# Optimization of Laser-Assisted Polypropylene Aluminum Joining
## 3 Results and Discussions
### Design of Experiment  
Table 3 is used as an input for the DoE based on the Box-Behnken method. A total number of 17 tests were performed for the DoE to find a regression model for the shear load or breaking force followed by optimization of the laser-engraving process concerning  
Figure 8: Simulation of two patterns under the same load and constraints; the example with the dot pattern (**a**); and the other with the line pattern (**b**), regarding Von Mises stress (MPa) distribution. Figure 7: Zoomed comparison between: with fixture (**a**) and without fixture (**b**), regarding Von Mises stress (MPa) distribution. the given parameters. The parameters and the corresponding levels (introduced in Table 2) were chosen based on the preliminary experiments and screening tests. Table 4 presents the ANOVA (Analysis of Variance) of the model. A quadratic regression after backward elimination of the high-order interactions and linear terms presents a good model to fit the data. For the given ANOVA table, R\({}^{2}\) is 0.8779, adjusted R\({}^{2}\) is 0.8046, and predicted R\({}^{2}\) is 0.6900. Equation (4) describes the actual regression equation of the model for the shear load. As observed, based on the \(p\)-values, speed, width of engraved lines, linear interaction of density and speed, and square terms of density and width are significant parameters. However, density is insignificant. Nevertheless, to respect the hierarchy in the model, density cannot be eliminated. Equation (4) shows the regression equation for the shear load based on the presented DoE. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **Sum of Squares** & **df** & **Mean Square** & **F-Value** & \(p\)**-Value** \\ \hline Model & 4.754 \(\times\) 10 [5] & 6 & 79,233.52 & 11.98 & 0.0005 & significant \\ \hline A-Density & 23,958.60 & 1 & 23,958.60 & 3.62 & 0.0862 & \\ \hline B-Speed & 70,706.80 & 1 & 70,706.80 & 10.69 & 0.0084 & \\ \hline C-Width & 2.299 \(\times\) 10 [5] & 1 & 2.299 \(\times\) 10 [5] & 34.77 & 0.0002 & \\ \hline AB & 46,074.62 & 1 & 46,074.62 & 6.97 & 0.0248 & \\ \hline A\({}^{2}\) & 53,877.15 & 1 & 53,877.15 & 8.15 & 0.0171 & \\ \hline C\({}^{2}\) & 45,043.39 & 1 & 45,043.39 & 6.81 & 0.0261 & \\ \hline Residual & 66,141.69 & 10 & 6614.17 & & & \\ \hline Lack of Fit & 25,524.91 & 6 & 4254.15 & 0.4190 & 0.8366 & not significant \\ \hline Pure Error & 40,616.78 & 4 & 10,154.19 & & & \\ \hline Cor Total & 5.415 \(\times\) 10 [5] & 16 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ANOVA for the reduced quadratic model of the shear load. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Run** & **Density (\%)** & **Speed (mm/s)** & **Width (mm)** & **Shear Load (N)** \\ \hline
1 & 25 & 32.5 & 0.2 & 1332 \\
2 & 25 & 5 & 0.1 & 878.9 \\
3 & 25 & 32.5 & 0.2 & 1211.7 \\
4 & 15 & 32.5 & 0.1 & 900 \\
5 & 35 & 32.5 & 0.3 & 1090 \\
6 & 35 & 5 & 0.2 & 822.7 \\
7 & 15 & 60  
\[\text{Shear load}=125.1+(38.3\text{ Density})-(6.3\text{ Speed})+(5827\text{ Width})+(0.39\text{ Density Speed})-\left(1.1\text{ Density}^{2}\right)-\left(10,329\text{ Width}^{2}\right) \tag{4}\]  
Figure 9 shows the normal plot of the residuals and the desirability ramp for the optimum parameters. As observed, the optimum condition that maximizes the shear load or breaking force is achieved at 27.3% density, 0.28 width, and 60 mm/s speed of the laser-engraving process. For the best combination of the parameters in DoE, a shear load of above 1400 N will be achieved. Table 5 shows the confirmation test for three runs at 25% density, 32.5 mm/s speed, and 0.2 mm width of the engraved lines. The three different runs with the given process parameters of laser engraving resulted in 1250.6, 1285, and 1176.2, which present a mean value of 1237.27. To sum up, considering miscellaneous parameters, including the \(p\)-values, R\({}^{2}\), the difference between predicted and adjusted R\({}^{2}\), normal distribution of the residuals, and a confirmation test, the model is confirmed.