# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 2 Theoretical Formulation
### Geometric Parameters  
The position vector of a point located on the element's mid-surface is approximated as follows:  
\[\mathbf{\underline{X}}=\sum_{k=1}^{n}\ \psi_{k}\left(\xi^{1},\ \xi^{2}\right)\ \mathbf{ \underline{X}}^{k}, \tag{1}\]  
where \(\mathbf{\underline{X}}^{k}\) is the position vector of the \(k\)th spectral node, \(\psi_{k}\left(\xi^{1},\ \xi^{2}\right)\) is the \(k\)th interpolation function of order \(p\) (i.e., isoparametric formulation is used). The total number of nodes \(n\) per element is given by \(n=\left(p+1\right)\left(p+1\right)\).  
In this formulation, the normal vector of a point located on the element's mid-surface is approximated as  
\[\mathbf{\underline{n}}=\sum_{k=1}^{n}\ \psi_{k}\left(\xi^{1},\ \xi^{2}\right)\ \mathbf{ \underline{n}}^{k}, \tag{2}\]  
where \(\mathbf{\underline{n}}^{k}\) is the unit normal vector of the \(k\)th spectral node.  
Figure 1: Shell element and its mid-surface.  
Figure 2: Finite element discretization of shell mid-surface.  
Now, using the position vector in Equation (1) and scaling the normal vector in Equation (2) by a factor of \(h/2\) (half of the thickness), the position of a typical point in the element \(\mathbf{\Phi}\) is given as  
\[\mathbf{X}=\sum_{k=1}^{n}\ \psi_{k}\left(\xi_{1},\ \xi_{2}\right)\ \left( \mathbf{X}^{k}+\xi^{3}\frac{h}{2}\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{    }}}}}}}}}}}}\mathbf{h}^{k} \right). \tag{3}\]  
In addition to the above approximations, the covariant basis vectors for each point in the element must be defined:  
\[\mathbf{g}_{i}=\frac{\partial\mathbf{X}}{\partial\xi^{i}}\equiv\mathbf{X}_{,i} \hskip 42.679134 pti=1,\,2,\,3. \tag{4}\]  
The covariant basis vectors in Equation (4), by means of Equation (3), can be separated as follows:  
\[\mathbf{g}_{\alpha} =\mathbf{a}_{\alpha}+\xi^{3}\frac{h}{2}\mathbf{\hat{n}}_{,\alpha}, \tag{5}\] \[\mathbf{g}_{3} =\frac{h}{2}\mathbf{\hat{n}}. \tag{6}\]  
where the \(\alpha\) subindex can take values of \(1,2\). Furthermore, the vector \(\mathbf{a}_{\alpha}\) is given by  
\[\mathbf{a}_{\alpha}=\mathbf{X}_{,\alpha}.\]  
A differential line element in the typical shell element, in terms of the curvilinear coordinates, can be expressed as  
\[\left\{\begin{array}{c}dX_{1}\\ dX_{2}\\ dX_{3}\end{array}\right\}^{T}=\left\{\begin{array}{c}d_{5}^{\alpha 1}\\ d_{5}^{\alpha 2}\\ d_{5}^{\alpha 3}\end{array}\right\}^{T}\left[\begin{array}{cc}\frac{ \partial X_{1}}{\partial\xi^{1}}&\frac{\partial X_{2}}{\partial\xi^{1}}&\frac {\partial X_{3}}{\partial\xi^{1}}\\ \frac{\partial X_{1}}{\partial\xi^{2}}&\frac{\partial X_{2}}{\partial\xi^{2}} &\frac{\partial X_{3}}{\partial\xi^{2}}\\ \frac{\partial X_{1}}{\partial\xi^{3}}&\frac{\partial X_{2}}{\partial\xi^{3}} &\frac{\partial X_{3}}{\partial\xi^{3}}\end{array}\right]=\left\{d\xi\right\}^ {T}\left[J\right], \tag{7}\]  
where \(\left[J\right]\) is the Jacobian matrix with determinant \(J\). In conjunction with the covariant basis vectors, a contravariant set of basis vectors \(\mathbf{g}^{i}\) is defined as follows:  
\[\mathbf{g}^{1}=\frac{\mathbf{g}^{2}\times\mathbf{g}^{3}}{J},\ \ \ \mathbf{g}^{2}=\frac{\mathbf{g}^{3}\times\mathbf{g}^{1}}{J},\ \ \ \mathbf{g}^{3}=\frac{\mathbf{g}^{1}\times\mathbf{g}^{2}}{J}.\]