 16b and Figure 17b, respectively. The model accuracies of the hybrid surrogates of all techniques for $f_1$ have similar values, but they are different for $f_2$. The 1-SA is the best prediction model, followed by the 2-MAs and 4-MAs techniques, respectively. The second test function is a complex problem, so adding one sampling point in each iteration is more stable and does not excessively disturb the process more than the proposed techniques. Nevertheless, the $R^2$ values of $f_2$ for all techniques are not very different. Therefore, the proposed techniques can still be available for the second test function.

From the results, the MA-MF-MO EGO, as the proposed approach, comparing again with the SA-MF-MO EGO approach, could be successfully verified by two mathematics test functions. The proposed techniques accomplish the goal of obtaining an optimal solution; the pareto solutions of all the techniques are rather similar, and the pareto solutions of the proposed techniques are better than those of the 1-SA technique under some objectives. The computational convergence rate of the proposed techniques explicitly converges faster than the 1-SA technique. Additionally, the addition of multiple sampling points can increase the diversity of data and enhance the model accuracy of the hybrid surrogate. Hence, the MA-MF-MO EGO can be deployed in the optimized design of the airfoil shape.

4. Airfoil Design Problem

A schematic representation of the process of this work is displayed in Figure 18. The EGO with multiple additional sampling points, under multi-fidelity and multi-objective conditions, was applied to the design of an airfoil.

4.1. Problem Statement

Design variables of the airfoil shape were parameterized by Class Shape Function Transformation (CST). The CST [26] was utilized to create an arbitrary shape in dimensional geometry such as an elliptic body, circular duct, cone, wedge, airfoil, etc. The CST is a polynomial function that is under the control of a weight factor and consists of a class function and a shape function. The CST equation is denoted as follows:

$${\displaystyle \frac{y}{c}}\equiv C\left({\displaystyle \frac{x}{c}}\right)S\left({\displaystyle \frac{x}{c}}\right)+{\displaystyle \frac{x}{c}}{\displaystyle \frac{\Delta Z_{te}}{c}}$$

(17)

where $y$ is the thickness of the airfoil, $c$ is the chord of the airfoil, $\Delta Z_{te}$ is the thickness at the trailing edge of the airfoil, and $x$ is any position of the airfoil in the horizontal direction. $C\left({\displaystyle \frac{x}{c}}\right)$ and $S\left({\displaystyle \frac{x}{c}}\right)$ are class and shape functions expressed as:

$$C\left({\displaystyle \frac{x}{c}}\right)\equiv {\left({\displaystyle \frac{x}{c}}\right)}^{N_1}{\left[1-{\displaystyle \frac{x}{c}}\right]}^{N_2} \mathrm{where} 0\le {\displaystyle \frac{x}{c}}\le 1$$

(18)

$$S\left({\displaystyle \frac{x}{c}}\right)\equiv {\displaystyle \sum _{i=0}^n\left[b_i\cdot K_{i,n}\left(\frac{x}{c}\right)\cdot {\left(1-\frac{x}{c}\right)}^{n-1}\right]} \mathrm{where} K_{i,n}={\displaystyle \frac{n!}{i!\left(n-i\right)!}}$$

(19)

where $N_1$ and $N_2$ are constant values that prescribe characteristics of the CST shape; $N_1$ and $N_2$ are set to be 0.5 and 1.0 for round nose airfoils. $b_i$ refers to the six design variables of the airfoil shape; then, $\mathit{b} ={\left[b_1,b_2,b_3,b_4,b_5,b_6\right]}^T$. $K_{i,n}$ refers to Bernstein binomial. The upper surface of the airfoil is governed by $b_1,b_2 \mathrm{and} b_3$. The lower surface of the airfoil is governed by $b_4,b_5 \mathrm{and} b_6$. The placement of $\mathit{b}$ on the airfoil and the design variables range are shown in Figure 19a,b and

Table 1. Design variable range of airfoil parameterization.

Design Variable	Design Range
$b_1$	0.10–0.18
$b_2$	0.05–0.15
$b_3$	0.05–0.15
$b_4$	−0.18–−0.01
$b_5$	−0.15–−0.05
$b_6$	−0.18–−0.02

, respectively.

