## 4 Algorithms of the Transition Piezoelectric Elements
### Strain-Displacement Matrix
#### 4.3.1 The Transition Element Based on the Classical Model  
The strain-displacement matrix for the classical transition model is assumed in the form corresponding to the three-dimensional elasticity and the hierarchical shell models, for the three-dimensional and thick- or thin-walled geometries, respectively. This means that the three-dimensional strain and stress state is applied for this classical model. This state is characterized by  
\[\mathbf{\epsilon}=\overset{\epsilon}{\mathbf{B}}\overset{\epsilon}{\mathbf{q}}^{hpq} \tag{43}\]  
and reflects the relation between the local strains \(\mathbf{\epsilon}\) and global nodal displacement dofs \(\overset{\epsilon}{\mathbf{q}}^{hpq}\). The local strain directions are coincident with two tangent (the first two or two longitudinal) and one normal (the third or transverse) directions within the structure. The strain-displacement matrix \(\overset{\epsilon}{B}\) will also be denoted as \(\overset{\epsilon}{\mathbf{B}}^{1}\equiv\overset{\epsilon}{\mathbf{B}}\) in our further considerations. The form of this matrix is derived from the standard approach applied to five-parameter thick-shell elements of the second order [63]. Here, we adopt it to the needs of the 3D-based hierarchical shell model. This form reads:  
\[\overset{\epsilon}{\mathbf{B}}=[...,\mathbf{B}_{k},...],\quad\overset{\epsilon}{\mathbf{ B}}^{T}=[...,\mathbf{B}_{k}^{T},...]^{T}. \tag{44}\]  
The above blocks \(\mathbf{B}_{k}\equiv\mathbf{B}_{k}^{1}\) of the strain-displacement matrix correspond to the element displacement dofs at the element nodes and are composed of the following components:  
\[\mathbf{B}_{k}=\left[ \begin{array}{cccc}B_{k,11}&B_{k,12}&B_{k,13}\\ B_{k,21}&B_{k,22}&B_{k,23}\\ B_{k,31}&B_{k,32}&B_{k,33}\\ B_{k,41}&B_{k,42}&B_{k,43}\\ B_{k,51}&B_{k,52}&B_{k,53}\\ B_{k,61}&B_{k,62}&B_{k,63}\end{array} \right]. \tag{45}\]  
Any term \(B_{k,mn}\), \(m=1,2,..,6\), \(n=1,2,3\) of each block \(\mathbf{B}_{k}\), where \(m\) corresponds to each of six components of the strain vector and \(n\) to each global direction of the displacement dof vector, can be determined as equal to:  
\[ \begin{split} B_{k,1n}&=\beta_{k,1}\theta_{1n};\quad B_{k, Ant}=\beta_{k,1}\theta_{2n}+\beta_{k,2}\theta_{1n}\\ B_{k,2n}&=\beta_{k,2}\theta_{2n};\quad B_{k,5n}=\beta_{k,2} \theta_{3n}+\beta_{k,3}\theta_{2n}\\ B_{k,3n}&=\beta_{k,3}\theta_{3n};\quad B_{k,6n}=\beta_{k,3} \theta_{1n}+\beta_{k,1}\theta_{3n},\end{split}  \tag{46}\]  
with the terms \(\theta_{pn}\) representing components of the transformation matrix \(\mathbf{\theta}\) which converts the local Cartesian directions \(p=1,2,3\) of the displacement dofs into the global ones. The terms \(\beta_{k,p}\) can be calculated from  
\[\beta_{k,p}=P_{k,1}A^{1p}+P_{k,2}A^{2p}+P_{k,3}A^{3p}, \tag{47}\]  
where the quantities \(P_{k,p}=\partial N_{k}/\partial\xi_{r}\) are the partial derivatives, with respect to the normalized directions \(r=1,2,3\), of the shape function \(N_{k}\) from the corresponding shape function matrix. The auxiliary quantities \(A_{rp}\) are equal to:\[A^{rp}=j^{r1}\theta_{1p}+j^{r2}\theta_{2p}+j^{r3}\theta_{3p}, \tag{48}\]  
with the terms \(j^{rn}\) representing terms of the inverse Jacobian matrix \(\mathbf{J}^{-1}\) and \(\theta_{np}\) being the components of the transposed transformation matrix \(\mathbf{\theta}^{T}\). The latter matrix transforms derivatives with respect to the global Cartesian directions into derivatives with respect to the local Cartesian directions. The terms of the transformation matrices can be obtained from:  
\[\mathbf{\theta}=\left[ \begin{array}{ccc}\theta_{11}&\theta_{12}&\theta_{13}\\ \theta_{21}&\theta_{22}&\theta_{23}\\ \theta_{31}&\theta_{32}&\theta_{33}\end{array} \right]=\left[ \begin{array}{ ccc}u_{1}&v_{1}&w_{1}\\ u_{2}&v_{2}&w_{2}\\ u_{3}&v_{3}&w_{3}\end{array} \right], \tag{49}\]  
where the unit vectors components \(u_{p}=U_{p}/U\), \(v_{p}=V_{p}/V\), \(w_{p}=W_{p}/W\), \(p=1,2,3\) can be obtained from the vectors \(\mathbf{U}\), \(\mathbf{V}\) and \(\mathbf{W}\) equal to:  
\[\mathbf{W}=\left[