Mesh adaptive computation of upper and lower bounds in limit analysis

Abstract

AbstractAn efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented, with a formulation that explicitly considers the exact convex yield condition. The approach consists of two main steps. First, the continuous problem, under the form of the static principle of limit analysis, is discretized twice (one per bound) using particularly chosen finite element spaces for the stresses and velocities that guarantee the attainment of an upper or a lower bound. The second step consists of solving the resulting discrete non‐linear optimization problems. These are reformulated as second‐order cone programs, which allows for the use of primal–dual interior point methods that optimally exploit the convexity and duality properties of the limit analysis model. To benefit from the fact that collapse mechanisms are typically highly localized, a novel method for adaptive meshing is introduced. The method first decomposes the total bound gap as the sum of positive contributions from each element in the mesh and then refines those elements with higher contributions. The efficiency of the methodology is illustrated with applications in plane stress and plane strain problems. Copyright © 2008 John Wiley & Sons, Ltd.