## 4 Experiments and Analysis
### Real-World Application of Airfoil Design
The problem is thus articulated as follows:  
\[\max_{\theta_{1},\theta_{2}\sim,\theta_{10}}C_{L/D}(M,Re,AoA,Hicks(\theta_{1}, \theta_{2},...,\theta_{10})) \tag{12}\]  
\[s.t.-0.001\leq\theta_{i}\leq 0.001,i=1.7 \tag{13}\]  
\[-0.006\leq\theta_{i}\leq 0.006,\ i=2.5 \tag{14}\]  
\[-0.009\leq\theta_{i}\leq 0.009,\ i=3.4 \tag{15}\]  
\[-0.002\leq\theta_{i}\leq 0.002,\ i=6.10 \tag{16}\]  
\[-0.007\leq\theta_{i}\leq 0.007,\ i=8.9 \tag{17}\]  
In the provided equation, \(C_{L/D}\) signifies the fitness value of the lift-to-drag ratio, while _Hicks_ denotes the ten decision variables formulated using a Hicks-Henne function. The parameters \(M,Re,AoA\) represent predefined values for the Mach number, Reynolds number, and angle of attack. Within this study, two algorithms, CALSAPSO and GPEME, are evaluated alongside the newly proposed algorithm. The primary metric for gauging their performance is the lift-to-drag ratio. Results of this analysis can be found in Table 6. Itis noteworthy that the algorithm presented in this paper outperforms the others, particularly in terms of the maximum, median, and average values of the lift-to-drag ratio. An instance of the symmetric airfoil NACA0012 is globally renowned and frequently employed as a benchmark for theoretical analyses and wind tunnel comparisons. Its geometry is parameterized at 0.5 before being simulated via the Xfoil software (0.0.8). It then undergoes an optimization process using ASAPSO, capped at 120 evaluations. In the provided visual, the initial airfoil is shown in green and its optimized counterpart in red, emphasizing the airfoil's smooth and inherently symmetrical contour. The comparison of the original airfoil geometry with ASAPSO airfoil geometry is given in Figure 12. From Figure 12, it can be illustrated that the proposed ASAPSO algorithm exhibits better optimization airfoil geometry compared to NACA0012. The integration of surrogate models with PSO can significantly enhance the efficiency of engineering design processes, particularly in fields like aerospace, automotive, and renewable energy. By optimizing complex systems such as airfoils or turbine blades, managers can achieve better performance outcomes, a reduced time-to-market, and lower development costs. By reducing the computational cost of simulations and improving the convergence speed of optimization algorithms, the proposed method can lower the overall cost of engineering projects. This makes advanced optimization techniques more accessible to smaller firms and industries with limited resources. The optimization of aerodynamic designs, such as airfoils, directly contributes to energy efficiency. For example, optimizing \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Best** & **Median** & **Mean** \\ \hline ASAPSO & **105.33** & **99.98** & **99.62** \\ CALSAPSO & 100.27 & 98.65 & 99.01 \\ GPEME & 96.38 & 94.86 & 94.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the lift-over-drag ratio on the aerodynamic airfoil design problem after 25 runs. Figure 12: Comparison of the original airfoil geometry with ASAPSO airfoil geometry. wind turbine blades can lead to increased energy capture, while optimizing aircraft wing designs can reduce fuel consumption. These improvements have significant environmental benefits, contributing to the reducing of carbon emissions and promoting sustainability.