Well-conditioned and optimally convergent second-order Generalized/eXtended FEM formulations for linear elastic fracture mechanics

The Generalized/eXtended Finite Element Method (G/XFEM) has been established as an approach to provide optimally convergent solutions for classes of problems that are challenging for the standard version of the Finite Element Method (FEM). For problems with non-smooth solutions, as those within the Linear Elastic Fracture Mechanics (LEFM) context, the FEM convergence rates are often not optimal and are bounded by the strength of the singularity in the analytical solution. This can be overcome by G/XFEM and many researches have focused on delivering first-order convergent solutions for LEFM problems. The difficulty in obtaining higher-order accurate approximations, however, relies mainly on also controlling the growth rate of the stiffness matrix condition number. In this paper, well-conditioned and optimally convergent second-order G/XFEMs are proposed for LEFM simulations by augmenting second-order Lagrangian FEM approximation spaces. More specifically, two strategies are proposed in order to accurately represent second-order discontinuous functions along a crack. Also, a third strategy that essentially improves the use of linear Heaviside functions in the sense that these enrichments no longer cause linear dependencies among the set of G/XFEM shape functions is proposed. The numerical experiments show the robustness of the formulations presented herein for a set of crack topologies.