Abstract：The generalized finite element method (GFEM) enriches the approximation space of the conventional finite element method (FEM). However, the extra degrees of freedom (DOFs) used in the traditional GFEM lead to the issue of linear dependence and the resulting stiffness matrix may be singular. The extra-dof-free GFEM which do not employ extra DOFs removes the issue of linear dependence completely and the conditioning of the stiffness matrix is restored. In this paper, the capability of the extra-dof-free GFEM for nearly incompressible analysis is investigated. Our numerical experiments clearly demonstrates that the extra-dof-free GFEM suffers from the volumetric locking, even though a quadratic approximation function is employed. Aiming at this issue, the techniques of assumed strain developed in FEM is introduced into the extra-dof-free GFEM. Two particular schemes, respectively, called selectively reduced integration (SRI) and mean dilatation (MD) are considered. In both schemes, the dilatational part of strains is assumed to be constant throughout the element. This constant is the dilatation at the center of the element in the first scheme, but in the second scheme it is the mean dilatation of the element. The modified strain matrices based on these two schemes are derived, i.e., the well-known B-bar matrix. Two benchmark examples are employed to investigate the performance of the proposed method in the nearly incompressible analysis. In the example of Cook's membrane, it is shown that volumetric locking of the extra-dof-free GFEM is increasingly severe when the Poisson ratio approaches 0.5. In contrast, the introduced assumed strain techniques can effectively handle volumetric locking and the membrane can always deform in a correct manner. In all the nearly incompressible tests, the accuracy and convergence of the extra-DOF-free GFEM are remarkably improved by the introduced SRI and MD techniques. But more than that, in such incompressible analysis, the extra-DOF-free GFEM also exhibits better robustness against mesh distortion than the conventional FEM. This is demonstrated by the investigation of the Cook's membrane example with distorted meshes.
2023, Vol. 44 
