The Generalized or eXtended Finite Element Method (GFEM) has been intensively developed in the last two decades and today is available in mainstream commercial Finite Element software like Abaqus, ANSYS, and LS-DYNA. The GFEM offers several advantages over the classical Finite Element Method (FEM) in modeling problems involving crack propagation, material discontinuities, and multiscale phenomena. This presentation reports on recent advances of Generalized Finite Element Methods for two application areas. The first one is the simulation of non-planar 3-D hydraulic fracture (HF) propagation. The near-wellbore reorientation of fractures towards the preferred fracture plane leads to complex fracture geometries which in turn induce a strong nonlinear pressure drop near the wellbore. Fracture conductivity is substantially reduced by the fracture tortuosity which constricts the flow of the fracturing fluid as well as hydrocarbons through the fractures. GFEM simulations of fracture re-orientation and interaction near a wellbore are presented.
Interactions among multiple spatial scales are pervasive in many engineering applications. Structural failure is often caused by the onset of localized damage like cracks or shear bands that are orders of magnitude smaller than the structural dimensions. In the second part of this talk, a GFEM based on the solution of interdependent macro/global and fine/local scale problems is presented. The local problems focus on the resolution of fine-scale features of the solution in the vicinity of regions with singularities or localized nonlinearities, while the global problem addresses the macro-scale behavior of the structure. Fine-scale solutions are accurately computed in parallel and embedded into the global solution space using the partition of unity method. Examples demonstrating that the GFEM enables accurate solution of problems involving nonlinear, multi-scale phenomena on macro-scale meshes that are orders of magnitude coarser than those required by the Finite Element Method are presented.