A unilateral L2L^{2}-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

AbstractWe consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic {L^{2}}-penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework of {W^{1,2}(0,T;L^{2})}.

Download Full-text

Irreversibility and alternate minimization in phase field fracture: a viscosity approach

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/s00033-020-01357-x ◽

2020 ◽

Vol 71 (4) ◽

Author(s):

Stefano Almi

Keyword(s):

Phase Field ◽

Field Variable ◽

Minimization Algorithm ◽

Vanishing Viscosity ◽

Vanishing Viscosity Limit ◽

Phase Field Models ◽

Iterative Minimization ◽

Viscosity Limit

Abstract This work is devoted to the analysis of convergence of an alternate (staggered) minimization algorithm in the framework of phase field models of fracture. The energy of the system is characterized by a nonlinear splitting of tensile and compressive strains, featuring non-interpenetration of the fracture lips. The alternating scheme is coupled with an $$L^{2}$$ L 2 -penalization in the phase field variable, driven by a viscous parameter $$\delta >0$$ δ > 0 , and with an irreversibility constraint, forcing the monotonicity of the phase field only w.r.t. time, but not along the whole iterative minimization. We show first the convergence of such a scheme to a viscous evolution for $$\delta >0$$ δ > 0 and then consider the vanishing viscosity limit $$\delta \rightarrow 0$$ δ → 0 .

Download Full-text

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

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2018057 ◽

2019 ◽

Vol 53 (2) ◽

pp. 659-699 ◽

Cited By ~ 5

Author(s):

S. Almi ◽

S. Belz ◽

M. Negri

Keyword(s):

Phase Field ◽

Phase Field Model ◽

Gradient Flow ◽

Time Discretization ◽

Field Variable ◽

Small Time ◽

One Step ◽

The One ◽

Discrete Finite Element ◽

Minimization Scheme

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.

Download Full-text