Convergence properties of the enriched Scaled Boundary Finite Element Method in fracture mechanics

Abstract

The standard Finite Element Method (FEM) yields optimal rates of convergence for smooth domains. The presence of cracks, corners or notches within the considered domain leads to low rates of convergence which are independent of the polynomial degree of the employed trial functions. In order to regain full rates of convergence, an appropriate adaptation and a skilful choice of test and trial space of the FEM is necessary. The Scaled Boundary FEM (SBFEM) has turned out to be an appropriate method for the treatment of 2D crack problems, in which the singularity is entirely located within the considered domain. However, in 3D problems, singularities are still present at the meeting points of the crack front and the discretized boundary. Then, the SBFEM suffers from the same drawbacks as the standard FEM. In order to dispel this inconvenience, an extension of the trial space with asymptotic crack tip displacement solutions is considered resulting in improved rates of convergence.