## 1 Introduction  
Selective laser melting (SLM), known as direct metal laser melting or laser powder bed fusion, provides significant potential for the direct fabricating of complex three-dimensional structures. Unlike selective laser sintering, SLM involves the complete melting and solidification of metal materials. Various thermal-physical phenomena and morphologies change when a laser beam irradiates the powder layer [1, 2]. Initially, part of the powder layer melts when the temperature reaches the melting point as the part has been heated. Subsequently, the molten part solidifies as the temperature falls when the laser beam moves away. During the SLM process, heat transfer mechanisms such as heat conduction between powder particles and heat loss by convection and radiation on the surface are involved. Fluid dynamics such as the surface tension effect, recoil pressure, and the Marangoni convection will occur [3]. Phase transformations, including melting and fusion of powder particles, formation and solidification of the molten pool, and evaporation at the liquid-gas interface, were considered by [4, 5]. [PERSON] et al. [6] and [PERSON] [7] also considered penetration phenomena such as spatial distribution and absorption of laser radiation in the powder. Furthermore, the transient physical behavior during the SLM process is significantly governed by interdependent process parameters: laser-related pa rameters, scan-related parameters, powder-related parameters, and temperature-related parameters [8]. In practice, the processing parameters applied to a specific case are provided by the material suppliers, obtained through trial procedures and experience. To reduce the cost of the trial process, numerical approaches can be used. Owing to the complexity of the SLM process, many numerical models with approximations and assumptions have been proposed to simplify and avoid time-consuming simulations. [PERSON] et al. [9] used two-dimensional finite element methods for heat conduction and elastic deformation to calculate the temperature and stress distributions of a single metallic layer formed on the powder bed, where the whole area was treated as continuous. [PERSON] et al. [10] proposed a model for coupled radiative and conductive heat transfer with an effective volumetric heat source estimated from laser radiation scattering and absorption in a powder layer. Differences in thermal conductivity between the powder bed and dense material were also considered. [PERSON] et al. [11] provided a non-linear three-dimensional transient model based on a sequentially coupled thermo-mechanical field analysis code developed in the ANSYS parametric design language (APDL). It revealed that the cyclic melting and cooling rates resulted in high von Mises stresses in the product. [PERSON] and [PERSON] [12] treated the powder bed as a continuum without the effect of discrete powder particles. The formation process of the continuous molten pool, where the thermo-capillary force and the recoil pressure are induced by evaporation, were studied by commercial finite volume method software. [PERSON] et al. [13] built fine-scale numerical models with ALE3D, where a laser-ray model was applied, and the powder layer was presumed to be homogeneous bulk materials with effective powder-layer material properties. [PERSON] and [PERSON] [14] proposed a volumetric heat source model that viewed the powder bed as an optical medium, and the laser heat source was assumed to be absorbed gradually along the depth of the powder layer, derived by the Monte Carlo method. A thermal model was developed on Ansys Additive Science to simulate the SLM of a single bead by [15]. The lack of fusion and keyhole defects, which were determined using the calculated melt pool dimensions, were discussed. In this paper, a transformation is introduced to describe the model of a powder bed and heat source in a coordinate system based on a constant moving heat source instead of a conventional fixed coordinate system.