2. Model problems

The following model problems are considered in this chapter: the linear static problem of elasticity, the linear electrostatic problem and the linear problem of stationary piezoelectricity. For each of the model problems, the appropriate finite element formulation is presented. For this purpose, the standard engineering matrix notation is applied.

#### 2.1. Elastostatics

Here, the problems of a three-dimensional (solid) and 3D-based shell or solid-to-shell bodies are considered. Such problems were presented in [1]. In that work, the local (strong) and variational (weak) formulations of the problems are given. These formulations take advantage of the former considerations from [2, 23, 24] and are repeated in [3]. Using the variational formulation presented therein, one can derive the global finite element equations of the problem under consideration and write them in the following form:

$$\mathbf{K}\_{M}\boldsymbol{q}^{\rho\_{\cdot}\ln p} = \mathbf{F}\_{V} + \mathbf{F}\_{S} \tag{1}$$

The nodal mass forces vector can be defined in the standard way

f S e ¼ ð1 0

normalized coordinate ξ<sup>3</sup> and the coordinates η<sup>i</sup>

matrix J (see [3, 9] again).

2.2. Electrostatics

with γ and b

e

f S e ¼ ð1 0 ð1 0 N<sup>T</sup> e

ð�ξ2þ<sup>1</sup> 0

ð�ξ2þ<sup>1</sup> 0

N<sup>T</sup> e

The element nodal forces vector due to the surface traction p can be defined in the following

N<sup>T</sup> e

corresponding to the bases and sides of the prismatic element. Above the element, bases and sides are defined with the normalized longitudinal coordinates ξj, j ¼ 1, 2, or the transverse

element [3, 9]. The term wspð ÞJ is the coefficient defined with the components of the Jacobian

The general formulations of the problems of electrostatics can be found in [25]. Here, classical linear dielectric models are applied to such problems. The local and variational formulations for this case was presented in [1] for any 3D or 3D-based geometry (bulky, symmetric-

In Eq. (5), K<sup>E</sup> represents the global characteristic matrix of dielectricity, while F<sup>Q</sup> stands for the global characteristic electric charges nodal vector. The vector w<sup>r</sup>, <sup>h</sup><sup>π</sup> is the unknown global nodal vector of electric potentials. This vector definition results from the applied r, hπ-approximation, where r and π represent the transverse and longitudinal orders of approximation. The global

thickness or transition ones). The corresponding finite element equations read:

chapter. The global matrix K<sup>E</sup> is the result of summation of the element contributions

ð�ξ2þ<sup>1</sup> 0

bT e γb e

relation between the electric field components and the nodal electric potentials (or shortly

denoting the electric (or permittivity) constants matrix and the matrix of the

potential vector is composed of the element potential vectors w

kE e ¼ ð1 0 ð1 0

and f represent the element shape functions matrix and the mass loading vector,

p wspð ÞJ dξ2dξ<sup>1</sup>

p wspð ÞJ dξ3dη<sup>i</sup>

f detð ÞJ dξ1dξ2dξ3, (3)

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Adaptive Modeling and Simulation of Elastic, Dielectric and Piezoelectric Problems

, i ¼ 1, 2, 3 tangential to the sides of the

<sup>K</sup>Ew<sup>r</sup>,h<sup>π</sup> <sup>¼</sup> <sup>F</sup><sup>Q</sup> (5)

, which are described later in this

detð ÞJ dξ1dξ2dξ<sup>3</sup> (6)

e

(4)

161

f e <sup>M</sup> ¼ ð1 0 ð1 0

where N e

respectively.

two forms

where K<sup>M</sup> is the global stiffness matrix, while F<sup>V</sup> and F<sup>S</sup> represent the global vectors of the volume and surface nodal forces. The vector qq,hp stands for the global displacement degrees of freedom (dof), corresponding to hpq approximation, and is composed of the element (local) displacement dof vectors q <sup>e</sup> of the elements <sup>e</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, E, where <sup>E</sup> is the total number of elements within an elastic body. These vectors are defined later in this chapter.

The global stiffness matrix is composed (aggregated) of the element stiffness matrices of the form

$$\stackrel{\epsilon}{\dot{\mathbf{k}}} = \int\_{0}^{1} \int\_{0}^{1} \int\_{0}^{-\zeta\_{2} + 1} \stackrel{\epsilon}{\mathbf{B}}^{T} \mathbf{D} \stackrel{\epsilon}{\mathbf{B}} \det(\mathbf{J}) \, d\xi\_{1} d\xi\_{2} d\xi\_{3} \tag{2}$$

where D denotes the elastic constants matrix, B e represents the strain-displacement matrix, and detð ÞJ is the Jacobian matrix determinant. The limits and coordinates of the integration correspond to the normalized coordinates ξi, i ¼ 1, 2, 3 of the prismatic elements applied in [1]. The specific forms of the strain-displacement matrix can be found in the works [9, 10, 12] for the 3D-based versions of the prismatic solid (and hierarchical shell), first-order shell and solidto-shell (and shell-to-shell) adaptive elements, respectively.

The nodal mass forces vector can be defined in the standard way

$$\stackrel{\epsilon}{f}\_M = \int\_0^1 \int\_0^1 \int\_0^{-\xi\_2 + 1} \stackrel{\epsilon}{N} f \det(f) \, d\xi\_1 d\xi\_2 d\xi\_3 \tag{3}$$

where N e and f represent the element shape functions matrix and the mass loading vector, respectively.

The element nodal forces vector due to the surface traction p can be defined in the following two forms

$$\begin{aligned} \stackrel{\varepsilon}{f}\_S &= \int\_0^1 \int\_0^{-\xi\_2 + 1} \mathbf{N}^T \mathbf{p} \text{ wsp}(\mathbf{J}) \, d\xi\_2 d\xi\_1 \\ \stackrel{\varepsilon}{f}\_S &= \int\_0^1 \int\_0^1 \mathbf{N}^T \mathbf{p} \text{ wsp}(\mathbf{J}) \, d\xi\_3 d\eta\_i \end{aligned} \tag{4}$$

corresponding to the bases and sides of the prismatic element. Above the element, bases and sides are defined with the normalized longitudinal coordinates ξj, j ¼ 1, 2, or the transverse normalized coordinate ξ<sup>3</sup> and the coordinates η<sup>i</sup> , i ¼ 1, 2, 3 tangential to the sides of the element [3, 9]. The term wspð ÞJ is the coefficient defined with the components of the Jacobian matrix J (see [3, 9] again).

#### 2.2. Electrostatics

the adaptive modeling and simulation of the elasticity problems, to the adaptive analysis of

The novelty of this particular chapter is the direct comparison of the robustness of the modeling and simulation algorithms of the coupled problem of piezoelectricity and the problems of

The following model problems are considered in this chapter: the linear static problem of elasticity, the linear electrostatic problem and the linear problem of stationary piezoelectricity. For each of the model problems, the appropriate finite element formulation is presented. For

Here, the problems of a three-dimensional (solid) and 3D-based shell or solid-to-shell bodies are considered. Such problems were presented in [1]. In that work, the local (strong) and variational (weak) formulations of the problems are given. These formulations take advantage of the former considerations from [2, 23, 24] and are repeated in [3]. Using the variational formulation presented therein, one can derive the global finite element equations of the prob-

where K<sup>M</sup> is the global stiffness matrix, while F<sup>V</sup> and F<sup>S</sup> represent the global vectors of the volume and surface nodal forces. The vector qq,hp stands for the global displacement degrees of freedom (dof), corresponding to hpq approximation, and is composed of the element (local)

The global stiffness matrix is composed (aggregated) of the element stiffness matrices of the

e

and detð ÞJ is the Jacobian matrix determinant. The limits and coordinates of the integration correspond to the normalized coordinates ξi, i ¼ 1, 2, 3 of the prismatic elements applied in [1]. The specific forms of the strain-displacement matrix can be found in the works [9, 10, 12] for the 3D-based versions of the prismatic solid (and hierarchical shell), first-order shell and solid-

BT e DBe

elements within an elastic body. These vectors are defined later in this chapter.

ð�ξ2þ<sup>1</sup> 0

<sup>K</sup><sup>M</sup> <sup>q</sup>q, hp <sup>¼</sup> <sup>F</sup><sup>V</sup> <sup>þ</sup> <sup>F</sup><sup>S</sup> (1)

detð ÞJ dξ1dξ2dξ<sup>3</sup> (2)

represents the strain-displacement matrix,

<sup>e</sup> of the elements <sup>e</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, E, where <sup>E</sup> is the total number of

this purpose, the standard engineering matrix notation is applied.

lem under consideration and write them in the following form:

k e ¼ ð1 0 ð1 0

to-shell (and shell-to-shell) adaptive elements, respectively.

where D denotes the elastic constants matrix, B

dielectric and piezoelectric phenomena.

160 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

pure elasticity and pure dielectricity.

2. Model problems

2.1. Elastostatics

displacement dof vectors q

form

The general formulations of the problems of electrostatics can be found in [25]. Here, classical linear dielectric models are applied to such problems. The local and variational formulations for this case was presented in [1] for any 3D or 3D-based geometry (bulky, symmetricthickness or transition ones). The corresponding finite element equations read:

$$\mathbf{K}\_E \boldsymbol{\varphi}^{\rho, h\pi} = \mathbf{F}\_Q \tag{5}$$

In Eq. (5), K<sup>E</sup> represents the global characteristic matrix of dielectricity, while F<sup>Q</sup> stands for the global characteristic electric charges nodal vector. The vector w<sup>r</sup>, <sup>h</sup><sup>π</sup> is the unknown global nodal vector of electric potentials. This vector definition results from the applied r, hπ-approximation, where r and π represent the transverse and longitudinal orders of approximation. The global potential vector is composed of the element potential vectors w e , which are described later in this chapter. The global matrix K<sup>E</sup> is the result of summation of the element contributions

$$\stackrel{\mathcal{e}}{\mathbf{k}}\_{E} = \int\_{0}^{1} \int\_{0}^{1} \int\_{0}^{-\xi\_{2} + 1} \stackrel{\mathcal{e}}{b} \stackrel{\mathcal{e}}{\gamma} \stackrel{\mathcal{e}}{\mathbf{b}} \det(\mathbf{J}) \, d\xi\_{1} d\xi\_{2} d\xi\_{3} \tag{6}$$

with γ and b e denoting the electric (or permittivity) constants matrix and the matrix of the relation between the electric field components and the nodal electric potentials (or shortly field-potential matrix). The specific form of the latter matrix in the case of the prismatic element can be found in the work [26].

The nodal electric charges vector of the element e has to be defined in a different way on the prismatic element bases and sides, that is,

$$\boldsymbol{\hat{f}}\_{\mathcal{Q}} = \int\_{0}^{1} \int\_{0}^{-\xi\_{2} + 1} \boldsymbol{n}^{T} \boldsymbol{c} \, \mathrm{wsp}(\mathcal{J}) \, d\xi\_{2} d\xi\_{1} \tag{7}$$

approximations, pq and πr, within both fields. Thanks to this, the corresponding adaptation

Adaptive Modeling and Simulation of Elastic, Dielectric and Piezoelectric Problems

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163

The presented elastic, dielectric and piezoelectric models are all based on the 3D-based approach, which results in the application of the three-dimensional or 3D-based degrees of freedom (dofs) only. The mechanical shell and transition models are also equipped with such dofs. This means that mid-surface degrees of freedom of the conventional shell and transition models are not applied. The so-called through-thickness dofs are employed instead. Also, some constraints are imposed on the three-dimensional displacements field of the shell and transition models so as to obtain the equivalence of the conventional and 3D-based descriptions. The related issues are presented in detail in the works [3, 5]. Analogously, in [7], the 3D-based hierarchy of dielectric models was proposed. It includes the three-dimensional and symmetric-thickness hierarchical models. Three-dimensional and 3D-based through-thickness dofs are employed in these models. In the latter work, also the 3D-based mechanical and dielectric models were combined, so as to obtain the 3D-based hierarchy of the piezoelectric models. This idea was also recalled in [1]. Note that all the presented 3D-based models, either elastic, dielectric or piezoelectric ones, can be

with 3D denoting three-dimensional elasticity, MI representing hierarchical shell models of higher order, RM being the first-order shell model corresponding to Reissner theory of shells and 3D=RM and MI=RM standing for the transition models of solid-to-shell or shell-to-shell character. The hierarchical shell and shell-to-shell models constitute two sub-hierarchies:

MI ¼ f g M2; M3; M4; … ,

where I represents the order of the hierarchical model MI. This order is equivalent to the order

where 3D represents three-dimensional theory of dielectricity, while EJ denotes the 3D-based

M ∈ M, M ¼ f g 3D; MI; RM; 3D=RM; MI=RM (11)

MI=RM <sup>¼</sup> f g <sup>M</sup>2=RM; <sup>M</sup>3=RM; <sup>M</sup>4=RM;… (12)

E∈E, E ¼ f g 3D; EJ (13)

EJ ¼ f g E1; E2; E3;… (14)

processes within both fields can be performed independently.

treated as the 3D models polynomially constrained through the thickness.

of polynomial constraints defining the transverse displacement.

Subsequently, the hierarchy E of 3D-based dielectric models E includes:

hierarchical models. The latter models constitute the following subhierarchy:

The mechanical hierarchy M of the 3D or 3D-based elastic models M reads:

3. The applied numerical techniques

3.1. Hierarchies of models

and

$$\boldsymbol{\hat{f}}\_{Q} = \int\_{0}^{1} \int\_{0}^{1} \boldsymbol{\mathfrak{n}}^{\prime} \boldsymbol{c} \, \mathrm{wsp}(\boldsymbol{J}) \, d\xi\_{3} d\eta\_{i} \tag{8}$$

where n <sup>e</sup> and c are the element shape functions vector and the scalar density of the surface electric charges.

#### 2.3. Stationary piezoelectricity

The local and variational formulations of linear piezoelectricity combine our former considerations concerning the linear elasticity and linear dielectricity [13, 27]. This approach was repeated in [1]. The corresponding finite element formulation can be written in the form a coupled system of equations. The coupling is represented by the matrix K<sup>C</sup> in the following way

$$\begin{aligned} \mathbf{K}\_M \mathfrak{q}^{\rho, hp} - \mathbf{K}\_\mathbb{C} \mathfrak{q}^{\rho, h\pi} &= \mathbf{F}\_V + \mathbf{F}\_{S\prime} \\ \mathbf{K}\_\mathbb{C}^T \mathfrak{q}^{\rho, hp} + \mathbf{K}\_E \mathfrak{q}^{\rho, h\pi} &= \mathbf{F}\_Q \end{aligned} \tag{9}$$

The coupling term can be called the global characteristic matrix of piezoelectricity, while the rest terms retain their previous meaning. The additional remark concerns special or simplified versions of the above equation. The inverse or direct piezoelectric problems can be considered here with the right-hand side terms equal to zero in the first and second equation, respectively. It is also worth mentioning that different pq and πr adaptive approximations of the vectorial displacement and scalar electric fields are proposed in (9), with the common h-approximation.

The global matrix of piezoelectricity introduced above can be obtained through the standard finite element summation procedure, where the following element contributions are employed

$$\mathbf{k}\_{\mathbb{C}}^{\varepsilon} = \int\_{0}^{1} \int\_{0}^{1} \int\_{0}^{-\xi\_{2} + 1} \mathbf{B}^{T} \mathbf{C} \overset{\varepsilon}{b} \det(\mathbf{J}) \, d\xi\_{1} d\xi\_{2} d\xi\_{3} \tag{10}$$

with C representing the piezoelectric (coupling) constants matrix.

The element contributions to the other terms of (9) are defined as before, that is, in accordance with (2)–(4) and (6)–(7). Note that the different shape functions matrices, N e and n e , for the displacements and potential fields are employed here due to the different orders of approximations, pq and πr, within both fields. Thanks to this, the corresponding adaptation processes within both fields can be performed independently.
