# Fabrication of Mesh Patterns Using a Selective Laser-Melting Process
## 3 Finite Element Analysis  
To evaluate the residual stress and transient temperature distributions according to scan track in the SLM process, finite element (FE) simulation has been conducted. The commercial finite element analysis code, ABAQUSTM, was used in this study. Figure 6 shows two different scan tracks, one is a straight scan track and another is a circular scan track. The solution processes of the finite element analysis involved two main steps. In the first step, a transient thermal analysis was carried out to generate the temperature history of the entire workpiece. In the second step, a mechanical analysis was conducted to calculate the residual stress and the load in this step is the temperature field file  
Figure 4: Schematic drawing of multi-track SLM with various amounts of laser energy heat input. Figure 5: Cross-section of the multiple parts obtained using various amounts of laser heat input (laser scan rate of 110 mm/s, powder layering thickness of 0.5 mm), with a laser power of **(a)** 100 W for low laser heat input, **(b)** 150 W for medium laser heat input, and (c) 200 W for higher laser heat input. generated in the previous step. The temperature history of all nodes generated in the thermal analysis was imported as a predefined field into the mechanical analysis. The powder bed was modeled with a dimension of \(2.5\times 2.5\times 0.5\) mm. Element type was an 8-noded heat transfer brick element (DC3D8) of \(60\times 60\times 60\)\(\mu\)m as shown in Figure 7. Total number of elements was 14,112 and total number of nodes was 16,641. To represent a moving heat source, heat flux was applied on the top surface of powder bed by using the DFLUX user subroutine in ABAQUS and the USDFLD user subroutine was used to consider the temperature dependent properties of powder and solid. In transient thermal analysis, the laser beam heat source was modeled as a Gaussian distribution heat flux on the powder bed surface, which is represented as Equation (1). The user subroutine DFLUX in ABAQUS was used to model the motion of the heat source according to a different scan track. \[I(r)=\frac{fAP}{\pi r_{0}^{2}}\textit{exp}\left(-f\frac{r^{2}}{r_{0}^{2}}\right) \tag{1}\]  
where \(I\) is the laser intensity, \(f\) is the distribution factor, \(A\) is the energy absorption coefficient, \(p\) is the power of the heat source, \(r_{0}\) is the radius of the heat source, and \(r\) is the radial distance of any point from the axis of the heat source. The laser power absorptivity is 0.4 [42]. A combined surface  
Figure 6: Two different scan tracks; (**a**) straight scan track, (**b**) circular scan track. Figure 7: Finite element (FE) simulation model. film condition (\(h_{c}\)) was applied as a film boundary condition to account for convection and radiation on all external surface. The combined surface coefficient is \(h_{c}=2.4\times 10^{-3}\varepsilon\mathrm{T}^{1.61}\), where \(\varepsilon\) is 0.85 emissivity [43]. The thermo-capillary phenomenon in a molten pool was considered by modifying the thermal conductivity by a factor of 2.5. As the density increases, the Young's modulus of material also increases. For an accurate FE analysis, the temperature dependent density and Young's modulus of the material need to be used [44]. The relevant temperature dependent thermal and mechanical properties for AISI 304 are obtained from literature [42], and [PERSON]'s ratio of 0.3 was used. The powder bed was considered as a mixture of solid and gas phase thus the density and conductivity of powder were considered as Equations (2) and (3) [45]. \[\rho_{powder}=\rho_{bulk}\times(1-\varnothing) \tag{2}\]  
\[k_{powder}=k_{bulk}\times(1-\varnothing) \tag{3}\]  
where \(\emptyset\) is the porosity of the powder and is assumed as 0.4 [42]. Figure 8 shows the transient temperature distributions for both straight and circular scan tracks. The color was set as grey if the temperature is higher than the melting point (1733 K); therefore, the gray region represents the molten pool area.