## 7 Analysis of Nonlinear Problems
### Numerical Results and Analysis  
The seismic responses of the saturated porous media are analyzed by the proposed method and the implicit Newmark method, respectively. Figure 10 shows the time history curves of horizontal and vertical displacements at measuring points D1-D4. The horizontal displacement and vertical displacement at each measuring point gradually increase with time and gradually decrease with burial depth, and reach the maximum of 0.1 m and 0.12 m, respectively, at the top of the embankment. In addition, the time history curves of both horizontal and vertical displacement show the stepwise increase in amplitude, respectively, at 6.2 s and 16.6 s.  
Figure 11 shows the excess pore pressure time history curves at measuring points W1-W4. These curves exhibit basically the same variation trends, and their peaks occur at basically consistent time points. The amplitude of pore pressure at each measuring point gradually increases with the increase in the burial depth. The excess pore pressure of the dense sand layer reaches a maximum value of 75 kPa.  
Moreover, Figures 10 and 11 compare the results of the proposed method and the implicit Newmark method. The comparison shows that the numerical results of each measuring point obtained by the two methods agree well, indicating the correctness of the proposed method.  
To observe the overall lateral deformation and settlement of the site, Figure 12 shows the contour plots of horizontal displacement of the site at 6.2 s, 16.6 s, and 30.0 s, as well as the contour plots of vertical displacement at 30 s. The deformation in the figure is magnified 30 times. As can be seen from the figure, the horizontal displacement at the nodes on the left and right sides of the embankment is the largest; the embankment gradually collapses under the earthquake, causing the embankment soil to move toward both sides, with the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Loose Sand Layer** & **Dense Sand Layer** \\ \hline Shear contraction coefficient \(c_{2}\) & 5 & 5 \\ Dilatancy coefficient \(d_{2}\) & 3 & 3 \\ Liquefaction coefficient \(l_{1}\) & 1 kPa & 1 kPa \\ Liquefaction coefficient \(l_{2}\) & 0.0 & 0.0 \\ Initial void ratio \(e\) & 0.6 & 0.6 \\ _Cs 1_ & 0.9 & 0.9 \\ _Cs2_ & 0.02 & 0.02 \\ _Cs 3_ & 0.7 & 0.7 \\ Standard pressure & 101 & 101 \\ \hline \hline \end{tabular}
\end{table}
Table 3: _Cont._  
Figure 10: Time history curves of horizontal and vertical displacements at measuring points D1–D4.  
maximum horizontal and vertical displacements reaching 0.13 m and 0.19 m, respectively, at 30 s.  
Figure 13 shows the contour plots of pore pressure ratios of the ground soil. The loose sand layer near the drainage boundary liquefies at 6.2 s and 16.6 s. As the decrease in the input acceleration, the pore water is drained upwards, and the soil particles are redistributed. The soil tends to a stable state.  
Figure 11: Pore pressure time history curves of the model at measuring points W1–W4.  
Figure 12: Contour plots of horizontal and vertical displacements of the model.  
Figures 12 and 13 also compare the contour plots of displacement and pore pressure ratio computed by the proposed method and the implicit Newmark implicit method, respectively. The soil deformation patterns obtained by the two methods are similar, and the magnitudes of displacement and pore pressure computed by the two methods are basically consistent, further validating the correctness of the proposed method.