American Institute of Mathematical Sciences

Advanced
January  2021, 14(1): 71-88. doi: 10.3934/dcdss.2020323

Adaptive time stepping in elastoplasticity

Sören Bartels and  Jakob Keck

Department of Applied Mathematics, Mathematical Institute, University of Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg i. Br., Germany

Dedicated to Alexander Mielke on the occasion of his 60th birthday

Received  March 2019 Revised  October 2019 Published  January 2021 Early access  April 2020

Using rate-independent evolutions as a framework for elastoplasticity, an a posteriori bound for the error introduced by time stepping is established. A time adaptive algorithm is devised and tested in comparison to a method with constant time steps. Experiments show that a significant error reduction can be obtained using variable time steps.

Keywords: Plasticity, time discretization, error control, adaptivity.
Mathematics Subject Classification: Primary: 65N12, 74C05.
Citation: Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323
References:
[1]

J. Alberty, C. Carstensen, S. A. Funken and R. Klose, Matlab implementation of the finite element method in elasticity, Computing, 69 (2002), 239-263.  doi: 10.1007/s00607-002-1459-8.

[2]

J. Alberty, C. Carstensen and D. Zarrabi, Adaptive numerical analysis in primal elastoplasticity with hardening, Computer Methods in Applied Mechanics and Engineering, 171 (1999), 175-204.  doi: 10.1016/S0045-7825(98)00210-2.

[3]

S. Bartels, Quasi-optimal error estimates for implicit discretizations of rate-independent evolutions, SIAM Journal on Numerical Analysis, 52 (2014), 708-716.  doi: 10.1137/130933964.

[4]

S. Bartels, Numerical Methods for Nonlinear Partial Differential Equations, vol. 47 of Springer Series in Computational Mathematics, Springer, 2015 doi: 10.1007/978-3-319-13797-1.

[5]

L. Gallimard, P. Ladevèze and J. P. Pelle, Error estimation and adaptivity in elastoplasticity, International Journal for Numerical Methods in Engineering, 39 (1996), 189–217. doi: 10.1002/(SICI)1097-0207(19960130)39:2<189::AID-NME849>3.0.CO;2-7.

[6]

W. Han and B. Reddy, Plasticity: Mathematical Theory and Numerical Analysis, Interdisciplinary Applied Mathematics, Springer, 1999.

[7]

D. Knees, On global spatial regularity in elasto-plasticity with linear hardening, Calc. Var. Partial Differential Equations, 36 (2009), 611-625.  doi: 10.1007/s00526-009-0247-0.

[8]

C. Kreuzer, C. A. Möller, A. Schmidt and K. G. Siebert, Design and convergence analysis for an adaptive discretization of the heat equation, IMA J. Numer. Anal., 32 (2012), 1375-1403.  doi: 10.1093/imanum/drr026.

[9]

A. Mielke, Evolution of rate-independent systems, Handbook of Differential Equations: Evolutionary Equations, II (2005), 461-559. 

[10]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Applications, vol. 193 of Applied Mathematical Sciences, Springer, 2015. doi: 10.1007/978-1-4939-2706-7.

[11]

R. H. Nochetto, G. Savaré and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Communications on Pure and Applied Mathematics, 53 (2000), 525-589.  doi: 10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M.

[12]

S. I. Repin and J. Valdman, Functional a posteriori error estimates for incremental models in elasto-plasticity, Cent. Eur. J. Math., 7 (2009), 506–519. doi: 10.2478/s11533-009-0035-2.

[13]

M. Sauter and C. Wieners, On the superlinear convergence in computational elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 200 (2011), 3646-3658.  doi: 10.1016/j.cma.2011.08.011.

[14]

A. Schröder and S. Wiedemann, Error estimates in elastoplasticity using a mixed method, Appl. Numer. Math., 61 (2011), 1031-1045.  doi: 10.1016/j.apnum.2011.06.001.

[15]

J. Simo and T. Hughes, Computational Inelasticity, Interdisciplinary Applied Mathematics, Springer, 1998.

[16]

G. Starke, An adaptive least-squares mixed finite element method for elasto-plasticity, SIAM Journal on Numerical Analysis, 45 (2007), 371–388, URL http://www.jstor.org/stable/40232933. doi: 10.1137/060652609.