The airfoil was operated at Reynolds number (Re) = 1,000,000 and Mach number (Ma) < 0.3, because there were experimental data for NACA0012 at Re = 1,000,000. Thus, the panel method and CFD solver can be validated against the experimental data. The properties of air are given in

Table 2. Properties of air at atmospheric pressure.

$\mathit{T}$ (°C)	$\mathit{\rho }$ (kg/m3)	$\mathit{\mu }$ (kg/ms)	$\mathit{\nu }$ (m2/s)	$\mathit{\gamma }$
15	1.2257	1.802 × 10−5	1.470 × 10−5	1.4

. The chord of the airfoil was 1 m and the aspect ratios were infinite; then, the velocity and the Ma number of the freestream could be estimated approximately as 14.735 m/s and 0.04331, respectively.

Drag and lift are the forces that can be representative of the performance of the airfoil and make the tradeoff for each other; this is because, as the angle of attack increases, the lift and the drag also increase. The minimization of the drag at the cruising speed and the minimization of the reciprocal square lift at the descent are the challenges that must be overcome in airfoil design. Hence, the objective functions of this work were expressed as [27]:

$$\mathit{min}: f_1(b)=C_d \mathit{at} C_l=0.5$$

(20)

$$\mathit{min}: f_2(b)=1/C_l^2 \mathit{at} \mathit{aoi}=5 \mathit{deg}$$

(21)

4.2. Design of Experiment

To generate design variables or input for evaluation, Latin Hypercube Sampling (LHS) [28], one of the most popular methods for DOE computer experiments, is used. Additionally, the number of initial sampling points is defined by users and LHS can generate multiple decimal places which fit for parameters of the airfoil. The LHS generates design variables randomly by normalizing such design variables with respect to the design range. Each design variable cannot be a repetitive value, because each design variable remembers the row and column in which the design variable is taken. Thus, the distribution of design variables is diverse and beneficial for the observed data and constructing the hybrid surrogate model to avoid local optimal solutions. This work used LHS in MATLAB. The low-fidelity data were set to be 200 initial sampling points of airfoil parameters, while the high-fidelity data were set to be 30. Figure 20a,b show the design variables of the high-fidelity data, the distribution between the design variables, the $b_1$ and $b_2$ of the high-fidelity data and the design variable, and the $b_5$ and $b_6$ of the low-fidelity data, respectively.

4.3. Aerodynamic Evaluation

This work estimated the auxiliary data with a quick calculation solver by using a panel method for the low-fidelity data and simulated a small sample size of the high-fidelity data by using CFD based on RANS.

4.3.1. High-Fidelity Aerodynamic Evaluation

High-fidelity aerodynamic evaluation of this work was conducted in CFD. The governing equation of the conservation of mass and linear momentum is expressed as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla \cdot (\rho \stackrel{\rightharpoonup }{\mathit{v}})=0$$

(22)

where ${\displaystyle \frac{\partial \rho }{\partial t}}$ is the rate of change of mass per unit volume over time, $\nabla$ is the del operator $\left(\nabla ={\displaystyle \frac{\partial }{\partial x}}\widehat{i}+{\displaystyle \frac{\partial }{\partial y}}\widehat{j}+{\displaystyle \frac{\partial }{\partial z}}\widehat{k}\right)$, and $\stackrel{\rightharpoonup }{\mathit{v}}$ is a velocity vector $\left(\stackrel{\rightharpoonup }{\mathit{v}}=u\widehat{i}+v\widehat{j}+w\widehat{k}\right)$.

$$\rho \left({\displaystyle \frac{\partial \stackrel{\rightharpoonup }{\mathit{v}}}{\partial t}}+\left(\stackrel{\rightharpoonup }{\mathit{v}}\cdot \nabla \right)\stackrel{\rightharpoonup }{\mathit{v}}\right)=\stackrel{\rightharpoonup }{k}-\nabla p+\mu \cdot \Delta \stackrel{\rightharpoonup }{\mathit{v}}$$

(23)

where $\rho$ is the mass per unit volume, ${\displaystyle \frac{\partial \stackrel{\rightharpoonup }{\mathit{v}}}{\partial t}}$ is the change in velocity over time, $\left(\stackrel{\rightharpoonup }{\mathit{v}}\cdot \nabla \right)\stackrel{\rightharpoonup }{\mathit{v}}$ is speed and direction of fluid, $\stackrel{\rightharpoonup }{k}$ is external forces, $\nabla p$ is the pressure gradient, and $\mu \cdot \Delta \stackrel{\rightharpoonup }{\mathit{v}}$ is the internal stress force.

