# Analytical Prediction of Balling, Lack-of-Fusion and Keyholing Thresholds in Powder Bed Fusion
## 4 Conclusions  
In this paper, analytical modeling methods are proposed to predict the defect generation in laser powder bed fusion additive manufacturing, without relying on iteration-based computation. The thresholds for the generation of lack-of-fusion, balling and keyholing defects were predicted under various process conditions. The molten pool dimensions in the additive manufacturing process were first calculated by a closed-form analytical thermal model. The heat input from laser source, heat loss at part boundaries and effect of powder bed porosity on thermal properties were considered in the thermal model. The obtained molten pool information was then employed in the calculation of the thresholds  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Power (W)**} & \multicolumn{6}{c}{**Scanning Speed (mm/s)**} \\ \cline{2-9}  & **200** & **400** & **600** & **800** & **1000** & **1200** & **1400** \\ \hline
20 & LOF & LOF & LOF & LOF & LOF & LOF & LOF \\
40 & LOF & LOF & LOF & LOF & LOF & LOF & LOF \\
60 & No & LOF & LOF & LOF & LOF & LOF & Balling \\
80 & No & LOF & LOF & LOF & Balling & Balling & Balling \\
100 & No & LOF & LOF & Balling & Balling & Balling & Balling \\
120 & Keyhole & No & LOF & Balling & Balling & Balling & Balling \\
140 & Keyhole & No & Balling & Balling & Balling & Balling & Balling \\
160 & Keyhole & No & Balling & Balling & Balling & Balling & Balling \\
180 & Keyhole & Keyhole & Balling & Balling & Balling & Balling & Balling \\
200 & Keyhole & Keyhole & Balling & Balling & Balling & Balling & Balling \\ \hline \hline \end{tabular}
\end{table}
Table 7: Process window of the Ti6 Al4V. LOF represents lack-of-fusion defect.  
Figure 9: Sensitivity of thresholds to scan speed.  
for defects. One criterion for the calculation of threshold for lack-of-fusion defect was presented; one criterion for balling and one criterion for keyholing were also proposed.  
The experimental investigations of Ti6 Al4V, Inconel 718 and 316L in powder bed fusion were employed to validate the proposed analytical methods. The predicted thresholds of these defects have good agreement with experimental results. The computational time for the presented models is less than five minutes on a personal computer with a Intel i5 processor running at 3.4 GHz. The balling threshold has negative relationships with both laser power and scan speed. The lack-of-fusion threshold has negative and positive relationships with power and speed, respectively. The keyholing threshold holds positive and negative relationships with power and speed, respectively. The acceptable prediction accuracy and high computational efficiency of the proposed methods will make them good tools for the optimization of process conditions in powder bed fusion. Furthermore, the presented models can be a good basis for future research on the prediction of defects of complex parts in additive manufacturing.  
Conceptualization, W.W. and J.N.; methodology, W.W.; software, W.W.; validation, W.W., J.N. and S.Y.L.; formal analysis, W.W.; investigation, W.W.; resources, S.Y.L.; data curation, W.W.; writing--original draft preparation, W.W.; writing--review and editing, W.W., J.N. and S.Y.L.; visualization, W.W.; supervision, S.Y.L.; project administration, S.Y.L.; funding acquisition, S.Y.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.