## 1. Introduction

In this chapter of the book we extend our hitherto propositions concerning 3D-based hierarchical models of liner elasticity onto 3D-based linear dielectric and piezoelectric media. In the case of hierarchical models of linear elasticity we apply 3D-elasticity model, hierarchical shell models, first-order shell model and the solid-to-shell or shell-to-shell transition models. In the case of dielectricity we utilize 3D-dielectricity model, and the 3D-based hierarchical dielectric models as well. The piezoelectric case needs combination of two mentioned mechanical and electric hierarchies, so as to generate the hierarchy of 3D-based piezoelectric models. Any combination of the mentioned elastic and dielectric models is possible. As far as the *hp*-discretization is concerned we extend the ideas of hierarchical approximations, constrained approximations and the transition approximations of the displacement field onto electric potential field of dielectricity or the coupled electro-mechanical field of piezoelectricity. The mentioned approximations allow *p*-adaptivity (three-dimensional or longitudinal), *q*-adaptivity (transverse one), *h*-adaptivity (three-dimensional or two-dimensional ones) and *M*-adaptivity (model adaptivity). The error assessment in three classes of problems is based on the equilibrated residual methods (ERM) applied to total and approximation error estimations. The modelling error is obtained as a difference of the previous two errors. The estimated error values are utilized for adaptivity control. The adaptive procedures for dielectricity and piezoelectricity are obtained through the generalization of the three- or four-step strategies applied so far to the elasticity case. The difficulties in generalization of the above-mentioned methods of hierarchical modelling, *hp*-approximations, error estimation and adaptivity control onto electrical and electro-mechanical problems are addressed in this chapter.

## 2. Considered problems

In the chapter we consider five problems. The first three correspond to stationary problems of mechanical, electric and electro-mechanical equilibrium of the elastic, dielectric and piezoelectric media, respectively. The last two problems deal with free vibration problems of linear elasticity and linear piezoelectricity.

### 2.1. Elastostatic problem

Let us start with the standard formulation of the linear elasticity [1]. The problem local equations include equilibrium, constitutive and geometric relations:

where *i,j,k,l* = 1,2,3 are the elasticity constants tensor, and the stress and strain tensors, respectively. The given vector of mass load is denoted as *f*_{i} , while *u*_{i} is the unknown vector of displacements. The above equations hold in volume *V* of the body.

The standard boundary conditions for stresses and displacements are:

where *n*_{j} denotes components of the normal to the body surface *S*, composed of its loaded *S*_{P} and supported *S*_{W} parts: *p*_{i} and *w*_{i} represent the given stresses and displacements on *S*_{P} and *S*_{W}, respectively.

The equivalent variational formulation results from minimization of the potential energy of the elastic body:

where *v*_{i} represent admissible displacements conforming to the displacement boundary conditions.

The above variational functional can be utilized in the derivation of the global finite element equations of the form:

where *q,hp*-approximation of displacements will be addressed in the next sections.

### 2.2. Electrostatic problem of dielectrics

The standard local formulation of linear dielectricity [2] consists of the Gauss law, here corresponding to the lack of volume charges, the constitutive relation and the electric field

Above, *i,j* = 1,2,3 stands for the dielectric constants tensor, while

The boundary conditions for the electric displacements and electric potential read:

where

The corresponding variational formulation which reflects minimization of the potential electric energy can be described as follows:

with

The finite element formulation can be expressed as follows:

where *ρ,hπ*-approximation of the potential will be explained later on in this chapter.

### 2.3. Stationary electro-mechanical problem

Formally, the local formulation of the piezoelectric problem of electro-mechanical equilibrium [3] can be treated as a combination of the linear elasticity and linear dielectricity Eqs. (1) and (5):

What couples both sets of equations are the modified constitutive relations, where the coupling piezoelectric constants tensor

The boundary conditions of the coupled problem are:

The variational functional of the electro-mechanical problem consists of the terms of functionals (3) and (7) completed with the terms describing the piezoelectric coupling through the tensor

The above variational formulation leads to the following finite element equations

where

The above finite element equations correspond to a very general case when both the direct and inverse piezoelectric phenomena are present. Substitution of the second Eq. (12) into the first one leads to the single combined equation from which the nodal displacements

### 2.4. Mechanical problem of free vibration

The local formulation of the free vibration problem of linear elasticity is composed of the following equations

where is a density of the elastic body, while *u _{i}* =

*a*sin

_{i}*ωt*, with

The boundary conditions are:

The variational formulation of the free vibration problem takes advantage of the Hamilton’s principle and reads

The finite element formulation derived from the above variational functional represents a set of uniform algebraic equations. Such a set possesses a solution if the following characteristic equation is fulfilled:

From this equation ** M** represents the mass (or inertia) matrix.

