Implementations for the GFEM global-local method on the INSANE platform - Mathematical determination of the stiffness parameter

Resumo

The Generalized Finite Element Method with global-local enrichments (GFEMgl) uses the idea of the standard Global-local Finite Element to build the functions that will enrich the approximate solution numerically. Its methodology relies on three stages: (1) a global solution of the entire domain, (2) a local solution where local phenomena are detailed described, and (3) a return to the global analysis incorporating the results from the local solution. A crucial aspect of this approach is the definition of the stiffness parameter used to link the global and local problems, traditionally determined empirically based on material properties and boundary conditions.
The central point of this work is the  mathematical formulation to determine this stiffness parameter, replacing the traditional empirical method. Based on the approach proposed by Birner and Schweitzer in a paper of 2019, the stiffness parameter is estimated through the relationship between the force and displacement vectors of the global problem, ensuring they are comparable in magnitude. This approach was implemented into the INSANE platform, an in-house code developed at UFMG.
To validate the proposal, simulations of crack propagation in a two-dimensional  linear elastic medium, considering Mode I and Mixed-Mode fracture,  were performed. The results were compared with analytical reference solutions, demonstrating that the mathematical approach is feasible and offers greater accuracy in modeling crack propagation.