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Solving primal plasticity increment problems in the time of a single predictor–corrector iteration


The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a well-established method for the solution of strictly convex block-separably nondifferentiable minimization problems. It achieves multigrid-like performance even for non-smooth nonlinear problems, while at the same time being globally convergent and without employing penalty parameters. We show that the algorithm can be applied to the primal problem of classical small-strain elastoplasticity with hardening. Numerical experiments show that the method is considerably faster than classical predictor–corrector methods. Indeed, solving an entire increment problem with TNNMG can take less time than a single predictor–corrector iteration for the same problem. At the same time, memory consumption is reduced considerably, in particular for three-dimensional problems. Since the algorithm does not rely on differentiability of the objective functional, nonsmooth yield laws can be easily incorporated. The method is closely related to a predictor–corrector scheme with a consistent tangent predictor and line search. We explain the algorithm, prove global convergence, and show its efficiency using standard benchmarks from the literature.

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  1. 1.

    A preprint version of this article [26] proposed to apply one multigrid iteration to the Schur complement system constructed in this section, which is positive-definite even if some degrees of freedom are truncated away. However, measurements showed that in this approach the cost of repeatedly computing the Schur complement dominates the cost of the multigrid iteration, without leading to a relevant improvement in the convergence speed.

  2. 2.

    Here the method assumes that \(\varphi \) is \(C^2\)-differentiable for all \(p \ne 0\). Generalizations appear straightforward, but seem to be absent from the literature.

  3. 3.

    Note that \(H_\nu ^\text {pc}\) differs from truncated tangent matrix \(H_\nu \) as defined in Sect. 4.3. While \(H_\nu \) has zero rows and columns for degrees of freedom where the dissipation is not differentiable, \(H_\nu ^\text {pc}\) keeps the elastic part there. However, as the corresponding degrees of freedom are held fixed in Step 1c there is no practical difference.

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  5. 5.

    This choice of accuracy is actually a form of cheating in favour of the predictor–corrector method. Textbook predictor–corrector methods solve the predictor problems up to machine accuracy, i.e., several orders of magnitude more precise than a multigrid method with the given termination criterion. Using a tighter termination criterion for the linear multigrd method would increase the run-time of the predictor–corrector method even more.


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Correspondence to Oliver Sander.

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Sander, O., Jaap, P. Solving primal plasticity increment problems in the time of a single predictor–corrector iteration. Comput Mech 65, 663–685 (2020). https://doi.org/10.1007/s00466-019-01788-y

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  • Plasticity
  • Small-strain
  • Increment problem
  • Multigrid
  • Nonsmooth