Truncated nonsmooth Newton multigrid for phase-field brittle-fracture problems, with analysis

Research output: Contribution to journalResearch articleContributedpeer-review



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.


Original languageEnglish
Pages (from-to)1059-1089
Number of pages31
JournalComputational Mechanics
Issue number5
Publication statusPublished - 20 May 2023

External IDs

ORCID /0000-0003-1093-6374/work/146166885
Scopus 85160245052
Mendeley 3dc219fe-f457-3f17-a7dc-b507b8c6f113



  • Brittle fracture, Convex analysis, Global convergence, Nonsmooth multigrid, Phase-field, Spectral strain decomposition

Library keywords