Shape-free finite element method: The plane hybrid stress-function (HS-F) element method for anisotropic materials

The sensitivity problem to mesh distortion and the low accuracy problem of the stress solutions are two inherent difficulties in the finite element method. By applying the fundamental analytical solutions (in global Cartesian coordinates) to the Airy stress function ϕ of the anisotropic materials, 8- and 12-node plane quadrilateral hybrid stress-function (HS-F) elements are successfully developed based on the principle of the minimum complementary energy. Numerical results show that the present new elements exhibit much better and more robust performance in both displacement and stress solutions than those obtained from other models. They can still perform very well even when the element shapes degenerate into a triangle and a concave quadrangle. It is also demonstrated that the proposed construction procedure is an effective way for developing shape-free finite element models which can completely overcome the sensitivity problem to mesh distortion and can produce highly accurate stress solutions.