Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics

• Published in: ESAIM. Mathematical modelling and numerical analysis 2019-03, Vol.53 (2), p.659-699
• Main Authors: Almi, S., Belz, S., Negri, M.
• Format: Article
• Language: English
• Subjects: 49S05; 74A45; Algorithms; Convergence; Gradient flow; Gradient flows; Optimization; phase-field fracture; Sobolev space
• ISSN: 0764-583X; 1290-3841
• DOI: 10.1051/m2an/2018057

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.