5. Conclusions

In this chapter, the algorithms of hierarchical modeling, hierarchical approximations, error estimation and adaptivity control, so far utilized successfully for elastic problems, are applied to the problems of dielectricity and piezoelectricity. To the best knowledge, no examples of such application had been reported in the existing literature.

This chapter shows how to assess effectivity of the algorithms of hierarchical approximations, equilibrated residual method of error estimation and three-step adaptive procedure, originally applied to elasticity and possible also in dielectricity and the coupled piezoelectric problems.

The observations from the tests concerning hierarchical hp-approximations allow for the following generalizations. The applied hierarchical approximations can be effective in modeling and simulation of all three classes of problems. The h- and p-convergence rate is the highest for the purely dielectric problems and the lowest for the analogous purely elastic ones. The convergence of the piezoelectric problems is located between the convergences of the corresponding pure problems. In the case of the pure problems, the convergence curves are monotonic, while in the case of the coupled problems of piezoelectricity, the loss of monotonicity can happen because of the sign change of the electromechanical potential energy.

The general conclusions concerning the applied error estimation method are as follows. In the pure problem of elasticity, the effectivities of the modeling, approximation and total errors are all close to 1.0, except for the cases of poor discretization (p ¼ 1, 2). In the pure dielectric problems, the values of the effectivities are between 0.9 and 1.0. For the cases of poor discretization, these values are close to 2.0. In the case of piezoelectricity, the indices are close to their values from the pure problems for low values of the approximation order, that is, for p ≤ 3 or/and π ≤ 3. For higher values of the approximation order, the corresponding effectivities are much higher than 1.0 for the approximation error. A bit smaller overestimation can be seen in the case of the total error. It can be concluded that for rich discretizations, some additional techniques are necessary in the case of piezoelectricity, so as to enforce values of the effectivities closer to 1.0.

Generalizations related to the applied adaptivity control algorithms can be formulated in the following way. For the analogous mechanical, electric and electromechanical problems, the three-step adaptive strategy leads to similar convergence results as it comes to the final mesh true error level, even though the hp-path in each of three cases can be different. The final error values are usually smaller or close to the admissible error level. In the case of piezoelectricity, too fine or/and too rich meshes may be generated, if overestimation, coming from the error estimation procedure, occurs.

The results concerning the effectivity of the application of the mentioned three algorithms to the dielectric and piezoelectric problems as well as the comparative effectivity analysis of the analogous mechanical, electric and electromechanical problems are unique—no other examples of such results can be found in the related literature.
