## 1 Introduction  
SLM has attracted significant industrial attention due to the possibility of using a large range of materials and producing complex geometrical products [1; 2]. However, the SLM process involves the couplings of multi-physics, such as interactions between laser and powder, fluid flow in melting pool, vaporization, and transformation of phase, etc. Experimental measurements allow the establishment of the relationship between input parameters (such as laser power, scanning speed, hatch spacing, etc.) and certain desired properties, while experimental studies are often costly and difficult. Thanks to the strong development of computing capabilities and numerical algorithms, the numerical method has now become an essential tool to predict mechanical consequences related to the parameters of manufacturing processes. The objective of numerical simulations is to design and optimize manufacturing processes and components; meanwhile, numerical simulation allows a better understanding of the physical phenomena behind the process. According to the modeling scale, the simulations can be composed of two kinds of models: (i) multi-physical simulations; (ii) thermo-mechanical simulations. The multi-physical models aim to compute more accurately the thermal and velocity fields by couplings of different scales of physics interactions, such as interactions between an electron or laser beam and powders [3], couplings between fluid and solid [4], etc. Numerous high-fidelity multi-physical models [5; 6; 7; 8; 9] have been proposed to model the realistic melting surface of the powder layer by tracking the solid-gas interface. Other models [10; 11; 12; 13] present the powder layer as a continuum medium to simplify the complexity, and an effective thermal conductivity and density are defined for the powder layer. The multi-physical simulations at the micro/mesoscale are extremely computationally expensive, and bridging the gap between the grain scale (micrometers) and the part scale (millimeters) is still a big challenge. In spite of the increases in computer efficiency, multi-physical simulation, starting from laser-powder interaction until the final part-scale distortions, is currently impossible. The current state-of-art shows that thermo-mechanical simulations have been carried out to predict residual stress and distortion in the part scale [14; 15; 16; 17]. As we can see from Figure 1, the thermo-mechanical simulations are performed in two steps. A heat transfer simulation is first carried out to determine the temperature distributions during the process, and the laser-powder interactions are simplified by a 2D heat source or 3D heat source [18]. Then, the temperature distributions are imposed as thermal loading in a mechanical simulation to predict the distortion and residual stresses. However, as the effect of fluid flows in the melting pool and the Marangoni effect are neglected in the thermal model, the prediction of temperature distributions would not be precise enough, which can result in the less accurate prediction of mechanical distortions and residual stresses. Therefore, correctly predicting the thermal results has become one of the key steps in thermo-mechanical simulations. Various numerical methods have been proposed to solve heat transfer equations, such as analytical solutions [19; 20; 21; 22; 23] and FE models [24; 25; 26; 27; 28; 29]. The comprehensive comparison of the analytical and FEM made by [PERSON] et al. [30] shows that good correlations of the melt pool width have been observed between numerical results and experimental measurements, while the comparison of melt pool depth is neglected. Similarly, [PERSON] et al. [27] employed the FE method to simulate the melting pool size in the selective laser melting (SLM) process, and the numerical results tended to overestimate the melt pool width, especially at high energy input. Therefore, a correction of effective absorptivity has been used to be incorporated with numerical results such that the prediction is in better agreement with measurements.