## 3 Model Parameterization and Multidisciplinary Optimization
### Multidisciplinary Optimization  
In the process of multidisciplinary optimization, the blade number of both T-100 and T-100 re design is fixed and not optimized in the 3-D optimization stage, because extra difficulty will be imported for the rotor shell cut which is essential for the CSM analysis.  
Regarding to the optimization variables, if a control point has variation in two coordinate directions, radial direction, and axial direction, it will have two variables in shape optimization process. Therefore, the total number of optimization variables for rotor optimization is 40, among which five variables for the solid hub back face profile, 11 for the hub and shroud meridional profiles, 14 for the beta-angle distribution, and 10 for the thickness distribution. In addition, four dependent variables whose positions are linked to one of the design variables are involved.  
Table 1 shows the design parameters of T-100 turbine and T-100 re turbine, and they are set in the optimization. It should be noted that the rotor rotational speed of T-100 re turbine is higher than that of original T-100 turbine, after the T-100 turbine was redesigned by the automated preliminary design proposed in Part I [20].  
The flow simulations are conducted using ANSYS CFX, a commercial CFD software. The total grid number of CFD is 478,613, as shown in Figure 5, in which the grid numbers of the nozzle, the rotor, and the diffuser are 156,925, 289,363 and 32,325, respectively. The RANS coupled with the SST turbulence model is solved with the first layer y+ less than 2. It is worth mentioning that the mesh number and the turbulence model used were verified by mesh independency and various turbulence models provided in ANSYS CFX, respectively. The details of verification can be found in Reference [12] published by our research group.  
In this paper, the software ANSYS Workbench is applied as the CSM solver. The total grid number of CSM is 555,663, as shown in Figure 6. The coupling of the CFD and the CSM solvers adopt a loose couple strategy, which at first evolves a CFD evaluation and then a CSM evaluation, and the aerodynamic change caused by the rotor deformation is not considered. That is to say, the effect of flow solid interaction was neglected because the blade of radial inflow turbines is relatively much more rigid than that of axial turbines. The deformation of the rotor blade is so small that its effect is negligible in this study for the aerodynamic performance.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Items** & **Unit** & **Value** \\ \hline Fluid & [-] & Air \\ Inlet total temperature & [K] & 1056.5 \\ Inlet total pressure & [kPa] & 580.4 \\ Mass flow rate & [kg/s] & 0.33 \\ Number of nozzle vanes & [-] & 19 \\ Number of rotor blades & [-] & 16 \\ Pressure ratio & [-] & 5.73 \\ Power & [kW] & 121 \\ Rotational speed (T-100) & [r/min] & 106,588 \\ Rotational speed (T-100 re) & [r/min] & 114,185 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design parameters of T-100 turbine and T-100 re turbine.  
Figure 5: Computational mesh of flow simulation. (**a**) Whole computation domain of flow area; (**b**) rotor inlet mesh.  
Figure 7 illustrates the convergence history of flow simulation for a single CFD evaluation. It can be seen that the higher convergence precision is obtained in a short time.