For each natural frequency

The second relation above is the normalization condition, completing

### 2.5. Coupled problem of free vibration

We start here with the local (strong) formulation of the undamped vibration problem of the piezoelectric medium

completed with the following boundary conditions of the coupled electromechanical field

In the case of stationary mass * u*) which are taken into account in the second problem of free vibration.

The local formulation of the mentioned free vibration problem of the piezoelectric can now be determined in the following way:

where the displacement and the coupled potential fields are: *u _{i}* =

*a*sin

_{i}*ωt*and

*ϕ*=

*α*sin

*ωt*, respectively, with

The above set of differential equations has to be completed with the boundary conditions of the form

The equivalent variational formulation of the problem reads

while the corresponding finite element equation describing free vibration of the initially stressed piezoelectric medium is

where

As the above set of linear algebraic equations is homogeneous, the solution to it can be obtained if and only if the following characteristic equation:

is fulfilled. This equation has been obtained after substitution of the second relation (23) into the first one. This allows to remove electric potential amplitudes

The corresponding

where the normalization condition, the same as in (17), has been applied.

## 3. Complexity of the modelling

There are three types of complexity considered in this chapter. The first one deals with physical complexity which consists in the presence of more than one physical phenomenon in the problem. The second complexity refers to geometry of the domain under consideration. The geometry is regarded as a complex one if more than one type of geometry is applied. One may deal with a three-dimensional geometry, thin-walled geometry and transition geometry, for example. The third type of complexity is model complexity. In this case, one employs more than one model for description of at least one physical phenomenon under consideration. Combination of these three types of complexity can be regarded as a unique feature of the presented research.

The examples of such complex modelling are electro-mechanical systems composed of geometrically complex elastic structures, joined with the geometrically complex piezoelectric actuators or sensors. In the general case of arbitrary geometry, such systems may require complex mechanical and electro-mechanical description.

### 3.1. Physical complexity

There are two physical sub-systems present in the considered electro-mechanical systems. The first sub-system concerns bodies subject to elastic deformation and representing structural or machine elements, while the second one concerns piezoelectric bodies acting as actuators or sensors, where the direct or inverse piezoelectric phenomena take place.

### 3.2. Geometrical complexity

In both, mechanical and piezoelectric, sub-domains we deal with the complex geometry of the structural and piezoelectric elements. In the case of the structural elements, they can be three-dimensional bodies, bounded with surfaces, thin- or thick-shell bodies [4] and solid-to-shell transition bodies. In the case of the piezoelectric members, they can be of three-dimensional, symmetric-thickness or transition character. The shell or symmetric-thickness elements are defined by means of the mid-surface and thickness concepts. In the case of both transition members, we deal with three-dimensional geometry, bounded with surfaces, apart from the boundary to be joined with the shell or symmetric-thickness elements. On this superficial boundary part the mid-surface and the symmetric thickness function have to be defined.

### 3.3. Model complexity

In the case of the mechanical sub-system complex geometry, the mechanical description may include: three-dimensional elasticity model, hierarchical shell models, the first-order shell model and the solid-to-shell or shell-to-shell transition models. The latter two models allow joining the first-order shell domains with the 3D elasticity and hierarchical shell ones, respectively. In the case of the piezoelectric sub-system, the dielectric model can either represent three-dimensional dielectricity or hierarchical symmetric-thickness dielectric models. The piezoelectric model can be any combination of the listed elastic and dielelectric models.

## 4. The applied methodology

There are five related aspects of the presented methodology of adaptive hierarchical modelling and adaptive

### 4.1. 3D-based approach

The applied 3D-based approach [5, 8] lies in application of only three-dimensional degrees of freedom (dofs) regardless of the applied mechanical or electric models. This means that the conventional mid-surface dofs of the shell models, i.e. mid-surface displacements, rotations and other generalized displacement dofs of the mid-surface, are replaced with the equivalent through-thickness displacement dofs similar to the three-dimensional dofs of the 3D elasticity model. Also the mid-surface dofs of the two-dimensional dielectric theory are replaced with the through-thickness electric potential dofs of the three-dimensional dielectrics.

