## 1 Introduction  
Laser powder bed fusion (LPBF) is an additive manufacturing technology for manufacturing metal and alloy parts. It is considered a promising manufacturing technology due to its high manufacturing flexibility [1]. An overhang structure inevitably appears in the design and manufacture of LPBF, bringing great difficulties to the manufacturing process [2]. In ideal conditions, there is no need to add support to the overhang structure because of the support of the powder bed during the LPBF process [3]. In reality, however, the conditions are very complex. Therefore, when the incline angle is too small or the overhanging length is too long, warping, cross formation, and a series of defects are often encountered [4]. The cause of the warping deformation is believed to be the lack of support of the overhang and the residual stress generated by the LPBF [5, 6]. The residual stress is thought to be caused by localized heating and cooling during the LPBF process [7]. Without a connection with the substrate, the heat cannot be dissipated in the overhanging area, and an overheating zone is formed during laser processing. This causes the building contour to exceed expectations, resulting in the formation of dress [8]. These defects could cause the failure of the part [9]. In order to ensure the forming size and performance of parts, many scholars have carried out a series of studies on overhang structures. [PERSON] [10] and [PERSON] [11]studied the influence of the incline angle. They found that at a smaller incline angle, a lower scanning speed and a larger laser energy caused the warping trend to become greater; meanwhile, the quality of the overhanging surface is in a relatively optimal state when the bending angle of the two overhanging surfaces is 80\({}^{\circ}\) (60\({}^{\circ}\) relative to the horizontal plane). [PERSON] carried out a series of studies on a spiderweb structure, and the result showed that when the incline angle is 0\({}^{\circ}\), many dross formations are produced in the bottom area of the spiderweb and its surface quality is greatly reduced [12]. Therefore, improving the formation quality of the overhang structure has become a focus of attention. [PERSON] found that the AlSi12 alloy could only reach 2 mm of overhanging length under the LPBF process without support [13]. [PERSON] used a layered control strategy to significantly increase the bridge-forming length by 5 mm [14]. In order to eliminate the warp and dross formation defects caused by the overhang structure, an external structure is required to support the overhang structure during LPBF in a general case [15; 16; 17]. [PERSON] analyzed the influence of the support profile on the dimensional stability of the overhang structure and the quality of the bottom surface [18]. This study found that the strong support profile had a higher dimensional stability, while the weak support profile had a better surface quality. Compared to the experimental research methods, the numerical analysis method fully solves the problems without consumption [19]. Additionally, the numerical analysis method can theoretically identify the variable process of temperature, stress, and deformation and is of great help in understanding the change of the physical field in the LPBF process. Many scholars have studied the formation process of the overhang structure by using a numerical simulation. [PERSON] investigated a novel reticulated shell structure with a large proportion of overhang structure and found a strut angle of 75 degrees demonstrated the best compressive behavior [20]. [PERSON] et al. investigated the manufacturing limit of the fully circular AlSi10 Mg overhang structure, which was 15 mm in diameter without added support. When a longer diameter was formed, the overhang surface produced significant cross formation defects [21]. [PERSON]. [PERSON] studied the effect of laser energy density on the process of the overhang structure by using a raytracing model.