# An Integrated Cellular Automata Model Improves the Accuracy of Secondary Fragmentation Prediction
## 2 Materials and Methods
### Stress Model  
In underground mining, the stresses in the rock mass are usually modeled as a continuum, but with block caving, continuous models do not adequately represent the loads and distributions associated with the broken material column [26; 27]. To address this limitation, the work done by [22] proposed a vertical stress model of granular material that can simulate static and dynamic flow conditions. This model was developed within a gravity flow simulator based on cellular automata to replicate both the scale of the problem and the flow conditions. To achieve this, the weight transition was applied through the cells of a cellular automata to simulate gravity flow in caving mines. To simulate stresses, the weight of a (granular) cell was randomly distributed among the lower cells, as shown in Figure 2.  
In Figure 2, a cell at level \(i\) distributed its weight, \(\mathrm{P_{i}}\), over a lower cell at level \(j\). Weight is a function of the specific gravity, \(\gamma_{i}\), and cell volume, V. In addition, shear forces (friction) and horizontal forces can be considered. However, in this work only vertical force transmission was modeled. The vertical forces in the model are the weight components (gravity) and the shear components (friction). Here, weight transmission is simplified by introducing a buoyancy parameter, E, which is defined between 0 and 1 and represents the fraction of the weight of a cell that is distributed to the lower cells. The weight of the cell distributed to the lower cells is represented by Equation (1):  
\[\gamma i-\gamma i\ \cdot(1-E)=\gamma i\ \cdot E \tag{1}\]  
Then, the weight of a cell at level \(j\) is its weight plus the weight fractions of the top nine cells (Equation (2)):  
\[P_{j}=\gamma_{j}V+E\sum_{i=1}^{9}P_{i}B_{i} \tag{2}\]  
Considering \(P_{j}\) as the weight of cell \(j\) and \(\beta_{i}\) as the weight fraction of cell \(i\) transmitted to cell \(j\), \(\beta_{i}\) is defined by the multinomial probability distribution in Equation (3):  
\[\beta(x_{j})=\frac{n!}{\prod_{j=1}^{k}x_{j}!}*\prod_{j=1}^{k}(p_{j})^{x_{j}} \tag{3}\]  
Figure 2: Profile view (2D) of weight transmission in a cell network [22]. Red arrows illustrate distribution of normal and shear forces.  
The proposed stress model is designed for granular material in a static state. However, as observed in previous research [26; 28] vertical stress tends to increase in areas of stagnation and decrease in areas of movement when gravity flow is initiated due to ore extraction. It is for this reason that a relaxation parameter R, between 0 and 1, is introduced to model the weight distribution between zones of movement and stagnation. The parameter R decreases the probability of weight distribution over the movement zone when one or more cells are in the stagnant zone, applying Equation (4).  
\[B_{k}=\frac{B_{i}R}{\sum_{i=1}^{9}B_{i}R} \tag{4}\]  
In Equation (4), \(R=R\) (i.e., 0 \(<\) R \(<\) 1, R must be calibrated) if the cell \(i\) is a movement cell, or R = 1 if not (stagnant cell). The weight distributed to the lower cells is defined by the distribution function \(B_{i}\), so the relaxation parameter multiplies \(B_{i}\times R\). The distribution function \(B_{i}\) is then updated to a new distribution function \(B_{k}\), as shown in Equation (4), maintaining the distribution without losing the transferred mass. In the 2D example shown in Figure 3, the weight of the top-center cell is transferred to the non-void cells below. Since one cell is in the movement zone (bottom-right cell), it will receive less weight.