The equivalence of the displacement mid-surface dofs and the through-thickness dofs can be expressed through:

where represents the local, tangent *t*_{m} stand for the *m*th power of the local, normal coordinate ( = 0 on the mid-surface) and the *m*th polynomial through-thickness function of this coordinate, respectively. The mid-surface and through-thickness displacement dof functions are denoted as and , respectively, with *m* = 1,2,…,*I* and

In the case of the electric potential field, the analogous equivalence can be seen in:

where *m*th (*m* = 1,2,…,*J*) mid-surface and through-thickness potential dof functions, while

### 4.2. Hierarchical models

In the case of the mechanical elastic models the hierarchy M of the 3D or 3D-based models

where 3*D* represents three-dimensional elasticity,

with

The hierarchical character of the above models results from the mentioned order

(30) |

Note that for the pure models

The hierarchy of 3D-based models possesses the following property:

guaranteeing that the solutions

where

In the case of the dielectric theories the hierarchy E of the 3D-based models

is composed of the three-dimensional theory 3D and the set:

of the 3D-based hierarchical dielectric models

The hierarchy can be ordered with respect to the order

and is characterized with the following property:

which says that the solutions *ϕ*^{J/L(E)} based on the subsequent models of the hierarchy give in the limit *ϕ*^{3D} of the highest model of the hierarchy, i.e. the model of three-dimensional dielectricity. In the case of the applied pure models, *D, L ≡ J*.

The norm applied for the model comparisons is based on the electrostatic energy

with ** E** being the electric displacement and electric field vectors, respectively.

### 4.3. Hierarchical approximations

The hierarchical approximations [6, 11] applied to the hierarchy of the 3D-based elastic models can be defined as follows [5, 8]:

(38) |

Above we applied the general notation,

The above approximations lead in the limit

Combination of the models defined with (30) and the corresponding approximations of (38) leads to the hierarchical numerical models of the following property:

As it can be seen above, the solutions to these models tend in the limit to the exact solution of the highest mechanical model

By analogy, the numerical approximations of the 3D-based dielectric models are:

Above, for the sake of further considerations, the longitudinal approximation order

while for the hierarchical numerical dielectric models, being the combination of the electric models (35) and the above approximations (41), one has

### 4.4. Error estimation

It is well known [17–19, 22] that the equilibrated residual methods of error estimation require solution of the following approximated local (element) problems of mechanical equilibrium:

where

The above three terms can be defined as follows

with

where denote element stiffness matrix, and element forces vectors due to mass, surface and inter-element stress loadings, while and represent solution and admissible displacement dof vectors in the local problem. The solution vector is then utilized in the error estimation. The collection of the element solutions obtained this way leads to the global error estimate which constitutes the upper bound of the true error [18, 19, 22].

The above approach can be extended onto the mechanical problem of free vibration by replacing the linear form of (44) by the virtual work of the inertia forces [26]. These forces are: , where is the element mass (or inertia) matrix, while the natural frequencies

In the case of electrostatic dielectricity, the local problems of the equilibrated residual method take the form [23]:

with the bilinear and linear forms, and , representing virtual electrostatic energy and the virtual work of external charges. The right term above is equal to the virtual work of the equilibrated charge flux . The indices

Taking into considerations the below definitions:

where ** γ** is the dielectricity constants matrix, one can transform (47) into the following local finite element equation

Above denote element dielectricity matrix, and element vectors due to surface and inter-element charges, while and are the solution and admissible electric potential dof vectors of the local problem. This element solution is applied to the error estimation of the dielectric problems. The global error estimate obtained with use of the above element solutions upper-bounds the true global error.

