Skip to main content
SpringerLink
Log in

Adaptive selective ES-FEM limit analysis of cracked plane-strain structures

  • Research Article
  • Published:
Frontiers of Structural and Civil Engineering Aims and scope Submit manuscript

Abstract

This paper presents a simple and efficient approach for predicting the plastic limit loads in cracked planestrain structures.We use two levels of mesh repartitioning for the finite element limit analysis. The master level handles an adaptive primal-mesh process through a dissipation-based indicator. The slave level performs the subdivision of each triangle into three sub-triangles and constitutes a dual mesh from a pair of two adjacent sub-triangles shared by common edges of the primal mesh. Applying a strain smoothing projection to the strain rates on the dual mesh, the incompressibility constraint and the flow rule constraint are imposed over the edge-based smoothing domains and everywhere in the problem domain. The limit analysis problem is recast into the compact form of a second-order cone programming (SOCP) for the purpose of exploiting interior-point solvers. The present method retains a low number of optimization variables. It offers a convenient way for designing and solving the large-scale optimization problems effectively. Several benchmark examples are given to show the simplicity and effectiveness of the present method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hill R. On discontinuous plastic states, with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids, 1952, 1(1): 19–30

    Article  MathSciNet  Google Scholar 

  2. Ewing D J F, Richards C E. The yield-point loading of singlynotched pin loaded tensile strips. Journal of the Mechanics and Physics of Solids, 1974, 22(1): 27–36

    Article  Google Scholar 

  3. Miller A G. Review of limit loading of structures containing defects. International Journal of Pressure Vessels and Piping, 1988, 32(1–4): 197–327

    Article  Google Scholar 

  4. Koiter W T. General theorems for elastic plastic solids. Progress in Solid Mechanics, Sneddon I N, Hill R, eds. Nord-Holland, Amsterdam, 1960, 1: 165–221

    MathSciNet  Google Scholar 

  5. Melan E. Theorie statisch unbestimmter Systeme aus ideal plastischem Baustoff. Sitzber. Akad. Wiss. Wien IIa, 1936, 145: 195–2182

    MATH  Google Scholar 

  6. Prager W, Hodge PG. Theory of Perfectly Plastic Solids. New York: Wiley, 1951, 3

    MATH  Google Scholar 

  7. Chakrabarty J. Theory of Plasticity. 3rd ed. Elsevier Butterworth- Heinemann, 2006, 4

    Google Scholar 

  8. Yan A M, Nguyen-Dang H. Limit analysis of cracked structures by mathematical programming and finite element technique. Computational Mechanics, 1999, 24(5): 319–333

    Article  MATH  Google Scholar 

  9. Vu D K. Dual Limit and Shakedown analysis of structures. Dissertation for the Doctoral Degree. Belgium: Université de Liège, 2001, 5

  10. Khan I A, Ghosh A K. A modified upper bound approach to limit analysis for plane strain deeply cracked specimens. International Journal of Solids and Structures, 2007, 44(10): 3114–3135

    Article  MATH  Google Scholar 

  11. Khan I A, Bhasin V, Chattopadhyay J, Singh R K, Vaze K K, Ghosh A K. An insight of the structure of stress fields for stationary crack in strength mismatch weld under plane strain mode–I loading–Part II: Compact tension and middle tension specimens. International Journal of Mechanical Sciences, 2014, 87: 281–296

    Article  Google Scholar 

  12. Le C V, Askes H, Gilbert M. A locking-free stabilized kinematic EFG model for plane strain limit analysis. Computers & Structures, 2012, 106–107: 1–8

    Article  Google Scholar 

  13. Tran T N, Liu G R, Nguyen-Xuan H, Nguyen-Thoi T. An edgebased smoothed finite element method for primal-dual shakedown analysis of structures. International Journal for Numerical Methods in Engineering, 2010, 82(7): 917–9386

    MATH  MathSciNet  Google Scholar 

  14. Nguyen-Xuan H, Rabczuk T, Nguyen-Thoi T, Tran T N, Nguyen- Thanh N. Computation of limit and shakedown loads using a nodebased smoothed finite element method. International Journal for Numerical Methods in Engineering, 2012, 90(3): 287–310

