## 7 Numerical Results
### Doubly-Curved Panel of Translation  
As last example, a CFCF doubly-curved panel of translation is considered (Figure 4d). Since its geometry is obtained by moving a parabola over another parabola defined by the same shape, this structure is named also as elliptic paraboloid. Its position vector can be written as follows  
\[\mathbf{r}(\alpha_{1},\alpha_{2})=\left(\frac{k\tan\alpha_{1}}{2}-\frac{k\tan^ {2}\alpha_{2}}{4}\sin\alpha_{1}\right)\mathbf{e}_{1}-\frac{k\tan\alpha_{2}}{2 }\,\mathbf{e}_{2}+\left(\frac{k\tan^{2}\alpha_{1}}{4}+\frac{k\tan^{2}\alpha_{ 2}}{4}\cos\alpha_{1}\right)\mathbf{e}_{3} \tag{101}\]  
in which the coordinates \(\alpha_{1},\alpha_{2}\) are defined in the intervals \(\alpha_{1}\in\left[\alpha_{1}^{0},\alpha_{1}^{1}\right]\) and \(\alpha_{2}\in\left[\alpha_{2}^{0},\alpha_{2}^{1}\right]\), assuming \(\alpha_{1}^{0}=\alpha_{2}^{0}=0.588003\), \(\alpha_{1}^{1}=\alpha_{2}^{1}=-0.588003\). On the other hand, \(k=\left(a^{2}-d^{2}\right)/b\) is the characteristic parameter of the two parabolas. It is well-known that a parabolic curve is completely defined by the definition of three points with abscissas \(a,c,d\) and the relative distance \(b\) between the points of coordinates \(a\) and \(d\). In particular, \(a\) and \(c\) represent the abscissas of the extreme points of the parabolic arch as it can be deducted from Figure 8b. The current geometry can be obtained by setting \(a=3\) m, \(b=1\) m, \(c=-3\) m, and \(d=0\) m. For extra details concerning the present shape, the book by [PERSON] et al. [2] can be taken into account as a reference. As it can be noted from Figure 5c, the structure is made of three layers. In particular, the lower ply is isotropic and fully metal (Stainless steel), and the upper one is completely ceramic (Silicon nitride). These two layers have the same reference thickness \(h_{0,1}=h_{0,3}=0.05\) m. On the other hand, the central core is graded between these two limit cases and its reference thickness is equal to \(h_{0,2}=0.2\) m. Thus, the overall shell thickness that can be assumed as a reference is \(h_{0}=0.3\) m. Varying the exponent \(p^{(2)}=p\in[0,\infty]\) of the volume fraction distribution, several mechanical configurations can be obtained. As far as the thickness profile is concerned, the thickness variation affects both the two principal coordinates \(\alpha_{1},\alpha_{2}\), as it can be easily observed in the cross-sections shown in Figure 4d. In particular, a quadratic variation is applied along \(\alpha_{1}\), whereas a sine-wave variation is applied on the second direction. Mathematically speaking, the thickness of the shell at issue is defined as follows  
\[h(\alpha_{1},\alpha_{2})=h_{0}\left(1+\frac{3}{4}\left(\frac{\alpha_{1}- \alpha_{1}^{0}}{\alpha_{1}^{1}-\alpha_{1}^{0}}\right)^{2}+\frac{1}{2}\sin( \pi\left(3\left(\frac{\alpha_{2}-\alpha_{2}^{0}}{\alpha_{2}^{1}-\alpha_{2}^{0} }\right)+1\right)\right)\right) \tag{102}\]  
Even in this circumstance, the three structural models previously employed are taken with and without the Murakami's function. The natural frequencies related to the \(\text{FSDT}_{RS}^{\chi=1.2}\), ED2\({}^{\chi=1.2}\), and ED3 are shown in Table 6, whereas the ones obtained by using the \(\text{FSDT}_{RS}^{\chi=1.2}\), ED2\({}^{\chi=1.2}\), and EDZ3 are written in Table 7, for several values of the parameter \(p\). For the sake of completeness, the 3D-FEM solutions, which are really close to the LGDQ ones, are also included in these tables. Figure 10: First six mode shapes for a CF doubly-curved shell of revolution with hyperbolic meridian with variable thickness assuming \(p=1\) as exponent. It should be specified that the finite element model is obtained by using 51,350 brick elements (676,302 degrees of freedom).