## 4 Numerical Study on Seismic Response of FSPM  
In this section, the time-stepping explicit algorithm derived above is applied to study the dynamic response of the FSPM in a rectangular domain under seismic loadings. The transmitting artificial boundary [17] is applied to simulate the wave propagation toward the far field infinite domain. The calculation model is shown in Figure 4. The dimension of the calculating domain is 80 m by 80 m. The top surface of the calculation model is set as the free and drainage boundary, and the left, right and bottom boundaries of the calculation model are all set as the transmitting artificial boundary. The four-node rectangular element is used for the finite element discretization of the calculation domain. Although high computational accuracy can be gained using a relatively small mesh size, this will lead to an increase of the number of finite element meshes. The computational effort and computer memory required will be enormous. From the associated consideration for the computational accuracy and effort, a finite element mesh size of 8 m by 8 m is adopted for the numerical study in this section. The strong motion record from the Loma Prieta earthquake is used as seismic excitation, input perpendicularly from the bottom boundary of the finite element model as a compressive wave and shear wave, respectively. The duration of the Loma Prieta earthquake record is 48.7 s; the displacement and velocity time history of the strong motion record are shown in Figures 5 and 6. The material properties of the FSPM are shown in Table 1. The time step \(\Delta t\) used for the calculation is 0.0001 s.  
Figure 3: Comparison of the numerical results with the analytical solution.  
Figure 2: Time history of the applied load.  
Figure 4: Calculation model for the seismic response of the FSPM.  
Figure 5: Displacement time history of the Loma Prieta earthquake record.  
Figure 6: Velocity time history of the Loma Prieta earthquake record.  
Figure 11: Solid skeleton velocity time history of the free surface in the vertical direction (compressive wave input).  
Figure 10: Pore fluid pressure time history of the central depth of the calculating domain (shear wave input).  
Figure 9: Solid skeleton displacement time history of the free surface in the horizontal direction (shear wave input).  
It is shown in Figures 7 and 9 that, when the seismic excitation is input perpendicularly from the bottom boundary of calculating domain, the profiles of the solid skeleton displacement time history of the free surface of the calculating domain are in accordance with that of the displacement time history of the incident seismic motion, and the peak displacement of the free surface has a roughly two-fold increase relative to the peak displacement of the incident seismic record. A similar trend can be observed for the solid skeleton velocity time history of the free surface of the calculating domain from Figures 11 and 12. The numerical results presented above are in accordance with the elastic wave theories [20]. This indicates that the time-stepping explicit algorithm proposed in the current study is applicable and effective for the numerical solution of the dynamic problems of the FSPM based on the \(\mathbf{u}\)-\(\mathbf{p}\) dynamic formulation.