## 3 Optimization Design Strategy
### CFD Analyses  
In this study, calculation of the optimization objectives and verification of the optimized result were all completed by CFD. Thus, the credibility of the optimization results was determined by the accuracy of the CFD analyses. Therefore, 3D steady incompressible Reynolds-Averaged Navier-Stokes (RANS) equation was used in the simulation of the mixed flow pump. The governing equations are defined as follow:  
\[\frac{\partial V_{j}}{\partial x_{i}}=0 \tag{4}\]  
\[\frac{\partial(V_{i}V_{j})}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial p}{ \partial x_{i}}+\phi\frac{\partial^{2}V_{i}}{\partial x_{i}x_{j}}+\frac{ \partial\tau_{ij}}{\partial x_{j}} \tag{5}\]  
where \(\tau_{ij}\) are the Reynolds stresses, \(V\) is the time-average velocity, and \(\theta\) is the kinematic viscosity. The RANS equation was solved by the Shear stress transport (SST) \(k-\omega\) turbulence model because this model has the advantages of accurately calculating the internal flow pattern of the mixed flow pump impeller [28] and shows less sensitivity to Y+ [29]. A high-resolution scheme was used to solve the convection-diffusion equation. All the walls in the computational domains were considered as no-slip. The boundary conditions of the outlet and inlet were set as the \"static pressure\" and \"mass flow rate\", respectively. \"Frozen rotor\" was adopted for data transfer between the rotating domain and the fixed domain, while \"none\" was adopted for data transfer between the fixed domains. The convergence criteria for all calculations was set as \(5\times 10^{-5}\).  
The meshes quality and quantity have a great influence on the accuracy of CFD analyses [30]. Compared with unstructured mesh, structured mesh has the advantages of controllable quantity and quality. Therefore, the discretization of the entire computational domain was accomplished by structured meshes, as shown in Figure 4. ANSYS ICEM  
Figure 3: Three-dimensional inverse design method flowchart.  
was employed in the mesh division of outlet and inlet pipes, and ANSYS TurboGrid was employed in the mesh division of vane diffuser and impeller. Figure 5 shows the result of the mesh independence check. The pump efficiency increased gradually with increasing number of meshes until \(4.71\times 10^{6}\), after which the pump efficiency remained basically unchanged. Therefore, the total number of meshes used in this study was \(4.71\times 10^{6}\). Meanwhile, the maximum Y+ on the impeller blade and hub was less than 30, as shown in Figure 4.  
The experimental performance and numerical performance of the baseline model were compared to illustrate the reliability of the CFD analysis; the results are shown in Figure 6. The experimental performance of the baseline model was tested by the Tianjin experimental test bench, as shown in Figure 7. In this test bench, the flow rate was measured by the intelligent electromagnetic flowmeter with the measurement error at \(<\pm 0.2\%\), the torque and rotation speed was measured by the intelligent torque speed sensor with the measurement error at \(<\pm 0.1\%\), and the head was measured by the intelligent differential pressure transmitter with the measurement error at \(<\pm 0.1\%\). The random uncertainty and  
Figure 4: Mesh of the computational domains.  
Figure 5: Mesh independence check.  
overall uncertainty of the test bench were less than 0.1% and 0.3%, respectively. Figure 6 shows that the experimental results of pump head and efficiency are in good agreement with the numerical results in the whole flow range. The maximum deviations of the pump head and efficiency did not exceed 4% and 2%, respectively. Thus, the CFD simulation used in this study has sufficient accuracy to ensure reliability of the optimization results.