## 3 Results and Discussions  
### RSM Depth Modeling  
With the depth of the hardened layer as the response variable, a multivariate regression model was constructed using data from BBD experiments at varying induction frequencies. Subsequently, the model's performance was evaluated through analysis of variance (ANOVA) [36], with the ANOVA results for the hardened layer depth detailed in Tables 4 and 5. The F-value, which represents the statistic of the F-test, indicates the ratio of the sum of squares of deviations between and within groups relative to the degrees of freedom, while the \(p\)-value indicates the significance of the factors. As shown in Table 4, at an induction frequency of 10 kHz, the factors of laser power (A), scanning speed (B), induction power (C), the quadratic terms B\({}^{2}\) and C\({}^{2}\), and the interaction term BC all have a significant effect on the depth of the hardened layer (\(p<0.0001\)), while the remaining factors are non-significant. Comparing mean square values, the process parameters affecting depth are ranked as C > B > A. The \(p\)-value for the lack-of-fit term is 0.5128 (\(p>0.05\)), indicating that the model effectively describes the relationship between the depth of the hardened layer and the influencing factors. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **Sum of Squares** & **Mean Squares** & **F-Value** & **\(p\)-Value** & **Significance** \\ \hline Model & 10.68 & 1.19 & 828.44 & \textless{}0.0001 & Yes \\ A-P & 0.6483 & 0.6483 & 425.51 & \textless{}0.0001 & Yes \\ B-v & 2.73 & 2.73 & 1908.39 & \textless{}0.0001 & Yes \\ C-IP & 7.10 & 7.10 & 4952.77 & \textless{}0.0001 & Yes \\ A\({}^{2}\) & 0.0039 & 0.0039 & 2.71 & 0.1436 & No \\ B\({}^{2}\) & 0.0915 & 0.0915 & 68.83 & \textless{}0.0001 & Yes \\ C\({}^{2}\) & 0.1603 & 0.1603 & 111.89 & \textless{}0.0001 & Yes \\ AB & 0.0002 & 0.0002 & 0.1291 & 0.7299 & No \\ AC & 0.0000 & 0.0000 & 0.0342 & 0.8585 & No \\ BC & 0.1002 & 0.1002 & 69.96 & \textless{}0.0001 & Yes \\ Residual & 0.0100 & 0.0014 & & & \\ Lack of fit & 0.0100 & 0.0033 & 2.35 & 0.5128 & No \\ Pure error & \textless{}0.0001 & \textless{}0.0001 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ANOVA of hardened layer depth predicted by RSM model (induction frequency = 10 kHz). In addition, the coefficient of determination of the RSM model has a value of 0.998 for R\({}^{2}\) and 0.985 for R2 adjust. After calculating Equations (1)-(3), the RMSE is 0.249 mm, the MAE is 0.019 mm, and the MAPE is 0.31%. The final response regression prediction of the quadratic polynomial model of the actual parameters is established as follows:  
\[ \begin{array}{l}D(mm)=12+0.003056\times P-8.35\times v+0.0646\times IP+5.6 \times 10^{-5}\times P\times v-\\ 9.2\times 10^{-7}\times P\times IP+0.0325\times v\times IP-3.38\times 10^{-7}\times P ^{2}+0.9985\times v^{2}-0.00136\times IP^{2}\end{array} \]  
Table 5 indicates that at an induction frequency of 15 kHz, laser power (A), scanning speed (B), induction power (C), and the quadratic term C\({}^{2}\) significantly influence the depth of the hardened layer (\(p\)! 0.0001). Additionally, the quadratic terms A\({}^{2}\) and B\({}^{2}\) and the interaction term AC have a notable impact (\(p\)! 0.05), while the other factors are non-significant. The ranking of the process parameters affecting depth remains consistent with that at 10 kHz: C? B? A. The \(p\)-value for the lack-of-fit term is 0.6217 (\(p\)? 0.05), indicating that the model effectively describes the relationship between the depth of the hardened layer and the influencing factors. In addition, the coefficient of determination of the RSM model has a value of 0.993 for R\({}^{2}\) and 0.955 for R2 adjust. After calculating Equations (1)-(3), the RMSE is 0.069 mm, the MAE is 0.054 mm, and the MAPE is 1.12%. The final response regression prediction of the quadratic polynomial model of the actual parameters is established as follows:  
\[