## 4 Reinforcement Springback Compensation Method
### Bending and Springback Compensation Method
#### 4.2.2 Numerical Analysis Method  
(1) Least square method. The least square method seeks the best function matching of data by minimizing the sum of squares of errors. The least square method can easily obtain unknown data and minimize the sum of squares of errors between the obtained data and the actual data. When used as an interpolation method, the least square method can also be used for curve fitting [37]. The theory of the least square method is simple, and the calculation is small. Even though the use of cubic spline curve or RBF (radial basis function) for curve fitting is popular today, the least square method is still widely used in polynomial curve or line fitting problems. When the least square method is used to solve the fitting curve function, the form of \(s(x)\) needs to be determined by observation, as follows:  
\[s(x)=a_{0}\phi_{0}(x)+a_{1}\phi_{1}(x)+\ldots+a_{n}\phi_{n}(x)(n<m) \tag{6}\]  
Among them, \(a_{0},a_{1},\ldots,a_{n}\) is the stay constant, and \(\phi_{0}(x)=1,\phi_{1}(x)=x,\phi_{2}(x)=x^{2},\ldots,\phi_{n}(x)=x^{n}\) is determined by observations of functional form. Using the least square method, the mode of matching function selected is very important. If the discrete data show exponential variation law, the mode of matching function in exponential form should be selected; if it shows the polynomial variation law, the mode of matching function in polynomial form should be selected. If the mode selected is wrong, the fitting effect will be poor. This is also an area that needs special attention when using the least square method for curve fitting. (2) Newton polynomial interpolation method. Polynomial interpolation is also widely used in data analysis and curve fitting. Most practical problems require the function \(y=f(x)\) to represent some kind of law. However, \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Bending** \\ **Angle** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Property of** \\ **Reinforcement** \\ \end{tabular} } & \multicolumn{2}{c}{**Bending Springback Adjustment**} & \multirow{2}{*}{
\begin{tabular}{c} **Diameter of** \\ **Support Shaft** \\ \end{tabular} } \\ \cline{3-3} \cline{5-4}  & & **Calculation Formula** & & \\ \hline
30\({}^{\circ}\) & & \(\Theta=0.006\)D + 0.274d & 0.3d & \\
45\({}^{\circ}\) & HPB235 & \(\Theta=0.022\)D + 0.436d & 0.55d & D = 5d \\
60\({}^{\circ}\) & HRB335 & \(\Theta=0.054\)D + 0.631d & 0.9d & \\
90\({}^{\circ}\) & HRB400 & \(\Theta=0.215\)D + 1.215d & 2.29d & \\
135\({}^{\circ}\) & & \(\Theta=0.006\)D + 0.274d & 3.31d & D = 2.5d \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Steel bar springback experience data sheet. the practical problems are relatively complicated and there are many influencing factors. In the actual process, only a series of scattered points in a certain interval \([a,b]\) can be obtained. In order to be studied, these scattered points need to be fitted to the corresponding curve so that an accurate functional relation \(f(x)\) can be obtained [38]. However, it is difficult to obtain the exact functional relation \(f(x)\), so a simple functional relation \(P(x)\) is usually used to replace it, and \(P(x_{i})=f(x_{i})\) is required to be valid for \(i=1\), \(2\), \(\ldots\), \(n\), as shown in Figure 17.  
The interpolation polynomial solved by the Newton interpolation method satisfies \(P_{n}(x_{j})=y_{j},j=0,1,\ldots,n\), and the interpolation polynomial solved is in the following form:  
\[ \begin{array}{ll}P_{n}(x)=&f(x_{0})+f[x_{0},x_{1}](x-x_{0})+f[x_{0},x_{1},x_ {2}](x-x_{0})(x-x_{1})+\ldots\\ &+f[x_{0},x_{1},\ldots,x_{n}](x-x_{0})\ldots(x-x_{n-1})\end{array}  \tag{7}\]  
where \(f[x_{0},x_{1},\ldots,x_{n}]\) is the average difference of order \(n\) of function \(f(x)\). (3) The cubic spline interpolation method. Spline interpolation is a kind of interpolation method commonly used in industrial design to obtain smooth curves.