## 3 Novel Approach to Computing Infill Criteria for MOEGO
### A Brief Description of the Monte Carlo Approach
The speedup ratio of the GMOEGO in comparison with the MOEGO algorithm is given in the sixth column of Table 2. It can be seen that benefit from the GPU implementation, speedups up to 13.734 times are achieved in comparison with the cost of the MOEGO algorithm based on 8 cores in parallel, which shows the effectiveness of the GPU architecture. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Test** & **Algorithm** & **IGD** & \begin{tabular}{c} **Total Cost** \\ **(Hours)** \\ \end{tabular} & \begin{tabular}{c} **MCHVI** \\ **Cost (Hours)** \\ \end{tabular} & \begin{tabular}{c} **SUR of Total Cost** \\ **(Times)** \\ \end{tabular} \\ \hline ZDT1 & \begin{tabular}{c} MOEGO \\ GMOEGO \\ \end{tabular} & \(1.817\times 10^{-5}\) & 4.952 & 4.511 & \\  & GMOEGO & \(2.080\times 10^{-5}\) & 0.944 & 0.703 & 5.246 \\ \hline ZDT2 & \begin{tabular}{c} MOEGO \\ GMOEGO \\ \end{tabular} & \(2.124\times 10^{-5}\) & 5.247 & 4.530 & \\  & GMOEGO & \(3.878\times 10^{-5}\) & 0.800 & 0.601 & 6.559 \\ \hline ZDT3 & \begin{tabular}{c} MOEGO \\ GMOEGO \\ \end{tabular} & \(1.594\times 10^{-4}\) & 4.249 & 2.474 & 4.634 \\  & GMOEGO & \(1.216\times 10^{-4}\) & 0.917 & 0.636 & 4.634 \\ \hline DTLZ2 & \begin{tabular}{c} MOEGO \\ GMOEGO \\ \end{tabular} & \(3.200\times 10^{-3}\) & 32.943 & 32.843 & 32.843 & 13.734 \\  & GMOEGO & \(3.070\times 10^{-3}\) & 2.397 & 2.365 & 13.734 \\ \hline DTLZ5 & \
\begin{tabular}{c} MOEGO \\ GMOEGO \\ \end{tabular} & \(1.260\times 10^{-4}\) & 28.014 & 26.138 & \\  & GMOEGO & \(1.290\times 10^{-4}\) & 2.353 & 2.272 & 11.906 \\ \hline DTLZ7 &
\begin{tabular}{c} MOEGO \\ GMOEGO \\ \end{tabular} & \(5.530\times 10^{-3}\) & 29.679 & 27.426 & \\  & GMOEGO & \(3.740\times 10^{-3}\) & 2.392 & 2.183 & 12.408 \\ \hline \end{tabular}
\end{table} 
Table 2: Results of the test functions. Figure 7: Pareto front comparisons: **(a)** ZTD1; **(b)** ZTD2; **(c)** ZTD3; **(d)** DTLZ2; **(e)** DTLZ5; **(f)** DTLZ7. ## 5 Aerodynamic Design Optimization

In order to investigate the performance of the proposed GMOEGO algorithm in real engineering optimization applications, an aerodynamic airfoil shape optimization case [46] is selected to test the algorithm. The baseline of the airfoil is RAE2822 under the conditions of \(Ma=0.75\), \(Re=6.5\times 10^{6}\), and \(\alpha=2.31^{\circ}\). The optimization target is to maximize the lift-drag ratios under two different Mach numbers: \(Ma=0.6\), \(Ma=0.75\). The optimization problem can be defined as:

\[\left. \begin{array}{c}Max.\quad\left(C_{L}/C_{D}\right)_{Ma=0.6}\\ \quad\left(C_{L}/C_{D}\right)_{Ma=0.75}\end{array} \right\} \tag{18}\] \[s.t.\left\{\left\{x_{i}\in\mathbf{X}|i=1,\;2,\;\cdots,\;n\right\} \right\}\]

in which \(C_{L}\) is the lift coefficient, \(C_{D}\) is the drag coefficient, \(\mathbf{X}\) is the design parameters of the Hicks-Henne parametrization method [47], and \(n\) is the number of control parameters. In this paper, \(n=14\) is set to parametrize the airfoil geometry. The Computational Fluid Dynamics (CFD) simulator used to calculate the aerodynamic coefficients is the open-source software SU2 [48]. In this work, the Jameson scheme is selected as the spatial discretization scheme, and the Spalart-Allmaras (SA) one equation turbulence model is used for turbulence closure. The computational O-type mesh of RAE2822 airfoil is shown in Figure 9.

During the optimizations, 43 initial samples are set by using the LHS method, and the maximum FEs of CFD evaluations is limited to 400. The computations are carried out on the computational platform of Intel Core i9-9900k CPU (8 cores) with GTX-1066 GPU. After 400 CFD evaluations, the Pareto front obtained by the proposed GMOEGO algorithm is illustrated in Figure 10. Three representative optimized solutions marked with Optimum 1-3 in Figure 10 are selected to show their aerodynamic performances. Figure 8: Wall-clock time history comparisons: (**a**) ZDT series functions; (**b**) DTLZ series functions.