ANSYS Fluent R1 2022 was used to evaluate $C_l$ and $C_d$. The CFD methodology began with shaping the geometry of the airfoil and air domain. An in-house Fortran code was used to generate each airfoil geometry. The domain was a C-mesh shape; the dimensions of the domain and the boundary conditions are shown in Figure 21a. The radius and downward length were set to be 10 times the chord. The arc, top, and bottom line were set to be the velocity inlet. The back line and airfoil entity were set to be the pressure outlet and the wall, respectively. The rest were set to be a part of the air domain. A quadrilateral mesh shape was chosen; the mesh around the airfoil should be condensed and maintained with approximately $y^{+}\approx 1$ because the entire phenomenon adjacent to the wall must be captured.

Mesh convergence was studied, as shown in

Table 3. Mesh convergence study.

Number of Mesh	${\mathit{C}}_{\mathit{l}}$	${\mathit{C}}_{\mathit{d}}$	$\mathit{\%}\mathit{d}\mathit{i}\mathit{f}\mathit{f} {\mathit{C}}_{\mathit{l}}$	$\mathit{\%}\mathit{d}\mathit{i}\mathit{f}\mathit{f} {\mathit{C}}_{\mathit{d}}$
40,000	0.52688	0.01343	0.1292%	1.2739%
160,000	0.52686	0.01334	0.1312%	0.6067%
360,000	0.52719	0.01329	0.0695%	0.2551%
640,000	0.52743	0.01327	0.0239%	0.0863%
1,000,000	0.52756	0.01326	-	-

. Five mesh numbers were considered. A total of 1,000,000 elements were found to be the finest and they used for a comparison with the other mesh numbers. The differences between $C_l$ and $C_d$ were less than 1%, except the $\%diff C_d$ of the 40,000 elements. This work uses a sufficient mesh number for saving computation time and maintaining an error of less than 1%. Hence, the mesh with 160,000 elements was selected.

The Spalart–Allmaras approach was chosen for the turbulence model. This model has one equation that solves the transport equation for the kinetic eddy turbulent viscosity and is suitable for aerodynamic flows. The properties of air and freestream velocity were applied, following Table 2. The coupled scheme was used for pressure–velocity coupling. The least squares cell-based gradient was used for spatial discretization. The second order and second-order upwind were implemented for discretizing the continuity, linear momentum, and turbulence equations, respectively. The residual target of all the governing equations was set to be 10⁻⁶. The computation time of the CFD model in this work was approximately 300 s. NACA0012 airfoil was provided to validate the CFD model against experimental data, as shown in Figure 22. Sheldahl et al. [29] tested various symmetrical airfoils such as NACA-0009, -0012, and -0015 through a 180-degree angle of attack, for use in vertical-axis wind turbines. The airfoils applied for the experiments in the Walter H. Beech Memorial Wind Tunnel at Wichita State University, which has a 7 × 10 ft test section. Figure 22a,b display $C_l$ and $C_d$ versus the angle of attack between the CFD data and the experimental data at a Reynolds number of one million. The angle of attack above 10 degrees is in the viscous region, and the airfoil has already stalled. The shear stress of this zone is dominant. The flow is not attached to the surface of the airfoil and, finally, causes separation and a disordered movement of flow behind the airfoil. Hence, the CFD model of this study had difficulty in predicting this phenomenon; the error between the CFD data and the experimental data were then too large. However, this zone is not appropriate for the analysis objectives of this work because those objectives are not commonly deployed in the stall zone. Therefore, the CFD data of this zone can be neglected. Those objectives are functional in the inviscid flow region or if the angle of attack is less than 10 degrees, as is generally used in the design process. In this zone, the flow is attached to the surface of the airfoil and has not separated yet, which keeps creating lift. The trends of both $C_l$ and $C_d$ of the CFD model are quite similar to those of the experimental data. Hence, the CFD model of this work was sufficient for use in the analysis as a high-fidelity evaluation.

