## 3 Numerical Model  
The numerical model used in this study is the FLAC program with the PM4 Sand model. FLAC is an explicit, finite difference program that performs Lagrangian analysis to calculate equilibrium equations and stress-strain relations. In the finite difference method (FDM), derivatives of governing equations of motion and constitutive relations are represented using physical variables (e.g., velocity and displacement) at discrete points (grid points in FLAC). The time\(-\)marching method is used in FLAC to solve these governing equations. The Lagrangian analysis updates the coordinates of the grid at each time step in the large strain mode of FLAC, which is most suitable for modeling the large deformations of soil induced by soil liquefaction. The numerical model was validated and calibrated using the results of the centrifuge test studying the uplift of the underground structure [1]. The layout and monitored points of the numerical model are shown in Figure 3. Three centrifuge tests (listed in Table 2) were selected for numerical model verification and PM4 Sand input parameter calibration. All tests consisted of one layer of Toyoura sand with different relative densities (_Dr_) and were subjected to a sinusoidal wave (20 cycles and 1.2 Hz). Time histories of six instruments (_DV1_, _P2_, PM2_, _PU1_, _A0_, _A2_, and _AM2_, shown in Figure 3) in the centrifuge test CASE 98\(-\)01 were selected for comparison. The numerical model (shown in Figure 4) was composed of four sizes of FDM grids. The 0.5 m \(\times\) 0.25 m and 0.5 m \(\times\) 0.5 m (width \begin{table}
\begin{tabular}{l l} \hline \hline
**Input Parameter** & **Description** \\ \hline Shear modulus coefficient (\(G_{o}\)) & \(G_{o}\) is used to estimate the small strain shear modulus. \\ Contraction rate parameter (\(l_{p\alpha}\)) & \(h_{p\alpha}\) controls the contraction rate of the plastic volumetric strain related to the cyclic resistance ratio (\(CRR\)) of soil. \\  & \(h_{o}\) relates to relative density (_Dr_) and controls the ratio of \\ Ratio of plastic modulus and elastic modulus (\(h_{o}\)) & plastic modulus to elastic modulus. Model default value \\  & provides reasonable modulus reduction and damping \\ Material constant of bounding stress & \(n^{b}\) controls the bounding surface (dilatancy), which is \\ ratio (\(n^{h}\)) & related to the peak effective friction angles. \\ Material constant of dilatancy stress & \(n^{d}\) controls the phase transformation where the \\ ratio (\(n^{d}\)) & contraction behavior transitions to the dilation behavior. \\ Maximum value of fabric–dilatancy & \(z_{max}\) is the maximum value of fabric–dilatancy tensor. \\ tensor (\(z_{max}\)) & \(z_{max}\) and \(A_{do}\) are used to control the buildup value of the \\ Material constant of dilatancy (\(A_{do}\)) & porewater pressure upon reversal of stress increment \\  & following a dilative tendency response. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key input parameters of the PM4 Sand model. Figure 2: Responses of cyclic undrained simple shear test of PM4 Sand model: (**a**) shear stress and shear strain and (**b**) stress path. height) grids were placed around the underground structure to accurately capture the soil movement. The 1.0 m \(\times\) 0.25 m and 1.0 m \(\times\) 0.5 m grids were placed at areas away from the underground structure (1.5 times the width of the underground structure = 7.5 m) to lower the number of FDM grids and speed up the FLAC calculation. In this study, Toyoura sand was modeled using the PM4 Sand model, and the underground structure was modelled as a rigid body using the elastic model with elastic moduli about 10 times the elastic moduli of Toyoura sand. After evaluating the purposes of the key inputs of the PM4 Sand model (listed in Table 1), three inputs were selected and adjusted to fit the centrifuge test results in this study: contraction rate parameter (\(h_{po}\)), ratio of plastic modulus to elastic modulus (\(h_{o}\)),