Equilibrated residual method for estimation of modelling and approximation errors in complex piezoelectric models

The article presents the equilibrated residual method (ERM) of error estimation in coupled problems in the case of complex piezoelectric models. These models include hierarchical, first-order, and transition models within the mechanical field of displacements, as well as hierarchical models within the electric field of potential. Three (classical, modified and enhanced) transition models are considered. The paper presents a variational formulation of the model problem of general piezoelectricity in the case of complex piezoelectric models and the finite element approximation of this problem. Next, the equilibration residual method for coupled problems of piezoelectricity and complex piezoelectric models is presented. The mechanical, electric and coupled parts of the modelling, approximation and total error estimators, and true errors are given. Effectivity indices (the ratio of estimated error to true error) are used to assess the quality of error estimation in the case of three error parts and three types of error for the complex models of piezoelectric plates. The effectivity results for simple piezoelectric models and uncoupled problems of elasticity and dielectricity are applied as references.