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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2013005685
pages 581-596


Varun Gupta
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, Urbana, Illinois 61801, USA
Dae-Jin Kim
Department of Architectural Engineering, Kyung Hee University, Engineering Building, 1 Sochon-Dong Kihung-Gu, Yongin, Kyunggi-Do, Korea 446-701
Armando Duarte
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, Urbana, Illinois 61801, USA


This paper presents an extension of a two-scale generalized finite element method (GFEM) to three-dimensional fracture problems involving confined plasticity. This two-scale procedure, also known as the generalized finite element method with global-local enrichments (GFEMgl), involves the solution of a fine-scale boundary value problem defined around a region undergoing plastic deformations and the enrichment of the coarse-scale solution space with the resulting nonlinear fine-scale solution through the partition-of-unity framework. The approach provides accurate nonlinear solutions with reduced computational costs compared to standard finite element methods, since the nonlinear iterations are performed on much smaller problems. The efficacy of the method is demonstrated with the help of numerical examples, which are three-dimensional fracture problems with nonlinear material properties and considering small-strain, rate-independent J2 plasticity.


  1. Babuška, I., Banerjee, U., and Osborn, J., Generalized finite element methods—Main ideas, results and perspective. DOI: 10.1142/S0219876204000083

  2. Babuška, I., Caloz, G., and Osborn, J., Special finite element methods for a class of second order elliptic problems with rough coefficients. DOI: 10.1137/0731051

  3. Babuška, I. and Melenk, J., The partition of unity method. DOI: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N

  4. Barros, F., Proenca, S., and Barcellos, C., Generalized finite element method in structural nonlinear analysis: A p-adaptive strategy. DOI: 10.1007/s00466-003-0503-7

  5. Belytschko, T. and Black, T., Elastic crack growth in finite elements with minimal remeshing. DOI: 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S

  6. Belytschko, T., Moës, N., Usui, S., and Parimi, C., Arbitrary discontinuities in finite elements. DOI: 10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M

  7. Bordas, S., Rabczuk, T., and Zi, G., Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. DOI: 10.1016/j.engfracmech.2007.05.010

  8. de Souza Neto, E. A., Peric, D., and Owen, D. R. J., Computational Methods for Plasticity: Theory and Applications.

  9. Duarte, C., Babuška, I., and Oden, J., Generalized finite element methods for three dimensional structural mechanics problems. DOI: 10.1016/S0045-7949(99)00211-4

  10. Duarte, C. and Kim, D.-J., Analysis and applications of a generalized finite element method with global-local enrichment functions. DOI: 10.1016/j.cma.2007.08.017

  11. Duarte, C. and Oden, J., An h-p adaptive method using clouds. DOI: 10.1016/S0045-7825(96)01085-7

  12. Düster, A. and Rank, E., A p-version finite element approach for two- and three-dimensional problems of the j2 flow theory with non-linear isotropic hardening. DOI: 10.1002/nme.391

  13. Elguedj, T., Gravouil, A., and Combescure, A., Appropriate extended functions for the X-FEM simulation of plastic fracture mechanics. DOI: 10.1016/j.cma.2005.02.007

  14. Elguedj, T., Gravouil, A., and Combescure, A., A mixed augmented lagrangian-extended finite element method for modelling elastic-plastic fatigue crack growth with unilateral contact. DOI: 10.1002/nme.2002

  15. Fan, R. and Fish, J., The rs-method for material failure simulations. DOI: 10.1002/nme.2134

  16. Galland, F., Gravouil, A., Malvesin, E., and Rochette, M., A global model reduction approach for 3d fatigue crack growth with confined plasticity. DOI: 10.1016/j.cma.2010.08.018

  17. Gendre, L., Allix, O., and Gosselet, P., A two-scale approximation of the Schur complement and its use for non-intrusive coupling. DOI: 10.1002/nme.3142

  18. Gupta, V., Kim, D.-J., and Duarte, C., Analysis and improvements of global-local enrichments for the generalized finite element method. DOI: 10.1016/j.cma.2012.06.021

  19. Hutchinson, J., Singular behaviour at the end of a tensile crack in a hardening material. DOI: 10.1016/0022-5096(68)90014-8

  20. Kim, D.-J., Duarte, C., and Proenca, S., Generalized finite element method with global-local enrichments for nonlinear fracture analysis.

  21. Kim, D.-J., Duarte, C., and Proenca, S., A generalized finite element method with global-local enrichment functions for confined plasticity problems. DOI: 10.1007/s00466-012-0689-7

  22. Kim, D.-J., Pereira, J., and Duarte, C., Analysis of three-dimensional fracture mechanics problems: A two-scale approach using coarse generalized FEM meshes. DOI: 10.1002/nme.2690

  23. Legrain, G., Moës, N., and Verron, E., Stress analysis around crack tips in finite strain problems using the extended finite element method. DOI: 10.1002/nme.1291

  24. Lubliner, J., Plasticity Theory.

  25. Melenk, J. and Babuška, I., The partition of unity finite element method: Basic theory and applications. DOI: 10.1016/S0045-7825(96)01087-0

  26. Moës, N., Dolbow, J., and Belytschko, T., A finite element method for crack growth without remeshing. DOI: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J

  27. Oden, J. and Duarte, C., Chapter: Clouds, Cracks and FEM‘s.

  28. Oden, J., Duarte, C., and Zienkiewicz, O., A new cloud-based hp finite element method. DOI: 10.1016/S0045-7825(97)00039-X

  29. O‘Hara, P., Duarte, C., and Eason, T., Generalized finite element analysis of three-dimensional heat transfer problems exhibiting sharp thermal gradients.

  30. Oskay, C. and Fish, J., On calibration and validation of eigendeformation-based multiscale models for failure analysis of heterogeneous systems. DOI: 10.1007/s00466-007-0197-3

  31. Pereira, J., Duarte, C., Guoy, D., and Jiao, X., Hp-Generalized FEM and crack surface representation for non-planar 3-D cracks. DOI: 10.1002/nme.2419

  32. Pereira, J., Kim, D.-J., and Duarte, C., A two-scale approach for the analysis of propagating three-dimensional fractures. DOI: 10.1007/s00466-011-0631-4

  33. Prabel, B., Combescure, A., Gravouil, A., and Marie, S., Level set X-FEM non-matching meshes: Application to dynamic crack propagation in elastic-plastic media. DOI: 10.1002/nme.1819

  34. Rabczuk, T., Areias, P. M. A., and Belytschko, T., A meshfree thin shell method for non-linear dynamic fracture. DOI: 10.1002/nme.2013

  35. Rahman, S. and Kim, J., Probabilistic fracture mechanics for nonlinear structures. DOI: 10.1016/S0308-0161(01)00006-0

  36. Rao, B. and Rahman, S., An enriched meshless method for non-linear fracture mechanics. DOI: 10.1002/nme.868

  37. Rice, J., McMeeking, R., Parks, D., and Sorensen, E., Recent finite element studies in plasticity and fracture mechanics. DOI: 10.1016/0045-7825(79)90026-4

  38. Rice, J. and Rosengren, G., Plane strain deformation near a crack tip in a power-law hardening material. DOI: 10.1016/0022-5096(68)90013-6

  39. Simo, J. and Hughes, T., Computational Inelasticity. DOI: 10.1007/b98904

  40. Simo, J. and Taylor, R., Consistent tangent operators for rate-independent elastoplasticity. DOI: 10.1016/0045-7825(85)90070-2

  41. Strouboulis, T., Babuška, I., and Copps, K., The design and analysis of the generalized finite element method. DOI: 10.1016/S0045-7825(99)00072-9

  42. Strouboulis, T., Copps, K., and Babuška, I., The generalized finite element method. DOI: 10.1016/S0045-7825(01)00188-8

  43. Sukumar, N., Moës, N., Moran, B., and Belytschko, T., Extended finite element method for three-dimensional crack modelling. DOI: 10.1002/1097-0207(20000820)48:11<1549::AID-NME955>3.0.CO;2-A

  44. Szabo, B. and Babuška, I., Finite Element Analysis.

  45. Torres, I. and Proenca, S., Generalized finite element method in nonlinear three-dimensional analysis. DOI: 10.1142/S0219876208001388

  46. Waisman, H., Fish, J., Tuminaro, R., and Shadid, J., Acceleration of the generalized global basis (ggb) method for nonlinear problems. DOI: 10.1016/