# Pore Size Distribution in Granular Material Microstructure
## 2 Synthetic Specim Generation
### Generation Method  
Packings classified as \"very loose random\" according to Table 1 were obtained via the algorithm developed in the previous study by [PERSON] et al. [27]. This method is initiated by randomly generating initial \((x,y,z)\) coordinates for every sphere according to a specific distribution. This determines the initial behavior of every sphere (e.g., collision, rotation, or relative stability with the other spheres). In order to generate denser specimens, an algorithm was developed to closely pack hard spheres through a series of steps, which can be divided into four main stages: initialization of sphere coordinates, particle expansion/contraction, overlap reduction, and vibration. Commonly, initialization is done by assigning spheres random positions within the domain, often encompassing an excessive number of anomalously large overlaps. A suitable alternative is to utilize a random loose packing [27] as the initial input for spheres coordinates. By starting the algorithm with an overlap-free packing, several unnecessary overlap reduction and vibration steps are eliminated, improving the overall efficiency. Post-initialization, the packing is densified by imposing a small (2.5% of \(radius_{particle}\)) uniform particle expansion. The overlap reduction stage ensues, according to the procedure proposed by [PERSON] et al. [29]. In each iteration, particles are moved in the direction of an overlap-minimizing vector, and the maximum global overlap (MGO) is calculated. This process is comparable to the particle rearrangements produced in DEM simulations by repulsion forces at particle contacts. Nonetheless, the efficiency is much higher since geometrically updating particle positions dispenses the use of a Newtonian time integration scheme [44], necessary in dynamic simulations. Cycling of the overlap-reduction phase is interrupted under two circumstances: the MGO decreases below the intermediate tolerance \(T_{i}\), or the number of iterations surpasses the established limit \(i_{\text{max}}\). In the first case, spheres are enlarged, and the system is once again subjected to a stage of overlap reduction. Conversely, when the cycling limit is exceeded (\(iteration>i_{\text{max}}\)), a slight contraction is applied to all particles. The packing density is then determined within a virtual container. If the desired density state has been reached, the simulation is stopped and sphere coordinates are recorded. Otherwise, the global overlap tolerance \(T_{i}\), initially set at 0.01, is decreased to accelerate the process, and the specimen is submitted to the overlap reduction process again. The density is computed inside a virtual rectangular container [19], so as to exclude the zone influenced by boundary effects. This zone was shown to dissipate at a distance no larger than 4 \(radius_{particle}\) from the walls [16]. Due to the quasi-deterministic nature of the overlap reduction process, particles can become jammed, thereby hindering convergence. Hence, a vibration stage was introduced post-particle expansion. This step, which was originally implemented after [PERSON] and [PERSON] [31], consists of moving particles in the direction of a random vector. The resulting perturbations promote local rearrangement that releases trapped particles and advances the packing towards its target state. The combination of these steps, allowed tangible improvement of the computational cost. A complete flowchart of the algorithm is shown in Figure 1.  
Most available geometric packing methods rely on periodic boundary conditions to avoid handling particle-wall interactions, thus restricting simulations to only producing cuboid specimens [45; 46; 47; 48]. The present study addresses this impediment by developing a particle-wall interaction scheme (Figure 2), which enables specimens to be generated within cylindrical containers with (\(\frac{radius_{\text{equatime}}}{radius_{\text{particle}}}\geq 10\)).