## 2 Formulations
\begin{array}{l}f_{1}\ =\ \frac{\partial\overline{N}_{xx}}{\partial x}+ \frac{1}{R}\frac{\partial\overline{N}_{\theta\theta}}{\partial x}+p_{1}{+} \frac{\partial\widetilde{N}_{xx}}{\partial x}+\frac{1}{R}\frac{\widetilde{N}_ {\theta\theta}}{\partial\theta},\\ f_{2}\ =\ \frac{\partial\overline{N}_{xx}}{\partial x}+\frac{1}{R}\frac{\partial \overline{N}_{\theta\theta}}{\partial\theta}+p_{2}{+}\frac{\partial\overline{ N}_{\theta\theta}}{\partial x}+\frac{1}{R}\frac{\partial\overline{N}_{\theta\theta}}{ \partial\theta},\\ f_{3}\ =\ -q+c_{1}(\frac{\partial\overline{N}_{xx}}{\partial x^{2}}+\frac{2}{R} \frac{\partial\overline{N}_{\theta\theta}}{\partial x}+\frac{1}{R}\frac{ \partial\overline{N}_{\theta\theta}}{\partial\theta}),\\ f_{4}\ =\ \frac{\partial\overline{N}_{xx}}{\partial x}+\frac{1}{R}\frac{ \partial\overline{N}_{\theta\theta}}{\partial x}+\frac{1}{R}\frac{\partial \overline{N}_{\theta\theta}}{\partial x}-c_{1}(\frac{\partial\overline{N}_{xx }}{\partial x}+\frac{1}{R}\frac{\partial\overline{N}_{\theta\theta}}{ \partial\theta}),\\ f_{5}\ =\ \frac{\partial\overline{N}_{xx}}{\partial x}+\frac{1}{R}\frac{ \partial\overline{N}_{\theta\theta}}{\partial\theta}+\frac{\partial\overline{ N}_{\theta\theta}}{\partial x}+\frac{1}{R}\frac{\partial\overline{N}_{\theta\theta}}{ \partial\theta}-c_{1}(\frac{\partial\overline{N}_{\theta\theta}}{\partial x}+ \frac{1}{R}\frac{\partial\overline{N}_{\theta\theta}}{\partial\theta}),\\ \end{array}  \tag{1}\]  
where  
\[\begin{array}{l}(\overline{N}_{xx},\widetilde{\overline{M}}_{xx}, \overline{P}_{xx})\ =\ \frac{h^{*}}{\int\frac{2}{h^{*}}}\ (\overline{Q}_{11}\alpha_{x}+\overline{Q}_{12}\alpha_{\theta}+ \overline{Q}_{16}\alpha_{x\theta})\Delta T(1,z,z^{3})dz,\\ \transverse displacement derivatives are referenced by [PERSON] et al. in 1978 [24]. \(k_{\alpha}\) is the advanced nonlinear shear coefficient, the advanced \(k_{\alpha}\) values are usually functions of \(h^{*}\), \(h_{3}\), \(c_{1}\), \(R_{n}\), and \(T\). In 1990, the GDQ method was presented by [PERSON] and [PERSON], then applied by [PERSON] in 2017 [18] and by [PERSON] and [PERSON] in 1997 [25]. The differential quadrature (DQ) method was presented by [PERSON] et al. in 1989 [26]. The GDQ method approximates the derivative of function \(f(x,\theta)\) at the coordinate of arbitrarily grid point \((x_{i},\theta_{j})\), \(i=1,2,\ldots\), \(N,j\ =\ 1,2,\ldots,M\), in the two-dimensional formulation can be given, e.g.,  
\[ \begin{array}{c}\frac{\partial f}{\partial x}\mid_{ij}\approx\sum\limits_{l =1}^{N}A^{(1)}_{i,l}f_{l,j^{\prime}}\ \frac{\partial f}{\partial\theta_{\mid,j^{\prime}}}\approx\sum\limits_{l =1}^{M}B^{(1)}_{j,m}f_{l,m},\ \ldots,\\ \frac{\partial^{d}f}{\partial x^{2}\partial\theta^{2}}\mid_{ij}\approx\sum \limits_{l=1}^{N}A^{(2)}_{i,l}\sum\limits_{m=1}^{M}B^{(2)}_{j,m}f_{l,m},\end{array}  \tag{2}\]  
where \(A^{(m)}_{i,l}\) and \(B^{(m)}_{j,m}\) denote the weighting coefficients for the \((m)\)-th order derivative of the function \(f(x,\theta)\) with respect to the \(x\) and \(\theta\) directions. Take the function \(f(x,\theta)\) for the component of middle-plane displacements \(u^{0}\), \(v^{0}\), \(w\) for the plate and \(u_{0}\), \(v_{0}\), \(w\) for the shell and shear rotations, \(\psi_{x}\) and \(\psi_{y}\) for the plate, and \(\phi_{x}\) and \(\phi_{\theta}\) for the shell in the places for the \((m)\)-th order derivative, and the non-dimensional parameters are used in the GDQ approaches, e.g., for the cylindrical shells, used as follows:  
\[X\ =\ x/L,\ U\ =\ u_{0}/L,\ V\ =\ v_{0}/R,\ W\ =\ w/h^{*}. \tag{3}\]  
The time sinusoidal displacement, shear rotations, temperature of thermal vibrations, and actuated magnetostrictive Terfenol-D control law are assumed in the simulation examples, e.g., for the shell as follows [17]:  
\[