## 2 Modeling Methodology
The interface model was applied to all contact surfaces of the foundation structure, as illustrated in Figure 3, and consisted of five key interaction zones: (1) the lateral interface between the pile cap and the uppermost soil layer, (2) the interface between the pile cap base and the second soil layer, (3) the interface between the piles and the second and third soil layers, and (4) the interface between the piles and the weathered rock. Each interface area is marked with a different color in Figure 3. The actual field soil conditions consisted of four different strata, but in this study, a single stratum was simulated for parametric study according to the difference in soil relative density, so the material properties of Interfaces 1, 2, 4, and 5 were entered identically. The layout for the interface model is depicted in Figure 3, and the primary input parameters for the interface model included normal and shear stiffness as well as the internal friction angle, as summarized in Table 3.  
Figure 3: Layout and input properties for interface model. Another critical aspect in the dynamic analysis of numerical models with a limited number of elements is the boundary condition. To mitigate errors arising from wave reflections at the model boundaries, a quiet boundary condition was applied to both lateral faces of the model along the \(x\)-axis. This approach ensures that seismic waves propagating towards the boundaries are absorbed rather than reflected, effectively simulating the response of a semi-infinite soil domain. The input seismic motion was applied to the model base as a velocity-time history in the x-direction, while responses in the y-direction were constrained. Figure 4 depicts the composition of model boundary conditions. The dynamic analysis procedures for the developed three-dimensional continuum model are outlined as follows. Initially, a geo-static equilibrium analysis was performed to verify the numerical stability including mesh convergence and static equilibrium of the model. Subsequently, dynamic analysis was conducted. The seismic loading applied at the model base propagated through the soil-pile-structure system, ultimately reaching the pile cap. The dynamic response was evaluated at the upper-leftmost point of the pile cap, as depicted in Figure 5. The extent and nature of soil-pile-structure interaction (SPSI) can significantly influence the superstructure response. While extensive research has been conducted on the effects of inertial and kinematic interactions within SPSI systems, most studies have focused on either one of these effects or have primarily examined pile responses (e.g., acceleration, displacement, and internal forces such as bending moments) rather than superstructure behavior. However, the ultimate objective of seismic design is to ensure serviceability of the superstructure. In pile foundation systems, the superstructure response is primarily governed by the pile cap response, which subsequently serves as the input load for structural analysis. Accordingly, this study focuses on investigating how inertial and kinematic interactions in soil-pile-structure systems differ between single piles and pile groups, and how these interactions influence the seismic response of the superstructure. Figure 4: Composition of model boundary conditions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **Properties** & **Interface 1** & **Interface 2** & **Interface 3** & **Interface 4** & **Interface 5** \\ \hline \multirow{3}{*}{Dr = 30\%} & friction angle (\({}^{\circ}\)) & 26 & 26 & 30 & 26 & 26 \\  & Stiffness\_n (MPa) & 1200 & 1200 & 6000 & 1200 & 1200 \\  & Stiffness\_s (MPa) & 1200 & 1200 & 6000 & 1200 & 1200 \\ \hline \multirow{3}{*}{Dr = 80\%} & friction angle (\({}^{\circ}\)) & 40 & 40 & 30 & 40 & 40 \\  & Stiffness\_n (MPa) & 3600 & 3600 & 6000 & 3600 & 3600 \\ \cline{1-1}  & Stiffness\_s (MPa) & 3600 & 3600 & 6000 & 3600 & 3600 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input properties of the interface model.