## 4 Numerical Results
For validation purposes, we analyzed the FG hollow cylindrical shells under a thermal loading, as a limit case of cylindrical panels. Therefore, by setting \(\theta_{0}=2\pi\) and enforcing the geometric and physical compatibility conditions at \(\theta=0\) and \(\theta=\theta_{0}=2\pi\), a FG cylindrical panel reverted to a \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**NURBS-Based Multi–Step Method**} & \multicolumn{3}{c}{**Newmark’s Method (Galerkin Scheme)**} \\ \hline \(\boldsymbol{N_{t}}\) & \(\boldsymbol{T^{*}}\) & \(\boldsymbol{U}\) & \(\boldsymbol{\Sigma_{\theta\theta}}\) & **CPU Time (s)** & \(\boldsymbol{T^{*}}\) & \(\boldsymbol{U}\) & \(\boldsymbol{\Sigma_{\theta\theta}}\) & **CPU Time (s)** \\ \hline
10 & 0.523 & 0.704 & \(-0.836\) & 0.0024 & 0.311 & 0.456 & \(-0.425\) & 9.400 \\
25 & 0.449 & 0.581 & \(-0.732\) & 0.0196 & 0.340 & 0.448 & \(-0.616\) & 18.106 \\
50 & 0.405 & 0.549 & \(-0.584\) & 0.0263 & 0.339 & 0.473 & \(-0.483\) & 37.406 \\
75 & 0.367 & 0.516 & \(-0.529\) & 0.0339 & 0.340 & 0.475 & \(-0.486\) & 53.471 \\
100 & 0.352 & 0.489 & \(-0.506\) & 0.0408 & 0.341 & 0.475 & \(-0.488\) & 69.603 \\
125 & 0.346 & 0.481 & \(-0.497\) & 0.0452 & 0.342 & 0.476 & \(-0.490\) & 81.531 \\
200 & 0.342 & 0.476 & \(-0.491\) & 0.0584 & 0.342 & 0.476 & \(-0.490\) & 146.73 \\
300 & 0.342 & 0.475 & \(-0.489\) & 0.0660 & 0.342 & 0.476 & \(-0.490\) & 343.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between a NURBS-based multi-step technique and the [PERSON]’s scheme \(\left(n=1,\,h_{c}/h_{p}=5,\,\theta_{0}=75^{\circ},\,\psi=0.03,\,\xi=0.5,\, \theta=0.50_{0},\,F_{0}=0.8,\tau=0.1\right)\). cylindrical shell. This case study is here selected in agreement with [53], where a semi-analytical finite element method was applied for the analysis of FG cylindrical shells under the following thermal boundary and initial conditions:  
at  
\[r=R_{i}:\ T(r,z,t)=T_{0}\Big{(}1-e^{-0.5t}\Big{)} \tag{23a}\]  
\[r=R_{0}:\ k\frac{\partial T}{\partial r}+h_{c}T=0 \tag{23b}\]  
at  
\[z=0,L: \tag{23c}\]  
\[T(r,z,t)=0 \tag{23d}\]  
\[T(r,z,0)=0, \tag{23e}\]  
\[\left.\frac{\partial T(r,z,t)}{\partial t}\right|_{t=0}=0. \tag{23f}\]  
As far as the material properties are concerned, they vary throughout the shell thickness from a ceramic-rich material, at the inner surface, to a metal-rich material, at the outer surface, in accordance with Equation (1). The metal and ceramic mechanical properties are listed in Table 4. The dimensionless temperature and displacement distributions across the shell thickness are depicted in Figure 4 for a varying material graded index \(n\), where the accuracy of the proposed method against the uncoupled theory proposed in [53] is confirmed by the very good agreement between results. \begin{table}
\begin{tabular}{c c c} \hline
**Material** & **Zirconia** & **Steel** \\ \hline \(k[\mathrm{W/cm^{\circ}C}]\) & 2.09 & 20 \\ \(\rho[\mathrm{kg/m^{3}}]\) & 5700 & 8166 \\ \(C_{\circ}[\mathrm{J/kg^{\circ}C}]\) & 531.9 & 325.35 \\ \(E[\mathrm{MPa}]\) & 168.06 & 207.79 \\ \(\alpha[1/^{\circ}C]\) & \(2.3\times 10^{-4}\) & \(1.5\times 10^{-5}\) \\ \(\
u\) & 0.298 & 0.3178 \\ \hline \end{tabular}
\end{table} 
Table 4: Material properties of ceramic (Zirconia) and metal (steel) [53]. Figure 4: (a,b): Transient dimensionless temperature distributions and displacement field across the thickness of an FG hollow cylinder subjected to a prescribed temperature at its inner surface \(\Big{(}R_{i}=4\)cm, \(R_{0}=6\)cm, \(L=20\)cm, \(\overline{T}=T/T_{0}\), \(\eta=0.5\Big{)}\). The effect of the material gradient index \(n\) on the through-the-thickness variation and time histories of the thermoelastic field variables is plotted in Figures 5 and 6 for FG cylindrical panels with piezoelectric layers under a thermal shock at the inner surface. The heat wave front can easily be seen in Figures 5a and 6a, where panels with a pure metal core (i.e., \(n=0\)) feature a higher value of heat wave speed compared to the ones with nonhomogeneous cores (i.e., \(n\
eq 0\)).