## 2 Numerical Investigation on the Correlation of the COR and Damping Parameter  
### Brief Introduction to 3 DEC  
In this study, a DEM code called 3 DEC was adopted to simulate the process of rockfall movement. The 3 DEC code [35] enables the numerical simulation of contacts between polyhedral and planar boundaries. In a discontinuous numerical model, the following two behaviors in discontinuous systems must be considered: (1) the behavior of blocks and (2) the behavior of discontinuities. Figure 1 shows the basic computational principle of 3 DEC. The force/displacement law is applied to identify all contact forces from known displacements. The equation of movement in Figure 1 provides grid-point accelerations, \(\bar{u}_{i}\), in the X, Y, and Z directions, which resulted from the summation of the forces built up at each grid point using Equations (1) and (2). The accelerations were numerically integrated to determine the velocities and integrated again to determine the displacements. The next time step was initiated again using this new set of displacements [9]. \[\bar{u}_{i}=\frac{\int_{\mathrm{s}}\sigma_{\mathrm{ij}}\eta_{\mathrm{f}}ds+F_{ \mathrm{i}}}{m}+g_{\mathrm{i}} \tag{1}\]  
\[F_{\mathrm{i}}=F_{\mathrm{i}}^{\mathrm{c}}+F_{\mathrm{i}}^{\mathrm{j}} \tag{2}\]  
where \(\sigma_{\mathrm{ij}}\) is the zone stress tensor; \(s\) is the surface enclosing the mass, \(m\), present at the grid point; \(\eta_{\mathrm{j}}\) is the unit normal to \(s\); \(F_{\mathrm{i}}\) is the resultant of all external forces applied to the grid point; and \(g_{\mathrm{i}}\) is the gravitational acceleration. \(F_{\mathrm{i}}^{\mathrm{l}}\) presents the summation of the external applied load, and \(F_{\mathrm{i}}^{\mathrm{c}}\) is the result of the subcontact forces that exists only for the grid point along the block point [9]. To simulate discontinuous behavior, the contact between these two blocks must be calculated (Figure 2). The common plane (c-p) was used to judge and compute contacts between blocks [36]. c-p was located between the centroids of these two blocks, with the unit normal vector pointing from one centroid to the other. When block contact occurs in 3 DEC, normal joint stiffness \(k_{\mathrm{n}}\) and normal contact damper \(\eta_{\mathrm{n}}\) are connected in parallel in the normal direction (Figure 3a). Furthermore, in the shear direction (Figure 3b), shear joint stiffness \(k_{\mathrm{s}}\) and shear contact damper \(\eta_{\mathrm{s}}\) are connected in parallel and connected to the Mohr-Coulomb sl  
Figure 1: Calculation cycle of a DEM method with deformable blocks [35]. Figure 2: Sketch of the common plane in a 3D DEM [36]. contact dampers are used to mitigate unwanted contact vibrations. Damping is partly due to energy loss as a result of slippage along the contacts of blocks within the system, internal friction loss in the intact material, and any resistance caused by air or fluids surrounding the structure [35]. Damping can be divided into global and local. Global damping is used in the numerical integral calculation of velocity, and local damping is used in the calculation of contact forces. Both the joint stiffness and the contact damper are numerical parameters. [PERSON] [36] and [PERSON] [37] discussed the detailed theory and computational algorithms. Two types of damping (mass-proportional and stiffness-proportional) are available in 3 DEC. Mass-proportional damping applies proportional force to absolute velocity and mass, but in the opposite direction to that of velocity. Stiffness-proportional damping applies proportional force to the incremental stiffness matrix multiplied by relative velocities or strain rates to contacts or stresses in zones. In 3 DEC, the two forms of damping may be used separately or in combination.