4.3.2. Low-Fidelity Aerodynamic Evaluation

The low-fidelity evaluation of this work was performed using the panel method. This method involves numerical computation, especially for aerodynamic flows based on potential flow. The governing equation is expressed as follows:

$$Q_P=V_{\infty }\mathit{cos}(\alpha )+V_{\infty }\mathit{sin}(\alpha )-{\displaystyle \sum _{j=1}^n\frac{{\gamma }_j}{2\pi }{\displaystyle \underset{j}{\int }\frac{\partial {\theta }_{ij}}{\partial n_i}d{S}_j}}$$

(24)

where $Q_P$ is the normal velocity, $V_{\infty }$ is the velocity field, and ${\gamma }_j$ is the vortex strength as a constant value. For flow over an airfoil, the panel method must follow the Kutta condition $({\gamma }_i+{\gamma }_n=0)$.

Java Foil version 1.8.0_291 was used to evaluate $C_l$ and $C_d$. The Java Foil program is based on the vortex panel method. Each airfoil of low-SAmpling points was discretized into 200 panels. The properties of air, velocity, and other relevant properties were defined as shown in Table 2. The computation time of the Java Foil program of this work is approximately 1 s. The NACA0012 airfoil was provided to validate the Java Foil program settings against experimental data, as shown in Figure 23.

Figure 23a,b represent $C_l$ and $C_d$ versus the angle of attack between the panel method and the experimental data at a Reynolds number of one million. The limitation of the panel method is the lack of flow separation modelling, which means it is impossible to estimate $C_l$ and $C_d$ correctly in a stall zone. The trend and value of $C_l$ and $C_d$ graphs of this zone are completely different and have an enormous error compared with the experimental data. To solve this problem, the boundary layer equation is added for calculating drag force. The $C_d$ data from the panel method is in good agreement, as shown in Figure 23b, when the angle of attack is less than 10 degrees. The robustness of the panel method is established to predict the precise lift force in the inviscid region, as shown in Figure 23a. Hence, the panel method of this work is adequate for quickly estimating the aerodynamic coefficients for low-fidelity evaluations.

4.4. The Hybrid Surrogate Model and Data Improvement

The hybrid surrogate model of this work involved the combination of the high-fidelity function and the low-fidelity function. The high-fidelity function was constructed using high-fidelity data through the Kriging model as the global model. The low-fidelity function was constructed using low-fidelity data through RBF as the local deviation. This work had two objective functions, so the hybrid surrogate models were proposed for Equations (20) and (21). Then, the acquisition function occupied these hybrid surrogate models following Equation (8). The EHVI function of this work is expressed as follows:

$$\mathit{max} \text{:} \mathit{EHVI}\left[f_1(b),f_2(b)\right]={\displaystyle \underset{-\infty }{\overset{f_1}{\int }}{\displaystyle \underset{f_2}{\overset{\infty }{\int }}HVI\left[f_1,f_2\right]}}\times {\varphi }_1(f_1){\varphi }_2(f_2)d{f}_{1}d{f}_2$$

(25)

Data improvement was conducted to maximize the value of hypervolume improvement. A Genetic Algorithm was used to determine the maximum uncertainty area of the EHVI function, because it is the first and fundamental optimization method required to solve the single objective. The population was initialized to 100, and the generation number was set to 50. The tournament search was selected for the selection process. The crossover method that was used was the blend crossover operator (BLX) method, with a rate of crossover of 0.9 [30]. A mutation rate value of 0.1 was assigned to prevent the replication of the next generation’s population. The total computation of the population required 5000 iterations. The hybrid surrogate model and data improvement process were performed by an in-house Fortran code.

