Variational formulations for TFEM of piezoelectricity

The paper presents a family of variational formulations of Trefftz finite elements wherein the assumed internal displacement and electric potential fields a priori fulfil the governing differential equations of the problem over the element sub-domain, while the inter-element continuity and the boundary conditions are enforced using a modified variational principle together with an independent frame field defined on each element boundary. It is based on four free energy densities, each with two kinds of independent variables as basic independent variables, i.e. ((delta),D),((epsilon),E), ((epsilon),D), and ((sigma),E). Based on the assumed intra-element and frame fields, an element stiffness matrix equation is obtained which is implemented into computer programs for numerical analysis. Some numerical examples are considered to show the application of the proposed formulation.