# Seismic Isolation via I-Shaped and T-Shaped Large-Scale Phononic Metamaterials
## 3 Results
### Transmission Calculations  
The calculation of the transmission spectrum was followed to have a complete numerical approach and to validate the results from the band structure calculations. Due to its high computational cost, there has been a selection from the studied structures to check for its seismic isolation effectiveness in combination with the relevant flat surface. From the presented results of the band structures, the I-shaped LSPM, where \(\rm L=9.0\) m and \(\rm w=4.0\) m, was the choice for this part. For this case, the two calculated PBGs were found within the lowest frequencies, between 1.68 and 3.10 Hz and between 3.26 and 3.74 Hz, and the first gap had the highest calculated width \(\Delta f/f_{g}=0.59\). The calculated transmission spectrum for this case is shown in Figure 7a. The results are in very good \begin{table}
\begin{tabular}{c c c c} \hline \hline
**LSPM** & **Varying Parameter** & **PBG Limits** & **PBG Width** \\ \hline \multirow{3}{*}{T-shaped cavity (L = 9 m, \(\rm w=2.5\) m, \(\rm w=6.5\) m)} & \multirow{3}{*}{1.0} & 2.42–3.07 Hz & 0.24 \\  & & 3.39–4.02 Hz & 0.17 \\ \cline{1-1} \cline{2-4}  & & 2.5 & 2.50–3.06 Hz & 0.20 \\ \cline{1-1} \cline{2-4}  & & 3.35–3.94 Hz & 0.16 \\ \cline{1-1} \cline{2-4}  & & 4.0 & 2.63–2.97 Hz & 0.12 \\ \cline{1-1} \cline{2-4} T-shaped cavity with & 2.5 & 3.09–3.46 Hz & 0.11 \\ \cline{1-1} \cline{2-4} coated steel core (L = 9 m, \(\rm w=2.5\) m, \(\rm w_{T}=6.5\) m) & 4.0 & 1.73–2.16 Hz & 0.22 \\ \cline{1-1} \cline{2-4}  & & 3.23–3.42 Hz & 0.06 \\ \cline{1-1} \cline{2-4}  & & 3.71–4.07 Hz & 0.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Collected results for the case of T-shaped LSPMs and the relevant with the coated steel core. Figure 6: The band structure of the T-shaped LSPM with coated steel core for the case where \(\rm L_{T}=4.0\) m (**top**) and the relevant mode shapes of the normalized displacement field for the lowest (\(\rm f_{\rm i}\)) and highest (\(\rm f_{\rm h}\)) frequencies of the first PBG (**bottom**), which are indicated with the blue asterisks in the band structure. agreement with the corresponding results for the band structure of Figure 3. A slight shift to higher frequencies must be attributed to the different mesh size that was selected for the calculation of the transmission as explained in the \"Methods\" section and the presence of the soil substrate of the half-space. For example, there is a pass band in the calculated band structure of the examined I-shaped LSPM between 3.10 and 3.26 Hz, as indicated from the limits between the first and second PBGs. The transmission calculation showed this passing zone within a frequency range between 3.6 and 3.8 Hz, as indicated in Figure 7a. Overall, the calculation of the transmission spectrum validated the band structure calculations, indicating the effectiveness of the proposed structure for surface wave attenuation. For a better understanding of the attenuation mechanism, three mode shapes are shown in Figure 7 that represent the absolute value of the vertical component of the displacement field, normalized to its maximum value. The selection of the mode shapes was from the middle cross-section of the numerical three-dimensional half-space that was used in the simulations and are focused on the right part where the LSPM is designed. The mode shapes of Figure 7b,d clearly show that for the selected frequencies \(\mathrm{f_{1}}=1.8\) Hz  
Figure 7: (**a**) The transmission relation for the I-shaped LSPM where \(\mathrm{L}=9.0\) m and \(\mathrm{w}=4.0\) m. (**b–d**) Mode shapes of the absolute value of the vertical component of the displacement field, normalized to its maximum value, for frequencies (**b**) \(\mathrm{f_{1}}=1.8\) Hz, (**c**) \(\mathrm{f_{2}}=3.4\) Hz, and (**d**) \(\mathrm{f_{3}}=5.3\) Hz. and \(\rm f_{3}=5.3\) Hz, there are modes passing the LSPM, whereas for the selected frequency \(\rm f_{2}=3.4\) Hz, the vertical component of the displacement field, shown in Figure 7c, is nearly zero at the surface after the LSPM, indicating its attenuation performance.