5. Result and Discussion

This work proposes the application of an MA-MF-MO EGO to the optimization of the design of an airfoil shape and for studying the influence of adding multiple sampling points. The 2-MAs and 4-MAs techniques are the main contributions of this work to compare the performance of an airfoil, the computational convergence rate, and the model accuracy of the hybrid surrogate against those of the 1-SA technique. The 1-SA technique involves the addition of only one sampling point per iteration and does not need a predicted sampling point from the hybrid surrogate model. The 2-MAs technique involves the addition of two additional sampling points simultaneously and needs one predicted sampling point. The 4-MAs technique involves the addition of four additional sampling points simultaneously and needs three predicted sampling points. The total additional sampling points are set to be 20. Hence, the 1-SA technique has 20 iterations, the 2-MAs technique has 10 iterations, and the 4-MAs technique has 5 iterations. The two objective values were obtained from

Table 4. The result of each additional sampling point technique.

Iteration	1-SA			Iteration	2-MAs			Iteration	4-MAs
	No.	${\mathit{f}}_1$	${\mathit{f}}_2$		No.	${\mathit{f}}_1$	${\mathit{f}}_2$		No.	${\mathit{f}}_1$	${\mathit{f}}_2$
1	31	0.01139	1.6500	1	31	0.01139	1.6500	1	31	0.01139	1.6500
2	32	0.01125	1.7062		32	0.01116	1.7799		32	0.01116	1.7799
3	33	0.01125	1.7842	2	33	0.01178	3.1482		33	0.01178	3.1482
4	34	0.01179	3.1652		34	0.01123	1.7216		34	0.01123	1.7216
5	35	0.01112	1.8513	3	35	0.01114	1.8973	2	35	0.01114	1.8973
6	36	0.01117	1.7714		36	0.01118	1.8939		36	0.01118	1.8939
7	37	0.01141	1.6862	4	37	0.01127	1.7248		37	0.01124	1.8710
8	38	0.01146	1.6975		38	0.01125	1.8548		38	0.01147	1.6949
9	39	0.01129	1.7757	5	39	0.01133	1.7361	3	39	0.01133	1.7789
10	40	0.01131	1.8065		40	0.01156	1.9199		40	0.01127	1.7257
11	41	0.01155	1.9227	6	41	0.01179	1.9892		41	0.01127	1.7083
12	42	0.01178	1.9862		42	0.01144	1.6858		42	0.01139	1.8710
13	43	0.01133	1.7091	7	43	0.01185	2.2461	4	43	0.01163	1.9358
14	44	0.01122	1.9588		44	0.01134	1.6910		44	0.01159	1.7430
15	45	0.01123	1.7830	8	45	0.01135	1.7509		45	0.01132	1.7083
16	46	0.01133	1.6973		46	0.01141	1.6825		46	0.01187	1.8537
17	47	0.01127	1.7729	9	47	0.01126	1.7691	5	47	0.01136	1.7578
18	48	0.01143	1.7570		48	0.01158	1.7391		48	0.01131	1.7152
19	49	0.01129	1.7253	10	49	0.01131	1.7340		49	0.01132	1.7240
20	50	0.01117	1.8046		50	0.01136	1.7708		50	0.01142	1.7282

for all additional sampling points. The 31st sampling point of each technique has the same value. Because this is the first additional sampling point of the data improvement process. The 32nd sampling point of 1-SA is different from those of the 2-MAs and 4-MAs techniques; this is because the 32nd sampling point of 1-SA is evaluated by CFD, while the 32nd sampling points of the 2-MAs and 4-MAs techniques are predicted by the hybrid surrogate model before being evaluated through CFD. The 31st, 32nd, and 33rd sampling points of the 2-MAs and 4-MAs techniques have the same values, because the 4-MAs technique needs 3 predicted sampling points from the hybrid surrogate model; thus, the 31st, 32nd, and 33rd sampling points of the 2-MAs technique can be used automatically for the 4-MAs technique. Theoretically, the 34th sampling points should be the beginning of different values for each technique, but the 2-MAs and 4-MAs techniques have different values from the 37th sampling point onward. This is because the 2-MAs and 4-MAs techniques are multiple-additional-point techniques. The difference between these two techniques is that they involve the addition of two and four sampling points at a time, respectively, but the concept is still the same.

