## 3 Optimization with Design of Computer Experiments
\begin{array}{c}\beta_{0}\\ \beta_{1}\\ \beta_{2}\\ \beta_{11}\\ \beta_{22}\\ \beta_{12}\\ \gamma\\ \delta_{1}\\ \delta_{2}\end{array} \right]\varepsilon=\left[ \begin{array}{c}\varepsilon_ {1}\\ \varepsilon_{2}\\ \vdots\\ \vdots\\ \varepsilon_{N}\end{array} \right]\]  
Using the least-squares method, the estimated efficient \(b\) of the coefficient can be obtained as  
\[b=\left(X^{T}X\right)^{-1}X^{T}Y \tag{11}\]  
Finally, the fitted regression model is  
\[Y=\;Xb \tag{12}\]  
Based on Equations (10) and (11), the two fitted response models are  
\[ \begin{split} I(x,z)=582.4-251.5x_{1}-419.7x_{2}-775.4x_{1}^{2}- 90.4x_{2}^{2}-243.7x_{1}x_{2}\\ -36.2z_{1}-41.3x_{1}z_{1}+68.8x_{2}z_{1}\\ R_{rms}(x,z)=349.08+91.9355x_{1}+22.6783x_{2}+13.32x_{1}^{2}+2.32x_{2}^{2}+6x_{1 }x_{2}\end{split}  \tag{13}\]  
Upon the two response models, analyses of variance were performed following the procedure introduced in [12,13]. The results reveal that the two RSM models with the R-squared (\(R^{2}\)) = 86.2% and 99% are adequate and satisfactory from the statistical point of view. They indicate that the proportion of the variance for the predicted response is represented by the input variables or variances in the regression model. The 3D response surfaces and contours of the two response models are plotted in Figures 11 and 12. From the graphical representations of the regression responses models, we can clearly observe the sensitivity relationships between the responses and the input coded variables. As shown in Figure 11a, the laser power density increases with the focal plane, shifting in a negative direction and decreasing in a positive direction. A plateau laser power exists within some optimal regions. Additionally, the RMS spot radius size increases with laser power, as shown in Figure 12a,b. On the other hand, the convection coefficient at the cooling water channel plate surfaces has little influence on both optical intensity and spot radius size, which mainly depend on laser power and focal plane position. The optimization solution for the objective function, as represented in Equation (7), was sought using MATLAB programming. The optimal design settings were found with the maximum laser power density \(I=5813\) kW by laser power \(P=2.28\) kW, focal location \(F_{z}=-0.1732\) mm, and cooling indicator \(h=1500\) W/m\({}^{2}\)K. After optimization, the STOP simulation model was again used to validate the optical performance with the predicted solution by the response models. The optical performances under the predicted optimal design settings were also compared with the original design settings, as shown in Table 4. The results show that the optical intensity can achieve a 31.3% increase while RMS spot radius size maintains 0.3 mm with 0.33% tolerance. It is shown that the optimal solution using STOP can improve the optical performance in delivering higher laser power density with a targeted spot size, which are important processing parameters for laser-aided additive manufacturing. The preliminary testing result showed the effectiveness of the optimized optical performance obtained from the STOP analysis in depositing the metal wires for LAAM applications. Based on the optimized design, a laser head was fabricated and tested. The setup of the laser head in metal LAAM processing is given in Figure 13a. The optical laser head was mounted on a robot arm. During laser material processing, the metal wire generated a melt-pool on the surface of the substrate, whereas the metal wire was coaxially fed into the melt-pool. Subsequently, the wire was melted and deposited onto the substrate successfully, as shown in Figure 13b. Figure 13c,d shows the samples produced using the develop laser head. The deposited material was stainless steel 316. This research focused on STOP simulation and using it to obtain the desired optical performance. An investigation of the material's mechanical properties was beyond the scope of this research.