
-
Previous Article
Rate-independent evolution of sets
- DCDS-S Home
- This Issue
-
Next Article
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. |





[1] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[2] |
Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313 |
[3] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[4] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[5] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[6] |
Cécile Carrère, Grégoire Nadin. Influence of mutations in phenotypically-structured populations in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3609-3630. doi: 10.3934/dcdsb.2020075 |
[7] |
Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493 |
[8] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
[9] |
Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 |
[10] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
[11] |
A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909 |
[12] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[13] |
Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044 |
[14] |
Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 |
[15] |
Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 |
[16] |
Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]