Volume 53, Number 2, March-April 2019
|Page(s)||659 - 699|
|Published online||01 May 2019|
Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics
Technische Universität München, Germany
2 Dipartimento di Matematica, Università di Pavia, Italy
* Corresponding author: email@example.com
Accepted: 25 September 2018
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.
Mathematics Subject Classification: 49S05 / 74A45
Key words: Gradient flows / phase-field fracture
© EDP Sciences, SMAI 2019
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