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Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems, with analysis

Carsten Gräser, Daniel Kienle, Oliver Sander

TL;DR

This work proves convergence of the solver to a solution of the nonsmooth Euler–Lagrange equations of the spatial problem for any load and initial iterate, and shows several crucial convexity and regularity properties of the models considered here.

Abstract

We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems with linear time complexity. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without regularization or the introduction of a local history field. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler--Lagrange equations of the spatial problem for any load and initial iterate. On the way, we show several crucial convexity and regularity properties of the models considered here. Numerical comparisons to an operator-splitting algorithm show a considerable speed increase, without loss of robustness.

Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems, with analysis

TL;DR

This work proves convergence of the solver to a solution of the nonsmooth Euler–Lagrange equations of the spatial problem for any load and initial iterate, and shows several crucial convexity and regularity properties of the models considered here.

Abstract

We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems with linear time complexity. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without regularization or the introduction of a local history field. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler--Lagrange equations of the spatial problem for any load and initial iterate. On the way, we show several crucial convexity and regularity properties of the models considered here. Numerical comparisons to an operator-splitting algorithm show a considerable speed increase, without loss of robustness.

Paper Structure

This paper contains 26 sections, 15 theorems, 88 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Phase-field models of brittle fracture
  3. Degraded elastic energy density
  4. Isotropic splitting
  5. Volumetric decompositions
  6. Spectral decomposition
  7. Crack surface density
  8. Discretization and the algebraic increment potential
  9. Time discretization
  10. Finite element discretization
  11. Properties of the discrete incremental potential
  12. Truncated Nonsmooth Newton Multigrid for brittle fracture
  13. Convergence results
  14. Smoothers for brittle fracture problems
  15. Subspaces of displacement degrees of freedom
  16. ...and 11 more sections

Key Result

Lemma 2.5

Let $\psi_0^+$ and $\psi_0^-$ be convex, non-negative, and differentiable with Lipschitz continuous gradients $\nabla\psi_0^+$ and $\nabla\psi_0^-$. Then $\psi$ satisfies item:density_lipschitz, item:density_strongly_convex, and item:density_coercive.

Figures (16)

  • Figure 1: Crack width definition for the AT-2 functional
  • Figure 2: Notched square with a vertical displacement load (left). Exploiting symmetry we only simulate on the upper half of the domain (right).
  • Figure 3: 2d example, isotropic energy split: Evolution at time steps 145 and 160 computed with the TNNMG algorithm, and displacement--force curves for the AT-1 (top) and AT-2 (bottom) models
  • Figure 4: 2d example, isotropic energy split: Iterations per time step, for grid sizes $h_1$, $h_2$, $h_3$
  • Figure 5: 2d example, isotropic energy split: Wall-time per degree of freedom per time step, for grid sizes $h_1$, $h_2$, $h_3$
  • ...and 11 more figures

Theorems & Definitions (33)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 23 more