## 2 Model Development
### Modeling of Pad Surface Topography  
The polishing tool used in this paper was a porous polyurethane polishing pad. The pad surface was distributed with circular micropores and 2 mm \(\times\) 1.5 mm diversion grooves. During the polishing process, the surface topography of the pad has a very important influence on the removal mechanism of the polishing material. Therefore, in order to better understand the material removal process, it is necessary to study the surface topography characteristics of polishing pads. In this research, a KEYENCE VH7000 digital microscope was used to observe the surface morphology of the polishing pad, and the image recognition method was used to analyze and measure the distribution of the porous features on the pad surface. Figure 2a shows the measurement results of the surface morphology of the polyurethane polishing pad. We observed randomly distributed pores of different sizes on the pad surface. Figure 2b shows the pore distribution on the pad surface treated by the image recognition method. After the statistical analysis of the size of the pore area, we concluded that the pore area on the pad surface in the observation area accounted for 35% of the total observation area. In addition, the pore size on the pad surface was statistically analyzed, demonstrating that the average pore size was 85 \(\upmu\)m, and the fitted distribution curve showed an exponential distribution. After the hypothesis test, we found that the distribution of the micropore diameter \(D_{h}\) on the pad surface was exponential, with a mean value of 85 \(\upmu\)m, that is, \(D_{h}\sim\epsilon(0.085\) mm). In this study, Matlab software was used to simulate the surface topography characteristics of the polishing pad. The simulation steps were as follows:  
Figure 1: Flowchart of the theoretical modeling process for CCOS. We estimated the average spacing \(\Delta_{h}\) between the micropores on the pad surface by observing the total number of micropores on the pad surface \(N_{h}\) and the total surface area of the polishing pad \(S_{p}\), as shown in Equation (1). \[\Delta_{h}\approx\sqrt{S_{p}/N_{h}} \tag{1}\]  
The \(d_{g}\) value was taken as the average spacing of uniformly distributed micropores, and the position of the center point of the micropores was expressed as \(G_{xyz}\). Since the distribution of micropores mostly presented a disordered random position distribution, the position of the center point of the micropores was randomly shifted in the \(x,y,\) and \(z\) directions of the spatial coordinates, as follows:  
\[G_{xyz}=\left\{ \begin{array}{c}G_{i,j,k}^{x}\\ G_{i,j,k}^{y}\\ G_{i,j,k}^{z}\end{array} \right\}=\left\{ \begin{array}{c}G_{0,0,0}^{x}+i \cdot d_{g}+R_{x}\\ G_{0,0,0}^{y}+j\cdot d_{g}+R_{y}\\ G_{0,0,0}^{z}+k\cdot d_{g}+R_{z}\end{array} \right\} \tag{2}\]  
where \(R_{x}\), \(R_{y}\), and \(R_{z}\) represent the random offset of the initial center point position of the micropore in the three directions of \(x,y,\) and \(z\), respectively. 2. After the random displacement of the center position of the micropore was determined, the hemispherical \"micropore\" was simulated at the new position, with the diameter of the micropore obeying an exponential distribution with an average value of 85 \(\upmu\)m, so as to simulate the pore distribution characteristics of the pad surface. 3. The statistical analysis of the distribution characteristics of the micropore diameter on the pad surface and the porosity of the pad surface was conducted. Compared with the measured data, if the error of the simulation result was less than 5%, the simulation met the requirements. Otherwise, we re-sampled and simulated the position of the central hole of the micropore until the micropore distribution characteristics met the requirements. 4.