R-adaptivity in Limit Analysis



Direct methods aim to find the maximum load factor that a domain made of a plastic material can sustain before undergoing full collapse. Its analytical solution may be posed as a constrained maximisation problem, which is computationally solved by resorting to appropriate discretisation of the relevant fields such as the stress or velocity fields. The actual discrete solution is though strongly dependent on such discretisation, which is defined by a set of nodes, elements, and the type of interpolation. We here resort to an adaptive strategy that aims to perturb the positions of the nodes in order to improve the solution of the discrete maximisation problem. When the positions of the nodes are taken into account, the optimisation problem becomes highly non-linear. We approximate this problem as two staggered linear problems, one written in terms of the stress variable (lower bound problem) or velocity variables (upper bound problem), and another with respect to the nodal positions. In this manner, we show that for some simple problems, the computed load factor may be further improved while keeping a constant number of elements.


  1. 1.
    Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University, PressCrossRefzbMATHGoogle Scholar
  2. 2.
    Cottereau R, Díez P (2015) Fast r-adaptivity for multiple queries of heterogeneous stochastic material fields. Comput Mech 66:601–612MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hambleton J, Sloan S (2013) A perturbation method for optimization of rigid block mechanisms in the kinematic method of limit analysis. Comput Geotech 48:260–271CrossRefGoogle Scholar
  4. 4.
    Kim J, Panatinarak T, Shontz SM (2013) A multiobjective mesh optimization framework for mesh quality improvement and mesh untangling. Int J Numer Methods Eng 94(7):20–42MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Krabbenhøft K, Lyamin AV, Hjiaj M, Sloan SW (2005) A new discontinuous upper bound limit analysis formulation. Int J Numer Methods Eng 63:1069–1088CrossRefzbMATHGoogle Scholar
  6. 6.
    Lyamin AV, Sloan SW (2002a) Lower bound limit analysis using non-linear programming. Int J Numer Methods Eng 55:576–611CrossRefzbMATHGoogle Scholar
  7. 7.
    Lyamin AV, Sloan SW (2002b) Upper bound limit analysis using linear finite elements and non-linear programming. Int J Numer Anal Methods Geomech 26:181–216CrossRefzbMATHGoogle Scholar
  8. 8.
    Lyamin AV, Sloan SW, Krabbenhøft K, Hjiaj M (2005) Lower bound limit analysis with adaptive remeshing. Int J Numer Methods Eng 63:1961–1974CrossRefzbMATHGoogle Scholar
  9. 9.
    Ma L, Klug WS (2008) Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics. J Comput Phys 227(11):5816–5835MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Muñoz JJ, Bonet J, Huerta A, Peraire J (2009) Upper and lower bounds in limit analysis: adaptive meshing strategies and discontinuous loading. Int J Numer Methods Eng 77:471–501MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Muñoz JJ, Bonet J, Huerta A, Peraire J (2012) A note on upper bound formulations in limit analysis. Int J Numer Methods Eng 91(8):896–908MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sloan SW, Kleeman PW (1995) Upper bound limit analysis using discontinuous velocitiy fields. Comput Methods Appl Mech Eng 127(5):293–314CrossRefzbMATHGoogle Scholar
  13. 13.
    Thoutireddy P, Ortiz M (2004) A variational r-adaption and shape-optimization method for finite-deformation elasticity. Int J Num Methods Eng 61:1–21MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zielonka MG, Ortiz M, Marsden JE (2008) Variational r-adaption in elastodynamics. Int J Num Methods Eng 74:1162–1197MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Department of Civil and Environmental EngineeringNorthwestern University (Previously at University of Newcastle, Newcastle, Australia)EvanstonUSA
  3. 3.University of NewcastleNewcastleAustralia

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