## 3 TIF of the SWPT
### Modeling Pressure Distribution  
According to the [PERSON] contact theory, a spherical elastic body in contact with a planar rigid body will produce a circular contact area, and the contact pressure distribution will be Gaussian with a central peak [26]. However, the polishing wheel also rotates when contacting the workpiece, which causes sliding friction and affects the shape and pressure distribution of the contact area. The flexible layer of the polishing wheel is very thin, which does not meet the conditions of the [PERSON] contact theory. Therefore, the [PERSON] contact theory cannot accurately predict the contact area shape and pressure distribution during polishing. So, we use the finite element method to study the contact state.  
The contact model between the polishing wheel and the workpiece is shown in Figure 3. When the polishing wheel is not rotating, the contact area is circular (Figure 3b, black dotted line). When the polishing wheel is rotating, friction resistance is generated, which causes the screwed-in part of the flexible polishing pad to be squeezed and the contact area to become larger, the screwed-out part of the flexible polishing pad is stretched and the contact area becomes smaller, and eventually the contact area becomes oval (Figure 3b red solid line).  
This paper uses the finite element method to study the dynamic contact characteristics between the polishing wheel and the workpiece. The finite element model consists of a polishing wheel and a workpiece. The polishing wheel consists of an external flexible polishing pad and an internal rigid hub. The flexible layer uses a polyurethane polishing pad with a thickness of 2 mm. The rigid hub is a spherical wheel with a diameter of 50 mm and a width of 20 mm, and the material is aluminum alloy. The workpiece is K9 glass with length, width and height of 80 mm, 30 mm and 15 mm, respectively.  
The 3D model is imported into the ABAQUS finite element software, and its element type is a 8-node 6-hedron, as shown in Figure 4. The material parameters of the model components are shown in Table 1. Among them, a rectangular coordinate system is established with the contact center point of the polishing wheel and the workpiece as the origin, and the center point Rp of the rigid hub rotation axis is used as the control reference point.  
Contact constraint: Establish a contact constraint between the flexible polishing pad and the workpiece and set the friction coefficient to 0.4.  
Boundary condition: The workpiece's six degrees of freedom are completely limited, and the rigid hub has only one degree of freedom for rotation and one degree of freedom for movement of the vertical workpiece surface. A coupling constraint is established between the inner surface of the flexible polishing pad and the outer surface of the rigid hub.  
Load: A pressure load of 20 N and a motion load of 2 rps are applied to the central control point Rp.  
Figure 3: Contact model. (**a**) 3D model, (**b**) contact area.  
After calculation by the ABAQUS software, the pressure distribution is obtained, as shown in Figure 5. It can be seen from the pressure distribution that the contact area is oval, symmetrical up and down, and asymmetrical left and right. This is because when the polishing wheel rotates, friction resistance is generated, which causes the screwed-in part of the flexible polishing pad to be squeezed and the contact area becomes larger, the screwed-out part of the flexible polishing pad is stretched, and the contact area becomes smaller. The point of maximum pressure is the origin of the coordinate system, because this point has the largest deformation.