# American Institute of Mathematical Sciences

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

## Adaptive time stepping in elastoplasticity

 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.

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.

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