Fine to coarse mesh transition in phase-field fracture simulations using the virtual element method

In this study, the virtual element method (VEM) is utilized to address fine-to-coarse mesh transitions in phase-field fracture simulations for brittle, homogeneous media. The VEM discretization of the phase-field brittle damage equation is proposed, where the consistency and stability matrices of the damage sub-problem are derived by treating it as a general second-order linear elliptic equation. A nodal average phase-field measure is introduced to compute the degraded stress field for the elasticity subproblem. This leads to an explicit dependence of the elasticity stability matrix on the phase-field variable. A refinement strategy based on the analytical displacement fields of linear elastic fracture mechanics (LEFM) is proposed to give some guidelines on the number and positioning of hanging nodes relative to the crack front. The proposed discretization strategy is benchmarked against numerical simulations using the finite element method (FEM), smoothed finite element method (SFEM), and experimental results to demonstrate its robustness. The coupled equations for damage and displacement field is solved using a staggered algorithm implemented in the commercial software Abaqus (Standard). A Static Adaptive Mesh Refinement (SAMR) strategy is also implemented in Abaqus (Standard) to highlight the ease with which VEM can be used in phase-field fracture simulations when the crack path is not known a priori. The versatility of the strategy can lead to the efficient treatment of hanging nodes in adaptive mesh refinement (AMR) and global-local approaches, as well as enable efficient and accurate phase-field fracture simulations in large-scale engineering structures.