Upper and lower bounds in limit analysis: Adaptive meshing strategies and discontinuous loading

SUMMARY Upper and lower bounds of the collapse load factor are here obtained as the optimum values of two discrete constrained optimisation problems. The membership constraints for Von Mises and MohrCoulomb plasticity criteria are written as a set of quadratic constraints, which permits to solve the optimisation problem using specific algorithms for Second Order Conic Program (SOCP). From the stress field at the lower bound and the velocities at the upper bound, we construct a novel error estimate, based on elemental and edge contributions to the bound gap. These contributions are employed in an adaptive remeshing strategy that is able to reproduce fan-type mesh patterns around points with discontinuous surface loading. The solution of this type of problems is analysed in detail, and from this study some additional meshing strategies are also described. We particularise the resulting formulation and strategies to two-dimensional problems in plane strain and we demonstrate the effectiveness of the method with a set of numerical examples extracted from the literature. Copyright c � 2007 John Wiley & Sons, Ltd.

[1]  L. Prandtl,et al.  Über die Härte plastischer Körper , 1920 .

[2]  Scott W. Sloan,et al.  Three-dimensional Mohr-Coulomb limit analysis using semidefinite programming , 2007 .

[3]  Jose Luis Silveira,et al.  An algorithm for shakedown analysis with nonlinear yield functions , 2002 .

[4]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[5]  John Lysmer,et al.  LIMIT ANALYSIS OF PLANE PROBLEMS IN SOIL MECHANICS , 1970 .

[6]  Scott W. Sloan,et al.  Lower bound limit analysis with adaptive remeshing , 2005 .

[7]  Pedro Díez,et al.  Adaptivity based on error estimation for viscoplastic softening materials , 2000 .

[8]  Ted Belytschko,et al.  Plane Stress Limit Analysis by Finite Elements , 1970 .

[9]  Kristian Krabbenhoft,et al.  A general non‐linear optimization algorithm for lower bound limit analysis , 2003 .

[10]  Scott W. Sloan,et al.  Upper bound limit analysis using linear finite elements and non‐linear programming , 2001 .

[11]  C. Martin,et al.  Lower bound limit analysis of cohesive‐frictional materials using second‐order cone programming , 2006 .

[12]  Wai-Fah Chen.  Limit Analysis and Soil Plasticity , 1975 .

[13]  Zhangzhi Cen,et al.  Numerical determination of limit loads for three-dimensional structures using boundary element method , 2004 .

[14]  Raúl A. Feijóo,et al.  An adaptive approach to limit analysis , 2001 .

[15]  A. Huerta,et al.  A unified approach to remeshing strategies for finite element h-adaptivity , 1999 .

[16]  S. Sloan,et al.  Upper bound limit analysis using discontinuous velocity fields , 1995 .

[17]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[18]  Scott W. Sloan,et al.  Lower bound limit analysis using non‐linear programming , 2002 .

[19]  Jaime Peraire,et al.  Mesh adaptive computation of upper and lower bounds in limit analysis , 2008 .

[20]  Héctor,et al.  Computation of upper and lower bounds in limit analysis using second-order cone programming and mesh adaptivity , 2004 .

[21]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[22]  E. Christiansen.  Limit analysis of collapse states , 1996 .

[23]  Scott W. Sloan,et al.  A new discontinuous upper bound limit analysis formulation , 2005 .

[24]  Knud D. Andersen,et al.  Computation of collapse states with von Mises type yield condition , 1998 .

[25]  Josep Sarrate Ramos,et al.  Management, design and developement of a mesh generation environment using open source software , 2009 .

[26]  Edmund Christiansen,et al.  Automatic mesh refinement in limit analysis , 1999 .