## 2 Finite Element Model Verification
### FEM Verification: Comparison of Test Data  
Using the ABAQUS finite element package, the vertical displacements of the upper part of the polishing pad and wafer are calculated. The analysis results are shown in Figures 11 and 12, showing similar behavior to the test result. The analysis result in Figure 11a confirms that model #1 shows the best behavior, as shown in the experimental result in Table 2. Tables 6 and 7 compare the values at r = 143 mm point where some manufacturers empirically remove the edge of the wafer by 7 mm [16]. As shown in Table 6, it can be confirmed that model #1 has the best performance in both experiments and analysis under the same pressure conditions and that both experimental and analytical results are advantageous in the order #1 \(>\) #6 \(>\) #2 \(>\) #3, #4, #5.  
As shown in models #1, #2, and #4 from \(x\) = 150 mm to \(x\) = 160 mm in Figure 11b, due to the change in adding a groove and widening the width of the groove, the area at the bottom of the retainer ring decreases and transferred pressure applied to the pad increases. As a result, the displacement of the pad located at the bottom of the wafer is increased, and finally, the displacement of the wafer increases, as shown in Figure 11a.  
Figure 11: Finite element analysis results of all models: (**a**) displacement of wafer, (**b**) displacement of pad.  
Figure 12: Finite element analysis results of model #4 for change in P\({}_{1}\): (**a**) displacement of pad, (**b**) displacement of pad.  
When comparing the results of model #2 and model #5 in Table 6, it can be seen that both the test and analysis results deteriorate due to the reduced inner diameter of model #5. This is presumed to be because the rebound effect increases as the retainer ring gets closer to the wafer. Likewise, when comparing the result of model #2 and model #6, it is estimated that the influence on the wafer edge is reduced when the position of the groove of model #2 is moved from the near side to the far side of the wafer. Table 7 and Figure 12 shows the displacements obtained by changing P\({}_{1}\) in model #4. It can be seen that decreasing the pressure at P\({}_{1}\) reduces the relative displacement. Since the pressure applied to the retainer ring is the same, it can be observed that the displacement distribution from \(x=150\) mm to \(x=160\) mm in Figure 12b is also same. Proper P\({}_{1}\) may help to reduce the rebound due to the large pressure on the retainer ring, however, excessive P\({}_{1}\) may cause excessive deformation of the wafer itself downward, as shown in Figure 12a. More detailed results for changing P\({}_{1}\) will be covered in Section 3.3.  
As such, it was confirmed that the FEM model and the existing test results showed a similar trend. Through this, the suggested finite element model was verified indirectly, and in the next section, based on the verified finite element model, in order to improve the non-uniformity of the wafer, the MRR will be predicted as the shape of the retainer ring changes through adding simple machining processes such as machining round shapes or tapering without significantly changing the best model, the current mass production model (model #1).  
Finite Element Analysis Result: Investigation of the Influence of Pressure Distribution and Shape of Retainer Ring  
In this section, based on the basic model (model #1), from an engineering point of view, the influence of the shape changes of the retainer ring is investigated by adding simple machining processes such as machining round sections or tapering without significantly changing the current structure of the retainer ring. In addition, the influence on the load applied to the wafer and retainer ring is investigated.