Estratégia de acoplamento multidimensional para problemas da mecânica da fratura linear elástica

Abstract

As an instance of the Partition of Unity Method (PUM), the Generalized Finite Element Method (GFEM) is a numerical technique that employs enrichment functions which, when multiplied by the Partition of Unity (PU) functions, expand the solution space of the analyzed problem. These enrichment functions are selected based on the specific characteristics of the problem under study and can also be numerically derived from local analyses, resulting in what is known as the GFEM with Global-Local strategy. Usually, the application of the GFEM is restricted to models where both the global and local problems use the same type of elements in the finite element mesh discretization. However, in various scenarios, combining finite elements of different dimensions becomes a computationally efficient approach to represent the complexity associated with the analysis of certain parts of the structure. Ensuring displacement compatibility and force equilibrium in these multidimensional models depends on an adequate choice of coupling methods at the interfaces between different types of elements. This work proposes an efficient coupling strategy for multidimensional analysis, integrating models with elements of different formulations at the interface. The approach employs an iterative process to compute the coefficient matrix of the deformable MPC (\textit{Multipoint Constraint}) method, implemented within the Global-Local analysis framework of the GFEM. This configuration enables the coupling to be performed only once, even in fracture mechanics applications. Consequently, crack propagation is confined to the local model, where the mesh is adapted exclusively in the region of interest, preserving the global model structure and eliminating the need for additional couplings. Two models and different simulations are presented to validate the proposed strategy, aiming for accurate simulation of defects with two and three-dimensional behavior in one-dimensional structures.