On crack simulation by mixed-dimensional coupling in GFEM with global-local enrichments

A strategy that combines the global-local version of the generalized finite element method (GFEM${\hspace{0.17em}}^{\text{gl}}$) with a mixed-dimensional coupling iterative method is proposed to simulate two-dimensional crack propagation in structures globally represented by Timoshenko-frame models. The region of interest called the local problem, where the crack propagates, is represented by a 2D elasticity model, where a fine mesh of plane stress/strain elements and special enrichment functions are used to describe this phenomenon accurately. A model of Timoshenko-frame elements simulates the overall behavior of the structure. A coarse mesh of plane stress/strain elements provides a bridge between these two representation scales. The mixed-dimensional coupling method imposes displacement compatibility and stress equilibrium at the interface between the two different element types by an iterative procedure based on the principle of virtual work. After establishing the constraint equations for the interface, the 2D elasticity model is related to the small-scale model by the global-local enrichment strategy of GFEM${\hspace{0.17em}}^{\text{gl}}$. In such a strategy, the numerical solutions of the local problem subjected to boundary conditions derived from the global-scale problem enrich the approximation of this same global problem in an iterative procedure. Each step of the crack propagation requires a new sequence of this iterative global-local procedure. On the other hand, the constraint equation for the interface is defined only once. The crack representation in a confined region by the global-local strategy avoids a remeshing that would require new constraint equations. Two numerical problems illustrate the proposed strategy and assess the influence of the analysis parameters.