## 1 Introduction
### State of the Art
#### 1.1.2 Transition Piezoelectric, Dielectric and Elastic Models And Elements
In the work of [PERSON] [34], in order to avoid the numerical locking, the method based on strain interpolation (with the points where the strain values are known) was applied instead of the reduced integration. In the work of [PERSON] [35], the opposite idea is applied, i.e., the elements consist of the larger solid part and smaller shell part. The latter part comprises one or two lateral sides. The top and bottom three-dimensional dofs and mid-surface dofs are applied within the solid and shell parts, respectively. Three-dimensional stress state is assumed throughout single-layered shells or each layer of composite shells. Yet another idea was suggested in Reference [36], by [PERSON], who proposed generalization of the solid-to-shell transition elements for the elements joining two-dimensional and three-dimensional elements. In this approach, the shell or two-dimensional parts consist of one or two lateral sides of the transition element. The longitudinal order of approximation can be higher than three. In the entire element, either three-dimensional or plane stress state can be applied. Adaptive Transition Elements for Elasticity  
In the case of the adaptive methods, finite elements of different sizes, orders of approximation (non-conforming approximations), and mechanical models are present in finite element meshes. In the case of different element sizes resulting from \(h\)-adaptivity based on remeshing, continuity of the field of unknowns may be based on introduction of the distorted elements on the boundaries between the mesh parts of various density. In the case of different sizes resulting from \(h\)-adaptivity based on local element refinement, continuity of the fields of unknowns requires introduction of the distorted elements [37] or is based on transition elements for monolithic [38] or multilayered [39] structures. Such transition elements deliver piece-wise linear [40] or approximate [41] continuity along the entire common edge or face, or at the common nodes only, e.g., within triangles [42], quadrilaterals [43] and hexahedrons [44]. Note that application of the smart idea of the constrained approximation, proposed by [PERSON] for 2D [1] and 3D [2], removes the necessity of introduction of the transition elements of this type as the continuity is automatically guaranteed by the constraints on the boundary between the elements of different sizes. As far as the different approximation orders of the elements are concerned, transition elements may be introduced with the consistent approximation on the boundary between specific 2D [20] or general 3D [45] elements, with the continuity guaranteed again on the entire common edge or face, or at the common nodes only. A smart alternative is the idea of 2D [1] and 3D [2] hierarchical shape functions, defined independently in the element vertices, on its edges and sides, and in the interior of the element, which allows for the same approximation order on the common edge or face of the neighboring elements and different orders in their interiors. In the case of different models of the neighboring elements, the continuity of the field of unknowns may require transition elements which conform the model of the largernumber of nodal dofs with some of these dofs constrained to zero on the boundary with the model of the smaller number of nodal dofs. The 3D-2D [46], 2D-2D [47], and 2D-1D [48] versions of this approach can be found. The alternative is the transition elements equipped with the degrees of freedom of both models (see comments in Reference [49]). Note that such transition elements are not necessary if the uni-dimensional, e.g., three-dimensional, approach is applied within each of the neighboring models. The idea lies in using the same, e.g., three-dimensional, dofs for each model. This usually requires internal constraints in the model simplified (e.g., 3D-based shell models) with respect to the three-dimensional one.