## 1 Introduction
### State of the Art
#### 1.1.2 Transition Piezoelectric, Dielectric and Elastic Models And Elements  
Transition Elements for Three Physical Classes of Problems  
It can be concluded from the previous subsection that the piezoelectric transition models and elements that serve joining the three-dimensional (or hierarchical shell) piezoelectric models with the first-order piezoelectric model are rare. The only works, which we have cited, concern the idea [15] and variational [11] and finite element [12] formulations of the same classical transition piezoelectric model and element. They guarantee continuity of the displacement and transverse deformation fields between the mentioned models. They also guarantee continuity of the electric potential between the piezoelectric models. In Reference [16], it has been demonstrated that, in the case of dielectric field of electric potential used in electrostatics, no transition elements between the three-dimensional model and symmetric-thickness hierarchical models of the first and higher orders are necessary if the consistent finite element approximations of non-adaptive or adaptive character are applied. This is because the same definition of electric field (intensity) is applied within both dielectric models. In the case of the elastic mechanical fields, two groups of the transition models can be distinguished. The first one guarantees continuity of the inconsistent generalized displacement fields of the neighboring models. In such inconsistent models, the kinematic unknowns are different or of different order, e.g., in the transverse direction. In the second group, additionally the definitions of the derivatives of the kinematic unknowns, e.g., strains and/or stresses, are different in the neighboring models and the corresponding finite elements. In this context, one may deal with the neighborhood of the three-dimensional elasticity and shell or plate theories, the three-dimensional theory, and beam or truss theories, as well as the shell/plate theories and the one-dimensional theories of beams or trusses. Due to the paper scope, only the first case will be elucidated here. A general review, including all three cases, can be found in the thesis [24]. Non-Adaptive Transition Elements for Elasticity  
We start with the transition elements of the classical (non-adaptive) finite element methods where the neighboring elements are joined through the sides of the same size and finite element interpolations on these sides are consistent. In addition, location of three-dimensional, shell/plate, and transition domains is fixed and strictly determined by the structure geometry. The first examples of the transition elements of this type can be attributed to [PERSON] [25], who took into account the consideration of [PERSON] [26] on thick shell elements and his own research [27] on general and axi-symmetric shell elements. The work, in Reference [25], of [PERSON] presents solid-to-shell elements which enable joining the thick-shell element with the linear, quadratic or cubic solid elements. The elements possess the bigger shell part equipped with the mid-surface dofs, and the smaller solid part limited to one side of the element and equipped with the tree-dimensional dofs. Plane stress is assumed in the entire transition element. Applications of the elements of this type to axi-symmetric [28] and three-dimensional [29] problems are presented. Extension onto thermo-mechanical analysis was done in Reference [30]. Note that the independent research of [PERSON] and [PERSON] [31], based on the mentioned work of [PERSON], led to the analogous transition elements. [PERSON] and [PERSON] [32] suggested application of the three-dimensional or plane stress states in the Surana's element depending on the geometrical shape of the domain the element is located in. [PERSON] and others [33] extended application of the Surana's elements onto laminated composite structures. Both, the shell and solid parts of the transition element are modeled as such composites.