Upper bound limit analysis using linear finite elements and non‐linear programming

TLDR

A new method for computing rigorous upper bounds on limit loads for one‑, two‑ and three‑dimensional continua is described. The method uses linear finite elements with kinematically admissible velocity discontinuities, formulating a convex nonlinear programming problem that minimizes dissipated power under linear and nonlinear constraints, solved efficiently by a two‑stage quasi‑Newton scheme. The approach yields mesh‑size‑independent iteration counts, requires no yield‑surface linearization, handles 3‑D geometries without special treatment, and is up to two orders of magnitude faster than equivalent linear‑programming schemes. © 2001 John Wiley & Sons, Ltd.

Abstract

Abstract A new method for computing rigorous upper bounds on the limit loads for one‐, two‐ and three‐dimensional continua is described. The formulation is based on linear finite elements, permits kinematically admissible velocity discontinuities at all interelement boundaries, and furnishes a kinematically admissible velocity field by solving a non‐linear programming problem. In the latter, the objective function corresponds to the dissipated power (which is minimized) and the unknowns are subject to linear equality constraints as well as linear and non‐linear inequality constraints. Provided the yield surface is convex, the optimization problem generated by the upper bound method is also convex and can be solved efficiently by applying a two‐stage, quasi‐Newton scheme to the corresponding Kuhn–Tucker optimality conditions. A key advantage of this strategy is that its iteration count is largely independent of the mesh size. Since the formulation permits non‐linear constraints on the unknowns, no linearization of the yield surface is necessary and the modelling of three‐dimensional geometries presents no special difficulties. The utility of the proposed upper bound method is illustrated by applying it to a number of two‐ and three‐dimensional boundary value problems. For a variety of two‐dimensional cases, the new scheme is up to two orders of magnitude faster than an equivalent linear programming scheme which uses yield surface linearization. Copyright © 2001 John Wiley & Sons, Ltd.