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Viscoelasticity with limiting strain
Adaptive time stepping in elastoplasticity
Department of Applied Mathematics, Mathematical Institute, University of Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg i. Br., Germany |
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.
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. |





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