## 3 Results and Discussion
### Correcting the Energy Input of the Primary Process: Simulation 11  
In the original implementation of the LPBF process, every layer of elements is deposited at the melting temperature of the metal, which for 17-4 PH stainless steel is 1405 \({}^{\circ}\)C. The energy associated with elevating a layer of elements from room temperature to the melting temperature can be calculated as follows:  
\[E_{simul}=\rho V\int\limits_{T_{room}}^{T_{m}}C_{p}dT \tag{8}\]  
where \(V\) is the volume of the part, and \(T_{room}\) and \(T_{m}\) are the room and the melting temperature respectively. For a 1e-3 m\({}^{3}\) cube, this energy equates to 6375 kJ. On the other hand, it is possible to calculate the total transferred energy for the actual LPBF process. The total energy is equal to  
\[E_{real}=\frac{\alpha P}{v}\frac{V}{hb} \tag{9}\]  
In this equation, \(\alpha\) is the absorptivity of the powder, \(P\) the laser power, \(v\) the velocity of the laser, \(V\) the volume of the part, \(h\) the hatch spacing and \(\delta\) the thickness of a deposited layer. For the earlier cube, this results in an energy of 1.6e4 kJ. This is considerably higher than the energy from the simulation. This can be corrected by equating Equation (7) to Equation (8) and adding a correction term:  
\[E_{real}=E_{simul}+\rho C_{p}^{m}V\Delta T \tag{10}\]  
where the term accounts for the overheating of each element layer, in which \(C_{p}^{m}\) is the heat capacity at the melting temperature. This overheating is therefore equal to  
\[\Delta T=\frac{\alpha P}{p\gamma hC_{p}^{m}}-\frac{1}{C_{p}^{m}}\int\limits_{T _{room}}^{T_{m}}C_{p}dT \tag{11}\]  
which is incidentally independent of the volume of the cube. The value of this overheating term is 1595 \({}^{\circ}\)C. Therefore, the temperature in each deposited layer is set to 3000 \({}^{\circ}\)C. To ensure no stresses build up in the part while it is still molten, the stress-free temperature is set to the melting temperature of the part, i.e., 1405 \({}^{\circ}\)C.  
The results of the entire simulation when using this correction term are shown in Figure 14A. When comparing the stresses displayed in this figure with the ones from the earlier simulation, shown in Figure 8A, the stress and displacement are comparable, which reinforces the effectiveness of the stress relaxation temperature as a measure to increase reproducibility for the geometrical precision of a part. Figure 14B shows the normal stress along the cantilever for simulation 1 and 11. The stress is plotted from the bottom of the cantilever to the top. This shows that, even though the behaviour of the stress profiles are similar for both, and the compressive stress at the bottom is similar, the tensile stress at the top of the cantilever with corrected energy input is about two percent higher.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & **Simulation 8** & & **Simulation 9** & & **Simulation 10** \\ \hline Temperature & & 400 \({}^{\circ}\)C & & 300 \({}^{\circ}\)C & & 280 \({}^{\circ}\)C \\ \hline  & Sequence A & Sequence B & Sequence A & Sequence B & Sequence A & Sequence B \\ \hline Displacement & & & & & & \\ in z (m) & 0.0102 & 0.0363 & 0.0102 & 0.0364 & 0.0102 & 0.0103 \\ \hline End stress (Pa) & \(1.00\times 10^{9}\) & \(1.28\times 10^{9}\) & \(1.39\times 10^{9}\) & \(1.28\times 10^{9}\) & \(1.00\times 10^{9}\) & \(1.10\times 10^{9}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Data from simulations 8, 9 and 10.  
The results from simulation 11 are mostly relevant for modelling the LPBF process itself. For high-fidelity simulations, the energy input ought to be corrected. However, since the simulations presented in this work are reduced fidelity simulations, the correction does not significantly increase the predictive power of the model. Additionally, since the stress relaxation works as expected, namely removing high residual stresses, the difference between the two simulations becomes negligible, both in terms of the deformation of the part and the residual stresses after the entire process chain.