The above three terms can be defined as follows
with being the elasticity constants matrix. Thus, Eq. (44) can be written in the language of finite elements:
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 . These forces are:
In the case of electrostatic dielectricity, the local problems of the equilibrated residual method take the form :
with the bilinear and linear forms,
Taking into considerations the below definitions:
where γ is the dielectricity constants matrix, one can transform (47) into the following local finite element equation
The adaptive strategy for the complex problems of elasticity is based on the Texas Three-Step Strategy . 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 . Thanks to this the 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.
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.
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 and of the elastic and dielectric models, as shown in (30) and (35). Because of this we can propose the definition
which determines the hierarchy P of piezoelectric models as composed of all combinations (M,E) of the elastic and dielectric models. Even though the following property:
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) , electrostatic and coupling parts of the co-energy: , i.e.
where I/K(M)) and E = E(ϕJ/L(E)), while C is the coupling (piezoelectric) constants matrix.
The most general case of the -approximation within the coupled field of displacements and electric potential may include totally independent and approximations within the mechanical and electric fields. However, such an approach requires the vice-versa projections of the displacements and potential solutions between the independent -meshes of each field. The approach results in the additional projection error which should be included in the error estimation. Because of this we propose the simplified approach which consists in application of the common mesh and the independent and approximations within the displacements and electric potential fields. This assumption leads to the following limit property of the coupled solution
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 - and -adaptive piezoelectric finite elements, presented in Figure 1. As said before, represents the assumed characteristic element size, common for both fields, while and denote independent longitudinal approximation orders within the displacements and electric potential fields. The transverse approximation orders and of the mechanical and electric fields, respectively, are equivalent to the independent hierarchical orders and of the elastic and dielectric models, i.e. .
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 , while the vertex, mid-edge, mid-base, mid-side and middle nodes of mechanical character are denoted as either or or , while the corresponding electric ones are marked with .
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:
The main disadvantage of the local solutions
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
The main problem with the piezoelectricity in the context of adaptivity control of the three- or four-step -adaptive procedure is that the convergence theorem for the finite element approximation of the coupled piezoelectric field is not at one’s disposal. In these circumstances we propose to use the -convergence exponents of the elasticity and dielectricity problems for the mechanical and electric components of the coupled field.
With this assumption in mind we generalize the adaptive scheme applied successfully for the complex elastic structures . It can also be adopted for the dielectricity problems. The generalization is presented in Figure 2 in the form of a block diagram, where represent the consecutive steps of the algorithm—the initial, modification, intermediate and final ones.
In the presented chapter we showed how to generalize hierarchical modelling, -adaptive hierarchical approximations, the equilibrated residual method of error estimation and the four-step adaptive procedure, originally applied to elasticity and possible also in dielectricity, for the coupled pieozoelectric problems.
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 -mesh and independent p and approximations for the displacement and electric potential fields.
In the error estimation, we propose to decouple the mechanical and electric fields in the local problems of the residual approach.
Our four-step -adaptive algorithm is based on the convergence theorems for the purely elastic and purely dielectric problems. These theorems are applied to the intermediate and final meshes generation.
The support of the Polish Scientific Research Committee (now the National Science Centre) under the research grant no. N N504 5153 040 is thankfully acknowledged.
– displacement amplitude vector components ,
– a node with displacement degrees of freedom (dofs),
– a bilinear form representing the element virtual electrostatic energy,
– a node with electric potential dofs,
– a bilinear form representing the element virtual strain energy,
c – the electric surface charge,
– components of the piezoelectricity tensor (i,k,l = 1,2,3),
– the piezoelectric constants matrix,
– components of the electric displacement vector ,
– the electric displacement vector,
– elasticity tensor components, i,j,k,l = 1,2,3,
– the elasticity constants matrix,
– the number of an element,
a dielectric model,
E – the hierarchy of dielectric models,
– electric field vector components ,
– the electric field vector,
– mass load vector components ,
– the mass load vector,
– the global mass forces vector,
– the global surface forces vector,
– the global charge vector,
– measure of the element size,
– a measure of the element size in the local problem,
– the order of the hierarchical mechanical model,
– the order of the hierarchical dielectric model,
– the global piezoelectric (coupling) matrix,
– the global dielectric matrix,
– the global geometric stiffness matrix,
– the global (constitutive) stiffness matrix,
– the number of a dof function,
– mechanical model,
– the hierarchy of mechanical models,
– the global mass (inertia) matrix,
– the number of a natural frequency,
– vector components of the normal to the body surface,
– the global number of dofs (degrees of freedom),
– the element approximation order or the longitudinal order of approximation for displacements,
– surface load vector components,
– the longitudinal order of approximation for element displacements,
– piezoelectric model,
– the hierarchy of piezoelectric models,
– the surface load vector,
– the transverse order of approximation for displacements,
– the global displacement (or displacement amplitude) dofs vector,
– the local problem transverse order of approximation for displacements,
– the surface of a body (or medium),
– the body surface part with given electric potential,
– the loaded part of the body surface,
– the electrically charged part of the body (or medium) surface,
– the body surface part with given displacements,
t – time,
– the mth polynomial through-thickness function,
– the vector of global displacements,
– the global displacement components, ,
– the displacement vector in the local (element) problem,
– acceleration vector components, ,
– admissible displacement (or its amplitude) vector components ,
– the admissible displacement vector in the local (element) problem,
– volume of a body (or medium),
– given displacement vector components on the body surface,
– global Cartesian coordinate vector,
– components of the dielectricity tensor ,
γ – the dielectricity constants matrix,
– the mid-surface electric potential dof function,
– strain tensor components ,
– the strain vector,
– normalized coordinates of an element ,
– the longitudinal approximation order of electric potential,
– the element longitudinal approximation order of electric potential,
– the transverse approximation order of electric potential,
– the element transverse approximation order of electric potential,
– initial stress tensor components (i,j = 1,2,3),
– stress tensor components (i,j = 1,2,3),
– the stress vector,
– the global electric potential (or potential amplitude) dofs vector,
– the electric potential,
– the mth through-thickness electric potential dof function,
– the given electric potential on the body surface,
– the admissible electric potential,
– a natural frequency,
– the nth natural frequency
© 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
767total chapter downloads
Login to your personal dashboard for more detailed statistics on your publications.Access personal reporting
Edited by Radostina Petrova
By Mahboub Baccouch
Edited by Radostina Petrova
By Árpád Veress and József Rohács
We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.More About Us