# Analytical Prediction of Balling, Lack-of-Fusion and Keyholing Thresholds in Powder Bed Fusion
## 1 Introduction  
Laser powder bed fusion (LPBF) additive manufacturing has been proved to have superior advantages in fabricating products with complex geometries when compared with traditional manufacturing techniques [1; 2]. However, the defects occurring in the additive manufacturing process have serious detrimental effects on the quality of the final products and will limit the wide application of this kind of manufacturing technique [3; 4]. Thus, good prediction methods for the occurrence of the defects will be very helpful to figure out the suitable process conditions to fabricate products without defects. Lack-of-fusion is a common defect in LPBF which is induced by incomplete melting of powders and incomplete overlap of adjacent molten pools [5; 6], which will finally lead to lack-of-fusion porosity in products. Balling behavior is another kind of defect which is mainly caused by the poor wetting between solidified molten pools and the solid substrate, which will lead to unstable and discontinuous melting tracks [7; 8]. Severe balling defect can cause high surface roughness and porosity in the final products. Keyholing behavior is a defectoccurring under very high laser energy-density regime. It is related to the evaporation of components in molten material, which will produce gas in the molten pool. Some gas may be trapped by the solidification front of the molten pool, which will lead to high porosity in the final product [1; 9]. This paper focuses on the prediction of the generation of the three kinds of defects in laser powder bed fusion. Experimental, numerical, and analytical methods have been developed by researchers to investigate the mechanisms of the generation of the defects in the additive manufacturing process. [PERSON] et al. [10] investigated the influence of process conditions on the melting tracks, molten pool shapes and porosity of Ti6 Al4V in selective laser melting with the help of SEM images and optical photographs. [PERSON] et al. [1] employed ultrahigh-speed X-ray imaging to study the keyhole formation mechanism in additive manufacturing. [PERSON] et al. [11] investigates the keyhole-induced porosity by the micro-CT analysis. [PERSON] et al. [12] used the technique of high-speed X-ray imaging and diffraction to study the in-situ formation process of keyhole pores and molten pools in laser powder bed fusion systems. [PERSON] et al. [13] employed X-ray computed tomography to study the keyhole-induced pores in powder bed fusion. These experimental techniques have offered some valuable information on the formation process of defects. However, the high cost of the experimental equipment and complex measurement procedures limit the wide adoption of these techniques. To avoid the inconvenience of the experimental measurements, some researchers employed numerical methods to study the defects formation and their influence on the performance of the products. [PERSON] et al. [13] developed a numerical model to study the keyhole pores formation and keyhole-induced porosity in powder bed fusion processes. [PERSON] et al. [14] proposed a finite element-based thermal model for the calculation of molten pool dimensions and employed the predicted molten pool to study the occurrence of lack-of-fusion defect. [PERSON] et al. [15] presented a numerical simulation strategy to study the temperature profile and molten pool size in powder bed fusion. The powder bed packing pattern was considered in the simulation by a discrete element method. [PERSON] et al. [16] employed a 3D powder-scale model to study the mechanisms of pores formation in powder bed fusion processes. Although numerical methods can avoid the high cost on experimental equipment, the high computational cost is still a fatal drawback. Analytical modeling methods have obtained significant interest from researchers due to their acceptable accuracy and high computational efficiency in the prediction of temperature distribution, molten pool dimensions and related defects in AM. [PERSON] et al.