## 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
The extreme displacement \(\delta^{200}_{1000}=0.1105\) m is reached, similar to the 200-sample calculation of the 125-variable case (\(\delta^{200}_{125}=0.1165\) m). The results and convergence of both random parameters are illustrated in Figure 12.  
Figure 11: Convergence of simulation-based settlement estimation; the Monte Carlo FE material model; 10,000 samples, \(n_{E}=125\). Figure 12: Convergence of simulation-based settlement estimation; the Monte Carlo FE material model; 200 samples, \(n_{E}=1000\). Summing up, all essential results of the considered computational cases ([PERSON]'s moduli modelled by sets of random variates) are presented in Table 2. With regard to FE models, the mean settlement is not affected by [PERSON]'s modulus variation, but the standard deviation of the central node settlement decreases with the random variable domain rise. The extreme settlements are also listed in Figure 8:0 It should be noted that if different sample sets are computed, the presented output is about to vary. This is due to extreme deflections since a larger number of analysed samples increases the probability of unrealistic settlement peaks occurrence. Similar to random field modelling, an approximation of settlement standard deviation versus the number of the assumed random [PERSON]'s modulus variables is performed (Figure 13). An assumption is made on the horizontal axis--a cubic root of the number of variables \(c_{E}=\sqrt[3]{n_{E}}\) (\(0<c_{E}<10\)) is adopted, not the direct number \(n_{E}\) (\(0<n_{E}<1000\)). The logarithmic function is approximated \(\sigma_{\delta_{n_{E}}}=-0.00339796\ln(c_{E})+0.00783447\) (see Figure 13). Note that the diagram resembles the one in Figure 5 in the analysis of FE models with the random field modelling of [PERSON]'s modulus \(E\). The similarity between the diagrams in Figures 5 and 13 is not accidental. An apparent relationship is observable between the random field regions with similar stiffness moduli \(E\) defined by the correlation decay parameter \(d\) (Figure 3) and the number of sub-blocks \(n_{E}\) that divide the entire FE model volume according to a given mode (Figure 9). Although it is impossible to analytically define these relationships, they are visible in terms of extreme case analysis. When defining the total block volume by a single random value (\(n_{E}=1\)), ideal homogenization of the material emerges, and it resembles the random field with infinitely strong correlation, \(d\to 0\). Both simulations imply a large result scatter and significant standard deviation of the investigated settlement. While the number of sub \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**200 FE Models Incorporated**} & \multicolumn{3}{c}{**10,000 FE Models Incorporated**} \\ \cline{2-7} \multirow{2}{*}{\(n_{E}\)} & **Mean** & **Standard** & **Extreme** & **Mean** & **Standard** & **Extreme** \\  & **Value** & **Deviation** & **Settlement** & **Value** & **Deviation** & **Settlement** \\  & \(\mu_{\delta_{n_{E}}}\)**[m]** & \(\sigma_{\delta_{n_{E}}}\)**[m]** & \(\beta_{n_{E}}^{ext}\)**[m]** & \(\mu_{\delta_{n_{E}}}\)**[m]** & \(\sigma_{\delta_{n_{E}}}\)**[m]** & \(\beta_{n_{E}}^{ext}\)**[m]** \\ \hline
1 & 0.1087 & 0.0091 & 0.1525 & 0.1097 & 0.0174 & 1.1505 \\
8 & 0.1079 & 0.0036 & 0.1233 & \multicolumn{3}{c}{calculations not performed} \\
125 & 0.1083 & 0.0024 & 0.1165 & 0.1083 & 0.0029 & 0.1535 \\
1000 & 0.1075 & 0.0005 & 0.1105 & \multicolumn{3}{c}{calculations not performed} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Soil block settlement parameters, the 200-item and the 10,000-item simulation procedure, and the Monte Carlo FE material model. Figure 13: Soil block settlement dispersion versus the cubic root of the number of random variables \(c_{E}=\sqrt[3]{n_{E}}\). blocks is equal to the number of finite elements (\(n_{E}=1000\)), the FE model resembles the white noise random field of extremely weak correlation (\(d\gg 5\)).