## 2 Problem Definition  
A literature review, regarding the topic of concern here, has pointed out that the interaction between a strip footing and voids displays interesting complexities. Several parameters affect the interaction of the foundation, the soil, and the voids/tunnels. These parameters are listed in Table 1, along with the papers where they were already accounted for. In Figure 1, the two-dimensional geometry of the specific problem considered in this work is depicted. Assuming the strip footing and the tunnel to extend infinitely in the out-of-plane direction, plane strain conditions are adopted in the analysis. The strip footing is assumed rigid, featuring a rough surface in contact with the heterogeneous soil. According to the proposed setting, the following dimensionless variables are adopted to parametrize the problem: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **[PERSON] et al.** & **[PERSON], [PERSON]** & **[PERSON]** (2015) & **[PERSON], [PERSON] and** & **[PERSON] et al.** (2019) \\  & **[(11)]** & **and [PERSON]** (2012) & **[PERSON]** (2018) & & **[PERSON]** (2018) & \\ \hline Internal friction angle & \(\surd\) & & & & \(\surd\) \\ Cohesion & \(\surd\) & \(\surd\) & & \(\surd\) & \(\surd\) \\ Depth of void & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ Eccentricity of void & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ Shape of void & & \(\surd\) & \(\surd\) & \(\surd\) & \\ Size of void & & \(\surd\) & \(\surd\) & \(\surd\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters investigated in recent research. * Horizontal seismic acceleration coefficient \(k_{h}\), which is equal to the ratio between the horizontal earthquake-induced ground acceleration and the gravity acceleration. * Ratio \(\alpha=W/B\) between the void width \(W\) and the foundation width \(B\). * Ratio \(\beta=H/B\) between the void height \(H\) and the foundation width \(B\). * Undrained shear strength \(c_{0}/\gamma B\) of the soil at the ground level, \(c_{0}\) being the cohesion of the soil, and \(\gamma\) the soil specific gravity. * Rate of change \(kB/c_{0}\) of the cohesion \(c\) with the depth, according to the law: \[c=c_{0}+kz\] (1) where \(c_{0}\) is, as stated above, the value of the cohesion at the free surface \(z=0\). For a homogeneous soil, \(k\) is null, whereas for heterogeneous soils, \(k\) is positive--additional details can be found in [41; 42]. * Internal friction angle \(\varphi\) of the soil in the drained state. * Depth \(D=Z/B\) of the void, \(Z\) being the burial depth of the upper side (roof) of the void. * Eccentricity \(S=X/B\) of the void, \(X\) being the horizontal distance of the center of the void from the centerline of the foundation. Accounting for all the parameters listed above, the (dimensionless) undrained seismic ultimate bearing capacity \(Q\) of a strip footing placed above the void is assumed to be:  
\[Q=\frac{q_{u}}{\gamma B}=f\bigg{(}\frac{c_{0}}{\gamma B},\frac{kB}{c_{0}}, \varphi,S,\ D,\ \alpha,\ \beta,k_{h}\bigg{)} \tag{2}\]  
To set the database for digging with the ML algorithms, FELA analyses have been carried out to determine \(Q\) using the Optum G2 software [43]. By exploiting mesh adaptivity, the code allows for the mesh to be progressively refined, namely reducing the characteristic size of the elements where plastic dissipation takes place. On its own, this mesh adaption allows for the bounds to be tightened on the ultimate bearing capacity \(Q\) generated by each analysis, as described below, without any (potentially not objective) user intervention. This approach is broadly used in geotechnical engineering to solve stability problems [11; 44] and ascertain the footing bearing capacity [18; 45; 46; 47; 48]. Through the FELA method, which is a mixture of the finite element and the classical limit analysis of plasticity, lower and upper bounds on the ultimate bearing capacity of the foundation can be obtained.