All the sampling points, including the initial and additional sampling points of the two objectives, are plotted as a pareto graph. Figure 24a, Figure 24b and Figure 24c show the development of the initial sampling process to the optimal sampling points of the 1-SA, 2-MAs, and 4-MAs techniques, respectively. Mostly, the additional sampling point values of all the approaches are less than the initial sampling point values, because the additional sampling points pass the optimization process. All techniques have five sampling points that are not dominated by the other sampling points. The pareto solutions of all techniques are somewhat identical, as shown in Figure 24d, which means that the increments (regardless of whether there are one, two or four sampling points in each iteration) have not had a significant impact on the accuracy of the hybrid surrogate model and enhance the optimal solution. $f_1$ and $f_2$ can be converted into $C_l$ and $C_d$, respectively, as shown in

Table 5. $C_l$ and $C_d$ of optimal sampling point.

No.	1-SA		No.	2-MAs		No.	4-MAs
	${\mathit{C}}_{\mathit{l}}$	${\mathit{C}}_{\mathit{d}}$		${\mathit{C}}_{\mathit{l}}$	${\mathit{C}}_{\mathit{d}}$		${\mathit{C}}_{\mathit{l}}$	${\mathit{C}}_{\mathit{d}}$
35	0.7349	0.01111	35	0.7259	0.01113	35	0.7260	0.01113
36	0.7513	0.01116	32	0.7495	0.01115	32	0.7496	0.01115
32	0.7655	0.01124	34	0.7621	0.01122	34	0.7621	0.01122
46	0.7675	0.01132	44	0.7690	0.01133	41	0.7651	0.01126
31	0.7785	0.01139	31	0.7785	0.01139	31	0.7785	0.01139

. The lowest $C_d$ for $f_1$ of the 1-SA, 2-MAs, and 4-MAs techniques are 0.01111, 0.01113, and 0.01113, respectively. The highest $C_l$ for $f_2$ of the 1-SA, 2-MAs, and 4-MAs techniques are all the same at 0.7785.

The optimal airfoil shapes with respect to the minimization of $C_d$ for the 1-SA, 2-MAs, and 4-MAs techniques are shown in Figure 25, compared with the lowest $C_d$ of the initial airfoil shape.

Figure 25 indicates that the proposed approach is successful in receiving the optimal airfoil shapes with respect to the first objective. The 1-SA, 2-MAs, and 4-MAs techniques have decreases of 0.715%, 0.536%, and 0.536% for $C_d$ when compared against the initial airfoil No.6, respectively. In terms of optimization interpretation, data improvement can search for the optimal point for $f_1$. However, a decrement of less than 1% of $C_d$ is insignificant. The reason for this is that the design range of this work is not wide enough to generate an initial sampling point. The LHS is a random process, so it is possible to meet the optimal airfoil shape at the beginning, but this is not usually encountered. In terms of physical meaning, the cause of drag for the airfoil is the separation of flow. The components of the total drag for a two-dimensional geometry are the friction drag and the form drag. The friction drag is related to shear stress and is dominant in the area adjacent to the wall, so the surface of the airfoil directly affects the friction drag. However, there is no design variable that relates to the surface of airfoil. Thus, the friction drag of this work is then constant, regardless of the changing shape of the airfoil. Form drag is related to the geometry that blocks the flow, so the shape of the airfoil directly affects the formation of drag, whereas the shape of the airfoil is governed by the design variables of this work. Therefore, total drag is caused only by form drag, and the minimization of $C_d$ causes a decrease in the thickness of the airfoil. The initial airfoil No.6 has been transformed by reducing its thickness to become the optimal shapes. However, the thickness is rarely different because the improvement of the airfoil shapes is less than 1%.

The optimal airfoil shapes with respect to maximization of $C_l$ for the 1-SA, 2-MAs, and 4-MAs techniques are shown in Figure 26, compared against the highest $C_l$ of the initial airfoil shape.

