## 4 Discussion  
Although the finite difference method is effective in solving the problem, its calculation speed is slow, as the calculation slows down when solving a large system of linear equations. If the calculation results of the finite difference method can be fitted by a simple equation, it will increase the computation speed significantly. A natural idea is fitting the surfaces in Figures 5 and 6 directly. However, different types of surface equations are necessary to fit the surfaces in Figures 5 and 6, and the fitting results will be unsatisfactory. Another idea is fitting the curves in Figures 5 and 6, which was also proven to be unsatisfactory. After careful consideration, it was decided that the quadratic and quartic coefficients of the different \(y\) values can be fitted by a type of \"Gaussian\" curve with high precision. The basic curve equation can be expressed as Equation (9). \[f(y/B)=A_{1}\exp\left(-\left(\frac{y/B-B_{1}}{C_{1}}\right)^{2}\right)+A_{2} \exp\left(-\left(\frac{y/B-B_{2}}{C_{2}}\right)^{2}\right) \tag{9}\]  
where \(A_{1}\), \(B_{1}\), \(C_{1}\), \(A_{2}\), \(B_{2}\), and \(C_{2}\) are the fitting coefficients. The fitting coefficients in Equation (9) are obtained using the numerical computation package Numpy through the Python language. The fitting results are shown in Figures 11 and 12, and Table 1. It can be seen that the fitting results of all the cases are highly satisfactory. In Figure 11, \(a_{20}\) is the quadratic coefficient of the initial stress distribution, and the quartic coefficient is zero. In Figure 12, \(a_{40}\) is the quartic coefficient of the initial stress distribution, and the quadratic coefficient is zero. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multicolumn{2}{c}{**Mode**} & \(\mathbf{A_{1}}\) & \(\mathbf{B_{1}}\) & \(\mathbf{C_{1}}\) & \(\mathbf{A_{2}}\) & \(\mathbf{B_{2}}\) & \(\mathbf{C_{2}}\) \\ \hline \multirow{3}{*}{1} & \(a_{2}\) & 0.01292 & 0.03105 & 0.02057 & 1.114 & \(-\)0.1145 & 0.6015 \\  & \(a_{4}\) & \(-\)0.07125 & 0.3252 & 0.2516 & \(-\)0.119 & 0.431 & 0.4751 \\ \cline{1-1}  & \(a_{2}\) & 0.3064 & 0.2181 & 0.2123 & 0.2661 & 0.4202 & 0.3696 \\ \cline{1-1}  & \(a_{4}\) & \(1.059\) & \(-\)0.0938 & 0.2506 & \(-\)0.1504 & 0.2766 & 0.5254 \\ \hline \end{tabular}
\end{table} 
Table 1: Results of fitting coefficients. Figure 10: Results of different strip widths. (**a**) Stress distribution in the middle of the strip. (**b**) Stress distribution along the edge of the strip. To test the fitting precision, the case of \(a_{20}=-30\) MPa, \(a_{40}=50\) MPa, \(B\)= 1250 mm is simulated as an example with the finite difference method and fitting equation, and the results are shown in Figure 13. It can be seen that the stress evolution regulation along the edge and at a quarter of the strip, calculated by the fitting equation, is in good agreement with that calculated by the finite difference method. To achieve accurate flatness prediction, two issues need to be studied, as shown in Figures 2 and 3. The coupling relationship between the entry and exit tensile stress distribution and the velocity in the deformation zone has been extensively studied by the authors in previous research [27]. The second issue, which is the mechanism of the flow variation of the external tensile stress and velocity distribution in the deformation zone, is the main focus of this study. According to the research results in this study, it is obvious that the tensile stress distribution in the upstream and downstream regions of the deformation zone is different from that in the entry and exit regions of the deformation zone. Specially, between the exit of the deformation zone and the downstream region, the velocity distribution gradually becomes more uniform, while the tensile stress distribution  
Figure 11: Fitting results of model 1. Figure 12: Fitting results of model 2. Figure 13: Comparison between finite difference method and fitting equation. (**a**) Edge, (**b**) Quarter. gradually stabilizes.