## 3 Results and Discussion
### Optimization of the SLM Processing Parameters  
In Table 5, the density results using different methods including the Archimedes methods and image analysis were presented. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{ \begin{tabular}{c} \(P\) \\ **(W)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} \(v\) \\ **(mm/s)** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} \(h\) \\ **(mm)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Energy Density** \\ **(J/mm\({}^{2}\))** \\ \end{tabular}  } & \multicolumn{4}{c}{\(\rho_{relative}\) **(\%)**} \\ \cline{5-6}  & & & & & **Archimedes Methods** & **Image** \\ \cline{5-6}  & & & & **1** & **4** & **4*** & **Analysis** \\ \hline
1 & 625 & 1400 & 0.35 & 1.28 & 95.67 & 9  
Taking into account the influence of three variables, laser power (\(P\)), scan speed (\(v\)), and hatching distance (\(h\)), the relative density can be expressed by Equation (9). \[\rho_{relative}=a_{0}+a_{1}P+a_{2}v+a_{3}h+a_{11}P^{2}+a_{22}v^{2}+a_{33}h^{2}+a _{12}Pv+a_{13}Ph+a_{23}vh \tag{9}\]  
The regression constants \(a_{0}\), \(a_{1}\)... \(a_{23}\) were calculated by Equation (7). Table 6 presents the results of \(a_{0}\), \(a_{1}\)... \(a_{23}\). Table 5 shows the measured results of relative density using the Archimedes methods. In Table 6, the value of the correlation coefficient \(R^{2}\) is provided as a measure of how closely the statistical model fits the data--the coefficient of determination varies between 0 and 1. \(F\) is used to perform the \(f\) test and test the regression significance of the model in Equation (9), using different combinations of the constants \(a_{0}\), \(a_{1}\)... \(a_{23}\). For this study, the threshold of \(F\) values 5.9988 at the 95% confidence level, which is determined by the freedom degree of regression and residual variation, which are 9 and 4, respectively, corresponding to the design in Table 5. According to \(f\) test theory, if the \(F\) value of regression model is larger than the threshold, the model indicates a highly significant regression. From Table 6, the \(F\) value of 8.9160 is larger than the tabled \(F_{9,4}\) value of 5.9988 at the 95% confidence level. Thus, the model indicates highly significant regression. Furthermore, the model has high values of \(R^{2}\). This indicates that the models fit the results data well and are useful for predictions and optimization of parameters. According to the relative density model, the optimum combination of SLM process parameters is \(P_{\rm o}\) = 651 W, \(v_{\rm o}\) = 500 mm/s, and \(h_{\rm o}\) = 0.378 mm. Figure 5a-f show the 3d response surfaces and the 2D contours reflecting the change of relative density with power and scan speed at some specified hatching distance. According to the contour in Figure 5d-f, if the hatching distance being given out of the experimental range of 0.3 mm \(<\) h \(<\) 0.4 mm, the optimum combinations of parameters corresponding to the highest predicted value of relative density would be in the different ranges of parameters. The relative density is higher for AlSi10 Mg both at the combination of a higher scan speed, a higher laser power, and a lower hatching distance and at the combination of a lower scan speed, a moderate power, and an optional hatching distance in the range of experimental parameters. This is because the relative density does not obviously change with the hatching distance at a lower scan speed and a moderate power in the range of the parameters of this investigation. The model would have two combinations of optimum parameters if a greater range of experimental hatching distance