*Article* **3D-Based Transition** *hpq***/***hp***-Adaptive Finite Elements for Analysis of Piezoelectrics**

**Grzegorz Zboi ´nski 1,2,\* and Magdalena Zieli ´nska 1**

> **\***


**Abstract:** This paper concerns the algorithm of transition piezoelectric elements for adaptive analysis of electro-mechanical systems. In addition, effectivity of the proposed elements in such an analysis is presented. The elements under consideration are assigned for joining basic elements which correspond to the mechanical models of either the first or higher order, while the electric model is of arbitrary order. In this work, three variants of the transition models are applied. The first one assures continuity of displacements between the basic models and continuity of electric potential between these models, as well. The second transition piezoelectric model guarantees additional continuity of the stress field between the basic models. The third transition model additionally enables continuous change of the strain state between the basic models. Based on the mentioned models, three types of the corresponding transition finite elements are introduced. The applied finite element approximations are *hpq*/*hp*-adaptive ones, which allows element-wise changes of the element size parameter *h*, and the element longitudinal and transverse orders of approximation, respectively, *p* and *q*, depending on the error level. Numerical effectiveness of the models and their approximations is investigated in the contexts of: ability to remove high stress gradients between the basic and transition models, and convergence of the numerical solutions for the model problems of piezoelectrics with and without the proposed transition elements.

**Keywords:** electro-mechanical systems; piezoelectrics; hierarchical models; first-order models; transition models; *hpq*/*hp*-approximations; adaptivity; stress gradients; convergence
