## 1 Introduction
### State of the Art
#### 1.1.1 Piezoelectric Models, Approximations and Finite Elements  
Low-Order Models and Elements for Monolithic Piezoelectrics  
Let us start this brief survey with the first-order and second-order models of piezoelectrics, where the model order concerns transverse displacement and electric potential fields. In Reference [3], the authors present classical (non-adaptive) finite element approximation and its features for the two-dimensional conventional (mid-surface unknowns) Reissner-Mindlin piezoelectric plate model including membrane effects. The finite element approximation is of mixed type with shear stresses treated as independent unknowns. The Kirhchoff piezoelectric plate model can be obtained as a special case. The latter model was also analyzed in Reference [4]. The simple flat shell element based on the conventional (mid-surface) displacements, rotations, and electric potential for plane stress and neglect of the in-plane electric field and electric displacements is presented in Reference [5]. The element is based on the assumed natural strain formulation in order to overcome numerical locking phenomena. Subsequently, in Reference [6], the Reissner-Mindlin mechanical model is combined with the first-order and second-order electric potential field in the transverse directions. Two formulations are applied, the displacement-like and hybrid with the electric displacements treated as independent unknowns. In Reference [7], the second-order transverse mechanical field is applied in the frame of three-dimensional elasticity for thin-walled piezoelectric plates. The electric potential field is also of the second-order in the transverse direction. The balanced reduced integration is applied as a numerical technique which serves removing the locking phenomena. The analytical conventional model of the Reissner-Mindlin plates with non-constant electric field distribution is presented in Reference [8]. Hierarchical Models and Elements for Monolithic Piezoelectrics  
The hierarchical two-dimensional conventional (mid-surface unknowns) models of piezoelectric plates were introduced and mathematically substantiated in Reference [9]. The mixed formulation is applied there with stresses and electric displacements completing displacements and electric potential. [PERSON] and co-workers [10] proposed hierarchic plate models up to the fourth-order. The formulation is displacement-based in the case of the higher-order models, and mixed in the case of the Reissner-Mindlin model. The models employ three-dimensional degrees of freedom (dofs) for the top and bottom surfaces, and conventional (mid-surface) higher-order dofs. In addition, a thermal field can be included into this formulation. In Reference [11], the 3D-based (three-dimensional unknowns only) piezoelectric hierarchy of solid, first-order (Reissner-Mindlin) and hierarchical shell, and solid-to-shell and shell-to-shell transition piezoelectric models are formulated. Their \(hpq\)-adaptive finite element approximations are presented in Reference [12]. The formulation is displacement-like with displacements and electric potential as unknowns of the problem. In the case of the electric field, this approach allows for the three-dimensional and 3D-based symmetric-thickness models of dielectricity. The approach is based on the 3D-based hierarchical models [13] and approximations [14] for elasticity, and the corresponding models [15] and approximations [16] for dielectricity. Models for Piezoelectric Composites  
In the case of laminated piezoelectric structures, it is worth mentioning the following works. In Reference [17], the authors presented Reissner-Mindlin-type composite plate model with piezoelectric layers. The direct piezoelectric and pyroelectric effects are taken into account. In turn, the work of Reference [18] proposed the plain strain shell model with higher-order shear and normal deformation effects for analysis of cylindrical laminated piezoelectric shells with actuating and sensing piezoelectric layers.