**1. Introduction**

More and more common application of piezoelectrics in contemporary technology requires more and more efficient and accurate methods of their modeling and analysis. In this work, we develop the new transition elements which possess unique set of features enabling joining mechanical shell models of the first order and higher orders. Such models are incompatible due to the assumptions of the plane stresses and no elongation of the normals to the shell mid-surface, both present in the first-order model. In order to join such models, the transition models are necessary. This refers also to the piezoelectricity case, when the mechanical field modeling needs simultaneous application of the first- and higher-order mechanical models.

Adaptive capabilities of *hpq*/*hp*-type are the second important feature of the new elements, where *h* is the element size parameter, while *p* and *q* are the element longitudinal and transverse orders of approximation. As our implementation of these capabilities is based on the standard *hpq*-approximation rules, presented, respectively, in Reference [1,2] for 2D and 3D, we will not elucidate this aspect in this paper.

#### *1.1. State of the Art*

To the best of the authors' knowledge, the transition piezoelectric elements have not been proposed yet, both in the classical (non-adaptive) and adaptive versions. Because of

**Citation:** Zboi ´nski, G.; Zieli ´nska, M. 3D-Based Transition *hpq*/*hp*-Adaptive Finite Elements for Analysis of Piezoelectrics. *Appl. Sci.* **2021**, *11*, 4062. https://doi.org/ 10.3390/app11094062

Academic Editor: Marek Krawczuk

Received: 9 April 2021 Accepted: 26 April 2021 Published: 29 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

this, our literature survey can only be focused on two topics closest to the paper theme. The first topic concerns mathematical models of piezoelectrics and their finite element approximations. The second topic is the elastic transition models and the corresponding finite element approximations, as well.

#### 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 piezoelectricity, 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 secondorder 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. Carrera 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 3Dbased 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. The work of Reference [6], mentioned in the context of the monolithic piezoelectrics, should be mentioned here as applied to tree-layered shell structures with two outer piezoelectric layers and an electrically neutral core. A similar sandwich plate structures are modeled as two Reissner-Mindlin piezoelectric layers and one neutral layer of the same character [19], with the mid-surface interdependent displacement dofs. The electric fields are of mixed character and are defined by the outer layers' mid-surface electric potentials and independent electric field mid-surface values of the outer layers. In order to remove locking, the reduced integration method is applied. The generalized analytical Reissner-Mindlin model of laminated piezoelectrics was presented in Reference [21]. It allows analyses of the problems where the electric potential is not prescribed through the thickness. The method assumes small thickness and thus allows decomposition of the three-dimensional Reissner-Mindlin description into two-dimensional in-plane and one-dimensional through-the-thickness problems. Subsequently, the already cited work of Reference [10] concerning hierarchical models of single layered piezoelectrics can also be applied to the numerical analysis of hierarchically modeled multilayered structures, with piezoelectric layers included. The recent example [22] of analysis of multilayered shells concerns the first-order model of the functionally graded layers. In turn, the piece-wise second-order finite element model for multilayered structures with delamination and debonding was proposed in Reference [23].

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 threedimensional 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 Surana [25], who took into account the consideration of Ahmad [26] on thick shell elements and his own research [27] on general and axi-symmetric shell elements. The work, in Reference [25], of Surana 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 Bathe and Bolourchi [31], based on the mentioned work of Ahmad, led to the analogous transition elements. Gmür and Schorderet [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. Liao 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. In the work of Reference [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 Dávila [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 Gong, 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 Demkowicz 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 larger number 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. The linear [50] or higher [51] order of approximation can be applied in the transverse direction on the common face of the three-dimensional and 3D-based shell elements. Even though this technique guarantees continuity of the field of unknowns, high gradients of the derivatives on the boundary between the models appear. If one wants to ge<sup>t</sup> rid of such gradients, the transition elements removing stress [52] and, additionally, strain, Reference [24] gradients are necessary between the basic models being joined.
