Robust preconditioning for a mixed formulation of phase-field fracture problems

TL;DR

This paper develops a robust solver and preconditioner for phase-field fracture problems in incompressible materials, effectively handling nonlinearities and volume locking effects through a mixed formulation and monolithic solution approach.

Contribution

It introduces a problem-specific preconditioner leveraging saddle-point structure for efficient solution of coupled fracture equations in incompressible materials.

Findings

Solver is robust against Poisson ratio variations

Preconditioner improves convergence in numerical experiments

Method effectively handles nonlinearities and crack irreversibility

Abstract

In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the nonlinearities in the partial differential equation and employing a primal-dual active set strategy for the crack irreverrsibility constraint. The linear system in each Newton step is solved iteratively with a flexible generalized minimal residual method (GMRES). The key contribution of this work is the development of a problem-specific preconditioner that leverages the saddle-point structure of the displacement and pressure variable. Four numerical examples in pure solids and pressure-driven fractures are conducted on uniformly and locally refined meshes to investigate the robustness of the solver concerning the Poisson ratio as well as the discretization and regularization parameters.