## 4 Numerical Results  
We start this section with a validation of the proposed approach, which is followed by a parametric study that investigates the sensitivity of the thermoelastic response of the panel to the mechanical, thermal, or geometrical input parameters. Unless otherwise mentioned, we consider Ti-6 Al-4V and ZrO\({}_{2}\) in the numerical examples to be the metal and ceramic phases, respectively (Table 1), and we introduce the following dimensionless parameters  
\[F_{\mathrm{o}}=\frac{t\hat{\alpha}_{m}}{h^{2}}, \tag{22a}\] \[\tau=\frac{\tau_{0}\hat{\alpha}_{m}}{h^{2}},\] (22b) \[T^{\star}=\frac{T-T_{\infty}}{T_{\infty}},\] (22c) \[U=\frac{u}{(1-v_{c})\alpha_{c}T_{\infty}R_{\mathrm{m}}},\] (22d) \[\xi=\frac{r-R_{i}}{R_{\mathrm{o}}-R_{i}},\] (22e) \[\Sigma_{ii}=\frac{(1+v_{c})}{E_{c}\alpha_{c}T_{\infty}}\sigma_{ii} \tag{22f}\]  
with  
\[i=r,\theta. \tag{22g}\]  
where \(\hat{\alpha}_{m}(=k_{m}/\rho_{m}c_{m})\) is the thermal diffusivity of the metal, and the following geometry and physical parameters are assumed: \(R_{m}=1\,(\mathrm{m})\,,\,h=0.35\,\mathrm{m}\,,\,T_{0}=T_{\infty}=300\)K, \(T_{i}=800\)K, \(\hat{h}_{c}=10\) W/m\({}^{2}\)K, \(t_{0}=2\)s. The piezoelectric layers are made of lead zirconate titanate (PZT-4) with the mechanical properties given in Table 2, in accordance with [30]. As a first example, we compare the CPU time necessary for a thermoelastic study of a FG cylindrical panel, as provided by a NURBS-based multi-step time integration scheme and a [PERSON]'s method. Table 3 gathers the convergence results for the dimensionless displacement, hoop stress, and temperature at point \(\xi=0.5\), \(\theta=0.5\theta_{0}\) versus the number of time steps \(N_{t}\). As visible in Table 3, a NURBS-based multi-step approach requires a lower CPU time compared to a classical [PERSON]'s method. Accordingly, Figures 2 and 3 depict the convergence results from a transient thermoelastic study of FG cylindrical panels with a piezoelectric layer under a thermal loading condition. In these figures, the variations along the thickness direction of the dimensionless temperature, radial displacement, and hoop stress components are reported versus the DQ number of grid points throughout the thickness \(\left(N_{\xi}^{e}=N_{\xi},\,e=c,p\right)\) and circumferential direction \(\left(N_{\theta}\right)\). It is worth observing that nine grid points in the radial direction per layer (i.e., \(N_{\xi}=9\)) and eleven grid points in the circumferential direction yielded acceptable results. Hence, the values of \(N_{\xi}=9,N_{\theta}=11\), and \(N_{t}=300\) are assumed hereafter within the numerical investigation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Ti-6 Al-4V** & **ZrO2** \\ \hline \(k[W/\text{mK}]\) & 18.1 & 2.036 \\ \(\rho[\text{kg}/\text{m}^{3}]\) & 4410 & 5600 \\ \(C[\text{J}]/\text{kgK}]\) & 808.3 & 615.6 \\ \(E[\text{GPa}]\) & 66.2 & 117 \\ \(a[1/\text{K}]\) & \(10.3\times 10^{-6}\) & \(7.11\times 10^{-6}\) \\ \(\
u\) & 0.322 & 0.322 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of ceramic (ZrO2) and metal (Ti-6 Al-4V) [11]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**115** & **\(\mathbf{d_{1}}\) (coul/N)** & \(-3.92\times 10^{-12}\) \\ \hline \(C_{22}\) [GPa] & 139 & \(V_{p}\) & 0.31 \\ \(C_{55}\) [GPa] & 25.6 & \(E_{p}\) [GPa] & 63 \\ \(C_{12}\) [GPa] & 74 & \(\rho_{p}[\text{kg}/\text{m}^{3}]\) & 7500 \\ \(e_{2}\) [coul/m\({}^{2}\)] & \(-5.2\) & \(C_{p}\) [J/kgK] & 350 \\ \(e_{3}\) [coul/m\({}^{2}\)] & 15.1 & \(\alpha_{pr}[1/\text{K}]\) & \(2.62\times 10^{-6}\) \\ \(e_{4}\) [coul/m\({}^{2}\)] & 12.7 & \(\alpha_{p\theta}[1/\text{K}]\) & \(1.97\times 10^{-6}\) \\ \(\mu_{2}[\text{nF}/\text{m}]\) & 6.5 & \(k_{pr}[\text{W}/\text{mK}]\) & 1.5 \\ \(\mu_{3}[\text{nF}/\text{m}]\) & 5.6 & \(k_{p\theta}[\text{W}/\text{mK}]\) & 2.1 \\ \(P_{3}\) (coul/m\({}^{2}\)K) & \(5.4\times 10^{-5}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of the lead zirconate titanate (PZT-4) [30].