Mixed finite element analysis of 2D problems based on analytical solutions of deferential equation

In this paper, by using the Hellinger-Reissner functional, an efficient eight-node plane element with two degrees of freedom in each node is constructed. To do this, two independent displacement and stress fields in the Hellinger-Reissner functional are used. To find the stress field, instead of using hypothetical functions, the analytical solution of the governing differential equation is used. In this method, by solving the governance compatibility equation, the Airy's analytical stress functions are available. Then, by applying these stress functions, the stress field within the element is obtained. Also, for the displacement field of element, the eight-node isoparametric element interpolation functions are used. Next, by minimizing the Hellinger-Reissner mixed functional relative to the independent stress and displacement fields, the stiffness matrix and the nodal force vector are made available. To show the accuracy and efficiency of the proposed element, various benchmark tests will be analyzed. In these tests, the ability of the element for analysis of the curved structures with coarse meshes is evaluated. Also with the help of irregular meshes, the sensitivity of the response to mesh distortion is studied. These tests show the high accuracy of the proposed element in the analysis of complicated problems.