## 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  
An alternative variant of the FE model is proposed to produce results of higher numerical quality. The model introduces a specified, relatively small number of random Young's moduli, assigned to smaller subsets of adjacent elements of the entire soil block. First, a subdivision of the analysed subsoil block of \(n_{r}=1000\) elements into eight equal sub-blocks was assumed (\(n_{E}=8\)), as presented in Figure 7.  
As a result, each sub-block consists of \(5\times 5\times 5=125\) elements. For each sub-block, a distinct random Young's modulus \(E\) is independently generated; in turn, the FE model is described by \(1000/125=8\) different Young's moduli. The analysis is limited to 200 FE models only. Satisfactory convergence of the solution is reached under the conditions of \(\Delta_{\mu}\), and \(\Delta_{\sigma}\) is set in the random field case (Figure 8). Figure 8: Convergence of simulation-based settlement estimation; the Monte Carlo FE material model, \(n_{E}=8\). Figure 7: The subsoil FE model; 8 distinct sub-blocks of \(E\) modulus, \(n_{E}=8\). Mean value and the standard deviation of the soil settlement are \(\mu_{\delta_{8}}^{200}=0.1079\) m and \(\sigma_{\delta_{8}}^{200}=0.0036\) m, respectively. An extreme displacement \(\delta_{8}^{200}=0.1233\) m was reached. Similar to the homogeneous subsoil block, the unjustified central node displacement peaks may also appear, following a large number of samples. However, contrary to the single variable model, the peaks occur only when small Young's moduli are generated for the majority of eight sub-blocks, and the probability of such a generation seems relatively low. To further refine the analysis, another subdivision of the analysed subsoil block is introduced. Here, the 1000-element block is divided into 125 equal sub-blocks (Figure 9), each consisting of \(2\times 2\times 2=8\) elements. This means an input of \(1000/8=125\) different Young's moduli into the global FE model (\(n_{E}=125\)). At first, based on 200 FE model samples, mean value \(\mu_{\delta_{125}}^{200}=0.1083\) m and standard deviation \(\sigma_{\delta_{125}}^{200}=0.0024\) m were estimated. An extreme displacement \(\delta_{125}^{200}=0.1165\) m occurs. The solution presented in Figure 10 features a small scatter in individual values of central node displacements. Although convergence holds for 200 MC samples, under the previous conditions, a broader scale analysis of 10,000 FE models is performed. The results and convergence of the crucial parameters of the random response are shown in Figure 11.  
Figure 10: Convergence of simulation-based settlement estimation; the Monte Carlo FE material model; 200 samples, \(n_{E}=125\). Figure 9: The subsoil FE model; 125 distinct sub-blocks of \(E\) modulus, \(n_{E}=125\). According to prior predictions, solutions arrive in the 10,000-sample computations with relatively large settlements (\(\delta^{10000}_{125}=0.1539\) m). Contrary to the single variable 10,000 FE model analysis (Figure 6b), the peaks do not disturb solution convergence, the mean value here \(\mu^{10000}_{i_{125}}=0.1083\) m is identical to the 200 FE sample case (\(\mu^{200}_{i_{025}}=0.1083\) m), and the standard deviation \(\sigma^{10000}_{i_{125}}=0.0029\) m differs slightly from the \(\sigma^{200}_{i_{125}}=0.0024\) m result. Finally, an FE model is assembled, where each of 1000 elements is given a unique, random Young's modulus (\(n_{E}=1000\)). Although a white noise-like modulus distribution does not properly characterize the soil medium (opposite to the homogeneous description), the calculations are performed to check if the FE model subdivision into smaller sub-volumes affects its computed response. A series of 200 samples is analysed, and the mean value \(\mu^{200}_{i_{1000}}=0.1075\) m and standard deviation \(\sigma^{200}_{i_{1000}}=0.0005\) m are obtained.