# Estimating Contact Force Chains Using Artificial Neural Network
## 2 Materials and Methods
### DEM Data  
The simulation was performed on a workstation with Intel Xeon Gold 6154 processors (3.0 GHz, Skylake-SP) with 18-core\(\times\)2-CPU and 192 GB of main memory. The computational efficiency mainly depends on the number of grains and the size of stored data. The computation time used for a complete simulation is about 4 days. Figure 2a-d show the evolution of particle displacement during the shearing process. We can see that, with the sketch of a simplified protractor on Figure 2d, the sample exhibits failure along a shear band oriented at 131\({}^{\circ}\) (rotating counter-clockwise) from the horizontal. The typical curves of deviatoric stress vs. axial strain and volumetric strain vs. axial strain are shown in Figure 2e, f, respectively. The peak deviatoric stress is about 1.2 MPa under the confining pressure of 300 kPa. Figure 3a-d show the evolution of contact networks within the sample at the axial strains of 0%, 5%, 10%, and 15%, respectively. It is seen that the distribution of contact networks is uniform and isotropic at the initial state. As shearing progresses, the contact force distribution shifts slightly from the isotropic state to the vertical-prevailing state. Consequently, the distribution frequency undergoes a significant loss in the horizontal direction and a distinct gain in the vertical direction. For the ease of quantifying the heterogeneous distribution of contact force networks, all the data were discretized into a set of equally spaced bins and characterized by an angular histogram, as shown in Figure 3e-h. It is seen that the evolution of the frequency of contact forces in each bin during the whole shearing test is clearly visualized. \begin{table}
\begin{tabular}{c c} \hline \hline
**Sample Parameters** & \\ \hline Size: height \(\times\) width, mm & 300 \(\times\) 150 \\ No. of spheres & 6300 \\ Friction coefficient & 0.3 \\ Contact model & Linear rolling resistance model \\ Elastic modulus, Pa & 3.5 \(\times\) 10\({}^{7}\) \\ Rolling resistance coefficient & 0.35 \\ Density, kg/m\({}^{3}\) & 2650 \\ Wall friction coefficient & 0 \\ Damping coefficient & 0.7 \\ \hline
**Membrane Parameters** & \\ \hline Number of balls & 400 \\ Friction coefficient & 0 \\ Normal stiffness, N/m & 1 \(\times\) 10\({}^{6}\) \\ Shear stiffness, N/m & 1 \(\times\) 10\({}^{6}\) \\ Tensile strength, N & 1 \(\times\) 10\({}^{200}\) \\ Shear strength, N & 1 \(\times\) 10\({}^{200}\) \\ Density, kg/m\({}^{3}\) & 1000 \\ Ball size, mm & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Biaxial sample properties. Based on this statistical analysis, an ANN model was constructed in each bin to identify the functional relationship between the contact force and the five particle-scale features. Before training the ANN model, the input and output datasets need some preprocessing to reduce the computational cost. The detailed procedure, using the contact force as an example, is described as follows. The same procedure applies to the processing of the five particle-scale features. For each record, firstly, all the contact force orientations are discretized into 180 equally spaced bins and characterized by an angular histogram. Therefore, for a given contact, its contact force and the average particle size, velocity and spin of the two associated particles are grouped accordingly. Secondly, the values of all the contact forces within the same bin are summed up. Lastly, the summed contact force value in each bin is normalized with respect to the maximum value of the summed contact force among all the bins so that they all range from 0 to 1. The above steps are performed iteratively until all the records are processed. Figure 4a compares the typical evolutions of the contact force and the coordination number within the bin of [79\({}^{\circ}\), 80\({}^{\circ}\)] throughout the shearing process. It can be seen that there is a strong positive relationship between them during the shearing process. Figure 4b shows the evolutions of other particle-scale features.