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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

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
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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 $
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Figure 3.  Illustration of the geometry used in Example 5.2
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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
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Figure 5.  Results of the tests for Examplek 5.2 using notation from Figure 2. The adaptive time stepping leads to an error reduction
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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
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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
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