# Robust Algorithm Software for NACA 4-Digit Airfoil Shape Optimization Using the Adjoint Method
## 4 Results
### Multipurpose Optimization of Airfoil  
The optimization process involves selecting the NACA4418 airfoil as the primary airfoil and using it to determine the final shape of the airfoil by minimizing the total drag coefficient and maximizing the lift coefficient based on design parameters and the primary airfoil's shape. Optimization is performed under viscous flow conditions as specified in Table 3. The profile design variables include the maximum thickness, maximum camber, and maximum camber position, and their values are adjusted to achieve the intended purpose and obtain the final shape of the profile. The lift and drag values are deliberately chosen in such a way that they cannot be achieved under the defined flight conditions. The purpose of this work is to show the ability of the algorithm in defining the design limits of the shape of the airfoil. which moves toward optimizing the shape as much as possible; after reaching the defined design limits, it continues to optimize so that it can display the most ideal and closest possible shape to the output, according to the optimization goals. Table 4 indicates that although the airfoil thickness and maximum camber decreased, and the camber location changed toward the end of the airfoil, only the drag coefficient was optimized. Therefore, it is not possible to design an airfoil that optimizes both conditions to meet them simultaneously under the specified flow conditions. In Figure 5, the convergence of the cost function, lift-to-drag ratio, and aerodynamic coefficients is illustrated. The graphs show that all parameters have converged to an acceptable value after 20 iterations. The aerodynamic coefficients shown in the figures include the viscous drag coefficient \(C_{dr}\), the pressure drag coefficient \(C_{dp}\), the lift coefficient \(C_{l}\), the total drag coefficient \(C_{d}\), and the lift-to-drag ratio \(C_{l}/C_{d}\). Although both optimization goals were not achieved, the changes in all design parameters and aerodynamic coefficients have become insignificant. The authors deliberately chose input values for maximum lift and minimum drag coefficients that were unattainable due to design limitations to evaluate the algorithm's performance. The results show that the algorithm increases the lift-to-drag ratio by reaching the maximum lift target value while minimizing drag, but both goals cannot be achieved simultaneously due to design constraints. Figure 6 illustrates that after fifteen cycles, the maximum thickness \(t\), the bend location \(m\), and the maximum bend \(p\) exhibit insignific \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Initial Value** & **Final Value** \\ \hline Maximum Thickness & \(t\) & 0.180 \(c\) & 0.143 \(c\) \\ Maximum Camber & \(m\) & 4.000\% & 3.986\% \\ Maximum Camber position & \(p\) & 0.400 & 0.397 \\ Lift Coefficient & \(C_{l}\) & 0.101 & 0.398 \\ Pressure Drag Coefficient & \(C_{dp}\) & 0.150 & 0.083 \\ Viscous Drag Coefficient & \(C_{dp}\) & 0.030 & 0.037 \\ Total Drag Coefficient & \(C_{d}\) & 0.179 & 0.120 \\ Lift-to-Drag Ratio & \(C_{l}/C_{d}\) & 0.564 & 3.328 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The initial and final values of the components that determine the airfoil geometry and aerodynamic coefficients were optimized to achieve maximum lift and minimum drag coefficients. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Ratio** \\ \hline Mach Number & M & 0.72 \\ Reynolds Number & Re & 3000 \\ Angle of Attack & \(\alpha\) & 2.8\({}^{\circ}\) \\ \hline Target Lift Coefficient & \(Cl_{\max}\) & 0.45 \\ Target Drag Coefficient & \(Cl_{\min}\) & 0.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The flow conditions utilized in the proposed multipurpose shape optimization for the NACA4418 airfoil. Figure 5: The figures illustrate various aspects of the optimization process of the multipurpose lift-to-drag ratio algorithm. (**a**) Changes in the cost function during the optimization process.