TY - JOUR

T1 - An augmented-Lagrangian method for the phase-field approach for pressurized fractures

AU - Wheeler, M. F.

AU - Wick, T.

AU - Wollner, W.

N1 - Funding Information: The third author thanks the Center for Subsurface Modeling at ICES, UT Austin, for hosting his research stay in July 2013 and the DAAD (German Academic Exchange Service) for a travel grant vital to the initialization of this collaboration. The second author is grateful to the Alexander von Humboldt foundation supporting since October 2013 other studies build upon the present work. Copyright: Copyright 2014 Elsevier B.V., All rights reserved.

PY - 2014/4/1

Y1 - 2014/4/1

N2 - In the modeling of pressurized fractures using phase-field approaches, the irreversibility of crack growth is enforced through an inequality constraint on the temporal derivative of the phase-field function. In comparison to the classical case in elasticity, the presence of the pressure requires the additional constraint and makes the problem much harder to analyze. After temporal discretization, this induces a minimization problem in each time step over a solution dependent admissible set. To avoid solving the resulting variational inequality corresponding to the first order necessary conditions, a penalization approach is used, commonly, to remove the inequality constraint. It is well-known that for large penalty parameters the algorithm suffers from numerical instabilities in the solution process. Consequently, to avoid such a drawback, we propose an augmented Lagrangian algorithm for the discrete in time and continuous in space phase-field problems. The final set of equations is solved in a decoupled fashion. The proposed method is substantiated with several benchmark and prototype tests in two and three dimensions.

AB - In the modeling of pressurized fractures using phase-field approaches, the irreversibility of crack growth is enforced through an inequality constraint on the temporal derivative of the phase-field function. In comparison to the classical case in elasticity, the presence of the pressure requires the additional constraint and makes the problem much harder to analyze. After temporal discretization, this induces a minimization problem in each time step over a solution dependent admissible set. To avoid solving the resulting variational inequality corresponding to the first order necessary conditions, a penalization approach is used, commonly, to remove the inequality constraint. It is well-known that for large penalty parameters the algorithm suffers from numerical instabilities in the solution process. Consequently, to avoid such a drawback, we propose an augmented Lagrangian algorithm for the discrete in time and continuous in space phase-field problems. The final set of equations is solved in a decoupled fashion. The proposed method is substantiated with several benchmark and prototype tests in two and three dimensions.

KW - Augmented Lagrangian

KW - Finite elements

KW - Iterative solution

KW - Phase-field

KW - Variational fracture

UR - http://www.scopus.com/inward/record.url?scp=84892887557&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2013.12.005

DO - 10.1016/j.cma.2013.12.005

M3 - Article

AN - SCOPUS:84892887557

VL - 271

SP - 69

EP - 85

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

ER -