## 2 Basic Theory of HSDTPEs
### Incompatible Polynomial Approximation over Each Triangular Element
\begin{array}{cccccc}1&x_{0}&y_{0}&x_{0}^ {2}&x_{0}y_{0}&y_{0}^{2}&x_{0}^{3}&x_{0}^{2}y_{0}&y_{0}^{2}x_{0}&y_{0}^{3}\\ \end{array} \right] \tag{4}\]  
where \(x_{0}=x-x_{i}\), \(y_{0}=y-y_{i}\), \((x_{i},y_{i})\) are the coordinates of the central point of element \(e_{i}\). It should be noted that only triangular elements, as well as second-order and third-order polynomial functions, are implemented in the paper, but actually, arbitrary shape of elements and arbitrary orders of polynomials can also be easily employed to derive high-order shear deformation plate elements in the present work. Figure 2: Triangular elements for the mid-plane of the plate. Substituting Equation (2) into Equation (1), the displacement approximation over element \(e_{i}\) can be further expressed as:  
\[\mathbf{u}^{e}=\left\{ \begin{array}{c}u\\ v\\ w\end{array} \right\}=\left[ \begin{array}{cccc}\mathbf{P}^{2}&\mathbf{0}&\alpha_ {z}\mathbf{P}^{2}&\mathbf{0}&g(z)\mathbf{P}_{x}^{3}\\ \mathbf{0}&\mathbf{P}^{2}&\mathbf{0}&\alpha_{z}\mathbf{P}^{2}&g(z)\mathbf{P}_{ y}^{3}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{P}^{3}\end{array} \right] \left\{ \begin{array}{c}\mathbf{a}^{u_{0}}\\ \mathbf{a}^{v_{0}}\\ \mathbf{a}^{\theta_{x}}\\ \mathbf{a}^{\theta_{y}}\\ \mathbf{a}^{w}\end{array} \right\}=\mathbf{N}\;^{e}\mathbf{a}\;^{e} \tag{5}\]  
where \(\alpha_{z}=-z-g(z)\), \(\mathbf{P}_{,x}^{3}=\frac{\partial\mathbf{P}^{3}}{\partial x}\), \(\mathbf{P}_{,y}^{3}=\frac{\partial\mathbf{P}^{3}}{\partial y}\),  
\[\mathbf{N}^{e}=\left\{ \begin{array}{c}\mathbf{N}\;^{u}\\ \mathbf{N}\;^{v}\\ \mathbf{N}\;^{w}\end{array} \right\}=\left[ \begin{array}{cccc}\mathbf{P}^{2 }&\mathbf{0}&\alpha_{z}\mathbf{P}^{2}&\mathbf{0}&g(z)\mathbf{P}_{x}^{3}\\ \mathbf{0}&\mathbf{P}^{2}&\mathbf{0}&\alpha_{z}\mathbf{P}^{2}&g(z)\mathbf{P}_{ y}^{3}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{P}^{3}\end{array} \right] \tag{6}\]  
\[\mathbf{a}^{e}=\left[ \begin{array}{cccc}a_{1}&a_{2}&\cdots&a_{34}\end{array}  \right]^{\mathsf{T}} \tag{7}\]  
The strain-displacement relations of the linear elastic problem are given by:  
\[\varepsilon_{x}=\frac{\partial u}{\partial x},\varepsilon_{y}=\frac{\partial v }{\partial y},\varepsilon_{z}=\frac{\partial w}{\partial z}\approx 0,\gamma_{xy}= \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x},\gamma_{yz}= \frac{\partial w}{\partial y}+\frac{\partial v}{\partial z},\gamma_{xz}= \frac{\partial u}{\partial z}+\frac{\partial w}{\partial x} \tag{8}\]  
Substituting Equation (5) into Equation (8), we have:  
\[\varepsilon=\mathbf{L}\mathbf{u}^{e}\mathbf{=L}\mathbf{N}^{e}\mathbf{a}^{e} \mathbf{=B}\mathbf{a}^{e} \tag{9}\]  
where \(\varepsilon=\left[\varepsilon_{x},\varepsilon_{y},\gamma_{xy},\gamma_{yz}, \gamma_{xz}\right]^{\mathsf{T}}\) is the strain vector and B is the strain matrix, where:  
\[\mathbf{B}=\mathbf{L}\mathbf{N}^{e} \tag{10}\]  
\(\mathbf{L}\) is a differential operator where:  
\[\mathbf{L}=\left[ \begin{array}{cccc}\frac{\partial}{\partial x}&0&0\\ 0&\frac{\partial}{\partial y}&0\\ \frac{\partial}{\partial y}&\frac{\partial}{\partial x}&0\\ 0&\frac{\partial}{\partial z}&\frac{\partial}{\partial y}\\ \frac{\partial}{\partial z}&0&\frac{\partial}{\partial x}\end{array} \right] \tag{11}\]  
For an isotropic linear elastic material, the stress-strain relations in element \(e_{i}\) are given by:  
\[\boldsymbol{\sigma}=\mathbf{D}\mathbf{B}\mathbf{a}^{e} \tag{12}\]  
where \(\boldsymbol{\sigma}=\left[\sigma_{x},\sigma_{y},\tau_{xy},\tau_{yz},\tau_{xz} \right]^{\mathsf{T}}\), the transverse stress \(\sigma_{z}\) is assumed to be ignored for plate structures, and the elasticity matrix is:  
\[\mathbf{D}=D_{0}\left[ \begin{array}{cccc}1&v&0&0&0\\ v&1&0&0&0\\ 0&0&\frac{1-v}{2}&0&0\\ 0&0&0&\frac{1-v}{2k}&0\\ 0&0&0&0&\frac{1-v}{2k}\end{array} \right] \tag{13}\]where \(D_{0}=\frac{E}{1-v^{2}}\), \(E\) is the elastic modulus and \(v\) is the Poisson ratio.