    Article  MATH  MathSciNet  Google Scholar 

  15. Nguyen-Xuan H, Thai C H, Bleyer J, Nguyen P V. Upper bound limit analysis of plates using a rotation-free isogeometric approach. Asia Pacific Journal on Computational Engineering, 2014, 1(1): 12

    Article  Google Scholar 

  16. Nguyen-Xuan H, Tran L V, Thai C H, Le C V. Plastic collapse analysis of cracked structures using extended isogeometric elements and second-order cone programming. Theoretical and Applied Fracture Mechanics, 2014, 72: 13–27

    Article  Google Scholar 

  17. Nagtegaal J C, Parks D M, Rice J R. On numerically accurate finite element solutions in the fully plastic range. Computer Methods in Applied Mechanics and Engineering, 1974, 4(2): 153–177

    Article  MATH  MathSciNet  Google Scholar 

  18. Sloan S W, Kleeman P W. Upper bound limit analysis using discontinuous velocity fields. Computer Methods in Applied Mechanics and Engineering, 1995, 127(1–4): 293–314

    Article  MATH  Google Scholar 

  19. Capsoni A, Corradi L. A finite element formulation of the rigidplastic limit analysis problem. International Journal for Numerical Methods in Engineering, 1997, 40(11): 2063–2086

    Article  MATH  Google Scholar 

  20. Christiansen E, Andersen K D. Computation of collapse states with von Mises type yield condition. International Journal for Numerical Methods in Engineering, 1999, 46(8): 1185–1202

    Article  MATH  MathSciNet  Google Scholar 

  21. Vu D K, Yan A M, Nguyen-Dang H. A primal-dual algorithm for shakedown analysis of structure. Computer Methods in Applied Mechanics and Engineering, 2004, 193(42–44): 4663–4674

    Article  MATH  Google Scholar 

  22. Krabbenhøft K, Lyamin A V, Hjiaj M, Sloan S W. A new discontinuous upper bound limit analysis formulation. International Journal for Numerical Methods in Engineering, 2005, 63(7): 1069–1088

    Article  MATH  Google Scholar 

  23. Vicente da Silva M, Antao A N. A non-linear programming method approach for upper bound limit analysis. International Journal for Numerical Methods in Engineering, 2007, 72(10): 1192–1218

    Article  MATH  Google Scholar 

  24. Lyamin A V, Sloan S W. Upper bound limit analysis using linear finite elements and nonlinear programming. International Journal for Numerical and Analytical Methods in Geomechanics, 2002, 26(2): 181–216

    Article  MATH  Google Scholar 

  25. Makrodimopoulos A, Martin C M. Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. International Journal for Numerical Methods in Engineering, 2006, 66(4): 604–634

    Article  MATH  Google Scholar 

  26. Makrodimopoulos A, Martin CM. Upper bound limit analysis using simplex strain elements and second-order cone programming. International Journal for Numerical and Analytical Methods in Geomechanics, 2007, 31(6): 835–865

    Article  MATH  Google Scholar 

  27. Borges L A, Zouain N, Costa C, Feijoo R. An adaptive approach to limit analysis. International Journal of Solids and Structures, 2001, 38(10–13): 1707–1720

    Article  MATH  Google Scholar 

  28. Lyamin A V, Sloan S W, Krabbenhoft K, Hjiaj M. Lower bound limit analysis with adaptive remeshing. International Journal for Numerical Methods in Engineering, 2005, 63(14): 1961–1974

    Article  MATH  Google Scholar 

  29. Ciria H, Peraire J, Bonet J. Mesh adaptive computation of upper and lower bounds in limit analysis. International Journal for Numerical Methods in Engineering, 2008, 75(8): 899–944

    Article  MATH  MathSciNet  Google Scholar 

  30. Munoz J, Bonet J, Huerta A, Peraire J. Upper and lower bounds in limit analysis: adaptive meshing strategies and discontinuous loading. International Journal for Numerical Methods in Engineering, 2009, 77(4): 471–501

    Article  MATH  MathSciNet  Google Scholar 

  31. Martin C M. The use of adaptive finite-element limit analysis to reveal slip-line fields. Géotechnique Letters, 2011, 1(April-June): 23–29

