*3.3. Particle Placement*

*η*

Next, the generated particles must be placed into the concrete volume. As mentioned, they are randomly distributed; however, their placement must satisfy some primary conditions. First, all particles must be located within the concrete volume, and secondly, none of them can overlap with each other. Moreover, the placement process should include that the distance *si*,*<sup>j</sup>* between the centers of adjacent particles must be greater than the sum of their radii because of the mortar film covering the aggregates (Figure 3): max min min 1.1 0.55

$$s\_{i,j} \geq \frac{d\_i + d\_j}{2} + \gamma\_{i,j} \tag{15}$$

**Figure 3.** Particle placement in concrete specimen.

In this study, the thickness of mortar film is equal to 5% of the sum of the diameters of adjacent particles [24]:

$$\gamma\_{i,j} = 0.05 \cdot \left( d\_i + d\_j \right) \tag{16}$$

Additionally, the distance between the particle and the specimen boundary must be at least equal to:

$$
\gamma\_i = 0.05 \cdot d\_i \tag{17}
$$

The above conditions imply that the coordinates of the mass center of the particle with diameter *d* located within the volume of the concrete plate with thickness *h* and length *l* must be generated in the following way:

$$\mathbf{x}\_{l} = \eta(\mathbf{x}\_{\text{max}} - \mathbf{x}\_{\text{min}}) + \mathbf{x}\_{\text{min}} = \eta(l - 1.1d\_{l}) + 0.55d\_{l} \tag{18}$$

$$y\_i = \eta (y\_{\text{max}} - y\_{\text{min}}) + y\_{\text{min}} = \eta (h - 1.1d\_l) + 0.55d\_l \tag{19}$$

where η is the uniformly distributed random number in the interval (0,1).

The algorithm of the particle placement can be summarized in three steps:

