1. Introduction

This chapter constitutes a continuation and extension of the previous work [1] on theoretical and implementation difficulties in application of the adaptive hierarchical modeling and hp-adaptive finite element analysis to elasticity, dielectricity and piezoelectricity. In the cited work, the 3Dbased elastic, dielectric and piezoelectric hierarchies of models were elucidated. These models are based on either three-dimensional theories or reduced models polynomially constrained through the thickness. In the mentioned work, also the hierarchical approximations for the three

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

classes of hierarchical models are presented. The rules for ordering the hierarchical models and approximations are described. Then, the a posteriori error estimation, based on the equilibrated residual method (ERM) applied to the three classes of problems, is presented. The similarities and differences between the element (local) problems necessary for the element error estimation for these three cases are addressed. Finally, the three- and four-step error-controlled adaptive procedures for the three classes of problems are proposed. The procedures require the global problem solution on the initial, modified, intermediate (h-refined) and final (p-enriched) meshes.

overview of the four mentioned issues is presented. The interested readers can find such

Adaptive Modeling and Simulation of Elastic, Dielectric and Piezoelectric Problems

http://dx.doi.org/10.5772/intechopen.72265

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The 3D-based hierarchical shell models utilizing three-dimensional degrees of freedom (dofs) and conforming to higher order shell theories were firstly proposed in [2] and repeated in [3]. The conventional hierarchical shells, employing mid-surface dofs, were proposed in [4]. The 3D-based approach was extended onto the first-order shell and shell-to-shell theories in [3, 5]. The latter works also extend the 3D-based hierarchical modeling onto 3D elasticity and solidto-shell transition models. The author of this chapter is not aware of any hierarchical models of linear dielectricity. Some hierarchic piezoelectric models were presented in [6] in the context of multilayered plate structures. Suggestions on introduction of the 3D-based hierarchical dielec-

The hierarchical and constrained approximations necessary for p- and h-adaptivity, respectively, are adopted in our work and were proposed in [8]. Hierarchical approximations for conventional shells were developed in [4], for 3D-based shells in [2, 9] and for complex structures in [5]. The last paper collects partial results presented in [9–12]. Classical and hierarchical approximations for piezoelectric problems were elaborated in [6, 13]. Hierarchical approximations for the complex 3D or 3D-based hierarchical models of dielectrics and piezo-

The general considerations on error estimation based on the equilibrated residual method can be found in [14]. Application of this method to 3D elasticity was described in [15]. The method was also applied to the hierarchical shells of conventional character in [16]. The analogous approach for the 3D-based first-order shells was developed in [17, 18]. The method was also utilized to error estimation in the 3D-based complex structures [19]. Application of the method

Finally, adaptivity control by means of the three-step strategy for simple structures was presented in [21]. Within this strategy, three subsequent meshes are generated—initial, intermediate (or h-refined) and target (or p-enriched) ones. The method was applied to adaptive analysis of conventional hierarchical models of shell- and plate-like structures in [16]. In that work, the third step is split into two, that is, q and p enrichments are performed in sequence. The original three-step strategy was then extended in [3] by addition of the fourth step in which the mesh is modified to get rid of the numerical consequences of the improper solution limit, numerical locking and edge effect. The model adaptivity is performed along with the h-step and p and q enrichments are performed simultaneously. Such a four-step adaptive strategy is applied to modeling and simulation of complex elastic structures in [19]. Adaptive simulation in electric or electromechanical problems is less advanced. Adaptivity for simple piezoelectrics was introduced in [22]. Application of the three- or four-step strategies to the

analysis of simple and complex dielectrics and piezoelectrics was suggested in [1].

The main novelty of the presented research consists in application of the chosen techniques of hierarchical modeling and approximation, error estimation and adaptivity control, effective in

overviews in some of the publications cited below.

tric and piezoelectric models were formulated in [1, 7].

to dielectric and piezoelectric problems was suggested in [1, 20].

electrics were proposed in the works [1, 7].

