## 3 Numerical Simulation of Warm Forging-Finishing of Spiral Bevel Gear  
### Numerical Simulation Parameter Settings  
With the rapid development of computer graphics, applied mathematics and modern mechanics, the finite element method (FEM) has become a recognized and effective numerical computation method which can be used to deal with the problems of continuum mechanics and structural science. The rigid-plastic finite element method, rigid-viscoplastic finite element method and elastic-plastic finite element method are the most commonly used methods for finite element numerical simulation of metal plastic forming. The rigid-plastic finite element method is used in this paper.  
DEFORM-3D software is a finite element simulation analysis system based on process simulation. The software has good stability, a simple operation and powerful functions. It can analyze the metal flow process and various parameters in the metal during the plastic deformation process and has been widely used in the field of forging-forming of complex parts.  
In the forging process, the metal material not only bears the deformation force but also has heat transfer and heat exchange with the surrounding environment, leading to a non-constant temperature environment in the deformation process of the metal material. Therefore, the effect of heat transfer should be taken into account in the numerical simulation of the forging process of spiral bevel gears. Due to the complex forming process of warm-forging spiral bevel gear, many factors should be considered in the modeling of warm-forging spiral bevel gear in DEFORM-3D software, including the forging temperature of the gear blank, the limitation of the model boundary conditions and the influence of the heat produced in the process of forging deformation. In order to ensure the accuracy and efficiency of the simulation, the finite element simulation of the warm-forging-finishing spiral bevel gear is simplified as follows:  
(1) In the process of warm-forging-finishing deformation of spiral bevel gear, the main deformation of the gear blank is plastic deformation, while the elastic deformation of the die and the workpiece is small, so the gear blank is set as a plastic body, and the die is set as a rigid body. In this experiment, the gear blank material is AISI-1045 [70-2000 F (293-1373 K)], and the die material is AISI-H-13.  
(2) The warm-forging-finishing model of spiral bevel gear is shown in Figure 5. In order to improve the efficiency and accuracy of the simulation and shorten the operation time, 1/9 of the preform is taken for numerical simulation, the tetrahedral mesh is used to divide the preform, the number of generated meshes is set to about 120,000 and the automatic mesh division is set, as shown in Figure 7.  
Figure 6: Finished product of forging numerical simulation.  
(3) In addition to the preheated temperatures of the die and forging, the heat loss of the gear blank and the heat exchange between the forging die during the forging process should also be taken into account. In order to make the simulation results more in line with the reality, the heat transfer option of the simulation model is turned on, the ambient temperature is set to 20 \({}^{\circ}\)C, the contact heat coefficient is set to 5 N/s/mm/C and the convection coefficient is set to 0.02 N/s/mm/C.