# Process Parameter Optimisation in Laser Powder Bed Fusion of Duplex Stainless Steel 2205
## 4 Optimization Procedure
\begin{split}\sigma_{0.2}f(\text{P}^{*},\,\text{S}^{*},\,\text{H}^{* })&=539.254+17.70\cdot\text{P}^{*}-2.90\cdot\text{S}^{*}+\,12.30 \cdot\text{H}^{*}-\\ -6.44\cdot\text{P}^{*2}-11.44\cdot\text{S}^{*2}&+13.5 6\cdot\text{H}^{*2}+24.38\cdot\\ \cdot\text{P}^{*}\cdot\text{H}^{*}+11.88\cdot\text{P}^{*}\cdot\text{S}^{*}- 14.38\cdot\text{S}^{*}\cdot\text{H}^{*}\\ \text{R}^{2}&=0.556\end{split}  \tag{5}\]  
\[ \begin{split}\sigma_{\text{uts}}f(\text{P}^{*},\,\text{S}^{*},\, \text{H}^{*})&=802.72+51.90\cdot\text{P}^{*}-19.50\cdot\text{S}^{* }-14.60\cdot\text{H}^{*}-\\ -34.01\cdot\text{P}^{*2}-24.01\cdot\text{S}^{*2}-22.51\cdot\text{H}^{ *2}+32.00\cdot\\ \cdot\text{P}^{*}\cdot\text{H}^{*}+\,31.50\cdot\text{P}^{*}\cdot\text{S}^{* }-25.75\cdot\text{S}^{*}\cdot\text{H}^{*}\\ \text{R}^{2}&=0.789\end{split}  \tag{6}\]  
\[ \begin{split}\epsilon_{\text{rf}}f(\text{P}^{*},\,\text{S}^{*}, \,\text{H}^{*})&=41.28+15.62\cdot\text{P}^{*}-1.06\cdot\text{S}^{ *}-2.37\cdot\text{H}^{*}-\\ -13.87\cdot\text{P}^{*2}-1.99\cdot\text{S}^{*2}+0.02\cdot\text{H}^{*2 }+0.513\cdot\\ \cdot\text{P}^{*}\cdot\text{H}^{*}+\,0.43\cdot\text{P}^{*}\cdot\text{S}^{* }-1.67\cdot\text{S}^{*}\cdot\text{H}^{*}\\ \text{R}^{2}&=0.994\end{split}  \tag{7}\]  
\[ \begin{split}\text{A}\,f(\text{P}^{*},\,\text{S}^{*},\,\text{H}^{* })&=75.85+23.66\cdot\text{P}^{*}-5.40\cdot\text{S}^{*}-4.55\cdot \text{H}^{*}-\\ -19.35\cdot\text{P}^{*2}-4.57\cdot\text{S}^{*2}-2.37\cdot\text{H}^{ *2}+2.48\cdot\\ \cdot\text{P}^{*}\cdot\text{H}^{*}+\,2.84\cdot\text{P}^{*}\cdot \text{S}^{*}-4.41\cdot\text{S}^{*}\cdot\text{H}^{*}\\ \text{R}^{2}&=0.980\end{split}  \tag{8}\]  
As an example, Figure 5 shows the response surfaces of the mechanical properties for the encoded H* = 0. In the graphs, the color is automatically adjusted for each result, varying from dark blue for the lowest value to red for the highest value in the studied range. As the surfaces for every response have been obtained, the optimal solution for manufacturing DSS-SLM steel can be found. The numerical optimization tool built into the program was used. In this case, the optimization algorithm seeks to maximize the quantitative value of the mechanical properties and the density. Then, a permissible range is delimited for each of the given responses, within which the optimal result will be found. These boundary conditions will need to be defined for each response of the experiment design. At the end of the calculation, the software shows response predictions for all output variables. In addition, it generates an extra graph \"desirability\", which is a response surface that depicts the maximized average value of all the properties. The higher value of this surface marks the optimal parameters. A color map is generated, ranging from dark blue, representing the area where it is not advisable to choose the parameters, to red, representing the optimal area for parameter selection. Figure 6 presents the predictions of the responses corresponding to the solution identified as optimal. In these graphs, the laser beam separation (H*) takes the value of 1 and the other two variable parameters are represented. These surfaces are visualized on a two-dimensional plane, relating the responses to the power (P*) and velocity (S*) values of the laser. Table 6 provides a summary of the print parameters selected in Figure 6, as well as the estimated resulting properties of the identified optimal solution. The decoded values of the parameters shown in Table 6 are P = 174.9 W, S = 639.6 mm/s, and H = 0.12 mm. For these parameters, the VED is 45.57 J/mm\({}^{3}\). Figure 5: Response surfaces for H* = 0. (**a**) Density; (**b**) Yield strength; (**c**) Ultimate tensile stress; (**d**) Area reduction; (**e**) Elongation at fracture.