Abstract: We consider the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the Mises condition. After discretization with the finite element method, using divergence-free elements for the plastic flow, the kinematic formulation turns into the problem of minimizing a sum of Euclidean vector norms, subject to a single linear constraint. This is a nonsmooth minimization problem, since many of the norms in the sum may vanish at the optimal point. However, efficient solution algorithms for this particular convex optimization problem have recently been developed.

The method is applied to test problems in limit analysis in two different plane models: plane strain and plates. In the first case more than 80 percent of the terms in the sum are zero in the optimal solution, causing severe ill-conditioning. In the last case all terms are nonzero. In both cases the algorithm works very well, and we solve problems which are larger by at least an order of magnitude than previously reported. The relative accuracy for the discrete problems, measured by duality gap and feasibility, is typically of the order 1.0E-8. The discretization error, due to the finite grid, depends on the nature of the solution. In the applications reported here it ranges from 1.0E-5 to 1.0E-2.

Keywords: Limit analysis, plasticity, finite element method, nonsmooth optimization.