Figure 26 indicates that the proposed approach of this work is successful in producing the optimal airfoil shapes with respect to the second objective. The 1-SA, 2-MAs, and 4-MAs techniques have increased $C_l$ by 22.21% when compared against initial airfoil No.18. In terms of physical meaning, the curvature of the airfoil indicates lift production because of the more curved condensed streamline of flow at the top of the airfoil. It causes the velocity of air at the top of airfoil to be greater than at the bottom of the airfoil. According to the Bernoulli principle, a greater velocity causes a lower pressure in a fluid. Hence, the net pressure acts vertically in the upward direction on an airfoil, and it produces lift. The curvature of the airfoil is controlled by the camber of the airfoil. The optimization strategy is to keep the camber line above the chord line. The camber developments of an initial airfoil in becoming an optimal airfoil are presented as a dashed black line and a dashed red line, respectively. The optimal camber line is not only greater than the chord line, it is also greater than the initial camber line and it remains convex so that it can produce lift. Thus, the optimal airfoil of all proposed techniques has then been greater than the initial airfoil. In terms of optimization interpretation, the increment of $C_l$ is severely significant, because, according to Figure 19, the lower boundary airfoil has low camber, and the upper boundary airfoil is a symmetric airfoil. The straight camber line is a symmetric airfoil and does not produce lift. The initial airfoil is then in poor shape and cannot satisfy the second objective.

The computational convergence rates of the 1-SA, 2-MAs, and 4-MAs techniques are the major elements for the analysis of the differences between techniques. The computational convergence rates of $f_1$ and $f_2$ are displayed in Figure 27 and Figure 28, respectively.

From Figure 27, sampling point No.6 and No.18 are close to the lowest $C_d$ for the initial sampling point. The computation has terminated when the additional sampling reaches the lowest $C_d$. The 1-SA, 2-MAs, and 4-MAs techniques converge at the 35th sampling point. They indicate that all techniques use only five additional sampling points to find a global solution, so 1-SA uses five iterations, 2-MAs uses three iterations, and 4-MAs uses only two iterations. Although the number of convergent additional sampling points of all the techniques is the same, the 4-MAs technique uses fewer iterations than the other techniques, and the $C_d$ values are rather close. Hence, if users have several tools, such as instruments or computers for the solving process, then they can perform these simultaneously. Multiple additional sampling points should then be considered for the increment of the computational convergence rate, which does not spoil the model accuracy of the hybrid surrogate or the optimal solution. Additionally, the computation is carried out until the total number of sampling points is equal to 50, because the data improvement process ensures that the 35th sampling point is not a local optimal solution. The data improvement process accordingly explores other uncertain areas further, which increases the diversity of the data for the hybrid surrogate model.

In Figure 28, sampling point No.18 is close to $f_2$ for the initial sampling point. The computation has terminated when the additional sampling responds to the minimization of $f_2$. The 1-SA, 2-MAs, and 4-MAs techniques converge at the 31st sampling point, and they indicate that all techniques can find a global solution by using only 1 additional sampling point, so 1-SA, 2-MAs, and 4-MAs use only one iteration, because the combinations of 30 values in the high-fidelity data and the trend of 200 values in the low-fidelity data are sufficient to enhance the accurate hybrid surrogate model for $f_2$. Further computations are conducted to prevent a local optimal solution for $f_2$. Hence, multiple additional sampling points should be suggested for the same reason as that mentioned for $f_2$. According to Table 5, the pareto solution of the 4-MAs technique obtained the last additional sampling at the 41st sampling point, while the 2-MAs and 1-SA techniques obtained the last additional sampling at the 44th and 46th sampling points, respectively. Therefore, the computational convergence rate of the 4-MAs technique is faster than those of the 2-MAs and 1-SA techniques. In other words, techniques with multiple additional sampling points have a computational convergence rate that is faster than techniques with a single additional sampling point for the EGO process.

The cross-validation process was conducted to determine the coefficient of determination $\left(R^2\right)$ of each hybrid surrogate model for the 1-SA, 2-MAs, and 4-MAs techniques. This work holds 10 sampling points that are independent from the 30 initial sampling points of the high-fidelity data, but which are still in the design range according to Table 1 for cross-validation. CFD was used to calculate $f_1$ and $f_2$. The prediction of the hybrid surrogate model of the 1-SA, 2-MAs, and 4-MAs techniques are compared against the CFD data, as shown in Figure 29a,b, Figure 30a,b and Figure 31a,b for $f_1$ and $f_2$, respectively.

