## 8 Conclusions  
From the present work, the following conclusions can be drawn:  
* A finite element model for the linear vibration analysis, using spectral/_hp_ elements based on a seven-parameter shell formulation, is extended to study functionally graded shells. * Comparisons with the numerical results of a simulation using solid elements verify the performance of the present formulation. In addition, the comparisons suggest that the present formulation shows a better behavior for the modeling of conical, cylindrical, and parabolic shells subjected to the boundary conditions considered herein. * In general, although the relative errors are between 6% and 8% obtained in the study of hemispherical shells, the present formulation has a good behavior for moderately thick to thin shells. It must be recalled that mechanical properties through the thickness of the 3D model are approximated using several layers in this direction. On the other hand, in the studies reported using the present formulation, the variation of the mechanical properties is evaluated using several Gauss-points while numerical integration through thickness is made, resulting in a closer approximation of them. For the aforementioned reasons, some differences between results, but a similar behavior, can be expected. * Studying the effect of the power-law index on the free vibration response is relevant since, for a given thick or thin shell and boundary conditions, its response may be stiffer or not as the power-law index increases. * Finally, the spectral/_hp_ seven-parameter formulation used in the present finite element model allows a straightforward implementation of the FGM behavior and the use of three-dimensional constitutive equations. These features allow the use of considerably fewer elements, and therefore computational time, to model FG shell structures, when compared with models made of solid elements. Conceptualization, C.E.V.M. and M.E.G.R. ; investigation, C.E.V.M. and N.F.S. ; software, C.E.V.M., M.E.G.R. and N.F.S. ; validation, C.E.V.M. ; formal analysis, C.E.V.M., M.E.G.R. and N.F.S. ; writing--original draft preparation, C.E.V.M. and M.E.G.R. ; writing--review and editing, C.E.V.M., M.E.G.R., N.F.S. and L.D.C.G. ; visualization, C.E.V.M. All authors have read and agreed to the published version of the manuscript. The APC was funded by Instituto de Innovacion, Ciencia y Emprendimiento para la Competitividad para el Estado de Guanajuato (IDEAGTO) grant number IDEAGTO/CONV/069/2023. Not applicable. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{6}{c}{**Vibration Mode**} \\ \cline{2-7}  & \(f_{1}\)–\(f_{2}\) & \(f_{3}\)–\(f_{4}\) & \(f_{5}\)–\(f_{6}\) & \(f_{7}\)–\(f_{8}\) & \(f_{9}\) & \(f_{10}\)–\(f_{11}\) \\ \hline \multirow{4}{*}{7-PL} & 213.72 Hz & 284.29 Hz & 340.23 Hz & 435.41 Hz & 471.54 Hz & 490.25 Hz \\  & & & & & & \\ \cline{1-1} \cline{2-7}  & 203.73 Hz & 266.47 Hz & 338.50 Hz & 416.22 Hz & 457.11 Hz & 474.15 Hz \\ \cline{1-1} \cline{2-7}  & 3D & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 22: Vibration modes for a hemispherical shell with LVT and CF boundary conditions, \(n=5\) (FG-I, \(r_{0}/h_{m}=10\), \(r_{0}=2\) m, \(\beta=2\), \(\varphi_{0}=\pi/2\), \(\varphi_{1}=\pi/6\)). Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available. The authors declare no conflict of interest. The following abbreviations and nomenclature are used in this manuscript: