## 6 Conclusions  
1. Under actual working conditions, the extrusion roller is mainly affected by the contact stress between the roller shaft and the roller sleeve, and the extrusion force between the roller sleeve and the material. Based on the compression and rebound characteristics of the material, the extrusion pressure on the surface of the extrusion roller is calculated, and the normal direction stress \(p_{\gamma}(\alpha,\lambda)=102.05\) MPa and the tangential stress \(p_{\gamma}(\alpha,\lambda)=2.31\) MPa.
2. According to the finite element analysis results of the extrusion roller, the maximum deformation of the extrusion roller occurs at the non-stepped end of the inner ring of the roller sleeve, and the value is 1.379 mm. The maximum equivalent stress of the extrusion roller is located at the step of the contact surface between the roller sleeve and the roller shaft, and the value is 651.03 MPa. The maximum stress is on the brink of the allowable yield strength of the roller sleeve, so it is necessary to optimize the design.  
Figure 19: Simulation nephograms of optimized roller sleeve: (**a**) equivalent stress nephogram; (**b**) deformation nephogram.  
Figure 20: Simulation nephograms of optimized roller shaft: (**a**) equivalent stress nephogram; (**b**) deformation nephogram.  
3. The initial optimization of the extrusion roller structure is carried out by the method of size optimization, and the scheme of setting the transition arc at the step of the contact surface between the roller shaft and the roller sleeve is proposed. The simulation results show that the maximum equivalent stress is reduced by 26%, and the maximum deformation is decreased by 20%, which initially achieves the purpose of improving the structural strength and reducing the deformation. However, the decreasing trend of the maximum equivalent stress in the optimization scheme did not stop, and the transition arc parameters can still be refined and optimized further. 4. The extrusion roller is further optimized by using the Direct Optimization module of ANSYS Workbench to obtain the optimal solution of the transition arc. By comparing the data before and after optimization, the contact stress of extrusion roller is reduced by 17%, the contact sliding distance is decreased by 83%, the maximum equivalent stress is reduced by 29%, and the maximum deformation is decreased by 28%. This scheme is expected to improve the bearing capacity of the extrusion roller, reduce the production cost of the roller press, and provide a reference basis for the design of the extrusion roller.  
Conceptualization, W.W. and F.P.; methodology, W.W. and Y.L.; software, F.P. and Y.L.; validation, B.C. and Y.X.; formal analysis, Y.X.; investigation, F.P. and W.W.; resources, Y.W.; data curation, Y.L.; writing--original draft preparation, F.P., Y.L. and W.W.; writing--review and editing, W.W.; visualization, B.C.; supervision, W.W.; project administration, W.W. and Y.W.; funding acquisition, W.W. All authors have read and agreed to the published version of the manuscript.  
This research was financially supported by the Jiangsu Key Laboratory of Precision and Micro-Manufacturing Technology (2019), Jiangsu \"Six Talent Peak\" Project (JXQC-022), and the Jiangsu University Students' Practice Innovative Training Project (202010298054Z).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.