# Performance Optimization Design of Diagonal Flow Fan Based on Ensemble of Surrogates Model
## 4 Ensemble of Surrogate Model and Multi-Objective Optimization  
### Parameterized Design of Blade  
[PERSON] [15; 16] first proposed the CST parameterization method in 2006. This method determines the basic shape of airfoil based on the class function and corrects its shape parameters using the shape function. The research shows that the CST parameterization method has the advantages of adjustability, fewer design variables, and a wide design range. The technology is combined here with the design of the middle arc of the blade of the small diagonal flow fan, as shown in Figure 7.  
As shown in Figure 7a, the three-dimensional diagram of the diagonal flow fan impeller in this paper is shown. In Figure 7b, the xOy coordinate system is constructed with \(M_{1}\) in a single blade as the starting point and \(M_{2}\) as the end point. The parameterized blade has fixed endpoints \(M_{1}\) and \(M_{2}\), so the blade chord length c remains unchanged, as shown in Figure 7c.  
Usually, the airfoil curve can be expressed by the CST parameterization method as:  
\[\frac{y}{c}=C\left(\frac{x}{c}\right)S\left(\frac{x}{c}\right)+\frac{x}{c} \frac{\Delta z_{te}}{c},0\leq\frac{x}{c}\leq 1 \tag{5}\]  
where \(C(x/c)\) and \(S(x/c)\) are the class function and shape function, respectively; \(c\) is the chord length of the airfoil; and \(\Delta z_{te}\) is the trailing edge thickness of the airfoil, where \(\Delta z_{t}\) is 0.  
\[C\left(\frac{x}{c}\right)=\left(\frac{x}{c}\right)^{N_{1}}\left(1-\frac{x}{c} \right)^{N_{2}} \tag{6}\]  
In Formula (6), the airfoil type is determined by \(N_{1}\) and \(N_{2}\). For the parametric design of airfoil, let \(N_{1}=0.75\) to ensure the circular leading edge; let \(N_{2}=0.75\), guaranteeing the circular airfoil trailing edge.  
The shape function \(S(x/c)\) controls the curve modeling between the two points of the leading edge and the trailing edge of the airfoil, generally expressed in the Bernstein  
Figure 7: Blades of diagonal flow fan and their parameterization. (**a**) Diagonal flow fan impeller. (**b**) Arc line in blade. (**c**) Blade parametric coordinates.  
polynomial form by introducing a weighting factor. The weighted sum of the n-order Bernstein polynomials is:  
\[\begin{split} S\left(\frac{x}{c}\right)=\sum\limits_{i=0}^{n}\left[ b_{i}\cdot K_{i,n}\cdot\left(\frac{x}{c}\right)^{i}\cdot\left(1-\frac{x}{c} \right)^{n-i}\right]\\ K_{i,n}=\frac{n!}{i!(n-i)!}\end{split} \tag{7}\]  
Among them, \(K_{i,n}\) is the binomial coefficient, \(b_{i}\) is the weight factor, and \(n\) is the polynomial order. Usually, the higher the order is, the higher the accuracy of the airfoil is. Considering the fitting accuracy and engineering application, \(n=3\) is selected, that is, the third-order polynomial only needs four variables.  
In summary, the arc of the small diagonal flow fan blade in this paper can be expressed as:  
\[\begin{split} a_{i}(x)&=C(x)\cdot S(x)\\ a_{i}(x)&=x^{0.75}\cdot\left(1-x\right)^{0.75} \cdot\sum\limits_{i=0}^{n}v_{i}x^{i}(1-x)^{n-i}\end{split} \tag{8}\]  
where \(v_{0}\), \(v_{1}\), \(v_{2}\), and \(v_{3}\) are the four design variables in this paper.  
In order to explore the accuracy of CST parametric design of the blade camber line, the y/c residual is introduced to characterize the fitting error:  
\[y/c\ Residuals=\frac{y/c-y/c^{\prime}}{y-c} \tag{9}\]  
As shown in Figure 8, the arc error analysis of the blade is fitted using the CST parametric design method. It can be seen from the diagram that the error is higher at the leading edge and the trailing edge, and the maximum error point appears near the trailing edge, reaching 0.00105. Therefore, when the maximum fitting error is only 0.1%, this paper suggests that the parametric design method meets the accuracy requirements.