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.
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 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.
and
can be converted into
and
, respectively, as shown in
Table 5. The lowest
for
of the 1-SA, 2-MAs, and 4-MAs techniques are 0.01111, 0.01113, and 0.01113, respectively. The highest
for
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
for the 1-SA, 2-MAs, and 4-MAs techniques are shown in
Figure 25, compared with the lowest
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
when compared against the initial airfoil No.6, respectively. In terms of optimization interpretation, data improvement can search for the optimal point for
. However, a decrement of less than 1% of
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
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
for the 1-SA, 2-MAs, and 4-MAs techniques are shown in
Figure 26, compared against the highest
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
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
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
and
are displayed in
Figure 27 and
Figure 28, respectively.
From
Figure 27, sampling point No.6 and No.18 are close to the lowest
for the initial sampling point. The computation has terminated when the additional sampling reaches the lowest
. 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
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
for the initial sampling point. The computation has terminated when the additional sampling responds to the minimization of
. 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
. Further computations are conducted to prevent a local optimal solution for
. Hence, multiple additional sampling points should be suggested for the same reason as that mentioned for
. 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
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
and
. 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
and
, respectively.
The
value of
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
is given by the 4-MAs, 1-SA, and 2-MAs, consecutively. All the
values of
are greater than 0.9, which means that the hybrid surrogate model for
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
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
are
,
, and
, respectively. The
values of
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
is achieved by 4-MAs, 1-SA, and 2-MAs, consecutively. All the
values of
are greater than 0.95, which means that the hybrid surrogate model estimates are exact for the same reason as those of
. The root mean square errors of the 1-SA, 2-MAs, and 4-MAs techniques for
are
,
, and
, respectively. However, the 4-MAs technique obtained the greatest
value for
and
, which implies that multiple additional sampling points increase the diversity of the dataset.