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corresponding admissible error value is far exceeded. The obtained solution for displacements is

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

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

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

such application had been reported in the existing literature.

Figure 26. hπ-Adaptive convergence in the coupled problem.

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

change of the electromechanical potential energy.

better than expected.

5. Conclusions

problems.

Grzegorz Zboiński1,2\*

