Interior-point methods for the phase-field approach to brittle and ductile fracture

The governing equations of the variational approach to brittle and ductile fracture emerge from the minimization of a nonconvex energy functional subject to irreversibility constraints. This results in a multifield problem governed by a mechanical balance equation and evolution equations for the internal variables. While the balance equation is subject to kinematic admissibility of the displacement field, the evolution equations for the internal variables are subject to irreversibility conditions, and take the form of variational inequalities, which are typically solved in a relaxed or penalized way that can lead to deviations of the actual solution. This paper presents an interior-point method that allows to rigorously solve the system of variational inequalities. With this method, a sequence of perturbed constraints is considered, which, in the limit, recovers the original constrained problem. As such, no penalty parameters or modifications of the governing equations are involved. The interiorpoint method is applied in both a staggered and a monolithic scheme for both brittle and ductile fracture models. In order to stabilize the monolithic scheme, a perturbation is applied to the Hessian matrix of the energy functional. The presented algorithms are applied to three benchmark problems and compared to conventional methods, where irreversibility of the crack phase-field is imposed using a history field or an augmented Lagrangian. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

The paper introduces an interior-point method to rigorously solve the variational inequalities in brittle and ductile fracture models, eliminating the need for penalty parameters or modifications of the governing equations. The method is applied to different fracture models, effectively stabilizing the numerical schemes.