## 2 Numerical Model Construction
### Rigid Block Contact Modeling in Particle Flow Algorithms  
The discrete element software PFC (6.0) effectively captures rockfill materials' mechanical properties and considers their particle shape effect [31]. During the construction of numerical specimens, the contact state between particles of the ball unit is easy to determine with high computational efficiency due to its simple shape. However, the closure between spherical particles is weak, which contradicts the physical properties of rockfill materials [32]. Although clump units can realistically simulate rockfill material particles, they require more computer memory and computation time [33]. The rblock unit maintains the realism of the clump unit. The computational efficiency is significantly improved by having only one contact between the rblock unit and other units, as determined by the outer circle [34]. In the discrete element method, the contact model serves as a mechanical criterion for the interaction between particles, determining the macroscopic and microscopic mechanical properties of the entire particle system. The rblock unit can interact with the ball, clump, and wall regions in 3D, employing a linear contact model that defines the behavior of an infinitesimal interface. This model decomposes the contact force into linear and damped components, where \(M_{\text{c}}\equiv 0\) and \(F_{\text{c}}=F_{\text{1}}+F_{\text{d}}\). The choice of the linear contact model is motivated by its suitability for meeting the contact requirements between the blocks and improving computational efficiency. The elements of the linear contact model consist of normal and tangential springs with stiffnesses \(k_{\text{n}}\) and \(k_{\text{s}}\), respectively; sliding elements with friction factors \(\mu\); normal and tangential dampers with damping coefficients \(\beta_{\text{n}}\) and \(\beta_{\text{s}}\); and pressure-bearing elements identifying the amount of interparticle overlap \(g_{\text{s}}\), as shown in Figure 2a. When the reference gap is zero, the hypothetical surface [35] coincides with the surface of the workpiece, and the relative displacement increment at the contact during the time step (\(\Delta\)t) is calculated from \(\Delta\delta_{\text{n}}\) and \(\Delta\delta_{\text{s}}\) in Equation (1):  
\[ \begin{split}\Delta\delta_{\text{n}}:=\alpha\Delta\delta_{\text {n}},\Delta\delta_{\text{s}}:=\alpha\Delta\delta_{\text{s}}\\ \text{with }\alpha=\begin{cases}\frac{g_{\text{s}}}{g_{\text{s}}-(g_{ \text{s}})_{0}},(g_{\text{s}})_{0}>0\text{ and }g_{\text{s}}>0\\ 1,\text{ otherwise}\end{cases}\end{split}  \tag{1}\]  
where \(g_{\text{s}}\) is the surface gap and (\(g_{\text{s}}\))\({}_{0}\) is the surface gap at the beginning of the time step. A local Cartesian coordinate system (\(x,y,z\)) is established at the contact point \(O\), where \(y\) is parallel to the contact normal, and \(x\) and \(z\) are mutually perpendicular following the right-hand rule with \(y\), as illustrated in Figure 2b. In this coordinate system, the contact force \(F_{\text{c}}\) in the linear contact model can be decomposed into the normal contact force \(F_{\text{n}}\) and the tangential contact force \(F_{\text{s}}\), as shown in Figure 2c. The following formula calculates the contact force \(F_{\text{c}}\). Figure 1: Virtual structural surface reconstruction process for blocks. (**a**) Three-dimensional scanning device; (**b**) original specimen; (**c**) surface point cloud; (**d**) point cloud noise reduction; (**e**) PFC particle modeling. \[F_{c}=F_{n}+F_{s} \tag{2}\]  
Among them:  
\[\left\{ \begin{array}{l}F_{n}=F_{n}^{l}+F_{n}^{d}\\ F_{s}=F_{s}^{l}+F_{s}^{d}\\ F^{l}=F_{n}^{l}+F_{s}^{l}\\ F^{d}=F_{n}^{d}+F_{s}^{d}\end{array} \right. \tag{3}\]  
where \(F^{l}\) is the linear contact force, decomposed into a normal force \(F_{n}^{l}\) and a tangential force \(F_{s}^{l}\); \(F^{d}\) is the viscous damping force, decomposed into a normal force \(F_{n}^{d}\) and a tangential damping force \(F_{s}^{d}\).