1.3. Novelty of the research

In this chapter, attention is paid to the effectivity of the algorithms for adaptive modeling and simulation of three considered classes of physical phenomena, that is, elasticity, dielectricity and piezoelectricity. Effectivity of hierarchical approximations within elastic, dielectric and piezoelectric media is compared. For this purpose, convergence curves for the analogous model problems within three mentioned classes of problems are generated and assessed. Also, the exemplary comparative results of the approximations are presented for these three classes of model problems. In the case of the error estimation, the global and local (element) effectivity indices for the total, approximation and modeling errors, where the latter is the difference of the former two, in the exemplary model problems of elasticity, dielectricity and piezoelectricity are calculated and compared. The exemplary distribution of the element error indicators and the global values of the error estimators for the model problems of three classes are presented and compared. In the case of the adaptive procedures, the model- and hpq-adaptive algorithms, where h represents the element size parameter, while p and q stand for the element longitudinal and transverse orders of approximation, are of our interest. These algorithms are controlled with the estimated values of the modeling, approximation and total errors. In order to check the effectivity of these algorithms, results necessary for the obtainment of the hp-adaptive convergence curves for the mentioned three model problems of elasticity, dielectricity and piezoelectricity are produced. The convergence is assessed in the context of obtainment of the target values of the errors in subsequent steps of the adaptive calculations for three classes of problems. Also, the comparative results of the adaptive solutions of the model problems of three classes are presented.

### 1.1. Research objectives

The main objective of this research is to demonstrate the effectivity of our generalizing algorithms [1] adapted, modified or developed for the problems of elasticity, dielectricity and piezoelectricity. Also, the issue of comparison of the corresponding effectivities for these three classes of problems is of our interest. In relation to these objectives, the presented general approach to adaptive modeling and simulation is numerically tested in the context of the approximation algorithms, error estimation algorithms and adaptivity control algorithms as well.

### 1.2. State-of-the-art issues

In this brief survey, the issues of hierarchical modeling, hierarchical approximations, error estimation and adaptivity control are addressed. The survey is limited to the numerical techniques used in this chapter and the papers directly utilized for this research—no general overview of the four mentioned issues is presented. The interested readers can find such overviews in some of the publications cited below.

The 3D-based hierarchical shell models utilizing three-dimensional degrees of freedom (dofs) and conforming to higher order shell theories were firstly proposed in [2] and repeated in [3]. The conventional hierarchical shells, employing mid-surface dofs, were proposed in [4]. The 3D-based approach was extended onto the first-order shell and shell-to-shell theories in [3, 5]. The latter works also extend the 3D-based hierarchical modeling onto 3D elasticity and solidto-shell transition models. The author of this chapter is not aware of any hierarchical models of linear dielectricity. Some hierarchic piezoelectric models were presented in [6] in the context of multilayered plate structures. Suggestions on introduction of the 3D-based hierarchical dielectric and piezoelectric models were formulated in [1, 7].

The hierarchical and constrained approximations necessary for p- and h-adaptivity, respectively, are adopted in our work and were proposed in [8]. Hierarchical approximations for conventional shells were developed in [4], for 3D-based shells in [2, 9] and for complex structures in [5]. The last paper collects partial results presented in [9–12]. Classical and hierarchical approximations for piezoelectric problems were elaborated in [6, 13]. Hierarchical approximations for the complex 3D or 3D-based hierarchical models of dielectrics and piezoelectrics were proposed in the works [1, 7].

The general considerations on error estimation based on the equilibrated residual method can be found in [14]. Application of this method to 3D elasticity was described in [15]. The method was also applied to the hierarchical shells of conventional character in [16]. The analogous approach for the 3D-based first-order shells was developed in [17, 18]. The method was also utilized to error estimation in the 3D-based complex structures [19]. Application of the method to dielectric and piezoelectric problems was suggested in [1, 20].

