TY - JOUR

T1 - On the BFGS monolithic algorithm for the unified phase field damage theory

AU - Wu, Jian-Ying

AU - Huang, Yuli

AU - Nguyen, Vinh Phu

PY - 2020/3/1

Y1 - 2020/3/1

N2 - Despite the popularity in modeling complex and arbitrary crack configurations in solids, phase-field damage models suffer from burdensome computational cost. This issue arises largely due to the robust but inefficient alternating minimization (AM) or staggered algorithm usually employed to solve the coupled damage–displacement governing equations. Aiming to tackle this difficulty, we propose in this work, for the first time, to use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm to solve in a monolithic manner the system of coupled governing equations, rather than the standard Newton one which is notoriously poor for problems involving non-convex energy functional. It is found that, the BFGS algorithm yields identical results to the AM/staggered solver, and is also robust for both brittle fracture and quasi-brittle failure with a single or multiple cracks. However, much less iterations are needed to achieve convergence. Furthermore, as the system matrix is less reformed per increment, the quasi-Newton monolithic algorithm is much more efficient than the AM/staggered solver. Representative numerical examples show that the saving in CPU time is about factor 3∼7, and the larger the problem is, the more saving it gains. As the BFGS monolithic algorithm has been incorporated in many commercial software packages, it can be easily implemented and is thus attractive in the phase-field damage modeling of localized failure in solids.

AB - Despite the popularity in modeling complex and arbitrary crack configurations in solids, phase-field damage models suffer from burdensome computational cost. This issue arises largely due to the robust but inefficient alternating minimization (AM) or staggered algorithm usually employed to solve the coupled damage–displacement governing equations. Aiming to tackle this difficulty, we propose in this work, for the first time, to use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm to solve in a monolithic manner the system of coupled governing equations, rather than the standard Newton one which is notoriously poor for problems involving non-convex energy functional. It is found that, the BFGS algorithm yields identical results to the AM/staggered solver, and is also robust for both brittle fracture and quasi-brittle failure with a single or multiple cracks. However, much less iterations are needed to achieve convergence. Furthermore, as the system matrix is less reformed per increment, the quasi-Newton monolithic algorithm is much more efficient than the AM/staggered solver. Representative numerical examples show that the saving in CPU time is about factor 3∼7, and the larger the problem is, the more saving it gains. As the BFGS monolithic algorithm has been incorporated in many commercial software packages, it can be easily implemented and is thus attractive in the phase-field damage modeling of localized failure in solids.

KW - BFGS method

KW - Damage

KW - Fracture

KW - Monolithic algorithm

KW - Phase-field

KW - Staggered solver

UR - http://www.scopus.com/inward/record.url?scp=85074171291&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2019.112704

DO - 10.1016/j.cma.2019.112704

M3 - Article

AN - SCOPUS:85074171291

VL - 360

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 112704

ER -