    Article  Google Scholar 

  32. Van-Phuc P, Nguyen-Thoi T, Nguyen C H, Le V C. An effective adaptive limit analysis of soil using FEM and second-order cone programming. The International Conference on Advances in Computational Mechanics (ACOME), 2012, 177–190

    Google Scholar 

  33. Le V C. A stabilized discrete shear gap finite element for adaptive limit analysis of Mindlin–Reissner plates. International Journal for Numerical Methods in Engineering, 2013, 96: 231–2467

    MathSciNet  Google Scholar 

  34. Nguyen-Xuan H, Liu G R. An edge-based finite element method (ES-FEM) with adaptive scaled-bubble functions for plane strain limit analysis. Computer Methods in Applied Mechanics and Engineering, 2015, 285: 877–905

    Article  MathSciNet  Google Scholar 

  35. Rabczuk T, Belytschko T. Adaptivity for structured meshfree particle methods in 2D and 3D. International Journal for Numerical Methods in Engineering, 2005, 63(11): 1559–1582

    Article  MATH  MathSciNet  Google Scholar 

  36. Wu C T, Hu W. A two-level mesh repartitioning scheme for the displacement-based lower-order finite element methods in volumetric locking-free analyses. Computational Mechanics, 2012, 50(1): 1–18

    Article  MathSciNet  Google Scholar 

  37. Nguyen-Xuan H, Wu C T, Liu G R. An adaptive selective ES-FEM for plastic collapse analysis. European Journal of Mechanics- A/ Solid, submitted, 2014, 8

    Google Scholar 

  38. Liu G R, Nguyen-Thoi T, Lam K Y. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. Journal of Sound and Vibration, 2009, 320(4–5): 1100–1130

    Article  Google Scholar 

  39. Chapelle D, Bathe K J. The inf-sup test. Computers & Structures, 1993, 47(4–5): 537–545

    Article  MATH  MathSciNet  Google Scholar 

  40. Andersen K D, Christiansen E, Conn A R, Overton M L. An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms. SIAM Journal on Scientific Computing, 2001, 22(1): 243–262

    Article  MathSciNet  Google Scholar 

  41. Andersen E D, Roos C, Terlaky T. On implementing a primal-dual interior-point method for conic quadratic programming. Mathematical Programming, 2003, 95(2): 249–277

    Article  MATH  MathSciNet  Google Scholar 

  42. Dorfler W. A convergent adaptive algorithm for Poisson’s equation. SIAM Journal on Numerical Analysis, 1996, 33(3): 1106–1124

    Article  MathSciNet  Google Scholar 

  43. Rivara M C, Venere M. Cost analysis of the longest-side (triangle bisection) refinement algorithms for triangulations. Engineering with Computers, 1996, 12(3-4): 224–234

    Article  Google Scholar 

  44. Funken S, Praetorius D, Wissgott P. Efficient implementation of adaptive p1-FEM in Matlab. Preprint, 2008 (www.asc.tuwien.ac.at/ preprint/2008/asc19x2008.pdf)

    Google Scholar 

  45. Mosek. The MOSEK optimization toolbox for MATLAB manual. Mosek ApS, Version 5.0 Edition.9, 2009 (http://www.mosek.com)

  46. Le C V, Nguyen-Xuan H, Askes H, Rabczuk T, Nguyen-Thoi T. Computation of limit load using edge-based smoothed finite element method and second-order cone programming. International Journal of Computational Methods, 2013, 10(1): 1340004

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Interdisciplinary Research in Technology, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh, Vietnam

    H. Nguyen-Xuan

  2. Institute of Structural Mechanics, Bauhaus-Universität Weimar, Weimar, 99423, Germany

    T. Rabczuk

Authors
  1. H. Nguyen-Xuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Rabczuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Rabczuk.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen-Xuan, H., Rabczuk, T. Adaptive selective ES-FEM limit analysis of cracked plane-strain structures. Front. Struct. Civ. Eng. 9, 478–490 (2015). https://doi.org/10.1007/s11709-015-0317-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11709-015-0317-7

Keywords

Search

Navigation