# An Improved Multi-Objective Grey Wolf Optimizer for Aerodynamic Optimization of Axial Cooling Fans
## 3 Results and Discussion
### Aerodynamic Optimization with IMOGWO  
#### 3.2.1 Definition of Optimization Problem  
The multi-objective optimization process coupled with the IMOGWO algorithm is presented in Figure 14. Firstly, 10 design variables are chosen and used to realize the blade parametrization, which is shown in Table 6, and the Latin hyper-cube sampling method is applied to generate 100 design samples that consist of the initial DoE (design of experiment) database. Then, each blade is meshed and evaluated by CFD, and ANSYS CFX(r) (V2022R1) is adopted in this paper.  
After all design samples have been evaluated, a response surface model (RSM) is built, and the IMOGWO is applied to search for the optimal solutions on the RSM model. Eight Pareto optimum sets are selected and validated by CFD again; if they meet the predefined objectives or reach the final number of iterations, the iterative loop will stop. Otherwise, the eight Pareto sets will be added to the existing samples, and a new RSM model will be regenerated. Through continuous iteration to improve the accuracy of the RSM model, it can contribute to the whole optimization process.  
Figure 15 shows the CFD mesh and boundary domains for the single passage. H-type mesh elements are generated using ANSYS TurboGrid(r) (V2022R1). An average y+ of less  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Design Variable** & **Description** & **Range** \\ \hline
1 & S\_mid & Sweep at mid span & \(-\)15\%\(\rightarrow\)+15\% \\
2 & S\_tip & Sweep at blade tip & \(-\)15\%\(\rightarrow\)+15\% \\
3 & T\_mid & Twist at mid span & \(-\)5\%\(\rightarrow\)+5\% \\
4 & T\_tip & Twist at blade tip & \(-\)5\%\(\rightarrow\)+5\% \\
5 & L\_mid & Lean at mid span & \(-\)30\%\(\rightarrow\)+30\% \\
6 & L\_tip & Lean at blade tip & \(-\)30\%\(\rightarrow\)+30\% \\
7 & Th\_mid & Thickness at mid span & \(-\)10\%\(\rightarrow\)+10\% \\
8 & Th\_tip & Thickness at blade tip & \(-\)10\%\(\rightarrow\)+10\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Definition of the design variables.  
Figure 14: The multi-objective optimization process based on the IMOGWO.  
than five is constrained on the blade surfaces and endwalls for low-Reynolds turbulence wall function modeling. The shear stress transport (k-\(\omega\) SST) turbulence model is used for turbulence closure with a high-resolution advection scheme and first-order discretization. Convergence is achieved when monitoring points achieve stability; specifically, the RMS residuals fall below 1 \(\times\) 10\({}^{-4}\) and the imbalances of energy, momentum, and mass in all domains are below 0.05% [34].  
There are two objectives in the optimization problem: pressure rise (\(P_{r}\)) and total pressure efficiency (\(Eff_{tp}\)). They are expressed as follows:  
\[P_{r}=Area\_avg(P_{out})-Area\_avg(P_{in}) \tag{29}\]  
\[Eff_{tp}=\frac{TP_{r}\times VFR}{Q_{input}}\times 100\% \tag{30}\]  
where \(TP_{r}=Mass\_avg(P_{out})-Mass\_avg(P_{in})\). \(P_{r}\) and \(TP_{r}\) are the pressure and total pressure difference, respectively, VFR represents the volume flow rate, and \(Q_{input}\) indicates the input shaft power.