### 4.5. Adaptive strategy

The adaptive strategy for the complex problems of elasticity is based on the Texas Three-Step Strategy [24]. The latter strategy consists in solution of the global problem thrice on the so-called initial, intermediate and final meshes. The intermediate mesh is obtained from the initial one based on the initial mesh estimated values of element errors and the *hp* convergence theory relating these errors to the discretization parameter *h*-adaptation is performed. The final mesh is obtained from the intermediate one in the analogous way through *p*-refinement. This process takes advantage of the intermediate mesh estimated errors and the relation between these errors and the discretization parameter *p*. The original strategy can be enriched with the fourth step [5, 27], called the modification one, which is performed on the initial mesh and is applied when the unpleasant numerical phenomena due to the improper solution limit, numerical locking or boundary layers appear in the mechanical problem. The purpose of this additional step is to get rid of the numerical consequences of the mentioned phenomena before the error-controlled *hp*-adaptivity is started. The control of this step is based on the sensitivity analysis of the local (element) problems solutions to these three phenomena.

The above four-step strategy can be easily extended onto the complex dielectric problems. It has to account only for the boundary layer phenomenon within the modification step, as two other phenomena do not appear in the problems of dielectrics. Other steps do not change.

## 5. Problems within the methodology

In this section of the chapter we consider the main difficulties in generalization of the hierarchical models, hierarchical approximations, equilibration residual method of error estimation and three- or four-step adaptive strategies, presented in the previous sections for the problems of elasticity and dielectricity, onto coupled problems of piezoelectricity.

### 5.1. Hierarchical model and approximation issues

The first task here is to compose the elastic and dielectric models, defined with (28) and (33), respectively, into one consistent hierarchy of the piezoelectric media. Our proposition on how to perform this task follows from the main feature of both component hierarchies which are characterized with the independently changing orders

which determines the hierarchy P of piezoelectric models *M,E*) of the elastic

is valid in the limit, the ordering of the coupled electro-mechanical solutions measured in the energy norm is not unique due to different signs of the strain (or elastostatic)

where ^{I/K(M)}) and * E = E(ϕ^{J/L(E)})*, while

**is the coupling (piezoelectric) constants matrix.**

*C*The most general case of the

It should be noticed that the monotonic character of the co-energy of the consecutive approximate solutions is not guaranteed here due to the coupled character of the electro-mechanical field.

Practical realization of the above concepts of hierarchical modelling and approximations is implemented by means of the **Figure 1**. As said before,

The normalized versions of the prismatic adaptive elements are presented above, where the 3D-based solid (or hierarchical shell), transition (an example of) and first-order shell mechanical elements are combined with the three-dimensional (or 3D-based symmetric-thickness) dielectric elements. In the figure the normalized coordinates are defined as

### 5.2. Problems within error estimation

In order to generalize the equilibrated residual method for piezoelectricity the local elastic (44) and dielectric (47) problems have to be replaced with the coupled stationary problems describing the electro-mechanical equilibrium:

where and , while the above coupling forms are determined as follows:

Definitions (45), (48) and (55) allow to rewrite (54) in the finite element language:

The main disadvantage of the local solutions and obtained from the above set of equations is that they do not lead to the upper bound of the global approximation error of the global problem (12) due to the coupled character of piezoelectricity. Note that the upper bound property is present in the cases of pure elasticity and pure dielectricity. In these circumstances we propose the simplified approach which consists in decoupling of the local mechanical and electric fields:

The simplified solutions of the above two decoupled problems is suggested for serving the approximated error assessment of the coupled field.

In the case of the coupled free-vibration problem we propose to apply the same approach as for the purely elastic vibrations. This means that the inertia forces introduced in Subsection 4.4 have to replace the volume and surface forces in the first equations (56) and (57). The element constitutive stiffness matrix in both equations has to be changed for the following sum: + , where stands for the element geometric stiffness matrix due to the initial stresses which correspond to the electro-mechanical equilibrium. Because of this one may skip the electric charges in the second equations (56) and (57).

### 5.3. Adaptivity control matters

The main problem with the piezoelectricity in the context of adaptivity control of the three- or four-step

With this assumption in mind we generalize the adaptive scheme applied successfully for the complex elastic structures [28]. It can also be adopted for the dielectricity problems. The generalization is presented in **Figure 2** in the form of a block diagram, where

## 6. Conclusions

In the presented chapter we showed how to generalize hierarchical modelling,

We suggest to combine the hierarchical models for complex elastic structures with the analogous models of dielectricity in order to obtain all coupled combinations of the component mechanical and electric models. In the case of the approximations, we suggest to apply the common *p* and

In the error estimation, we propose to decouple the mechanical and electric fields in the local problems of the residual approach.

Our four-step