## 2 Methodology
### Dynamic Equilibrium  
The dynamic equilibrium equations of the SSI are as follows [28; 29; 30]:  
\[\mathbf{M}\ddot{\mathbf{V}}+\dot{\mathbf{C}}\dot{\mathbf{V}}+\mathbf{K}\mathbf{ V}=-\mathbf{M}^{\prime}\ddot{\mathbf{U}}_{\star}\] (1a) where \[\mathbf{M}= \begin{bmatrix}M_{\text{b}}&M_{\text{b}}&M_{\text{b}}\\ M_{\text{b}}&M_{\text{b}}+M_{\text{b}}&S_{0}+M_{\text{b}}\\ M_{\text{b}}&S_{0}+M_{\text{b}}&I_{0}+I_{\text{b}}\end{bmatrix} ,\,\mathbf{C}= \begin{bmatrix}c_{\text{b}}&0&0\\ 0&c_{\text{hh}}&c_{\text{hm}}\\ 0&c_{\text{hh}}&c_{\text{mm}}\end{bmatrix} ,\,\mathbf{K}= \begin{bmatrix}k_{ \text{b}}&0&0\\ 0&k_{\text{hh}}&k_{\text{hm}}\\ 0&k_{\text{mh}}&k_{\text{mm}}\end{bmatrix} ,\,\mathbf{M}= \begin{bmatrix}M_{ \text{b}}&M_{\text{b}}\\ M_{\text{0}}+M_{\text{b}}&S_{0}+M_{\text{b}}\\ S_{0}+M_{\text{b}}&I_{0}+M_{\text{b}}\end{bmatrix} ,\,\mathbf{V}= \begin{cases}u_ {\text{b}}\\ u_{0}\\ \varphi_{0}H^{\prime}/a\end{cases}  \tag{1b}\]  
The symbol \(S_{0}\) is the area moment of the foundation with respect to the axis \(z=c\), \(I_{0}\) and \(I_{\text{b}}\) are the inertia moments of the foundation and the superstructure, respectively, and we have \(I_{\text{b}}=M_{\text{b}}H^{\prime}\) for an oscillator here, for simplicity. The parameter \(H^{\prime}=H+c\) is the structure height from the structure top to the foundation bottom. Vector \(\mathbf{U}_{\star}=\left\{u_{\star},\varphi_{\star}H\right\}^{\text{T}}\) is the foundation input motion, corresponding to the dynamic response of the excavation where the foundation is placed, and \(\mathbf{U}_{\text{0}}=\left\{u_{0},\varphi_{0}H\right\}^{\text{T}}\) is the relative displacement of the foundation with respect to the foundation input motion [28; 29; 30], so we have \(\mathbf{U}=\mathbf{U}_{\star}+\mathbf{U}_{\text{0}}\). The physical meaning of \(\mathbf{U}\), \(\mathbf{U}_{\star}\) and \(\mathbf{U}_{\text{0}}\) in SSI is illustrated in Figure 2.  
Foundation impedances are complex-valued and are shown by  
\[\mathbf{K_{0}}(\omega)= \begin{bmatrix}K_{\text{hh}}(\omega)&K_{\text{hm}}( \omega)\\ K_{\text{mh}}(\omega)&K_{\text{mm}}(\omega)\end{bmatrix} = \begin{bmatrix}k_{ \text{hh}}(\omega)&k_{\text{hm}}(\omega)\\ k_{\text{mh}}(\omega)&k_{\text{mm}}(\omega)\end{bmatrix} +\text{i}\omega \begin{bmatrix}c_{\text{hh}}(\omega)&c_{\text{hm}}(\omega)\\ c_{\text{mh}}(\omega)&c_{\text{mm}}(\omega)\end{bmatrix}  \tag{2}\]  
Figure 2: The foundation motion \(\mathbf{U}\), foundation input motion \(\mathbf{U}_{\star}\) and foundation relative motion \(\mathbf{U}_{\text{0}}\). The real and the imaginary parts act as the spring and dashpot coefficients of the site, with the elements \(K_{\rm hh}\) of horizontal impedance, \(K_{\rm hhm}=K_{\rm mh}\) of coupled horizontal-rocking impedance and \(K_{\rm mm}\) of rocking impedance. Foundation impedances are obtained using a high-precision indirect boundary element method (IBEM), combined with the non-singular Green's functions of distributed loads presented in [24]. The foundation impedances in Equation (1) involve both translation and rotation, and the elements are already normalized by \(H^{\prime}/a\) to be unified to the same units. The boundary element method (BEM) needs element meshing only at boundaries, and it does not need artificial boundaries like the Finite Element Method (FEM) because it uses Green's functions to automatically satisfy the radiation conditions at infinity. The calculation on Green's functions is carried out in the frequency domain by Fourier transformation, and its logic is similar to the \"displacement method\" in the _Structural Mechanics_ textbook. The Green's functions correspond to the displacement or traction responses of some point in the half-space, subjected to some unit-distributed load, then the foundation impedances are obtained by a series of fictitious distributed loads acting on the foundation-soil boundary. The value of the fictitious loads, as well as the foundation impedances, are all found by the continuous conditions on the boundary.