# Evolutionary Algorithm to Optimize Process Parameters of AI/Steel Magnetic Pulse Welding
## 2 Materials and Methods
### Results and Discussion
#### 2.3.1 Effects of Process Parameters  
An analysis of variance (ANOVA) was used to analyze the impacts of parameters on the maximum load and weld length of the joint. Table 7 presents ANOVA results for the mean values of maximum load. The table includes the following symbols: (DF) for degrees of freedom, (Adj SS) for adjusted sum of squares, (Seq SS) for the sequential sum of squares, (Adj MS) for adjusted mean squares, (R-Sq) for coefficient of determination, 'F' for the variance or Fisher ratio of any factor, (Seq SS) for the pure sum of squares of any factor, and 'P' for critical probability. A _p_-value below 0.05 indicates that the model is statistically significant [33]. The results indicate that parameter C has a significant effect on the maximum load, with a contribution of 48.7%. Parameter F follows with a contribution of 10.6%, and parameter G has a contribution of 0.5%. On the other hand, interaction terms such as C \(\times\) G, G \(\times\) G, and G \(\times\) F have negligible effects on the maximum load. The coefficient of determination (R\({}^{2}\)) for the developed prediction models is 96.02%. MPW is a high-velocity collision welding process. Based on the ANOVA results, a quadratic polynomial model for maximum load was developed using coded units (i.e., \(-\)1, 0, +1). The model can be represented by the following equation:  
\[ \begin{array}{c}\text{Maximumload}=-10.78-0.0152\text{C}+3.08\text{G}-0.52 1\text{F}-0.000015\text{C}^{2}-3.868\text{G}^{2}-0.0135\text{F}^{2}\\ +0.00342\text{G}+0.000160\text{C}F+0.060\text{G}F\end{array}  \tag{1}\]  
Table 8 presents the ANOVA results for weld length. The results indicate that parameters C and F have significant effects on weld length. Specifically, parameter C has the greatest contribution at 55.1%, followed by parameter F at 17%, and parameter G at 0.2%. Additionally, the interaction terms C \(\times\) F and C \(\times\) G influence weld length. The contribution of C \(\times\) F is 4%, while C \(\times\) G contributes 0.7%. The developed prediction models have a coefficient of determination (R\({}^{2}\)) of 95.81%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline Model & 9 & 4.98741 & 0.55416 & 101.40 & 0.000 \\ Linear & 3 & 3.02280 & 1.00760 & 184.37 & 0.000 \\ C & 1 & 2.45984 & 2.45984 & 450.09 & 0.000 \\ G & 1 & 0.02748 & 0.02748 & 5.03 & 0.049 \\ F & 1 & 0.53548 & 0.53548 & 97.98 & 0.000 \\ Square & 3 & 1.93055 & 0.64352 & 117.75 & 0.000 \\ Interaction & 3 & 0.03405 & 0.01135 & 2.08 & 0.167 \\ Error & 10 & 0.05465 & 0.00547 & & \\ Total & 19 & 5.04206 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: ANOVA table for maximum load. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & _p-Value_ \\ \hline Model & 9 & 0.568492 & 0.063166 & 140.12 & 0.000 \\ Linear & 3 & 0.417941 & 0.139314 & 309.04 & 0.000 \\ C & 1 & 0.316236 & 0.316236 & 701.52 & 0.000 \\ G & 1 & 0.001237 & 0.001237 & 2.75 & 0.129 \\ F & 1 & 0.100468 & 0.100468 & 222.87 & 0.000 \\ Square & 3 & 0.122914 & 0.040971 & 90.89 & 0.000 \\ Interaction & 3 & 0.0276 The ANOVA results led to a quadratic polynomial model for weld length using coded units (\(-1,0,+1\)). The model is represented by the following equation:  
\[ \begin{split}\text{Weldlength}=2.07-0.00568\text{C}+1.005\text{G}- 0.244\text{F}-0.000002\text{C}^{2}-1.008\text{G}^{2}+0.00133\text{F}^{2}\\ -0.00058\text{C}G+0.000583\text{C}F+0.0361\text{G}F\end{split}  \tag{2}\]  
The peak current and frequency are closely related to the force required for collision with workpieces. If there is insufficient force, the welding may not be of high quality. Figure 3(a,c) show the effects of peak current and frequency. As the peak current and frequency increase, the maximum load also increases. The gap between workpieces has a role in accelerating the outer workpiece during welding.