SUMMARYThis paper describes a new formulation, based on linear ÿnite elements and non-linear programming, for computing rigorous lower bounds in 1, 2 and 3 dimensions. The resulting optimization problem is typically very large and highly sparse and is solved using a fast quasi-Newton method whose iteration count is largely independent of the mesh reÿnement. For two-dimensional applications, the new formulation is shown to be vastly superior to an equivalent formulation that is based on a linearized yield surface and linear programming. Although it has been developed primarily for geotechnical applications, the method can be used for a wide range of plasticity problems including those with inhomogeneous materials, complex loading, and complicated geometry.

SUMMARYA new upper bound formulation of limit analysis of two-and three-dimensional solids is presented. In contrast to most discrete upper bound methods the present one is formulated in terms of stresses rather than velocities and plastic multipliers. However, by means of duality theory it is shown that the formulation does indeed result in rigorous upper bound solutions. Also, kinematically admissible discontinuities, which have previously been shown to be very efficient, are given an interpretation in terms of stresses. This allows for a much simpler implementation and, in contrast to existing formulations, extension to arbitrary yield criteria in two and three dimensions is straightforward. Finally, the capabilities of the new method are demonstrated through a number of examples.

The application of conic programming to some traditionally difficult plasticity problems is considered. Convenient standard forms for conic programming of both limit and incremental elastoplastic analysis are given. The types of yield criteria that can be treated by conic programming is discussed and it is shown that the three-dimensional Mohr-Coulomb criterion can be cast as a set of conic constraints, thus facilitating efficient treatment by dedicated algorithms. Finally, the performance of a number of mixed finite elements is evaluated together with a state-of-the-art second-order cone programming algorithm.

SUMMARYA new method for computing rigorous upper bounds on the limit loads for one-, two-and threedimensional 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.

SUMMARYThe problem of small-deformation, rate-independent elastoplasticity is treated using convex programming theory and algorithms. A finite-step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the Lagrangian associated with a convex mathematical program. Next, an algorithm, based on the classical primal-dual interior point method, is developed. Several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem. The resulting method is compared to state-of-the-art elastoplastic procedures for which both similarities and differences are found. Finally, a number of examples are solved, demonstrating the capabilities of the algorithm when applied to standard perfect plasticity, hardening multisurface plasticity, and problems involving softening.

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SUMMARYRecently, Krabbenhøft et al. (Int. J. Solids Struct. 2007; 44:1533-1549) have presented a formulation of the three-dimensional Mohr-Coulomb criterion in terms of positive-definite cones. The capabilities of this formulation when applied to large-scale three-dimensional problems of limit analysis are investigated. Following a brief discussion on a number of theoretical and algorithmic issues, three common, but traditionally difficult, geomechanics problems are solved and the performance of a common primaldual interior-point algorithm (SeDuMi (Appl. Numer. Math. 1999; 29:301-315)) is documented in detail. Although generally encouraging, the results also reveal several difficulties which support the idea of constructing a conic programming algorithm specifically dedicated to plasticity problems.

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