## 3 Novel Approach to Computing Infill Criteria for MOEGO
### A Brief Description of the Monte Carlo Approach
The comparison of aerodynamic performance between the baseline and the selected optimized airfoils is listed in Table 3. It can be seen in the table that for optimum 1, the lift-drag ratio is increased by 7.14% for objective 1 and 22.57% for objective 2 when compared to the baseline. For optimum 2, the optimized ratios are 26.26% and 21.10%, respectively. For optimum 3, the lift-drag ratio of objective 1 is increased to 39.04%, and objective 2\({}^{\prime}\)s improvement of lift-drag ratio is 16.73%. The geometry shape of the optimized airfoil and the baseline are compared in Figure 11. Figure 12 illustrates the comparison of \(C_{p}\) distributions of two objectives, and the corresponding \(C_{p}\) comparisons are depicted in Figure 13. It can be learned that for objective 1, the shock wave of optimum 3 is weaker than that of optimum 1 and 2, which results in a significantly increasing lift-drag ratio (39.04%). For objective 2, the increase in lift-drag ratios is mainly due to the increase in lift coefficients. It can also be seen that different points on the Pareto front relate to the different aerodynamic performances of airfoils. Designers can select a specific airfoil depending on specific application scenarios and requirements. Figure 10: Optimization results of the GMOEGO algorithm. Figure 9: Computational mesh of RAE2822. Figure 11: Comparison of shape between optimized airfoil and baseline. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Airfoil** & **Number of Objective** & \(\mathbf{C_{L}}\)/\(\mathbf{C_{D}}\) \\ \hline Baseline & \(1\) & \(53.81\) \\
2 & \(61.86\) \\ \hline Optimum 1 & \(1\) & \(57.65\) (+\(7.14\%\)) \\
2 & \(75.90\) (+\(22.57\%\)) \\ \hline Optimum 2 & \(1\) & \(67.94\) (+\(26.26\%\)) \\
2 & \(2\) & \(74.91\) (+\(21.10\%\)) \\ \hline Optimum 3 & \(1\) & \(74.82\) (+\(39.04\%\)) \\
2 & \(72.21\) (+\(16.73\%\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of aerodynamic performance between baseline and optimized airfoils. Figure 12: Comparison of pressure coefficient distribution between optimized airfoil and baseline. Furthermore, in order to investigate the acceleration effect of the proposed GMOEGO algorithm in real applications, the aerodynamic airfoil shape optimization case above is also optimized by the counterpart referenced MOEGO algorithm. The computational platform is Intel Core i9-9900k CPU (8 cores), and all the other setups remain the same, as mentioned at the beginning of this section. The optimized Pareto fronts obtained by MOEGO and GMOEGO algorithms for comparison is in Figure 14. It can be seen that both algorithms could obtain similar Pareto fronts. Figure 15 shows the comparison of the wall-clock time histories of the two algorithms, and the corresponding comparison results are listed in Table 4. The comparison shows that, benefiting from the Monte Carlo-based GPU-acceleration technique, a significantly speedup ratio, up to 7.27 times, is achieved. The fourth column of Table 4 shows the wall-clock time cost of MCH



## 6 Conclusions

In this paper, the GMOEGO algorithm has been successfully developed for MOO problems. The outstanding hypervolume-based EI criterion is adopted as the multi-objective infill criterion computed in a novel Monte Carlo-based way of high efficiency. The time consumption of massive computing associated with Monte Carlo-based HVI approximation (MCHVI) is greatly alleviated by parallelizing in the GPU. The algorithm presented has been successfully validated by a set of typical mathematical test functions. Positive speedups up to 13.734 times can be achieved when compared with the traditional Monte Carlo-based MOEGO algorithm. The presented algorithm has also been successfully applied to an aerodynamic optimization problem, and a 7.27-times speedup is achieved with limited expensive-to-evaluate CFD evaluations, which shows the potential capability of the algorithm to deal with real engineering MOO problems in view of accuracy and efficiency. Conceptualization, S.X., J.Z. and H.C.; methodology, S.X. ; software, S.X. and J.Z.