A novel FEM by scaling the gradient of strains with factor α (αFEM)

This paper presents a novel finite element method of quadrilateral elements by scaling the gradient of strains and Jacobian matrices with a scaling factor α (αFEM). We first prove that the solution of the αFEM is continuous for αϵ[0, 1] and bounded from both below and above, and hence is convergent. A general procedure of the αFEM has been proposed to obtain the exact or best possible solution for a given problem, in which an exact-α approach is devised for overestimation problems and a zero-α approach is suggested for underestimation problems. Using the proposed αFEM approaches, much more stable and accurate solutions can be obtained compared to that of standard FEM. The theoretical analyses and intensive numerical studies also demonstrate that the αFEM effectively overcomes the following well-known drawbacks of the standard FEM: (1) Overestimation of stiffness matrix when the full Gauss integration is used; (2) Instability problem known as hour-glass locking (presence of hour-glass modes or spurious zero-energy modes) when the reduced integration is used; (3) Volumetric locking in nearly incompressible problems when the bulk modulus becomes infinite.