# Seismic Isolation via I-Shaped and T-Shaped Large-Scale Phononic Metamaterials
## 2 Materials and Methods
The excitation (E) was considered as a vertical prescribed displacement on a line segment equal to the lattice constant \(\alpha\) at the left side of the surface of the half-space, whereas the LSPM was considered at the other side of the surface. The excitation at the surface of the half-space is expected to generate both surface and bulk waves, and therefore the examined LSPM was at such a distance from the source that the bulk waves could be considered negligible. Six repetitive unit cells constitute the examined LSPM, as shown in Figure 2b. For better understanding of the presented numerical scheme, Figure 2c shows, as an example, the unit cell of the I-shaped LSPM that was considered for this analysis. The numerical data of the components of the displacement field were collected at the detection point, which is symbolized as D.  
The transmission spectrum of the structure was calculated from the numerical data collected at the detection point D of the half-space, normalized to the relevant data at the same point of the half-space without the presence of the LSPM. The vertical component of the displacement field was considered for the calculation of the transmission spectrum, as it is well known that the vertical component of the surface wave causes most of the damage in a building. In more detail, the collected data in the frequency domain at point D of the vertical component of the displacement field, \(u_{z(LSPM)}\), under the presence of the LSPM, were divided by the relevant data at point D, \(u_{z(GS)}\), for the flat ground surface (GS), without the presence of the LSPM in the numerical domain. The transmission coefficient was calculated from the ratio \(u_{z(LSPM)}/u_{z(GS)}\) at each examined frequency. A second detection point was also considered next to point D, at a distance \(\alpha/2\), to ensure the validity of the calculations. The results from both detection points were almost the same, and therefore only the results from point D are presented. For the meshing of the structure according to the available computational resources, a maximum element mesh size of 2.5 m was used. This choice provided satisfactory results for the examined frequency range, which enlightened the effectiveness and potential of the studied structure, as will be analyzed in the following sections of the manuscript. The materials used in the calculations were commonly employed for such structures, as documented by the relevant literature; such materials are considered linear, homogeneous, and isotropic. A sandy type of soil was considered, which has [PERSON]'s modulus \(\mathrm{E}=20\) MPa, Poisson ratio \(\mathrm{v}=0.3\), and density \(\mathrm{\rho}=1800\) kg/m\({}^{3}\). For the cases where a steel core is considered, the relevant parameters for steel were [PERSON]'s modulus \(\mathrm{E}=207\) GPa, Poisson ratio \(\mathrm{v}=0.3\), and density \(\mathrm{\rho}=7784\) kg/m\({}^{3}\), and for the rubber coating, the parameters for rubber were [PERSON]'s modulus \(\mathrm{E}=120\) kPa, Poisson ratio \(\mathrm{v}=0.47\), and density \(\mathrm{\rho}=1300\) kg/m\({}^{3}\). The chosen materials for the trenches filled with a coated core are materials that are used in several kinds of construction. Steel is a material that is used very often in almost all kinds of buildings and other critical infrastructures, such as bridges, as it is known for its long-term durability. Furthermore, rubber is also known for its uses as a vibration absorber and has also been used a lot in the literature as a constitutive material and in real constructions. Sandy-type soil was selected as a well-known type of soil that can represent many realistic scenarios of places where a possible application of the studied metamaterials could also be applied. It should be noted at this point that a selection of a different type of soil, which would represent another type of ground material, is expected to affect only the PBGs' limits in a way similar to the wave velocity values of the selected type of soil.