Gradient projection method for enforcing crack irreversibility as box constraints in a robust monolithic phase-field scheme

Abstract

A phase-field monolithic scheme based on the gradient projection method is developed to model crack propagation in brittle materials under cyclic loading. As a type of active set method, the gradient projection method is particularly attractive to enforce the irreversibility condition imposed on the phase-field variables as bound constraints, or box constraints. This method has the advantages of allowing the rapid change of active constraints during iterations and computing the projected gradient with a negligible cost. The gradient projection method is further combined with the limited-memory BFGS (L-BFGS) method to overcome the convergence difficulties arising from the non-convex energy functional. A compact representation of the BFGS matrix is adopted as the limited-memory feature to avoid the storage of fully dense matrices, making this method practical for large-scale finite element simulations. By locating the generalized Cauchy point on the piecewise linear path formed by the projected gradient, the active set of box constraints can be determined. The variables in the active set, which are at the boundary of the box constraints, are kept fixed to form a subspace minimization problem. A primal approach and a dual approach are presented to solve this subspace minimization problem for the remaining free variables at the generalized Cauchy point. Several two-dimensional (2D) and three-dimensional (3D) examples are provided to demonstrate the capabilities of the proposed monolithic scheme, particularly in enforcing the phase-field irreversibility during crack propagation under cyclic loading. In these numerical examples, the proposed monolithic scheme is combined with an adaptive mesh refinement technique to alleviate the heavy computational cost incurred by the fine mesh resolution required around the crack region. The proposed method is further compared with two other phase-field solving techniques regarding the convergence behavior. To ensure a fair comparison, the same problem settings and implementation techniques are adopted. The proposed monolithic scheme provides a unified framework to overcome the numerical difficulties associated with the non-convex energy functional, effectively enforce the phase-field irreversibility to ensure the thermodynamic consistency, and alleviate the heavy computational cost through adaptive mesh refinement in 2D and 3D phase-field crack simulations.