Published paperLe, Canh V., Gilbert, Matthew and Askes, Harm (2009) SUMMARYThe meshless Element-Free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation which involves approximating the displacement/velocity field using the moving least squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing directly displacements at the nodes. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non-smooth minimization problem can be transformed into a form suitable for solution using Second-Order Cone Programming (SOCP). The procedure is applied to several benchmark problems and is found in practice to generate good upper bound solutions for benchmark problems.

The yield-line method of analysis is a long established and extremely effective means of estimating the maximum load sustainable by a slab or plate. However, although numerous attempts to automate the process of directly identifying the critical pattern of yield-lines have been made over the past few decades, to date none has proved capable of reliably analysing slabs of arbitrary geometry. Here, it is demonstrated that the discontinuity layout optimization (DLO) procedure can successfully be applied to such problems. The procedure involves discretization of the problem using nodes inter-connected by potential yield-line discontinuities, with the critical layout of these then identified using linear programming. The procedure is applied to various benchmark problems, demonstrating that highly accurate solutions can be obtained, and showing that DLO provides a truly systematic means of directly and reliably automatically identifying yield-line patterns. Finally, since the critical yield-line patterns for many problems are found to be quite complex in form, a means of automatically simplifying these is presented.

Published paperLe, C.V., Nguyen-Xuan, H. and Nguyen-Dang, H. (2010) in which the pressure is equilibrated by corner loads only, ensuring that exact equilibrium relations associated with a uniform pressure can be obtained. Once the displacement or moment fields are approximated and the bound theorems applied, limit analysis becomes a problem of optimization. In this paper, the optimization problems are formulated in the form of a standard second-order cone programming which can be solved using highly efficient interior point solvers.The procedures are tested by applying it to several benchmark plate problems and are found good agreement between the present upper and lower bound solutions and results in the literature.Key words: Limit analysis, upper and lower bounds, displacement and equilibrium models, criterion of mean, second order cone programming

Published paperLe, Canh V., Nguyen-Xuan, H., Askes, H., Bordas , Stéphane P. A., Rabczuk, T. and Nguyen-Vinh, H. (2010) SUMMARYThis paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method (CS-FEM) with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged.