International Journal of Fracture

, Volume 203, Issue 1–2, pp 3–19 | Cite as

Goal-oriented error estimation and mesh adaptivity in 3d elastoplasticity problems

  • S. Sh. Ghorashi
  • T. RabczukEmail author


We propose a 3D adaptive method for plasticity problems based on goal-oriented error estimation, which computes the error with respect to a prescribed quantity of interest. It is a dual-based scheme which requires an adjoint problem. The computed element-wise errors at each load/displacement increment are utilized for the mesh adaptivity purpose. Mesh adaptivity procedure is performed based on refinement and coarsening by introducing hanging nodes in quadrilateral and hexahedral elements in 2d and 3d, respectively. Several numerical simulations are investigated and the results are compared with available analytical solutions, existing experimental data and results of mesh adaptivity based on other conventional error estimation methods.


Goal-oriented error estimation Mesh adaptivity Elastoplasticity Von-Mises stress 



The research has been supported by the German Research Foundation (DFG) through Research Training Group 1462, which is gratefully acknowledged by the authors.


  1. Areias P, Rabczuk T (2010) Smooth finite strain plasticity with non-local pressure support. Int J Numer Methods Eng 81(1):106–134Google Scholar
  2. Areias P, Rabczuk T (2013) Finite strain fracture of plates and shells with configurational forces and edge rotations. Int J Numer Methods Eng 94(12):1099–1122CrossRefGoogle Scholar
  3. Babuška I, Rheinboldt WC (1978) A-posteriori error estimates for the finite element method. Int J Numer Methods Eng 12(10):1597–1615CrossRefGoogle Scholar
  4. Bangerth W, Hartmann R, Kanschat G (2007) Deal.ii—a general-purpose object-oriented finite element library. ACM Trans Math Softw (TOMS) 33(4). doi: 10.1145/1268776.1268779
  5. Bangerth W, Heister T, Heltai L, Kanschat G, Kronbichler M, Maier M, Turcksin B, Young TD (2015) The deal.II library, version 8.2. Arch Numer Softw 3Google Scholar
  6. Bangerth W, Rannacher R (2003) Adaptive finite element methods for differential equations. Birkhäuser, BaselCrossRefGoogle Scholar
  7. Becker R, Rannacher R (1996) A feed-back approach to error control in finite element methods: basic analysis and examples. East West J Numer Math 4:237–264Google Scholar
  8. Becker R, Rannacher R (1998) Weighted a posteriori error control in fe methods. In: Bock HG et al (eds) ENUMATH 95. World Scientific Publ., Singapore, pp 621–637Google Scholar
  9. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620CrossRefGoogle Scholar
  10. Budarapu PR, Gracie R, Yang SW, Zhaung X, Rabczuk T (2014) Efficient coarse graining in multiscale modeling of fracture. Theor Appl Fract Mech 69:126–143CrossRefGoogle Scholar
  11. Budarapu PR, Gracie R, Bordas SPA, Rabczuk T (2014) An adaptive multiscale method for quasi-static crack growth. Comput Mech 53(6):1129–1148CrossRefGoogle Scholar
  12. Chen C, Mangasarian OL (1995) Smoothing methods for convex inequalities and linear complementarity problems. Math Program 71(1):51–69CrossRefGoogle Scholar
  13. Frohne J, Heister T, Bangerth W (2016) Efficient numerical methods for the large-scale, parallel solution of elastoplastic contact problems. Int J Numer Methods Eng 105(6):416–439CrossRefGoogle Scholar
  14. Gao X-L (2003) Elasto-plastic analysis of an internally pressurized thick-walled cylinder using a strain gradient plasticity theory. Int J Solids Struct 40(23):6445–6455CrossRefGoogle Scholar
  15. Ghorashi SSh, Mohammadi S, Sabbagh-Yazdi S-R (2011) Orthotropic enriched element free galerkin method for fracture analysis of composites. Eng Fract Mech 78(9):1906–1927CrossRefGoogle Scholar
  16. Ghorashi SSh, Valizadeh N, Mohammadi S (2012) Extended isogeometric analysis for simulation of stationary and propagating cracks. Int J Numer Methods Eng 89(9):1069–1101CrossRefGoogle Scholar
  17. Ghorashi SSh, Amani J, Bagherzadeh AS, Rabczuk T (2014) Goal-oriented error estimation and mesh adaptivity in three-dimensional elasticity problems. WCCM XI–ECCM V–ECFD VI, Barcelona, SpainGoogle Scholar
  18. Ghorashi SSh, Valizadeh N, Mohammadi S, Rabczuk T (2015) T-spline based XIGA for fracture analysis of orthotropic media. Comput Struct 147:138–146 CIVIL-COMPCrossRefGoogle Scholar
  19. Ghorashi SSh, Lahmer T, Bagherzadeh AS, Rabczuk T. Goal-oriented error estimation and mesh adaptivity in elasticity problems with heterogeneous material distribution. Eng Geol (submitted)Google Scholar
  20. González-Estrada OA, Nadal E, Ródenas JJ, Kerfriden P, Bordas SPA, Fuenmayor FJ (2014) Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Comput Mech 53(5):957–976CrossRefGoogle Scholar
