## 6 Validation of Method and Comparison of Computational Efficiency  
**Example 1**. : _The computational model for a one-dimensional (1D) semi-infinite domain of saturated porous media is shown in Figure 3. The left and right sides of the model are impermeable boundaries, and their horizontal displacement is fixed. The bottom of the model is assigned a viscoelastic artificial boundary, and the top surface of the model is a free drainage boundary applied with a constant uniform load of 1.0 Pa. The material parameters of the model listed in Table 1[21; 48] as Material No. 1 are taken from the reference [21; 48]. The analytical solutions, as the reference solutions, for the 1D problem of saturated porous media proposed by [PERSON] et al. [21]._  
In Figure 4, the numerical solutions and analytical solutions are compared at three dimensionless time parameter \(\tau\) = 20, 40, and 60, respectively. The numerical solutions of the displacement and pore pressure of the 1D soil column are in relatively good agreement with the analytical solutions. In addition, the numerical and analytical solutions of the displacement and pore pressure dynamic time-history responses at nodes 5 m and 45 m from the center of the load are compared in Figure 4, which shows consistency between the two results. The comparison results verify the accuracy of proposed method. Figure 3: Numerical model for 1D saturated porous media. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Basic Material Parameters**} \\ \hline
**Parameter** & **Value** \\ \hline \(E_{S}\) & 3000 Pa & [PERSON]’s modulus \\ \(\rho\) & 0.306 kg/m\({}^{3}\) & Density of two-phase media \\ \(\rho\)/ & 0.2977 kg/m\({}^{3}\) & Density of fluid-phase \\ \(n\) & 0.333 & Porosity \\ \(v\) & 0.2 & Poisson’s ratio \\ \(\overline{k}\) & 0.004883 m\({}^{3}\)s/kg & Dynamic permeability coefficient \\ \(\lambda\) & 833.3 Pa & Lame’s constants of the soil \\ \(G\) & 1250 Pa & Shear modulus \\ \hline \multicolumn{3}{c}{Variable material parameters considered in the example} \\ \hline Parameter Parameters & No. 1 & No. 2 \\ \hline \(K_{f}\) & 0.3999 \(\times\) 10\({}^{5}\) Pa & 0.6106 \(\times\) 10\({}^{5}\) Pa & Bulk modulus of pore fluid \\ \(K_{s}\) & \(\infty\) & 0.5005 \(\times\) 10\({}^{4}\) Pa & Bulk modulus of soil skeleton \\ \(Q_{b}\) & 0.1201 \(\times\) 10\({}^{6}\) Pa & 0.1385 \(\times\) 10\({}^{5}\) Pa & Compressibility coefficient of \\  & & & pore fluid \\ \hline \multicolumn{3}{c}{Wave propagation velocity in saturated soil} \\ \hline \multirow{2}{*}{Parameter} & Actual wave velocity \\ \cline{2-2}  & No. 1 & No. 2 \\ \hline \(Cp\) & 635.12 m/s & 176.15 m/s & \(P\) wave velocity \\ \(Cs\) & 63.92 m/s & 89.69 m/s & S wave velocity \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of linear elastic saturated soil. Figure 4: Comparison of numerical and analytical solutions for the 1D model of saturated porous media. **Example 2**. : _Computational efficiency is compared in this example. A 2D computational model with size of 50 m \(\times\) 50 m is shown in Figure 5. The top surface of the model is set as a drainage boundary and is subjected to a uniformly distributed sinusoidal load, and each of the other three sides are set as artificial boundaries. Three target nodes (1, 2, and 3) are selected for analysis. The material properties are listed in Table 1 as Material No. 2. [PERSON]'s method and Newmark method are used to compare the responses and computational efficiency with the proposed method, respectively._  
Figure 6 shows the displacement and pore pressure time-history curves of the target nodes. The red and blue curves in the figure represent the results obtained from the proposed method and the [PERSON]'s method [40], respectively. The results of the two methods have a good consistence. The consistency further validates the proposed method.