## 4 Optimization Methods  
### Bezier Surface Modeling Method  
The Bezier surface modeling method means using control points to describe the geometry of the blade. The significant advantage is that the high-order continuity of control points can guarantee the smoothness of the blade surface [16]. This paper first adopts a novel modeling method, as shown in Figure 6, by cutting the compressor blade from the trailing edge, the entire suction surface and pressure surface from a Bezier surface, thereby further reducing the number of design variables. The Bezier surface corresponds to the original blade at the four vertices. The Bezier surface modeling method is to superimpose the Bezier surface based on the original blade by changing the active points. The process of Bezier surface modeling is divided into two steps. First, the arc length of the original compressor profile is parameterized and converted into the dimensional arc length in the calculation plane. The chord-wise and span-wise directions of the compressor are dimensionless to the unit, respectively. The formula is as follows:  
\[\xi_{i,j}=\frac{\sum_{m=1}^{i}l_{m}}{L_{j}} \tag{1}\]  
\[\gamma_{i,j}=\frac{\sum_{n=1}^{j}l_{n}}{L_{i}} \tag{2}\]  
where \(L_{j}\) corresponds to the arc length of the radial _jth_ section line; \(l_{m}\) refers to the _mth_ chord length of the radial _jth_ section line; \(\xi_{i,j}\) is the dimensionless horizontal coordinates; \(L_{i}\) corresponds to the arc  
Figure 5: Calculated and measured compressor performance maps. (**a**) Adiabatic efficiency at the different normalized mass flow; (**b**) total pressure ratio at the different normalized mass flow. Figure 6: Bezier surface modeling of a compressor blade. length of the radial _ith_ section line, \(l_{n}\) is the _nth_ chord length on the radial ith section line; \(\gamma_{i,j}\) is the dimensionless ordinate coordinates. Second, based on the original compressor profile, the Bezier function is used to apply the fluctuation amount A. The value of fluctuation amount A in the circumferential direction is assigned to the original blade to obtain an optimized blade on the calculation plane. The formula is as follows:  
\[A=\sum_{k=0}^{n}\left(\sum_{l=0}^{m}P_{k,l}B_{l}^{m}(v)\right)B_{k}^{n}(u) \tag{3}\]  
\[B_{l}^{m}(v)=C_{k}^{n}v^{l}(1-v)^{l-m} \tag{4}\]  
\[B_{k}^{n}(u)=C_{k}^{n}u^{k}(1-u)^{n-k} \tag{5}\]  
\[C_{k}^{n}=\frac{n!}{(n-k)!k!}\ 0\leq k\leq n \tag{6}\]  
In the Formula (3), \(P_{k,l}\) corresponds to the control points of the Bezier surface, as can be seen in Figure 6; \(B_{l}^{m}(v)\), \(B_{k}^{n}(u)\) correspond to Bernstein functions determined by the Formulas (4) and (5), respectively. The v and u are two coordinate axes of the Bezier surface calculation plane; \(C_{k}^{n}\) is the combinatorial number determined by the Formula (6). The traditional aerodynamic optimization of the compressor mainly changes the different span-wise profile and three-dimensional sweep-lean characteristics of compressors, resulting in too many design variables, increased design space, and too much calculation cost, which leads to a \"dimensional disaster.\" It does not meet the actual fast application of the engineering project, thus the optimization fails. In this paper, in order to decrease the number of design variables, the Bezier surface modeling method was selected to optimize the suction surfaces and pressure surfaces of rotors and stators of the compressor, and the \(6\times 3\) order Bezier surface was used for modeling. According to the engineering philosophy of repeated-stage design, the design variables are further reduced. The fluctuation amount A of the second-stage is the same as the first stage, and the fluctuation amount A of rotors and stators are optimized respectively. The amplitude of fluctuation amount A is plus or minus 2 mm, considering that the geometric changes near the leading edge and the trailing edge have a greater impact on the flow field than the middle chord.