## 7 Numerical Results
### Doubly-Curved Shell of Revolution with Hyperbolic Meridian
Excellent agreement with the reference solutions is achieved in each circumstance. The LGDQ results are obtained by setting \(I_{N}=I_{M}=31\) and \(I_{PN}=I_{PM}=16\), and the natural frequencies are written for different values of \(p\). It should be noted that Tables 4 and 5 show the first 18 natural frequencies. In fact, some couples of natural frequencies are equal due to the symmetry of the structure, in terms of both geometry and materials. For the sake of clarity, the variation of the first six natural frequencies as a function of the exponent \(p\) are depicted in graphical form in Figure 9 for all the considered structural models, with and without the Murakami's function. As in the previous case, a remarkable difference in the structural response can be obtained for lower values of the parameter \(p\). Finally, the first six mode shapes for the doubly-curved shell of revolution with hyperbolic meridian with variable thickness are shown in Figure 10 assuming a linear variation for the volume fraction distributions of the two constituents in the FGM layers (\(p=1\)). Figure 8: Meaning of the geometric parameters required to sketch two different curves: (**a**) hyperbola; (**b**) parabola. _Appl. Sci._**2017**, \(7\), 131  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & & & & & & & & & & & \\ \hline \multirow{2}{*}{\(f\)} & \multicolumn{2}{c}{\(p=0\)} & \multirow{2}{*}{\(p=0\)} & \multirow{2}{*}{\(p=0.2\)} & \multirow{2}{*}{\(p=0.5\)} & \multirow{2}{*}{\(p=1\)} & \multirow{2}{*}{\(p=2\)} & \multirow{2}{*}{\(p=6\)} & \multirow{2}{*}{\(p=128\)} & \multirow{2}{*}{\(p=\infty\)} & \multirow{2}{*}{\(p=\infty\)} \\  & 3D-FEM & & & & & & & & & & 3D-FEM \\ \hline  
\hline \multicolumn{10}{c}{**ED3**} \\ \hline \multirow{2}{*}{\(f\)} & \multicolumn{2}{c}{\(p=0\)} & \multirow{2}{*}{\(p=0\)} & \multirow{2}{*}{\(p=0.5\)} & \multirow{2}{*}{\(p=1\)} & \multirow{2}{*}{\(p=2\)} & \multirow{2}{*}{\(p=6\)} & \multirow{2}{*}{\(p=128\)} & \multirow{2}{*}{\(p=\infty\)} & \multirow{2}{*}{\(p=\infty\)} \\  & 3D-FEM & & & & & & & & 3D-FEM \\ \hline
1-2 & 258.759 & 258.962 & 240.684 & 224.866 & 210.933 & 198.484 & 185.699 & 177.382 & 176.921 & 176.810 \\
3-4 & 300.373 & 300.814 & 281.158 & 264.034 & 248.782 & 234.891 & 220.0_Appl. Sci._**2017**, \(7\), 131  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & & & & & & & & & & & \\ \hline \multirow{2}{*}{\(f\)} & \multicolumn{2}{c}{\(p=0\)} & \multirow{2}{*}{\(p=0\)} & \multirow{2}{*}{\(p=0.2\)} & \multirow{2}{*}{\(p=0.5\)} & \multirow{2}{*}{\(p=1\)} & \multirow{2}{*}{\(p=2\)} & \multirow{2}{*}{\(p=6\)} & \multirow{2}{*}{\(p=128\)} & \multirow{2}{*}{\(p=\infty\)} & \multirow{2}{*}{\(p=\infty\)} \\  & 3D-FEM & & & & & & & & & 3D-FEM \\ \hline  
\hline \multicolumn{10}{c}{**EDZ3**} \\ \hline \multirow{2}{*}{\(f\)} & \multicolumn{2}{c}{\(p=0\)} & \multirow{2}{*}{\(p=0\)} & \multirow{2}{*}{\(p=0.5\)} & \multirow{2}{*}{\(p=1\)} & \multirow{2}{*}{\(p=2\)} & \multirow{2}{*}{\(p=6\)} & \multirow{2}{*}{\(p=128\)} & \multirow{2}{*}{\(p=\infty\)} & \multirow{2}{*}{\(p=\infty\)} \\  & 3D-FEM & & & & & & & 3D-FEM \\ \hline
1-2 & 258.759 & 258.865 & 240.627 & 224.839 & 210.923 & 198.480 & 185.681 & 177.334 & 176.870 & 176.810 \\
3-4 & 300.373 & 300.592 & 281.015 & 263.948 & 248.733 & 234 Figure 9: Variation of the first six natural frequencies for a CF doubly-curved shell of revolution with hyperbolic meridian with variable thickness as a function of the exponent \(p\).