## 3 Mixed FE-GDQ Numerical Formulation  
A set of rectangular-quadratic elements, \(N_{e}\), is considered to discretize the x-y plane of the domain. Each element should be differentiable at both in-plane and transverse displacements for solving the governing equations. Hence, the displacement components are approximated as:  
\[ \begin{array}{l}u(x,y,z,t)=\sum\limits_{j=1}^{N}\varphi_{j}(x,y)U_{j}(z,t),\\ v(x,y,z,t)=\sum\limits_{j=1}^{N}\varphi_{j}(x,y)V_{j}(z,t),\ \ \ \ j=1,2,\ldots,N\\ w(x,y,z,t)=\sum\limits_{j=1}^{N}\varphi_{j}(x,y)W_{j}(z,t)\end{array}  \tag{10}\]  
where \(N\) stands for the total number of nodes in the discretized x-y plane, and \(\varphi_{j}(x,y)\) denotes the global Lagrange interpolation functions. By combination of Equations (3) and (8), and integrating by parts in the \(z\)-direction, the following governing equations per each node \(i\) (\(i=1,2,\ldots,N\)) are obtained:  
\[ \begin{array}{l}\delta U_{i}:\sum\limits_{j=1}^{N}A^{ij}C_{55}(z_{j})\frac{ \partial^{2}U_{j}}{\partial z^{2}}+\ \sum\limits_{j=1}^{N}\frac{\partial C_{55}(z_{j})}{ \partial z}\frac{\partial U_{j}}{\partial z}+\sum\limits_{j=1}^{N}\left(D^{ ij}C_{55}(z_{j})-D^{ji}C_{12}(z_{j})\right)\frac{\partial W_{j}}{\partial z}\\ +\sum\limits_{j=1}^{N}\frac{\partial C_{44}}{\partial z}(z_{j})\,W_{j}-\sum \limits_{j=1}^{N}\left(E^{ji}C_{12}(z_{j})+E^{ij}C_{66}(z_{j})\right)U_{j}-\ \sum\limits_{j=1}^{N}\left(F^{ij}C_{11}(z_{j})+H^{ji}C_{66}(z_{j})\right)V_{j} \\ -P_{x}\sum\limits_{j=1}^{N}D_{b}{}^{ij}V_{j}-P_{y}\sum\limits_{j=1}^{N}B_{b}{} ^{ij}V_{j}-P_{xy}\sum\limits_{j=1}^{N}S_{b}{}^{ij}V_{j}-P_{xy}\sum\limits_{j=1 }^{N}K_{b}{}^{ij}V_{j}=0\\ \\ \delta V_{i}:\sum\limits_{j=1}^{N}A^{ij}\ C_{44}(z_{j})\frac{ \partial^{2}V_{j}}{\partial z^{2}}+\ \sum\limits_{j=1}^{N}\frac{\partial C_{44}(z_{j})}{ \partial z}\frac{\partial V_{j}}{\partial z}+\sum\limits_{j=1}^{N}\left(D^{ ij}\,C_{44}(z_{j})-D^{ii}\,C_{12}(z_{j})\right)\frac{\partial W_{j}}{\partial z}\\ +\sum\limits_{j=1}^{N}\frac{\partial C_{44}}{\partial z}(z_{j})\,W_{j}-\sum \limits_{j=1}^{N}\left(E^{ji}\ C_{12}(z_{j})+E^{ij}C_{66}(z_{j})\right)U_{j}-\ \sum\limits_{j=1}^{N}\left(F^{ij}\,C_{11}(z_{j})+H^{ji}\,C_{66}(z_{j})\right) V_{j}\\ -P_{x}\sum\limits_{j=1}^{N}D_{b}{}^{ij}V_{j}-P_{y}\sum\limits_{j=1}^{N}B_{b}{} ^{ij}V_{j}-P_{xy}\sum\limits_{j=1}^{N}S_{b}{}^{ij}V_{j}-P_{xy}\sum\limits_{j=1 }^{N}K_{b}{}^{ij}V_{j}=0\\ \\ \delta W_{i}:\sum\limits_{j=1}^{N}A^{ij}\,C_{11}(z_{j})\,\frac{ \partial^{2}W_{j}}{\partial z^{2}}+\sum\limits_{j=1}^{N}\left(B^{ij}C_{12}(z_ {j})-B^{ij}C_{55}(z_{j})\right)\frac{\partial U_{j}}{\partial z}\\ +\sum\limits_{j=1}^{N}\left(D^{ij}C_{12}(z_{j})-D^{ji}\,C_{44}(z_{j})\right) \frac{\partial V_{j}}{\partial z}-\sum\limits_{j=1}^{N}\ B^{ij}\frac{\partial C_{12}(z_ {j})}{\partial z}\ \ U_{j}\\ +\sum\limits_{j=1}^{N}D^{ij}\frac{\partial C_{12}(z)}{\partial z}\ \ V_{j}+\sum\limits_{j=1}^{N}\left(F^{ij}\,C_{44}(z_{j})+H^{ij}C_{55}(z_{j})+ A^{ij}\frac{\partial C_{11}(z)}{\partial z}\ \right)W_{j}\\ -P_{x}\sum\limits_{j=1}^{N}D_{b}{}^{ij}W_{j}-P_{y}\sum\limits_{j=1}^{N}B_{b}{} ^{ij}W_{j}-P_{xy}\sum\limits_{j=1}^{N}S_{b}{}^{ij}W_{j}-P_{xy}\sum\limits_{j=1 }^{N}K_{b}{}^{ij}W_{j}=0\\ \text{where,}\end{array}  \tag{13}\]  
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