[17]

U. Stefanelli, A variational principle for hardening elastoplasticity, SIAM J. Math. Anal., 40 (2008), 623-652.  doi: 10.1137/070692571.

show all references

References:
[1]

J. Alberty, C. Carstensen, S. A. Funken and R. Klose, Matlab implementation of the finite element method in elasticity, Computing, 69 (2002), 239-263.  doi: 10.1007/s00607-002-1459-8.

[2]

J. Alberty, C. Carstensen and D. Zarrabi, Adaptive numerical analysis in primal elastoplasticity with hardening, Computer Methods in Applied Mechanics and Engineering, 171 (1999), 175-204.  doi: 10.1016/S0045-7825(98)00210-2.

[3]

S. Bartels, Quasi-optimal error estimates for implicit discretizations of rate-independent evolutions, SIAM Journal on Numerical Analysis, 52 (2014), 708-716.  doi: 10.1137/130933964.

[4]

S. Bartels, Numerical Methods for Nonlinear Partial Differential Equations, vol. 47 of Springer Series in Computational Mathematics, Springer, 2015 doi: 10.1007/978-3-319-13797-1.

[5]

L. Gallimard, P. Ladevèze and J. P. Pelle, Error estimation and adaptivity in elastoplasticity, International Journal for Numerical Methods in Engineering, 39 (1996), 189–217. doi: 10.1002/(SICI)1097-0207(19960130)39:2<189::AID-NME849>3.0.CO;2-7.

[6]

W. Han and B. Reddy, Plasticity: Mathematical Theory and Numerical Analysis, Interdisciplinary Applied Mathematics, Springer, 1999.

[7]

D. Knees, On global spatial regularity in elasto-plasticity with linear hardening, Calc. Var. Partial Differential Equations, 36 (2009), 611-625.  doi: 10.1007/s00526-009-0247-0.

[8]

C. Kreuzer, C. A. Möller, A. Schmidt and K. G. Siebert, Design and convergence analysis for an adaptive discretization of the heat equation, IMA J. Numer. Anal., 32 (2012), 1375-1403.  doi: 10.1093/imanum/drr026.

[9]

A. Mielke, Evolution of rate-independent systems, Handbook of Differential Equations: Evolutionary Equations, II (2005), 461-559. 

[10]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Applications, vol. 193 of Applied Mathematical Sciences, Springer, 2015. doi: 10.1007/978-1-4939-2706-7.

[11]

R. H. Nochetto, G. Savaré and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Communications on Pure and Applied Mathematics, 53 (2000), 525-589.  doi: 10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M.

[12]

S. I. Repin and J. Valdman, Functional a posteriori error estimates for incremental models in elasto-plasticity, Cent. Eur. J. Math., 7 (2009), 506–519. doi: 10.2478/s11533-009-0035-2.

[13]

M. Sauter and C. Wieners, On the superlinear convergence in computational elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 200 (2011), 3646-3658.  doi: 10.1016/j.cma.2011.08.011.

[14]

A. Schröder and S. Wiedemann, Error estimates in elastoplasticity using a mixed method, Appl. Numer. Math., 61 (2011), 1031-1045.  doi: 10.1016/j.apnum.2011.06.001.

[15]

J. Simo and T. Hughes, Computational Inelasticity, Interdisciplinary Applied Mathematics, Springer, 1998.

[16]

G. Starke, An adaptive least-squares mixed finite element method for elasto-plasticity, SIAM Journal on Numerical Analysis, 45 (2007), 371–388, URL http://www.jstor.org/stable/40232933. doi: 10.1137/060652609.

[17]

U. Stefanelli, A variational principle for hardening elastoplasticity, SIAM J. Math. Anal., 40 (2008), 623-652.  doi: 10.1137/070692571.

