## 2 Numerical Modelling and Validation
### Heat Source Modelling  
Thermal effects can cause distortions and defects such as strength and contact loss between the layers or main material/substrate as the DED process has focused thermal energy used to fuse materials by melting [27]. Therefore, they should be evaluated and taken into account during the design for components manufactured by DED, in order to prevent any strength loss caused by residual strains and stresses on the component after the process. Melting pool control and DED bead geometries should also be defined properly. Modelling of thermo-mechanical behaviour is a significant part of the numerical analysis of the DED process. DED simulation has been performed supposing a welding approach. It is carried out on Simufact Welding Software as it provides possibilities to model heat treatment processes, variations of cooling and unclamping setups, as well as mechanical loading of welded structures. Additionally, it has a capability to define residual stresses and map thermal process effects on the part using flow curves of materials. 316L material is described as DED material with its chemical composition in Table 1 and its flow curves (see Figure 2) from software material database. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Element** & **C** & **Co** & **Cr** & **Mn** & **Mo** & **N** & **Ni** & **P** & **S** & **Si** \\ \hline \% Mass portion as fixed value & 0.018 & 0.19 & 16.63 & 1.57 & 2.05 & 0.0153 & 11.18 & 0.04 & 0.002 & 0.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Chemical composition of DED material (316L). Figure 1: Design variables. Figure 2: Flow Curves of DED Material (316L). The heat source is described in the numerical model as a volume and surface distributed heat source, which consists of a Gauss surface distribution of the heat input and a constant heat distribution cylinder. An illustration of the described heat source model can be seen in Figure 3.  
For the general Gaussian distribution, the surface flux \(Q_{r}\) (W\(\cdot\)mm\({}^{-2}\)) is expressed in the first equation [28];  
\[Q_{r}=Q_{0}e^{-Cr^{2}} \tag{1}\]  
where \(Q_{0}\) (W\(\cdot\)mm\({}^{-2}\)) is the maximum flux, \(r\) (mm) is the radial distance from the centre of the heat source, and \(C\) (mm\({}^{-2}\)) is the distribution width coefficient. For the particular case of laser beam welding, the heat flux is expressed as shown in the second equation:  
\[Q_{r}=\frac{2\eta_{e}P_{l}}{\pi r_{f}^{2}}e^{\frac{-2\gamma^{2}}{\gamma}} \tag{2}\]  
where \(\eta_{e}\) is the process efficiency, \(P_{l}\) is the power in (W), and \(r_{f}\) is the focus radius [28]. A DED simulation model has been performed based on the experimental laser parameters. Heat source parameters used in the analysis are displayed in Figure 4.  
Figure 4: Heat source parameters. Figure 3: Heat source modelling. Dilution is one of the most critical parameters for DED process performance. It is a dimensional parameter and is defined by the ratio between the melt pool penetration depth below the substrate and the sum of melt pool penetration depth and deposited height for a single track. It is calculated as shown in Equation (3);  
\[D_{r}\left(\%\right)=\left(D/(D+H)\right)\times 100 \tag{3}\]  
where \(D_{r}\left(\%\right)\) is the dilution percentage or dilution ratio, \(D\) is melt pool penetration depth below the substrate, and \(H\left(\text{mm}\right)\) is deposited single height. These definitions are shown in Figure 1:0  
Dilution simulation has been performed with three different Heat Source Powers, 1100, 1300, and 1500 Watt. On the other hand, 2 different scan velocities have been preferred as 8 and 12 mm/sn, keeping the dead bead height at a constant value of 5 mm. Brick elements are used for the CAE model in order to validate the heat source approach for the DED process. The advantage of using this type of element is the increase in the accuracy of the discretization. Some results for dilution are shown in Figure 6 under 1500 Watt, 12 mm/sn condition.