## 2 SLM Process Modeling in Moving Coordinate System
### Volumetric Heat Source Models  
Heat source models used in the SLM process are based on either the geometrically modified approach or the absorption coefficient profile approach. In the geometrically modified method, different geometric shapes are used to simulate the actual shape of the heat source. On the other hand, the heat source models are not constrained in specific geometries using the absorption coefficient profile approach, where a general form of the heat source is formed by a two-dimensional Gaussian distribution on the top surface while the laser energy is absorbed along the depth of the powder bed based on the absorption coefficient functions.  
In this study, we will study two types of volumetric heat source models, which are the double-ellipsoidal shape heat source model and the ray-tracing method heat source model, as shown in Figure 2. They belong to the geometrically modified approach and the absorption coefficient profile approach, respectively.  
Figure 1: Schematic diagram of thermal behavior in SLM process.  
Ray-tracing method heat source model (I)  
The ray-tracing model uses an absorption coefficient profile approach, where the laser energy is absorbed gradually along the depth of the powder bed. This method accounts for the optical properties of the powder bed, leading to a more accurate representation of laser energy distribution within the material. It can capture complex interactions between the laser and the powder, resulting in more precise temperature gradients and molten pool shapes. As shown in Figure 2a, [PERSON] and [PERSON] [14] applied a powder-bed model with randomly distributed particles and calculated the absorption coefficient profile function by Monte Carlo ray-tracing simulations. The volumetric heat source model can be represented as Equation (3) with a moving coordinate system  
\[Q(\xi,y,z)=\frac{2P}{\pi r_{0}{}^{2}}exp\bigg{[}-2\frac{\xi^{2}+y^{2}}{{r_{0} }^{2}}\bigg{]}\cdot f(z) \tag{3}\]  
where \(P\) is the power of the stationary laser source (W), \(f(z)=d\beta/dz\) is the absorption coefficient function \((1/\upmu\text{m})\) derived by the Monte Carlo ray-tracing simulation, and \(r_{0}\) is the radius of the laser beam \((\upmu\text{m})\).  
* Double-ellipsoidal shape heat source model (II)  
The double-ellipsoidal shape model uses a geometrically modified approach, where the heat source is represented by a fixed geometric shape (double-ellipsoid). While simpler and computationally efficient, it may not well capture the detailed absorption and scattering effects of the laser within the powder bed. This can lead to less accurate predictions of the temperature distribution and molten pool dimensions. As shown in Figure 2b, the ellipsoidal distribution is introduced by [PERSON] et al. [19] with semi-axes \(a\), \(b\), and \(c\) and the center at the reference coordinate origin. The volumetric heat source model can be described as Equation (4) in a moving coordinate system  
\[Q(\xi,y,z)=\frac{6\sqrt{3}P\beta}{bc\tau\sqrt{\pi}}\left\{\begin{array}{l} \frac{f_{r}}{a_{r}}exp\bigg{[}-3\bigg{(}\frac{\xi^{2}}{a_{r}^{2}}+\frac{y^{2} }{b^{2}}+\frac{z^{2}}{c^{2}}\bigg{)}\bigg{]}for\ \xi>0\\ \\ \frac{f_{r}}{a_{r}}exp\bigg{[}-3\bigg{(}\frac{\xi^{2}}{a_{r}^{2}}+\frac{y^{2} }{b^{2}}+\frac{z^{2}}{c^{2}}\bigg{)}\bigg{]}for\ \xi<0\end{array}\right. \tag{4}\]  
Figure 2: Schematic diagram of volumetric heat source models. (**a**) ray-tracing method; (**b**) double-ellipsoidal shape.  
where \(f_{f}=2a_{f}/(a_{f}+a_{r})\), \(f_{r}=2a_{r}/(a_{f}+a_{r})\), and \(f_{f}+f_{r}=2\).