Mixed Finite Element Formulation for 2D Problems Analysis Based on Analytical Solutions of Deferential Equation

In this paper, a high-order eight-node element based on the analytical response of the governing differential equation is proposed for the analysis of plane structures. The formulation of the proposed element is based on the Hellinger-Reisner mixed functional and the analytical response of the compatibility equation governing plane problems. It is worth noting that in order to formulate finite elements with the Hellinger-Reisner functional, two independent stress and displacement fields are required. For this purpose, Airy stress functions are first made available by the analytical solution of the compatibility equation. By utilizing these stress functions, the stress field within the element is obtained. Also, the quadratic displacement field of the isoparametric element is used for intra-element displacement. By applying the Hellinger-Reisner mixed functional and stationary of this functional relative to the independent stress and displacement fields, the stiffness matrix, and the element node force vector are made available. Finally, with various numerical tests, the accuracy and efficiency of the proposed element are evaluated. These tests prove the high accuracy of the proposed element in the analysis of plane structures.