The $R^2$ value of $f_1$ for the 1-SA, 2-MAs, and 4-MAs techniques are equal to 0.9211, 0.9071, and 0.9495, respectively. Figure 29a, Figure 30a and Figure 31a indicate that the highest precision prediction of $f_1$ is given by the 4-MAs, 1-SA, and 2-MAs, consecutively. All the $R^2$ values of $f_1$ are greater than 0.9, which means that the hybrid surrogate model for $f_1$ predicts well. This is because the number of values in the high-fidelity data for the construction of the hybrid surrogate model is adequate, and the trend of low-fidelity function from 200 sampling points enables the hybrid surrogate model to be more accurate. Furthermore, the Kriging model and RBF contribute to building the hybrid surrogate model through providing a proper number of initial sampling points. These results also specify that the multiple additional sampling point technique should be conducted to operate in the EGO process because 2-MAs and 4-MAs have $R^2$ values close to 1-SA. Thus, no matter how many sampling points are added in each iteration, the model accuracy of the hybrid surrogate is not impacted. The root mean square errors of the 1-SA, 2-MAs, and 4-MAs techniques for $f_1$ are $\pm 0.000135$, $\pm 0.000134$, and $\pm 0.000112$, respectively. The $R^2$ values of $f_2$ for the 1-SA, 2-MAs, and 4-MAs techniques are equal to 0.9611, 0.9600, and 0.9691, respectively. Figure 29b, Figure 30b and Figure 31b indicate that the highest precision prediction of $f_2$ is achieved by 4-MAs, 1-SA, and 2-MAs, consecutively. All the $R^2$ values of $f_2$ are greater than 0.95, which means that the hybrid surrogate model estimates are exact for the same reason as those of $f_1$. The root mean square errors of the 1-SA, 2-MAs, and 4-MAs techniques for $f_2$ are $\pm 0.206$, $\pm 0.209$, and $\pm 0.191$, respectively. However, the 4-MAs technique obtained the greatest $R^2$ value for $f_1$ and $f_2$, which implies that multiple additional sampling points increase the diversity of the dataset.

6. Conclusions

Multi-additional sampling point is a popular technique used in EGO approach for an aerodynamic shape design. The investigation of the influence of adding multi-sampling points in the MF-MO EGO process is proposed. The addition of 2-MAs and 4-MAs techniques in each iteration are used to study and compared against 1-SA technique. The performance of the system, the computational convergence rate and the hybrid surrogate model accuracy are the main elements to use for the comparisons. According to the results, the following conclusions can be written:

• All proposed techniques accomplish to obtain 5 optimal airfoil shapes for each technique. The decrease of $C_d$ is 0.715%, 0.536% and 0.536% when compares against the lowest $C_d$ of initial airfoil for 1-SA, 2-MAs, and 4-MAs, respectively, the increase of $C_l$ is 22.21% when compares against the highest $C_l$ of initial airfoil for all 1-SA, 2-MAs, and 4-MAs, respectively.
• The computational convergence rate of $f_1$ is terminated at 35th sampling point for all proposed techniques. The 1-SA uses 5 iterations, the 2-MAs uses 3 iterations, and the 4-MAs uses 2 iterations. The computational convergence rate of $f_2$ is terminated at 31st sampling point for all proposed techniques. The solution is found in 1 iteration. The pareto-solution of 4-MAs technique has obtained the last additional sampling at 41st while 2-MAs and 1-SA technique have obtained at 44th and 46th, respectively. The results show that the computational convergence rate can be accelerated by adding multiple sampling points in each iteration, but multi-additional sampling point does not impact principally the hybrid surrogate model accuracy or the optimal solution.

The $R^2$ of $f_1$ for 1-SA, 2-MAs, and 4-MAs techniques are equal to 0.9211, 0.9071, and 0.9495, respectively. And the $R^2$ of $f_2$ for 1-SA, 2-MAs, and 4-MAs techniques are equal to 0.9611, 0.9600, and 0.9691, respectively. The results show that the increment of sampling point for updating the hybrid surrogate model can enhance the accuracy of the model. However, the 4-MAs technique has obtained the greatest $R^2$ for $f_1$ and $f_2$, so multiple additional sampling point technique makes diversity of data and enhances slightly the hybrid surrogate model accuracy.