TY - JOUR

T1 - Matrix-free multigrid solvers for phase-field fracture problems

AU - Jodlbauer, D.

AU - Langer, U.

AU - Wick, T.

N1 - Funding Information: This work has been supported by the Austrian Science Fund (FWF) grant P29181 ‘Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems’. We thank the anonymous referees for their critical questions that helped to improve the paper.

PY - 2020/12/1

Y1 - 2020/12/1

N2 - In this work, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model with geometric multigrid methods. Using a standard matrix-based approach within the Finite Element Method requires lots of memory, which eventually becomes a serious bottleneck. A matrix-free approach overcomes this problem and greatly reduces the amount of required memory, allowing to solve larger problems on available hardware. One key challenge is concerned with the crack irreversibility for which a primal–dual active set method is employed. Here, the active set values of fine meshes must be available on coarser levels of the multigrid algorithm. The developed multigrid method provides a preconditioner for a generalized minimal residual (GMRES) solver. This method is used for solving the linear equations inside Newton's method for treating the overall nonlinear-monolithic discrete displacement/phase-field formulation. Several numerical examples demonstrate the performance and robustness of our solution technology. Mesh refinement studies, variations in the phase-field regularization parameter, iterations numbers of the linear and nonlinear solvers, and some parallel performances are conducted to substantiate the efficiency of the proposed solver for single fractures, multiple pressurized fractures, and a L-shaped panel test in three dimensions.

AB - In this work, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model with geometric multigrid methods. Using a standard matrix-based approach within the Finite Element Method requires lots of memory, which eventually becomes a serious bottleneck. A matrix-free approach overcomes this problem and greatly reduces the amount of required memory, allowing to solve larger problems on available hardware. One key challenge is concerned with the crack irreversibility for which a primal–dual active set method is employed. Here, the active set values of fine meshes must be available on coarser levels of the multigrid algorithm. The developed multigrid method provides a preconditioner for a generalized minimal residual (GMRES) solver. This method is used for solving the linear equations inside Newton's method for treating the overall nonlinear-monolithic discrete displacement/phase-field formulation. Several numerical examples demonstrate the performance and robustness of our solution technology. Mesh refinement studies, variations in the phase-field regularization parameter, iterations numbers of the linear and nonlinear solvers, and some parallel performances are conducted to substantiate the efficiency of the proposed solver for single fractures, multiple pressurized fractures, and a L-shaped panel test in three dimensions.

KW - Geometric multigrid

KW - Matrix-free

KW - Phase-field fracture propagation

KW - Primal–dual active set

UR - http://www.scopus.com/inward/record.url?scp=85091586758&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1902.08112

DO - 10.48550/arXiv.1902.08112

M3 - Article

AN - SCOPUS:85091586758

VL - 372

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 113431

ER -