## 2 Simplified Model of MF  
The 3D view of an MF is illustrated in Figure 1. Compared to the traditional foundation, which is made of concrete and longitudinal and transversal rebars, the MF contains heavy cores with cubic shapes that are coated with soft material in addition to the concrete and rebars. It is assumed that the MF is subjected to dynamic excitations in \(X\) and \(Y\) directions. Therefore, it can be divided into \(N\) unit cells in \(X\), \(Y\), and \(N\)-layers in \(Z\) directions. Assuming that the model is continuous, isotropic, made by materials with perfectly plastic behaviour, small deformation and insignificant damping, the governing equation of an inhomogeneous solid is written as [35]:  
\[\rho(\mathbf{r})\frac{\partial^{2}\mathbf{u}}{\partial t^{2}}=\
abla\{[ \lambda(\mathbf{r})+2\mu(\mathbf{r})](\
abla.\mathbf{u})\}-\
abla\times[\mu( \mathbf{r})\
abla\times\mathbf{u}] \tag{1}\]  
where \(\rho\) is the mass density, \(\mathbf{u}=(u_{x},u_{y},u_{z})\) is the displacement vector, \(t\) is the time parameter, \(\mathbf{r}=(x,y,z)\) the coordinate vector, \(\lambda\) and \(\mu\) are [PERSON]'s constants and \(\
abla\) is the Laplace operator. Decoupling the out-of-plane modes from those of in-plane ones, Equation (1) is rewritten as [35]:  
\[\rho(\mathbf{r})\frac{\partial^{2}u_{j}}{\partial t^{2}}=\frac{\partial}{ \partial x_{j}}\left[\lambda(\mathbf{r})\frac{\partial u_{l}}{\partial x_{j} }\right]+\frac{\partial}{\partial x_{l}}\left[\mu(r)\left(\frac{\partial u_{ l}}{\partial x_{j}}+\frac{\partial u_{lj}}{\partial x_{l}}\right)\right] \tag{2}\]  
Figure 1: Three-dimensional view of the proposed MetaFoundation. where \(u_{j}=u_{j}\) (\(x\), \(y\)) is the in-plane displacement vector, \(j\), \(l=1\), \(2\) and \(\mathbf{r}=(x,y)\). Using the Floquet-Bloch theorem for the periodic configuration of unit cells in uniaxial direction, the solution of Equation (2) is given as:  
\[u(x,t)=e^{i(qx-\omega t)}u \tag{3}\]  
where \(q\) is the wave vector in reciprocal space, \(\omega\) is the circular frequency. The wave amplitude is shown by \(u\), which is a periodic function and is expressed as:  
\[u(x)=u(x+A) \tag{4}\]  
where \(A\) is the unit cell size (shown in Figure 1). The stop band can be calculated through the dispersion analysis of the periodic structure. Two different approaches can be employed to conduct the dispersion analysis and obtain the stop band. The first approach is the 3D finite element modelling of the MF, which requires potential workload and high computational effort due to the complexity in modelling. The second approach is the use of the simple model of MF, which is represented by a set of masses and springs. Since the unit cells of each layer operate in parallel under seismic excitation, they can be represented by an equivalent unit cell through dynamic condensation. Figure 2 shows the simplified mass-spring model of an MF with \(N\)-layers in the \(Z\) direction. The total mass of the concrete in each layer is shown by the external mass (\(m_{1}\)), and the total mass of the metamaterials in each layer is shown by the internal mass (\(m_{2}\)). The stiffness corresponding to each layer of foundation and soft material is represented by springs whose stiffnesses are shown by \(k_{1}\) and \(k_{2}\), respectively. The stiffness corresponding to the soft material for each unit cell is obtained by:  
\[k_{2}=\frac{E_{SM}A_{SM}}{t_{SM}-(|u_{2}-u_{1}|)} \tag{5}\]  
where \(E_{SM}\) and \(t_{SM}\) are the modulus of elasticity and thickness of the soft material, respectively. The area of the soft material perpendicular to the applied wave direction is represented by \(A_{sm}\); \(u_{1}\) and \(u_{2}\) are displacements of external and internal masses. Note that, in this study, the stiffness corresponding to the soft material will be updated during the analysis based on the relative displacement of the internal and external masses. Therefore, more accurate seismic responses will be obtained.