## 3 Preliminary Optimal Design Method
### Optimization Problem  
The preliminary design method suffers from a lack of reliable design guidelines and extremely complex relations between aerodynamic performance and design parameters. These problems can only be settled with an optimization algorithm rather than repeated trial and error adjustment [5,21]. To avoid the well-known phenomenon of local extreme value, a GA was applied to explore the global optimal solution. Here, arithmetic crossover, roulette wheel reproduction, and non-uniform mutation operators were applied in the optimization process. The design adiabatic efficiency was selected as fitness. For each generation, population size was 200 and the number of generations was 100. Details of the GA, and its application, and validation in compressor optimization can be found in our other studies [31, 32, 33]. The current preliminary optimal design problem is illustrated as follows:  
Objective: To maximize design adiabatic efficiency \(\eta_{\text{ad}}^{*}\)  
\[\max f=\eta_{\text{ad}}^{*}, \tag{7}\]  
\[\eta_{\text{ad}}^{*}=\frac{(P_{2}/P_{1})^{(\gamma-1)/\gamma}-1}{(T_{2}/T_{1})- 1}, \tag{8}\]  
where the superscript \(*\) denotes the design condition, the superscript \(\gamma\) is the air heat capacity ratio, \(T\) is total temperature, and \(P\) is total pressure. Constraints: To keep total pressure ratio \(\pi\) equal to design total pressure ratio \(\pi^{*}\)  
\[\pi=\pi^{*}, \tag{9}\]  
and flow rate \(\dot{m}\) equal to design requirement \(\dot{m}^{*}\)  
\[\dot{m}=\dot{m}^{*}, \tag{10}\]As demonstrated in Section 2, mass flow rate and total pressure ratio are input as design requirements in the input module and remain unchanged in the following preliminary optimal design process. Then the constraints can be handled in the optimization algorithm and finally fulfilled. Design variables: Six types of design parameters, including aspect ratio \(\lambda\), solidity \(\sigma\), reaction \(\Omega\), rotation speed \(\omega\), outlet axial Mach number \(M_{\text{out}}\), and inlet radius ratio \(\tau_{\text{in}}\), were used in the preliminary design phase. The definition of \(\lambda\), \(\Omega\), \(\sigma\), and \(\tau_{\text{in}}\) are:  
\[\lambda=\frac{l}{c}, \tag{11}\]  
where \(l\) is blade height; \(c\) is blade chord length. \[\sigma=\frac{c}{s}, \tag{12}\]  
where \(s\) is the blade pitch. \[\Omega=1-\frac{(C_{2\theta}+C_{1\theta})(C_{2\theta}-C_{1\theta})}{2\omega(C_ {2\theta}r_{2}-C_{1\theta}r_{1})}, \tag{13}\]  
\[\tau_{\text{in}}=\frac{r_{\text{in,hub}}}{r_{\text{in,tip}}}, \tag{14}\]  
where \(r_{\text{in,hub}}\) is the compressor inlet hub radius and \(r_{\text{in,tip}}\) is the compressor inlet tip radius. We referred to Wisler et al. 's [PERSON] et al. (2018) research for the range of solidity, aspect ratio, reaction, and outlet axial Mach number selected in the present work. For IGV aspect ratios of 4-stage, 5-stage, 6-stage, and 7-stage compressors, the upper bounds were 1.9, 2.0, 2.1, and 2.2, and the lower bounds were 1.45, 1.55, 1.65, and 1.75, respectively. The upper bound was 1.45 and the lower bound was 1.00 for the outlet stator aspect ratio of four types of compressors. A linear variation was employed for aspect ratio upper bounds and lower bounds of each blade row along row-wise pattern from IGV to outlet stator of four types of compressors. The upper and lower bounds of solidity, reaction, and outlet Mach number were kept the same for four types of compressors (Table 1), respectively, which were basically consistent with the bounds of corresponding variables in [PERSON] et al. 's work. Importantly, the rotation speed and the inlet radius ratio can affect the magnitude of the relative Mach number. To keep the relative Mach number below 1.4, and to meet the applicable range of the shock loss model adopted [PERSON] et al. (2018), the upper bounds of the rotation speed were linearly decreased from 6500 rpm to 5000 rpm. Similarly, the lower bounds were linearly decreased from 5500 rpm to 4000 rpm as the number of stages increased.