## 3 Results and Discussion
### Development of a Mathematical Model for Flange-Forming Force  
The flange-forming force of the product after the flange forming of C45 billet tubes is influenced by the parameters \(H0/D0\), \(S0/D0\), and \(\alpha\). The objective is to establish a mathematical relationship between these three parameters and the flange-forming force (\(P_{M}\)) of the product. This allows us to analyze the influence trend and evaluate the impact of these parameters on the objective function. To achieve this, we employ the least squares method, which involves building a polynomial model of degree 3 between the flange-forming force (\(P_{M}\)) and the parameters \(H0/D0\), \(S0/D0\), and \(\alpha\). Higher-order polynomials are not suitable for this purpose, as they may yield numerous extremes and lack practicality. The model is represented by the following Equation (12):  
\[ \begin{split} P_{M}=a_{1}\cdot\left(\frac{H0}{D0}\right)^{3}+a_{2 }\cdot\left(\frac{S0}{D0}\right)^{3}+a_{3}\cdot\left(\alpha\right)^{3}+a_{4} \cdot\left(\frac{H0}{D0}\right)^{2}\cdot\frac{S0}{D0}+a_{5}\cdot\left(\frac{S0 }{D0}\right)^{2}\cdot\alpha\\ +a_{6}\cdot\left(\alpha\right)^{2}\cdot\frac{H0}{D0}+a_{7}\cdot \left(\frac{H0}{D0}\right)^{2}+a_{8}\cdot\left(\frac{S0}{D0}\right)^{2}+a_{9} \cdot\left(\alpha\right)^{2}+a_{10}\cdot\frac{H0}{D0}+a_{11}\cdot\frac{S0}{D0} +a_{12}\end{split}  \tag{12}\]  
Using the least squares method and solving the equation with Matlab software, we obtain the values for the 12 coefficients as follows:  
\[ \begin{split} P_{M}=185.081\cdot\left(\frac{H0}{D0}\right)^{3}-109 21\cdot\left(\frac{S0}{D0}\right)^{3}-0.005\cdot\left(\alpha\right)^{3}+15.860 \cdot\left(\frac{H0}{D0}\right)^{2}\cdot\frac{S0}{D0}\\ -4.396\cdot\left(\frac{S0}{D0}\right)^{2}\cdot\alpha+0.001\cdot \left(\alpha\right)^{2}\cdot\frac{H0}{D0}-390.375\cdot\left(\frac{H0}{D0} \right)^{2}+8665.1\cdot\left(\frac{S0}{D0}\right)^{2}\\ +0.121\cdot\left(\alpha\right)^{2}+236.545\cdot\frac{H0}{D0}-2124 \cdot\frac{S0}{D0}+137.633\end{split}  \tag{13}\]  
To assess the reliability of the regression function (Equation (13)), we conduct an ANOVA analysis. The results are shown in Table 10, indicating that the regression coefficient R\({}^{2}\) is 0.98 and the Fisher coefficient \(F\) is 1.6741. Both values are lower than the critical value of 8.7 (based on the Fisher criteria), demonstrating high confidence in the mathematical model's dependence on the parameters \(H0/D0\), \(S0/D0\), and \(\alpha\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Factor**} & **Degrees of Freedom** & **Sum of Squares** & **Mean Squared** & **Residual Variance** & **Repeated** & \multirow{2}{*}{**R\({}^{2}\)**} & \multirow{2}{*}{**F\(=S_{du}^{2}/S_{II}^{2}\)**} & \multirow{2}{*}{**F\({}_{0.05}(12,3)\)**} \\  & **S/N** & (\(\bar{\text{SN}}_{i}-\bar{\text{SN}}\)) & **SN** & (\(S_{du}^{2}\)) & & (\(S_{II}^{2}\)) & & \\ \hline \(H0/D0\) & 3 & 166.228 & 55.409 & & & & & \\ \hline \(S0/D0\) & 3 & 512.787 & 170.929 & & & & &By employing the Matlab software, a graphical representation (Figure 8) of the relationship between the flange-forming force and the forming process parameters (\(H0/D0\), \(S0/D0\), and \(\alpha\)) for the flange forming of C45 billet tubes can be readily generated. The graphical analysis reveals that the flange-forming force exhibits direct proportionality with the pitch angle of the bottom of the punch (\(\alpha\)), while displaying an inverse relationship with both \(H0/D0\) and \(S0/D0\). Moreover, this study encompasses a comparative assessment between the flange-forming force values derived from the formulated mathematical model (Equation (11)) and the corresponding experimental outcomes. For this evaluation, three sets of randomized parameters are employed for both simulation and testing. Employing the input parameters (\(H0/D0=0.6\); \(S0/D0=0.2\); and \(\alpha=0^{\circ}\)), the simulated and experimental profiles presented in Figure 9a exhibit a close similarity to the shape of the flange after the forming process for C45 billet tubes.