## 5 Parameter Optimization  
### Response Surface Methodology  
Response surface methodology (RSM) is a mathematical approach that combines experimental design and statistical analysis and is widely used in different fields to study the effects of multiple parameters on target parameters. RSM establishes a quadratic response surface regression model by fitting mathematical relationships between multiple influences and response variables to determine the optimal combinations, thus obtaining an optimal response within the planning horizon of the experimental design and with good robustness [41, 42]. The flow chart of the response surface methodology optimization is shown in Figure 12.  
Box-Behnken Design (BBD) is a widely used RSM for optimizing and modeling the effects of multiple factors on a response variable in an experiment, which was introduced by [PERSON] and [PERSON] in 1960. The BBD test is based on the levels of three factors: low (\(-1\)), middle (\(0\)), and high (\(+1\)). The BBD test avoids the potential for experiments to take maximum or minimum values for all factors by arranging the different levels of the factors in a series of experimental points that are concentrated on the middle and edges and excluded from the vertices (i.e., the extreme corners) of the factor space. The center point is repeated in the experiment and is used to estimate the experimental error and the stability of the model. This design can effectively reduce the number of experiments and lower the experimental risk while providing sufficient data for constructing highly accurate quadratic response surface models. With BBD, the main,  
Figure 12: The flow chart of the response surface methodology optimization.  
interaction, and secondary effects of factors can be systematically investigated to optimize the response variables. Figure 13 shows a 3-factor BBD. Because BBD usually have fewer design points, the running cost is lower than that of a central composite design (CCD) with the same number of factors. Therefore, the BBD is proposed to be chosen for the optimized design in this paper.  
In order to obtain a response surface model between the volume ratio and the mechanism parameters, the order of the response surface model needs to be chosen appropriately. Although the first-order polynomial linear response surface model has a relatively simple model, it can only reflect the linear relationship between the input and the output, and it is difficult to represent the nonlinear relationship. The high-order polynomial response surface model has a higher fitting accuracy, but the model is too complex and it takes a longer time to fit the response surface. Therefore, a multivariate quadratic response surface model is used in this paper, which ensures the accuracy and complete fitting of the nonlinear relationship between the volume ratio and the mechanism parameters while minimizing the complexity of the response surface model. The general equation of the multivariate quadratic response surface model is:  
\[y=\beta_{0}+\sum_{i=1}^{k}\beta_{i}x_{i}+\sum_{i=1}^{k}\beta_{ii}x_{i}^{2}+ \sum_{i=1}^{k-1}\sum_{j=i+1}^{k}\beta_{ij}x_{i}x_{j}+\varepsilon \tag{15}\]  
where \(x\) and \(y\) are the factor variable and response value, respectively, and \(k\) is the number of factors. \(\beta_{0}\) is the constant coefficient, \(\beta_{1},\beta_{2},\cdots,\beta_{k}\) and \(\beta_{11},\beta_{22},\cdots,\beta_{kk}\) are the coefficients of the primary and secondary terms of RSM, and \(\beta_{12},\beta_{23},\cdots,\beta_{(k-1)k}\) is the coefficient of the cross-effect term. The approximation error \(\varepsilon\) is generally equal to zero when the engineering accuracy requirements are met.