Numerical modelling of fatigue crack growth using XFEM - Ferreira, João Guilherme Gaspar Cordeiro (2015)
The Finite Element Method (FEM) is one of the most efficient tools used, in computational solid mechanics, for the numerical solution of Partial Differential Equations (PDE). This numerical technique has been extensively used in the past decades for finding approximate solutions to PDE in both engineering and science fields. A main feature of the FEM is the subdivision of a continuum into a discrete set of elements, being these elements connected by a topological map, usually referred to as the finite element mesh. The FEM can generally be used to model and predict the behaviour of continuous structures. However, problems arise when FEM is used on a domain with a discontinuity (like a crack). In this case, it is usual to use adaptive mesh refinement around the discontinuity. This process works ne, but has a very high computational cost. Alternatively, the eXtended Finite Element Method (XFEM) is a numerical method for modelling strong and weak discontinuities using local enrichment. It is a FEM generalization that enables the incorporation of local enrichment of approximation spaces. This enrichment is done through the partition of unity concept by adding special functions to the finite element approximation. For crack modelling in isotropic linear elasticity, the Heaviside function is used to enrich the completely cut elements and an asymptotic function is used to enrich the crack tip elements. This enrichment creates new degrees of freedom that must be integrated into the analysis during a post-processing step. This enables the domain to be modelled without explicitly meshing the crack surfaces and without a remeshing process for the crack propagation. In this context, this work addresses the main concepts of FEM and XFEM, the creation of a pedagogical XFEM software (with its numerical implementation process and software manual) and the differences between a standard FEM implementation and a XFEM program. Finally, some numerical results of the XFEM application are presented.