## 2 Random Spatial FE Models of the Subsoil
### The Monte Carlo FE Material Model
#### 2.2.2 The FE Models Described with Several Uncorrelated Random Variables
It is evident that the cases corresponding to the real-life soil conditions are included between these extremes. Since random field computations are difficult, they can be represented by accurate block division into sub-volumes of single Young's moduli. Note that the determined logarithmic approximations (Figures 5 and 13) can be related to the soil model presented in the paper only. However, it is supposed that the observed relation can be estimated in any other FE model of a geotechnical structure resting on random subsoil or the FE models of slopes of a complex multi-layered geological structure with uncertain parameters, e.g., prone to stability loss. The optimal number of singular variables to be randomly generated in a simple approach \(n_{E}\) can be devised. The data collected in Figure 13 lead to an assumption that the refined FE model with \(n_{E}=125\) variables may be regarded as the finest one. If \(n_{E}=8\) is assumed, the generated FE model becomes too similar to a model defined by a homogeneous field. The results show unjustified large jumps, so to reach the convergence of the MC solution, too many samples are required, discarding the analysis from real-life engineering problems. For \(n_{E}=125\), the results are practically independent of the number of FE models applied in the computational series, and the result scatter is realistic. It balances the complexity of model generation and output reliability well. Nevertheless, accepting \(n_{E}=125\) is arbitrary and strongly related to the analysed case. However, the diagram in Figure 13 shows that for such a value, the variability range of FE model parameters is small enough to produce engineering-sound solutions, decisive when investigating real-life problems. In a general case, for any FE model, the number of sub-blocks with the same material parameters \(n_{E}\) should be specified. The easiest way to define this number is based on the percentage of the number of sub-blocks \(p_{E}\) (\(0.0<p_{E}<1.0\)) to the total number of the finite elements in the model \(n_{el}\):  
\[n_{E}=p_{E}\cdot n_{el} \tag{3}\]  
For example, in the considered FE model of the block involving \(n_{el}=1000\) finite elements, the optimal number of different random values generated for the task should be \(n_{E}=0.125\cdot 1000=125\) (\(p_{E}=0.125\)). In turn, the entire volume is divided into 125 sub-blocks, each including \(1000/125=8\) finite elements. In the case of an irregular mesh of finite elements, specific random variables are linked with relevant volumes to represent their contribution to the entire modelled structure. Such computations, however, require additional, pre-generated external procedures assigning a certain number of [PERSON]'s moduli to precisely set sub-volumes, matching the FE model syntax. A regular mesh of finite elements, usually applicable in subsoil models, corresponds to the presented approach. Unfortunately, no clear distinction arises on the FE model definition, and the model adoption is intuitive. Despite this, such a simplified random variable approach is dedicated to advanced geotechnical engineering design to replace the three-dimensional random field analysis.