## 3 Results and Discussion
### Effect of Varying Upsetting Ratios and Punch's Pitch Angle on Flange-Forming Force  
The study investigated the influence of initial geometry parameters, including upsetting ratios (\(H0/D0\), \(S0/D0\)) and punch's pitch angle (\(\alpha\)), on the flange-forming force in the flange forming of C45 billet tubes. Table 5 presents the input and output parameters for the flange-forming process. The experimental design adhered to the planning method based on Taguchi [62], using the L16 orthogonal array to study three factors simultaneously (\(H0/D0\), \(S0/D0\), and \(\alpha\) \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Strain Rate**} & **Experiment** & **Simulation** & **Deviation** \\ \cline{2-3}  & \(P_{E}\) & \(P_{S}\) & \(\Delta P\) \\  & **(ton)** & **(ton)** & **(\%)** \\ \hline \(0.1\) S\({}^{-1}\) & & 26.5 & 21.36 \\ \hline \(10\) S\({}^{-1}\) & 33.4 & 38.6 & 14.54 \\ \hline Equation (1) & & 34.8 & 4.19 \\ \hline \end{tabular}
\end{table} 
Table 4: Flange-forming force obtained from simulations using flow models. Figure 5: The deformed shape obtained from both simulation and experiment: (**a**) strain rate of \(0.1\) S\({}^{-1}\), (**b**) strain rate of \(10\) S\({}^{-1}\), (**c**) the simulation results considering both elevated temperature and strain rates equation, and (**d**) flange-forming force graph from simulation along with the corresponding experimental data. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Geometrical Parameters** & **Level** & **Flange-Forming Force** \\ \hline \(H0/D0\) & 0.4; 0.6; 0.8; 1.0 \\ \hline \(S0/D0\) & 0.2; 0.25; 0.3; 0.35 \\ \hline \(\alpha\) & \(0^{\circ};5^{\circ};10^{\circ};15^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 5: The input and output parameters for the flange-forming process. with four levels for each factor. Table 6 shows the structure and options for conducting the experiments. Simulation results are analyzed statistically through the signal-to-noise ratio (S/N) calculation. S/N ratio represents the signal mean (S) divided by the standard deviation (N) and gauges the impact of input factors on the objective function. Optimization of the S/N ratio minimizes the loss function, enhancing system quality. The S/N ratio is tailored to the objective function, categorized into \"Larger is better\" (Equation (3)), \"Normal is best\" (Equation (4)), and \"Smaller is better\" (Equation (5)), with the latter chosen for the study's purpose:  
\[\mathrm{S/N}=-10 log(\frac{1}{n}\mathrm{\sum}\frac{1}{y_{i}{}^{2}}) \tag{3}\]  
\[\mathrm{S/N}=-10 log(\frac{-^{2}}{y}/D^{2}) \tag{4}\]  
\[\mathrm{S/N}=-10 log(\frac{1}{n}\mathrm{\sum}y_{i}{}^{2}) \tag{5}\]  
Here, \"S/N\" symbolizes the signal-to-noise ratio, a standard metric for gauging data reliability in experimental analysis. The variable \"\(y_{i}\)\" signifies the measured value from the \(i\)-th trial, obtained by averaging multiple measurements within that trial. The variable \"\(n\)\" denotes the total count of conducted tests or experiments. The (S/N) ratio (Equation (5)) is computed for each trial, signifying product deformation degree. The objective function is the flange-forming force (P) post-forging, influenced by \(H0/D0\), \(S0/D0\), and \(\alpha\), each with four levels detailed in Table 7. Forming force and S/N ratios are tabulated in Table 8. The mean signal/noise ratios per factor level are determined as Equation (6), displayed in Table 9. Figure 6 illustrates the impact levels of \(H0/D0\), \(S0/D0\), and \(\alpha\) on flange force:  
\[m_{ji}=\frac{1}{3}\mathrm{\sum}_{i=1}^{3}\big{(}\big{(}S/N)_{j}\big{)}_{i} \tag{6}\]  
\begin{table}