# Bridging Macro-Microscopic Parameters: Correspondence and Calibration Approaches
## 2 Principle of Simulation  
Owing to the incapability of PFC2D to directly simulate forces on the walls, in conventional uniaxial compression tests and direct shear tests within PFC2D, it is necessary to vary the wall velocity so as to induce alterations in the wall-particle distance, thereby influencing the particles in the vicinity of the wall. By monitoring the displacement of the wall at each computational time step, a relatively stable force exerted by the wall on the neighboring particles can be attained, and this process is termed servo-control.  
During the servo-control mechanism, the equation for wall velocity is as follows:  
\[v^{(w)}=G(\sigma^{measure}-\sigma^{required})=G\Delta\sigma^{(w)} \tag{1}\]  
Here, \(v^{(w)}\) represents the scalar wall velocity, \(\sigma^{measure}\) denotes the absolute stress measured by the wall, \(\sigma^{required}\) is the absolute stress that the wall needs to apply, \(\Delta\sigma^{(w)}\) signifies the difference between the current stress experienced by the wall and the required stress, and \(G\) represents the servo-control parameter.  
The calculation per unit time step is as follows:  
\[\Delta t=\beta\sqrt{M/K} \tag{2}\]  
Here, \(M\) denotes the total mass of particles, \(K\) stands for the total stiffness of particles, and \(\beta\) represents an adjustment coefficient. The unit time step \(\Delta t\) indicates the interval at which forces and displacement states of particles and walls within the model are updated.  
Collisions between particles and walls are computed using contacts, where each particle near the wall generates a separate contact with the wall. The contact stiffness \(k_{n}\) is utilized to calculate the stiffness when particles collide with the wall.  
\[k_{n}=\frac{k_{n}{}^{[1]}k_{n}{}^{[2]}}{k_{n}{}^{[1]}+k_{n}{}^{[2]}} \tag{3}\]  
In this equation, \(k_{n}{}^{[1]}\) represents the wall stiffness, while \(k_{n}{}^{[2]}\) represents the stiffness of the particle in the current contact. Therefore, the expression for the incremental force on the wall due to wall movement within a unit time step is as follows:  
\[\Delta F^{(w)}=\overline{k_{n}}Nv^{(w)}\Delta t \tag{4}\]  
Here, \(\overline{k_{n}}\) represents the average stiffness of contacts, \(N\) denotes the number of contacts, and \(\Delta t\) is the unit time step. Consequently, the increment of wall stress within a unit time step is given by the following equation:  
\[\Delta\sigma^{(w)}=\frac{\Delta F^{(w)}}{A}=\frac{\overline{k_{n}}Nv^{(w)} \Delta t}{A} \tag{5}\]  
where A denotes the wall area. Substituting Equation (1) into the previous equation yields the following:  
\[\Delta\sigma^{(w)}=\frac{\overline{k_{n}}N\Delta\sigma^{(w)}\Delta t}{A} \tag{6}\]  
where \(k_{n}\), \(N\), \(\Delta\sigma^{(w)}\), \(\Delta t>0\) and \(\Delta\sigma^{(w)}\) is not always equal to zero.  
\[G=\frac{A}{\overline{k_{n}}N\Delta t} \tag{7}\]  
In practical servo-control processes, directly using the servo-control parameter G often leads to opposite adjustments between the wall in the previous and subsequent time steps,making it challenging to achieve equilibrium. Therefore, the servo-control parameter G is typically multiplied by a relaxation factor \(\gamma\) (\(0<\gamma<1\)). When performing numerical simulations for uniaxial compression experiments and direct shear experiments, the wall velocity providing stable pressure is determined by Equation (1), where the servo-control parameter should be determined by the final equation.