Problems of Hierarchical Modelling and hp-Adaptive Finite Element Analysis in Elasticity, Dielectricity and Piezoelectricity

In this chapter, we consider theoretical and implementation difficulties in application of the hierarchical modelling and hp-adaptive finite element approach to elasticity, dielectricity and piezoelectricity. The main feature of the applied methodology is its generalizing character which is reflected by application of the same or analogous algorithms to three mentioned physical problems, including multi-physics problem of piezoelectricity, simple and complex physical description as well as simple and complex geometries. In contrast to the most common approaches dealing with a single physical phenomenon, described by a single physical model, within a single geometrical part, this chapter presents the ideas which brake and overcome such a simplicity. This presented chapter generalizes author's hitherto accomplishments, in hierarchical models and hp-approximations of linear elasticity, onto dielectricity and piezoelectricity. The same refers to error estimation and adaptivity control. In this context, the main similarities and differences of three physical problems are of interest in this work.