## 4 Algorithms of the Transition Piezoelectric Elements
### Kinematic Constraints within the Elements
#### 4.8.1 The Classical and Modified Elements
The first two differences, after division by the thickness \(t\), give rotations of the normal to the mid-surface, while the third difference determines a transverse elongation of the corresponding normal. Note that only the latter difference needs to be constrained (equal to zero) in accordance with the last Equation (27). Because of this, multiplication of the penalty stiffness by the diagonal matrix \(\mathbf{Z}\) is necessary. This matrix possesses the following structure:  
\[\overset{\epsilon}{\mathbf{Z}}=\left[ \begin{array}{cc}\mathbf{Z}^{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{Z}^{2}\end{array} \right]. \tag{72}\]  
The component block \(\mathbf{Z}^{1}\) is composed of the diagonal blocks \(\mathbf{Z}_{k}\) and \(\mathbf{Z}_{l}\) corresponding to the degrees of freedom \(k\) and \(l\):  
\[\mathbf{Z}^{1}\ =\ \text{diag}\left[...,\mathbf{Z}_{k},...,\mathbf{Z}_{l},...\right], \quad\mathbf{Z}_{k}=\text{diag}\left[0,0,0\right],\quad\mathbf{Z}_{l}=\text{diag} \left[0,0,1\right], \tag{73}\]  
where the degrees of freedom \(k\) and \(l\) correspond to the sums and differences of the top and bottom displacement dofs. The second component block has the simpler form  
\[\mathbf{Z}^{2}\ =\ \text{diag}\left[...,\mathbf{Z}_{m},...\right],\quad\mathbf{Z}_{m}= \text{diag}\left[0,0,0\right], \tag{74}\]  
which results from the fact that, for the degrees of freedom \(m\), other than the sums and differences of the displacement dofs, the penalty terms are not necessary. The respective penalty stiffness \(\overset{\epsilon}{\mathbf{k}}_{P}\), present in (66), is defined in the directions coincident with the constraint directions of which the first two are tangent and the third one is perpendicular to the mid-surface. The stiffness form corresponds to the displacement dofs that represent the mentioned sums and differences of the top and bottom dofs. The definition of the penalty stiffness reads:  
\[\overset{\epsilon}{\mathbf{k}}_{P}=\int_{\mathcal{R}_{\epsilon}}\overset{ \epsilon}{\mathbf{N}}^{T}\mathbf{n}P\mathbf{n}^{T}\overset{\epsilon}{\mathbf{N}}\, \text{cof}(\mathbf{I})\ d\xi_{3}d\eta_{i}. \tag{75}\]  
The penalty parameter has to be larger than the terms of \(\overset{\epsilon}{\mathbf{k}}_{M}\) and the terms of \(\overset{\epsilon}{\mathbf{k}}_{C}\), i.e., \(P\gg\max\{(k_{M})_{ij};\ i,j=1,2,...,n_{M}\}\), where \((k_{M})_{ij}\) represent terms of the stiffness matrix and \(n_{M}\) stands for the corresponding number of displacement dofs. Additionally, one has\(\max\{(k_{C})_{ij};\,i=1,2,...,n_{M};j=1,2,...,n_{E}\}\), with \((k_{C})_{ij}\) being terms of the piezoelectricity matrix and \(n_{E}\) denoting the corresponding number of electric potential dofs. The above form (75) of the penalty matrix is suitable for the geometry I of the classical and modified transition elements. In the above equation, \(R_{e}\) stands for the element side being the subset of the boundary \(R\) between the first-order and transition model: \(R_{e}\subset R\). The normal vector \(\mathbf{n}\) is defined as perpendicular to the mid-surface \(S_{m}\), i.e., \(\mathbf{n}\perp S_{m}\). It also approximately satisfies the condition \(\mathbf{n}\in R_{e}\) - approximately due to the finite element approximation of the element geometry. Note that two tangential directions and the third normal direction are coincident with the local directions present in the constraint Equation (27). On the element level, they are defined by the relations (49) and (50). Note also that the presence of the vector \(\mathbf{n}\) in the above definition results in imposition of the penalty terms in the third (normal) direction only, i.e., the constraints are applied in this direction only. The shape function matrix terms are non-zero on the boundary \(R_{e}\). Hence, the non-zero penalty terms appear for the element nodes placed on this boundary. In the case of the classical and modified transition elements of the geometry II, the above definition has to take into consideration the fact that the constraints (27) are imposed on the element vertical edge \(L_{e}\) only.