## 2 Analysis and Optimization of TIF Shape
### Modeling and Optimization of TIF
The analysis results are shown in Figure 3.  
According to the simulation results of the pressure distribution in Figure 3, the following can be known: when E = 0 mm, that is, when the pressure application point coincides with the center of the polishing disk, the peak value of the contact pressure is located at the pressure application point, and the pressure distribution gradually decreases towards the outside in a quasi-Gaussian shape. When E \(\
eq\) 0 mm, that is, when the pressure application point does not coincide with the center of the polishing disk, the pressure peak point deviates from the center of the polishing disk following the pressure application point and moves in the X+ direction. Meanwhile, with the increase in the moving distance, the overturning moment on the polishing disk increases, and the pressure distribution in some contact areas in the X-direction gradually decreases or even disappears. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameters** & **Pressure Output Shaft** & **Metal Disks** & **Pitch Layer** & **Workpiece** \\ \hline Material & Stainless Steel & Aluminum alloy & Pitch & JGS2 \\ Young modulus & 193 GPa & 72 GPa & 0.23 GPa & 74 GPa \\ Poisson radio & 0.28 & 0.33 & 0.40 & 0.17 \\ Density & 7.93 g/cm\({}^{3}\) & 2.75 g/cm\({}^{3}\) & 1.20 g/cm\({}^{3}\) & 2.2 g/cm\({}^{3}\) \\ Size & \(\Phi\)8 \(\times\) 20 mm & \(\Phi\)30 \(\times\) 8 mm & \(\Phi\)30 \(\times\) 2 mm & 50 \(\times\) 50 \(\times\) 20 mm \\ Mesh type & C3D8R & C3D8R & C3D8R & C3D8R \\ Mesh size & 0.5 mm & 0.5 mm & 0.5 mm & 1 mm \\ \hline \end{tabular}
\end{table} 
Table 1: Modeling parameters. Figure 2: Finite element analysis model. (**a**) Composition of the finite element analysis model. (**b**) Model with the pressure application point coinciding with the selected center of the polishing disk. (**c**) Model with the pressure application point separated from the rotation center of the polishing disk. In order to further analyze the influence of the position of the pressure application point on the form of the TIF, we obtained the polishing speed distribution V within the contact area according to Equation (3), as shown in Figure 4.  
\[V(x,y)=\omega\sqrt{x^{2}+y^{2}} \tag{3}\]  
where \(V(x,y)\) represents the magnitude of the velocity value at any contact point within the contact area between the polishing disk and the polished element, \(\omega\) represents the rotational speed of the polishing disk, and \(\sqrt{x^{2}+y^{2}}\) represents the distance of this point relative to the center of rotation. Figure 4: Normalized velocity distribution in the contact area of 30 mm diameter polishing disk (\(\omega=1\) rps). Figure 3: Results of contact pressure analysis. (**a**) Pressure distribution when E = 0 mm. (**b**) Pressure distribution when E = 2.5 mm (\(\frac{1}{2}\)D, D = 30 mm). (**c**) Pressure distribution when E = 5 mm (\(\frac{1}{6}\)D). (**d**) Pressure distribution when E = 7.5 mm (\(\frac{1}{4}\)D). On the premise of knowing the pressure distribution and velocity distribution, we obtained the TIF at different positions of the pressure application point through the Preston equation, as shown in Figure 5. Meanwhile, in order to compare the morphological differences in the TIFs at different positions of the pressure application point more intuitively, we intercepted the cross-sections of the TIF in the X-direction, normalized each cross-section, and then carried out a comparative analysis. The results are shown in Figure 6.  
It can be seen from Figures 5 and 6 that as the pressure application point deviates from the center of the polishing disk and moves in the X+ direction, the removal amount in the X- direction gradually decreases, while the removal amount in the X+ direction gradually increases. The TIF gradually changes from an annular shape to a quasi-Gaussian shape with a single peak. When E = \(\frac{1}{2}\)D, the removal amount of the TIF in the second and third quadrant regions has been reduced to a negligible level.