Finally, adaptivity control by means of the three-step strategy for simple structures was presented in [21]. Within this strategy, three subsequent meshes are generated—initial, intermediate (or h-refined) and target (or p-enriched) ones. The method was applied to adaptive analysis of conventional hierarchical models of shell- and plate-like structures in [16]. In that work, the third step is split into two, that is, q and p enrichments are performed in sequence. The original three-step strategy was then extended in [3] by addition of the fourth step in which the mesh is modified to get rid of the numerical consequences of the improper solution limit, numerical locking and edge effect. The model adaptivity is performed along with the h-step and p and q enrichments are performed simultaneously. Such a four-step adaptive strategy is applied to modeling and simulation of complex elastic structures in [19]. Adaptive simulation in electric or electromechanical problems is less advanced. Adaptivity for simple piezoelectrics was introduced in [22]. Application of the three- or four-step strategies to the analysis of simple and complex dielectrics and piezoelectrics was suggested in [1].

### 1.3. Novelty of the research

classes of hierarchical models are presented. The rules for ordering the hierarchical models and approximations are described. Then, the a posteriori error estimation, based on the equilibrated residual method (ERM) applied to the three classes of problems, is presented. The similarities and differences between the element (local) problems necessary for the element error estimation for these three cases are addressed. Finally, the three- and four-step error-controlled adaptive procedures for the three classes of problems are proposed. The procedures require the global problem solution on the initial, modified, intermediate (h-refined) and final (p-enriched) meshes. In this chapter, attention is paid to the effectivity of the algorithms for adaptive modeling and simulation of three considered classes of physical phenomena, that is, elasticity, dielectricity and piezoelectricity. Effectivity of hierarchical approximations within elastic, dielectric and piezoelectric media is compared. For this purpose, convergence curves for the analogous model problems within three mentioned classes of problems are generated and assessed. Also, the exemplary comparative results of the approximations are presented for these three classes of model problems. In the case of the error estimation, the global and local (element) effectivity indices for the total, approximation and modeling errors, where the latter is the difference of the former two, in the exemplary model problems of elasticity, dielectricity and piezoelectricity are calculated and compared. The exemplary distribution of the element error indicators and the global values of the error estimators for the model problems of three classes are presented and compared. In the case of the adaptive procedures, the model- and hpq-adaptive algorithms, where h represents the element size parameter, while p and q stand for the element longitudinal and transverse orders of approximation, are of our interest. These algorithms are controlled with the estimated values of the modeling, approximation and total errors. In order to check the effectivity of these algorithms, results necessary for the obtainment of the hp-adaptive convergence curves for the mentioned three model problems of elasticity, dielectricity and piezoelectricity are produced. The convergence is assessed in the context of obtainment of the target values of the errors in subsequent steps of the adaptive calculations for three classes of problems. Also, the comparative results of the adaptive solutions of the model

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

The main objective of this research is to demonstrate the effectivity of our generalizing algorithms [1] adapted, modified or developed for the problems of elasticity, dielectricity and piezoelectricity. Also, the issue of comparison of the corresponding effectivities for these three classes of problems is of our interest. In relation to these objectives, the presented general approach to adaptive modeling and simulation is numerically tested in the context of the approximation algorithms, error estimation algorithms and adaptivity control algorithms as

In this brief survey, the issues of hierarchical modeling, hierarchical approximations, error estimation and adaptivity control are addressed. The survey is limited to the numerical techniques used in this chapter and the papers directly utilized for this research—no general

problems of three classes are presented.

1.1. Research objectives

1.2. State-of-the-art issues

well.

The main novelty of the presented research consists in application of the chosen techniques of hierarchical modeling and approximation, error estimation and adaptivity control, effective in the adaptive modeling and simulation of the elasticity problems, to the adaptive analysis of dielectric and piezoelectric phenomena.

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 pure elasticity and pure dielectricity.
