We study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the minimization of the phase-field variable; the advantage of this choice, from a computational stand point, is in the efficiency of the numerical implementation. Irreversibility is then recovered a posteriori by a simple pointwise truncation. We exploit a time discretization procedure, with either a one-step or a multi (or infinite)-step alternate minimization algorithm. We prove that the time-discrete solutions converge to a unilateral L2-gradient flow with respect to the phase-field variable, satisfying equilibrium of forces and energy identity. Convergence is proved in the continuous (Sobolev space) setting and in a discrete (finite element) setting, with any stopping criterion for the alternate minimization scheme. Numerical results show that the multi-step scheme is both more accurate and faster. It provides indeed good simulations for a large range of time increments, while the one-step scheme gives comparable results only for very small time increments.
Accepté le :
DOI : 10.1051/m2an/2018057
Keywords: Gradient flows, phase-field fracture
Almi, S. 1 ; Belz, S. 1 ; Negri, M. 1
@article{M2AN_2019__53_2_659_0,
author = {Almi, S. and Belz, S. and Negri, M.},
title = {Convergence of discrete and continuous unilateral flows for {Ambrosio{\textendash}Tortorelli} energies and application to mechanics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {659--699},
year = {2019},
publisher = {EDP Sciences},
volume = {53},
number = {2},
doi = {10.1051/m2an/2018057},
mrnumber = {3945577},
zbl = {1421.49033},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2018057/}
}
TY - JOUR AU - Almi, S. AU - Belz, S. AU - Negri, M. TI - Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 659 EP - 699 VL - 53 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2018057/ DO - 10.1051/m2an/2018057 LA - en ID - M2AN_2019__53_2_659_0 ER -
%0 Journal Article %A Almi, S. %A Belz, S. %A Negri, M. %T Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 659-699 %V 53 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2018057/ %R 10.1051/m2an/2018057 %G en %F M2AN_2019__53_2_659_0
Almi, S.; Belz, S.; Negri, M. Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 659-699. doi: 10.1051/m2an/2018057
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