Figure 1.  Numerical approximation of the solution for Example 5.1 at times $ t = 8, 10, 13, 15 $. The grey shading corresponds to the norm of plastic strain $ p $. For visibility purposes, we only applied 4 red refinements as opposed to our actual tests, where 6 red refinements were used. The radially symmetric nature of the problem allows us to save computation time by only computing the solution for a quarter of the ring
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Results of the tests using Example 5.1. Logarithmic plot of the error term $ \eta_{{\rm tot}}^2 $ as a function of $ K^{-1} \sim \tau $ for the method with constant time steps (star) and the adaptive algorithm with $ \theta = 1 $ (cross). The remaining graphs (circle, diamond and square) consider the total number of computed steps $ K_{{\rm tot}} $ instead of the number of time steps $ K $ and show the results for different values of $ \theta $ (see Algorithm 4.1). In the case of constant time steps, we have $ K = K_{{\rm tot}} $. The triangle in the bottom right references the slope of a quadratic function. An error reduction by the adaptive method is observed. The difference is less significant when considering $ K_{{\rm tot}} $, but increases for smaller values of $ \theta $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Illustration of the geometry used in Example 5.2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Numerical approximation of the solution for Example 5.2 at time $ T = 10 $. The original triangulation of three halved squares was red refined 6 times and the grey shading corresponds to the norm of plastic strain $ p $. The increasing force causes a curved fissure to develop across the domain
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Results of the tests for Examplek 5.2 using notation from Figure 2. The adaptive time stepping leads to an error reduction
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Results of the tests for Example 5.3 using notation from Figure 2. A stronger improvement of adaptivity is observed than in Examples 5.1 and 5.2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Development of error function $ \varepsilon_{\tau} $ in time. The algorithm with constant time steps was used on Examples 5.1 (top-left), 5.2 (top-right) and 5.3 (bottom-left). The plot on the bottom-right shows the adaptive method for Example 5.1 and with $ {\varepsilon_{{\rm max}}} = 3.16\cdot 10^{-5} $. The graphs start at $ t > 0 $, because the error in the left out parts is almost zero. Similar results are observed for Examples 5.1 and 5.2, where after a certain amount of time, $ \varepsilon_{\tau} $ starts to monotonically increase. In Example 5.3, a single peak occurs during the step containing the time $ t_p \approx 138.32 $, when plastic strain first emerges
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Patrick Henning, Mario Ohlberger. Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 119-150. doi: 10.3934/dcdss.2015.8.119
[2]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014
[3]

Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic and Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003
[4]

Matthieu Hillairet, Alexei Lozinski, Marcela Szopos. On discretization in time in simulations of particulate flows. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 935-956. doi: 10.3934/dcdsb.2011.15.935
[5]

Christian Meyer, Stephan Walther. Optimal control of perfect plasticity part I: Stress tracking. Mathematical Control and Related Fields, 2022, 12 (2) : 275-301. doi: 10.3934/mcrf.2021022
[6]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677
[7]

Changbing Hu, Kaitai Li. A simple construction of inertial manifolds under time discretization. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 531-540. doi: 10.3934/dcds.1997.3.531
[8]

John B. Little. The ubiquity of order domains for the construction of error control codes. Advances in Mathematics of Communications, 2007, 1 (1) : 151-171. doi: 10.3934/amc.2007.1.151
[9]

Max Gunzburger, Sung-Dae Yang, Wenxiang Zhu. Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 569-587. doi: 10.3934/dcdsb.2007.8.569
[10]

Peter E. Kloeden, Björn Schmalfuss. Lyapunov functions and attractors under variable time-step discretization. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 163-172. doi: 10.3934/dcds.1996.2.163
[11]

Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429
[12]

T. Colin, Géraldine Ebrard, Gérard Gallice. Semi-discretization in time for nonlinear Zakharov waves equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 263-282. doi: 10.3934/dcdsb.2009.11.263
[13]

Pierluigi Colli, Shunsuke Kurima. Time discretization of a nonlinear phase field system in general domains. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3161-3179. doi: 10.3934/cpaa.2019142
[14]

Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial and Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311
[15]

Tomáš Roubíček. Thermodynamics of perfect plasticity. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 193-214. doi: 10.3934/dcdss.2013.6.193
[16]

Ahmad Ahmad Ali, Klaus Deckelnick, Michael Hinze. Error analysis for global minima of semilinear optimal control problems. Mathematical Control and Related Fields, 2018, 8 (1) : 195-215. doi: 10.3934/mcrf.2018009
[17]

Evelyn Herberg, Michael Hinze. Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022013
[18]

Ulisse Stefanelli. Analysis of a variable time-step discretization for a phase transition model with micro-movements. Communications on Pure and Applied Analysis, 2006, 5 (3) : 659-673. doi: 10.3934/cpaa.2006.5.659
[19]

Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353
[20]

Luis C. García-Naranjo, Mats Vermeeren. Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics. Journal of Computational Dynamics, 2021, 8 (3) : 241-271. doi: 10.3934/jcd.2021011

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (211)
  • HTML views (300)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy