## 3 Simulation Results and Discussion
### Influence Analysis and Scheme Optimization  
Based on the computation results in Table 4, the sensitivity analysis was carried out by the range analysis method. The influences of the stone columns' design parameters on the peak values of the ground acceleration and the EXPPR in the surrounding soils were \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**PGA/(m\(\cdot\)s\({}^{-2}\))**} & \multicolumn{4}{c}{**EXPPR**} \\ \cline{3-5}  & & **2 m** & **7 m** & **12 m** \\ \hline FF-0 & 1.55 & 0.91 & 0.96 & 1.00 \\ SC-1 & 1.86 & 0.88 & 0.90 & 0.84 \\ SC-2 & 2.64 & 0.32 & 0.34 & 0.32 \\ SC-3 & 2.91 & 0.17 & 0.19 & 0.17 \\ SC-4 & 2.80 & 0.23 & 0.26 & 0.24 \\ SC-5 & 1.96 & 0.83 & 0.85 & 0.78 \\ SC-6 & 2.75 & 0.27 & 0.30 & 0.28 \\ SC-7 & 2.53 & 0.39 & 0.42 & 0.39 \\ SC-8 & 2.84 & 0.19 & 0.22 & 0.20 \\ SC-9 & 2.07 & 0.78 & 0.80 & 0.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Peak values of ground acceleration and EXPPR of simulation models. Figure 4: Time histories of excess pore pressure ratio. evaluated. Three mean values corresponding to three levels were calculated firstly for each factor on the basis of a particular set of computation results. The set is defined to be the group of results calculated from the schemes that have the same level for a certain factor, e.g., the three mean values for diameter corresponding to levels 1-3 are calculated from the computation results of SC-1/2/3, SC-4/5/6, SC-7/8/9, respectively. Then, the maximum difference (MD) of mean values for each factor is obtained, which could reflect the impact of the factor on the target index. Table 5 is the range analysis table of ground acceleration. It can be concluded that the influence of the permeability is the most significant, followed by the stone column's replacement ratio, while the diameter and shear stiffness of the stone column have little effect on the ground acceleration. When the permeability ratio of the stone columns rises from 1 cm/s (L1) to 5 cm/s (L2) and 10 cm/s (L3), the corresponding PGA goes up significantly, from 1.96 m/s\({}^{2}\) to 2.64 m/s\({}^{2}\) and 2.86 m/s\({}^{2}\), respectively. When the replacement ratio increased from 10.1% (L1) to 12.9% (L2) and 15.7% (L3), the PGA climbed up slightly from 2.40 m/s\({}^{2}\) to 2.48 m/s\({}^{2}\) and 2.58 m/s\({}^{2}\), respectively. The sensitivity of the EXPPR to various factors at different depths of 2 m, 7 m, and 12 m presents the same characteristics, for which only the range analysis results at the depth of 7 m are displayed hereby (see Table 6). The table indicates that the EXPPR is the most sensitive to the permeability ratio, which is followed by the replacement ratio, and the increase of the column diameter and shear stiffness can hardly reduce the peak value of EXPPR. When the permeability ratio of the stone columns rises from 1 cm/s (L1) to 5 cm/s (L2) and 10 cm/s (L3), the corresponding mean values of peaks of EXPPR drop from 0.85 to 0.35 and 0.22, respectively. A conspicuous decline in the EXPPR can be seen when the permeability ratio increases from 1 cm/s (L1) to 5 cm/s (L2), whereas a relatively small decline appears when permeability ratio continues increasing from of 5 cm/s (L2) to 10 cm/s (L3). With regard to the impact of the replacement ratio, when it increases from 10.1% (L1) to 12.9% (L2) and 15.7% (L3), the mean values of peaks of EXPPR decrease slightly from 0.53 to 0.47 and 0.43, respectively. The design of stone columns is optimized to achieve the objective of enhancing the mitigation effect while reducing the economic cost. Therefore, the EXPPR, which indicates the risk of liquefaction, is firstly taken as the major index. From the range analysis above, it can be proved that the permeability ratio of the stone columns is the most sensitive factor for the EXPPR of the site. When the permeability ratio of the stone columns ascended from 1 cm/s to 5 cm/s, the peak value of the EXPPR in the surrounding soils descended from 0.85 to 0.35, and the liquefaction risk has been greatly reduced.