Solving primal plasticity increment problems in the time of a single predictor-corrector iteration. (English) Zbl 1477.74121

Summary: 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.


74S99 Numerical and other methods in solid mechanics
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)


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[1] Aifantis, Ec, On the microstructural origin of certain inelastic models, J Eng Mater Tech, 106, 326-330 (1984)
[2] Alberty, J.; Carstensen, C.; Zarrabi, D., Adaptive numerical analysis in primal elastoplasticity with hardening, Comput Methods Appl Mech Eng, 171, 175-204 (1999) · Zbl 0956.74049
[3] Caddemi, S.; Martin, Jb, Convergence of the Newton-Raphson algorithm in elastic-plastic incremental analysis, Int J Numer Methods Eng., 31, 1, 177-191 (1991) · Zbl 0825.73982
[4] Carstensen, C., Domain decomposition for a non-smooth convex minimization problem and its application to plasticity, Numer Linear Algebra Appl, 4, 3, 177-190 (1997) · Zbl 0890.49015
[5] Carstensen, C., Numerical analysis of the primal problem of elastoplasticity with hardening, Numer Math, 82, 577-597 (1999) · Zbl 0947.74061
[6] Chen, Y.; Davis, Ta; Hager, Ww; Rajamanickam, S., Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate, ACM Trans Math Softw, 35, 3, 22:1-22:14 (2008)
[7] Christensen, Pw, A nonsmooth Newton method for elastoplastic problems, Comput Methods Appl Mech Eng, 191, 1189-1219 (2002) · Zbl 1021.74049
[8] Drusvyatskiy D, Paquette C (2018) Variational analysis of spectral functions simplified. J. Convex Anal 25. arXiv preprint: arXiv:1506.05170 · Zbl 1398.49012
[9] Ebobisse, F.; Neff, P., Existence and uniqueness for rate-independent infinitesimal gradient plasticity with isotropic hardening and plastic spin, Math Mech Solids, 15, 6, 691-703 (2010) · Zbl 1257.74023
[10] Ekeland, I.; Temam, R., Convex analysis and variational problems (1999), Delhi: SIAM, Delhi · Zbl 0939.49002
[11] Geiger, C.; Kanzow, C., Theorie und numerik restringierter optimierungsaufgaben (2002), Berlin: Springer, Berlin · Zbl 1003.90044
[12] Gräser C (2011) Convex minimization and phase field models. Ph.D. thesis. Freie Universität Berlin
[13] Gräser, C.; Kornhuber, R., Multigrid methods for obstacle problems, J Comp Math, 27, 1, 1-44 (2009) · Zbl 1199.65401
[14] Gräser, C.; Sack, U.; Sander, O.; Barth, Tj; Griebel, M.; Keyes, De; Nieminen, Rm; Roose, D.; Schlick, T., Truncated nonsmooth Newton multigrid methods for convex minimization problems, Domain decomposition methods in science and engineering XVIII, 129-136 (2009), Berlin: Springer, Berlin
[15] Gräser, C.; Sander, O., Polyhedral Gauss-Seidel converges, J Numer Math, 22, 3, 221-254 (2014) · Zbl 1310.65072
[16] Gräser C, Sander O (2017) Truncated nonsmooth Newton multigrid methods for block-separable minimization problems. In: arXiv e-prints. To appear in IMA J Numer Anal arXiv:1709.04992 [math.NA] · Zbl 1183.65076
[17] Gruber, Pg; Valdman, J., Solution of one-time-step problems in elastoplasticity by a slant Newton method, SIAM J Sci Comput, 31, 2, 1558-1580 (2009) · Zbl 1186.74025
[18] Han, W.; Reddy, Bd, Plasticity (2013), Berlin: Springer, Berlin
[19] Lee, Y-J; Wu, J.; Xu, J.; Zikatanov, L., Robust subspace correction methods for nearly singular systems, Math Models Methods Appl Sci, 17, 11, 1937-1963 (2007) · Zbl 1151.65096
[20] Lewis, As, Convex analysis on the Hermitian matrices, SIAM J Optim, 6, 1, 164-177 (1996) · Zbl 0849.15013
[21] Martin, J.; Caddemi, S., Sufficient conditions for convergence of the Newton-Raphson iterative algorithm in incremental elastic-plastic analysis, Eur J Mech A Solids, 13, 3, 351-365 (1994) · Zbl 0809.73078
[22] Neff, P.; Sydow, A.; Wieners, C., Numerical approximation of incremental infinitesimal gradient plasticity, Int J Numer Methods Eng, 77, 414-436 (2009) · Zbl 1155.74316
[23] Pipping, E.; Sander, O.; Kornhuber, R., Variational formulation of rateand state-dependent friction problems, Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 95, 4, 377-395 (2015) · Zbl 1322.74054
[24] Rencontré, L.; Bird, W.; Martin, J., Internal variable formulation of a backward difference corrector algorithm for piecewise linear yield surfaces, Meccanica, 27, 13-24 (1992) · Zbl 0759.73061
[25] Rockafellar, Rt; Wets, Rj-B, Variational analysis (2010), Berlin: Springer, Berlin
[26] Sander O (July 2017) Solving primal plasticity increment problems in the time of a single predictor-corrector iteration. In: arXiv e-prints. arXiv:1707.03733 [math.NA]
[27] Simo, Jc; Hughes, Tjr, Computational inelasticity (1998), Berlin: Springer, Berlin
[28] Stein E, Wriggers P, Rieger A, Schmidt M (2002) Benchmarks. In: Stein E (ed)Errorcontrolled adaptive finite elements in solid mechanics. Wiley. Chap. 11, pp 385-404
[29] Wohlmuth, B.; Krause, R., Monotone methods on nonmatching grids for nonlinear contact problems, SIAM J Sci Comput, 25, 1, 324-347 (2003) · Zbl 1163.65334
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