  21. Jia Y, Anitescu C, Ghorashi SSh, Rabczuk T (2014) Extended isogeometric analysis for material interface problems. IMA J Appl Math 80(3):608–633. doi: 10.1093/imamat/hxu004
  22. Kelly DW, De SR Gago JP, Zienkiewicz OC, Babuška I (1983) A posteriori error analysis and adaptive processes in the finite element method: part i error analysis. Int J Numer Methods Eng 19(11):1593–1619CrossRefGoogle Scholar
  23. Khazal H, Bayesteh H, Mohammadi S, Ghorashi SSh, Ahmed A (2016) An extended element free Galerkin method for fracture analysis of functionally graded materials. Mech Adv Mater Struct 23(5):513–528. doi: 10.1080/15376494.2014.984093
  24. Ladevéze P, Chamoin L (2010) Calculation of strict error bounds for finite element approximations of non-linear pointwise quantities of interest. Int J Numer Methods Eng 84(13):1638–1664CrossRefGoogle Scholar
  25. Nguyen VP, Rabczuk T, Bordas SPA, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79(3):763–813CrossRefGoogle Scholar
  26. Nguyen-Thanh N, Valizadeh N, Nguyen MN, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T (2015) An extended isogeometric thin shell analysis based on Kirchhoff–Love theory. Comput Methods Appl Mech Eng 284:265–291CrossRefGoogle Scholar
  27. Pannachet T, Díez P, Askes H, Sluys LJ (2010) Error assessment and mesh adaptivity for regularized continuous failure models. Comput Methods Appl Mech Eng 199(1720):961–978CrossRefGoogle Scholar
  28. Prudhomme S, Oden JT (1999) On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput Methods Appl Mech Eng 176(14):313–331CrossRefGoogle Scholar
  29. Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Meth Eng 61(13):2316–2343CrossRefGoogle Scholar
  30. Rabizadeh E, Saboor Bagherzadeh A, Rabczuk T (2015) Adaptive thermo-mechanical finite element formulation based on goal-oriented error estimation. Comput Mater Sci 102:27–44CrossRefGoogle Scholar
  31. Rannacher R, Suttmeier F-T (1997) A feed-back approach to error control in finite element methods: application to linear elasticity. Comput Mech 19(5):434–446CrossRefGoogle Scholar
  32. Rannacher R, Suttmeier F-T (1998) A posteriori error control in finite element methods via duality techniques: application to perfect plasticity. Comput Mech 21(2):123–133CrossRefGoogle Scholar
  33. Rannacher R, Suttmeier F-T (1999) A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput Methods Appl Mech Eng 176(14):333–361CrossRefGoogle Scholar
  34. Sauter M, Wieners C (2011) On the superlinear convergence in computational elasto-plasticity. Comput Methods Appl Mech Eng 200(49–52):3646–3658CrossRefGoogle Scholar
  35. Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New YorkGoogle Scholar
  36. Stein E, Rüter M, Ohnimus S (2007) Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity. Comput Methods Appl Mech Eng 196(3740):3598–3613 (Special Issue Honoring the 80th Birthday of Professor Ivo Babuska)CrossRefGoogle Scholar
  37. Suttmeier F-T (2010) On plasticity with hardening: adaptive finite element discretization. Int Math Forum 52:2591–2601Google Scholar
  38. Theocaris PS, Marketos E (1964) Elastic-plastic analysis of perforated thin strips of a strain-hardening material. J Mech Phys Solids 12(6):377–380CrossRefGoogle Scholar
  39. Whiteley JP, Tavener SJ (2014) Error estimation and adaptivity for incompressible hyperelasticity. Int J Numer Methods Eng 99(5):313–332CrossRefGoogle Scholar
  40. Yang SW, Budarapu PR, Mahapatra DR, Bordas SPA, Zi G, Rabczuk T (2015) A meshless adaptive multiscale method for fracture. Comput Mater Sci 96:382–395CrossRefGoogle Scholar
  41. Zaccardi C, Chamoin L, Cottereau R, Ben Dhia H (2013) Error estimation and model adaptation for a stochastic–deterministic coupling method based on the arlequin framework. Int J Numer Methods Eng 96(2):87–109CrossRefGoogle Scholar
  42. Zhuang X, Augarde CE, Mathisen KM (2012) Fracture modeling using meshless methods and level sets in 3d: framework and modeling. Int J Numer Methods Eng 92(11):969–998CrossRefGoogle Scholar
  43. Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 24(2):337–357CrossRefGoogle Scholar
  44. Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Int J Numer Methods Eng 33(7):1331–1364CrossRefGoogle Scholar
  45. Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity. Int J Numer Methods Eng 33(7):1365–1382CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Research Training Group 1462Bauhaus-Universität WeimarWeimarGermany
  2. 2.Division of Computational MechanicsTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Faculty of Civil Engineering, Ton Duc Thang UniversityHo Chi Minh CityVietnam
  4. 4.Institute of Structural MechanicsBauhaus-Universität WeimarWeimarGermany

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