## 4 Numerical Validation
### Regression Model for Parameter Optimization  
In order to validate the regression model and optimize the analysis parameters, the parameter ranges are adjusted, and a new regression model is constructed considering the sensitivity analysis results in Section 4.1 and the LPBF process conditions. The convection coefficient, which has a very small sensitivity compared to other parameters, is fixed at 10 W/m\({}^{2}\)K [4; 9; 10], and the range of the emissivity is corrected to 0.37-0.45 considering the experimental value of 316L stainless steel powder layer [37]. The conductivity of 316L stainless steel powder at 295 K is about 0.2-0.25 W/mK [26], so that the conductivity coefficient is set to 0.014-0.018. The OPD range is modified from 120 \(\upmu\)m, which is the effective penetration depth in [PERSON] et al. [4], to 170 \(\upmu\)m.  
Forty-one analysis cases are generated by Box-Behnken design with the modified parameter range, as shown in Table 4, and a regression model for parameter optimization is constructed. Figure 8 shows the sensitivity for the modified parameter range. The porosity, OPD, and absorptivity have a dominant influence on the melt pool size, while the sensitivity of the emissivity and conductivity coefficient is relatively very small.  
For validation of the regression model, three arbitrary analysis parameter cases listed in Table 5 are selected within the parameter range in Table 4. By comparing the melt pool size between the thermal analysis model and the regression model, it can be confirmed whether the regression model accurately traces the thermal analysis model. The melt pool sizes by thermal analysis and the regression model are compared in Figure 9, and the errors are listed in Table 6. The largest errors of melt pool depth and width are 2.49% and 4.72%, respectively, but most cases show errors of about 1% or less. Therefore, it can be judged that the regression model has sufficient accuracy to be used for optimization of the thermal analysis parameters for calibration of the thermal analysis model for LPBF process simulation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Analysis Parameter** & **Min.** & **Max.** \\ \hline Emissivity, \(\varepsilon\) & 0.37 & 0.45 \\ Conductivity coefficient, \(\alpha\) & 0.014 & 0.018 \\ Porosity, \(\varphi\) & 0.36 & 0.44 \\ OPD (\(\upmu\)m), \(H\) & 102 & 170 \\ Absorptivity, \(A\) & 0.468 & 0.572 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Modified range of analysis parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Analysis Parameter** & **Case 1** & **Case 2** & **Case 3** \\ \hline Emissivity, \(\varepsilon\) & 0.42 & 0.4 & 0.38 \\ Conductivity coefficient, \(\alpha\) & 0.017 & 0.016 & 0.016 \\ Porosity, \(\varphi\) & 0.39 & 0.41 & 0.39 \\ OPD (\(\upmu\)m), \(H\) & 150 & 160 & 130 \\ Absorptivity, \(A\) & 0.57 & 0.47 & 0.53 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Analysis parameter case for validation.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Process Parameter**} & \multicolumn{2}{c}{**Process Parameter**} & \multicolumn{2}{c}{**Process Parameter**} & \multicolumn{2}{c}{**Process Parameter**} \\  & \multicolumn{2}{c}{**Case 1**} & \multicolumn{2}{c}{**Case 2**} & \multicolumn{2}{c}{**Case 3**} & \multicolumn{2}{c}{**Case 4**} \\ \hline Analysis & Depth & 0.18\% & 0.66\% & 0.41\% & 0.59\% \\ parameter case 1 & Width & 0.61\% & 1.04\% & 1.19\% & 0.34\% \\ \hline Analysis & Depth & 2.17\% & 2.21\% & 1.82\% & 2.49\% \\ parameter case 2 & Width & 1.63\% & 4.72\% & 0.13\% & 0.62\% \\ \hline Analysis & Depth & 1.19\% & 0.14\% & 0.32\% & 0.18\% \\ parameter case 3 & Width & 0.16\% & 0.81\% & 0.32\% & 0.40\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Relative error in melt pool size between analysis model and regression model.  
Figure 8: Sensitivity of analysis parameters for modified range to: (**a**) melt pool depth; (**b**) melt pool width.