Generalized mixed element method of plane problems in cartesian coordinate and its applications

In finite element methods, constructing elements in Cartesian coordinates facilitates the introduction of interpolation functions with mechanical properties. This article will start from the generalized mixed variational principle and establish an 8-node mixed finite element method based on the expression of rational finite elements in the Cartesian coordinate system, in which the stress interpolation function satisfies the equilibrium equation of two-dimensional problems. This element provides convenience for the introduction of stress boundary conditions and can solve both displacement and stress variables simultaneously. The numerical analysis of several classic examples shows that the stress interpolation function satisfying the equilibrium equation can improve the accuracy of numerical results for displacement and stress fields. Especially for the sparse mesh model, the accuracy of the shear stress results is significantly better than that of the conventional displacement element.