# Seismic Isolation via I-Shaped and T-Shaped Large-Scale Phononic Metamaterials
## 2 Materials and Methods  
The proposed structures consist of rectangular trenches in the ground surface that form I-shaped or T-shaped cavities. Figure 1 shows the unit cell of each structure together with its geometric parameters. Each cubic unit cell has edge \(\alpha=10\) m, which is the lattice constant of the structure. Figure 1a shows the I-shaped LSPM, and Figure 1b shows a top view of it with all relevant geometric parameters. The length of each I-shaped trench is symbolized as L, and its width is \(w\). The case where the I-shaped trench is filled with a coated heavy core is shown in Figure 1c. The rubber coating has thickness d. The relevant T-shaped LSPM is shown in Figure 1d,e. The horizontal part of the T-shaped structure has geometric parameters, which are symbolized as \(\rm{L_{T}}\) and \(\rm{w_{T}}\) for its length and width, respectively. The case where a heavy T-shaped core is filling the cavity is shown in Figure 1f. The width of the rubber coating is d. For better understanding of the materials used, the inset with the representation of each color is also included in Figure 1.  
For the calculation of the band structure of each proposed geometry, the Bloch theorem together with the finite element method, which is included in the commercial software package COMSOL(5.5) Multiphysics, were used. Bloch-Floquet periodic boundary conditions were considered at the vertical sides of each unit cell, whereas free boundary conditions were used at the top and bottom sides of each unit cell. Thus, the real eigenvalues of each structure were calculated along the high-symmetry directions, \(\rm{\Gamma-X-M-\Gamma}\), of the first Brillouin zone, as shown in Figure 1g. The calculated real eigenvalues together with the relevant values of the eigenvector \(\rm{\stackrel{{\rightarrow}}{{k}}}=\left(k_{x  
eigenvalues with the relevant values of the eigenvector are represented with black asterisks in the band structure diagrams which will be presented in the following sections. The discretization of the geometry, which was used for the calculations of the band structures, consisted of a customized four-node free tetrahedral standard mesh with a minimum element size of 0.1 m, a maximum element size of 1 m, and a maximum element growth rate of 1.45. This choice led to good convergency, providing an error estimate of less than \(10^{-7}\), as shown in the relevant convergence plots during the solution process, and thus provided accurate results, up to the maximum studied frequency in this study, which was 6 Hz. For a more realistic numerical approach to the effectiveness of the proposed structures, the transmission spectrum was calculated for the one that provided a better solution for surface wave attenuation. The calculation of the transmission was performed in a suitable three-dimensional half-space numerical domain, as shown in Figure 2. The size of the computational half-space was \(50\alpha\) along the x direction and \(25\alpha\) along the z direction. The thickness of the computational domain along the y direction was equal to the lattice constant \(\alpha\). At the vertical sides and the bottom of the half-space, perfectly matched layers (PMLs) were considered to avoid reflections from the boundaries. The PML layer in the half-space was selected at \(5\alpha\) to be comparable to half of the largest wavelength in the simulation. This choice eliminated the reflections from the boundaries as will be presented in the relevant mode shapes from the middle cross-section of the numerical three-dimensional half-space and provided reliable results with good agreement when compared to the relevant calculations for the band structure. At the top of the half-space free boundaries were adopted to mimic the real ground surface during the wave propagation. Continuity periodic boundary conditions were considered along the y direction. The top view of the numerical half-space is presented in Figure 2b.