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.
Highlights
In Reference [16], it has been demonstrated that, in the case of dielectric field of electric potential used in electrostatics, no transition elements between the three-dimensional model and symmetric-thickness hierarchical models of the first and higher orders are necessary if the consistent finite element approximations of non-adaptive or adaptive character are applied
We will use piezoelectric transition elements combining the ideas present in the following works concerning the uncoupled problem of elasticity
Due to basic character of the presented research, numerical effectiveness of these models and their finite element approximations is limited to two crucial aspects, namely ability to remove high stress gradients between the basic and transition models, and convergence of the numerical solutions for the model problems of piezoelectrics
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. We develop the new transition elements which possess unique set of features enabling joining mechanical shell models of the first order and higher orders. Such models are incompatible due to the assumptions of the plane stresses and no elongation of the normals to the shell mid-surface, both present in the first-order model. In order to join such models, the transition models are necessary This refers to the piezoelectricity case, when the mechanical field modeling needs simultaneous application of the first- and higher-order mechanical models. For 2D and 3D, we will not elucidate this aspect in this paper
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