## 5 Optimization of Variable Density Distribution of Micro-Textured Cutter  
Through the simulation of the stress field in milling titanium alloy with variable density micro-textured ball-end milling cutter, it is concluded that the instantaneous stress field and the maximum stress value of the cutter are directly affected by the different texture distribution densities on the rake face of the cutter. Therefore, it is necessary to establish the relationship between the different distribution densities of the micro-textures and the instantaneous stress field of the cutter, so as to optimize the texture density in the cutter-chip compact contact area and to obtain the best combination of texture distribution density, which provides a new concept for the design of micro-textured cutter. In this paper, a genetic algorithm was used to optimize the variable density distribution of micro-textured cutter. The instantaneous stress field of the cutter was taken as the optimization objective and the texture density \(X_{1}\) in the first area, \(X_{2}\) in the second area, and \(X_{3}\) in the third area of the cutter-chip compact contact area were taken as the optimization variables. When the genetic algorithm was adopted for optimization, the objective function should be established first. In this paper, the instantaneous stress field of a micro-textured cutter with a variable density was taken as the optimization objective, so a prediction model of the instantaneous stress field of the cutter was established as the objective function of the optimization model. The mathematical model was used to  
Figure 8: The curve of stress field and deformation of variable density micro-textured cutter with respect to time. (**a**) equivalent stress changes with time; (**b**) equivalent displacement varies with time. establish the instantaneous stress model of the variable density micro-textured cutter with respect to the variables \(X_{1}\), \(X_{2}\) and \(X_{3}\):  
\[\sigma=C\cdot X_{1}^{\alpha_{1}}\cdot X_{2}^{\alpha_{2}}\cdot X_{3}^{\alpha_{3}}, \tag{25}\]  
where \(C\) denotes the correlation coefficient of the prediction model and \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) denote the undetermined indices of the related independent variables. The logarithm of the two sides of the Equation (25) is  
\[\text{lg}\sigma=\text{lg}C+\alpha_{1}\text{lg}X_{1}+\alpha_{2}\text{lg}X_{2} +\alpha_{3}\text{lg}X_{3}. \tag{26}\]  
If \(y=\text{lg}\sigma\), \(\alpha_{0}=\text{lg}C\), \(x_{1}=\text{lg}X_{1}\), \(x_{2}=\text{lg}X_{2}\), \(x_{3}=\text{lg}X_{3}\), Equation (26) is transformed into a linear equation as follows:  
\[y=\alpha_{0}+\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}. \tag{27}\]  
According to Equation (27) and the stress field simulation data of the variable density micro-textured ball-end milling cutter, a multiple linear regression equation was established by the least-square method:  
\[\left\{ \begin{array}{l}y_{1}=\alpha_{0}+\alpha_{1}x_{11}+\alpha_{2}x_{12}+ \alpha_{3}x_{13}+\varepsilon_{1}\\ y_{2}=\alpha_{0}+\alpha_{1}x_{21}+\alpha_{2}x_{22}+\alpha_{3}x_{23}+ \varepsilon_{2}\\ \ldots\ldots\\ y_{9}=\alpha_{0}+\alpha_{1}x_{91}+\alpha_{2}x_{92}+\alpha_{3}x_{93}+ \varepsilon_{9}\end{array} \right., \tag{28}\]  
where \(\varepsilon_{i}\) denotes a random error. The stress field simulation data of the variable density micro-textured ball-end milling cutter was substituted into Equation (28).