## 3 Finite Element Analysis
### Simulation Results and Discussion  
It is generally accepted that the deformation characteristics of the workpiece can be reflected by grid variation in finite element analysis. Accordingly, the simulation results of mesh variation on the billet symmetry plane under TU and CEBU are shown in Figure 3. Due to the friction effect at the billet end, the grids flow downwardly and radially in TU, which leads to bulging generated at the billet center. Meanwhile, the size of the bulge increases as the upsetting progresses. In CEBU, however, the deformation is initially concentrated at the cone end, and the frictional effect at the billet center is largely dispersed. Correspondingly, the bulging is first formed at the bottom of the cone end, and then disappears when the reduction ratio reaches 50%. Compared to TU, CEBU achieves a much smaller bulge. The variation of equivalent strain on the billet symmetry plane during upsetting is shown in Figure 4. At the beginning of upsetting, the strain is mainly concentrated at the billet ends and is radially distributed in CEBU. However, a more uniform strain field is generated in TU due to a higher friction effect on the billet ends. When the reduction ratio reaches 25%, three characteristic deformation zones have been formed in TU, which are the billet end, the billet lateral surface, and the billet center. Correspondingly, two fan-shaped high-strain zones are formed at the billet ends in CEBU. As the reduction progresses, the two fan-shaped strain zones move closer together, resulting in a large compaction effect at the billet center. In contrast to CEBU, two larger rigid deformation zones are generated at the billet ends in TU, which is the reason for the formation of a small and uniform strain field at the billet center. The results indicate that, compared with TU, much smaller rigid deformation zones can be generated in CEBU. In addition, the compression effect at the billet center is much larger in CEBU than in TU. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **CEBU** & & **TU** \\ \hline \(d\) & 25 mm & \(d\) & 25 mm \\ \(h\) & 40 mm & \(h_{t}\) & 62.5 mm \\ \(h_{c}\) & 16.25 mm & & \\ \(\alpha\) & 15\({}^{\circ}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of finite element analysis models. Figure 2: Finite element analysis model. Corresponding to the billet radiuses illustrated in Figure 4, the variation of deformation parameters during upsetting is shown in Figure 5. In CEBU, the radius of the billet end, the cone end bottom, and the billet center are taken as the deformation parameters. As shown in Figure 5a, as the reduction ratio increases, the radius of the billet end is first increased linearly and then changed slightly as the reduction ratio increases from 20% to 51%. This is because most of the load is absorbed by the plastic deformation of the billet cone end when the reduction ratio is below 51%. Accordingly, the radius of the billet cone end bottom is greater than that of the billet center. However, when the reduction ratio increases, the radius of the billet cone end is higher than that of the billet center. Figure 4: Equivalent strain on billet symmetry plane (reduction ratio 5%, 25%, 50%). Figure 3: Variation of the grid on billet symmetry plane (reduction ratio 25%, 50%, 70%). is above 51%, the radius of the billet center is inversely larger than that of the billet cone end bottom as the reduction ratio increases. The same value can be achieved for the two radiuses when the reduction ratio is 51%, indicating that bulging has been eliminated. As shown in Figure 4, the difference between \(r_{1}\) and \(r_{2}\) can be treated as a bulging index. The results of the bulging index variation with the reduction ratio in TU and CEBU are proposed in Figure 5b. It clearly shows that the bulging size can increase as the reduction ratio improves in TU. However, in CEBU, the bulging is first formed at the cone end bottom and then at the billet center.