Virtual Element Methods for Phase Field Fracture: Mesh Transitions, Composite Damage, and Stabilization-Free Formulations

Abstract

Accurate simulation of crack propagation requires discretization methods that combine geometric flexibility with computational efficiency. Although the finite element method (FEM) and its variants remain the widely used discretization technique for fracture problems, its limitations become pronounced in complex geometries and adaptive refinement simulations due to restrictive element shapes (triangular/quadrilateral). Regularized continuum damage models, particularly phase-field fracture methods, amplify these issues by requiring fine meshes to capture crack-tip nonlinearities accurately.

The Virtual Element Method (VEM), a powerful generalization of FEM supporting arbitrary polygonal/polyhedral elements, has recently emerged as a promising alternative. Yet, its full potential for fracture modeling remains underdeveloped. This thesis addresses critical gaps in VEM fracture simulation through three novel contributions. The first contribution introduces a novel VEM-based mesh transition strategy for phase-field fracture simulations. A robust, lowest-order VEM discretization of the phase-field damage equation is proposed, explicitly accounting for the coupling between mechanical deformation and damage through a nodal-averaged phase-field measure. Using analytical displacement fields from linear elastic fracture mechanics (LEFM), guidelines for positioning and the number of hanging nodes in polygonal transition elements with respect to the crack front are established. A Static Adaptive Mesh Refinement (SAMR) strategy is 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 second contribution extends the developed VEM framework to capture complex damage mechanisms in composite structures. A novel VEM discretization is proposed, combining cohesive zone models for interfacial delamination with the phase-field fracture method for bulk matrix damage. This coupled formulation effectively simulates simultaneous fracture phenomena typical in composites such as: interfacial debonding, crack penetration and deflection at interfaces, and crack kinking. The methodology uniquely showcases VEM's flexibility by allowing mismatched meshes at interfaces and representing complex inclusions with single polygonal elements containing numerous nodes, thus significantly reducing mesh generation complexity and computational costs compared to traditional methods.

Lastly, this thesis addresses the fundamental limitation of stabilization-dependent formulations in standard VEM formulations. A novel stabilization-free, lowest-order VEM discretization for the brittle phase-field fracture equation is proposed. The virtual element space is strategically enriched with higher-order polynomial functions for gradient projection, eliminating problem-dependent stabilization terms that compromise accuracy in traditional VEM. Numerical experiments reveal that traditional stabilized VEM formulations can introduce unintended oscillations in the total energy evolution. In contrast, the proposed stabilization-free formulation eliminates these oscillations, resulting in smooth and stable energy evolution. The developed methods are rigorously validated through benchmark tests involving mode-I and mixed-mode fracture conditions on Lloyd-iterated Voronoi and non-convex meshes, demonstrating remarkable robustness in crack-path predictions and global mechanical response accuracy. The proposed frameworks in this thesis are implemented within the commercial software Abaqus (Standard) using user subroutine (UEL), which can lead to large-scale commercial adoption. The thesis thus significantly advances VEM's capabilities, offering versatile, efficient, and accurate fracture modeling techniques directly applicable to real-world engineering scenarios.