WIAS Preprint No. 2707, (2020)

Fully discrete approximation of rate-independent damage models with gradient regularization



Authors

  • Bartels, Sören
  • Milicevic, Marijo
  • Thomas, Marita
  • Weber, Nico

2010 Mathematics Subject Classification

  • 35K86 74R05 49J45 49S05 65M60 65M12

Keywords

  • Partial damage, damage evolution with gradient regularization, semistable energetic solutions, numerical approximation, iterative solution

DOI

10.20347/WIAS.PREPRINT.2707

Abstract

This work provides a convergence analysis of a time-discrete scheme coupled with a finite-element approximation in space for a model for partial, rate-independent damage featuring a gradient regularization as well as a non-smooth constraint to account for the unidirectionality of the damage evolution. The numerical algorithm to solve the coupled problem of quasistatic small strain linear elasticity with rate-independent gradient damage is based on a Variable ADMM-method to approximate the nonsmooth contribution. Space-discretization is based on P1 finite elements and the algorithm directly couples the time-step size with the spatial grid size h. For a wide class of gradient regularizations, which allows both for Sobolev functions of integrability exponent r ∈ (1, ∞) and for BV-functions, it is shown that solutions obtained with the algorithm approximate as h → 0 a semistable energetic solution of the original problem. The latter is characterized by a minimality property for the displacements, a semistability inequality for the damage variable and an energy dissipation estimate. Numerical benchmark experiments confirm the stability of the method.

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