Upper bound limit analysis using simplex strain elements and second-order cone programming

Makrodimopoulos, A. and Martin, C.M. (2007) Upper bound limit analysis using simplex strain elements and second-order cone programming. International Journal for Numerical and Analytical Methods in Geomechanics, 31 (6), 835-865. (doi:10.1002/nag.567).

Abstract

In geomechanics, limit analysis provides a useful method for assessing the capacity of structures such as footings and retaining walls, and the stability of slopes and excavations. This paper presents a finite element implementation of the kinematic (or upper bound) theorem that is novel in two main respects. First, it is shown that conventional linear strain elements (6-node triangle, 10-node tetrahedron) are suitable for obtaining strict upper bounds even in the case of cohesive-frictional materials, provided that the element sides are straight (or the faces planar) such that the strain field varies as a simplex. This is important because until now, the only way to obtain rigorous upper bounds has been to use constant strain elements combined with a discontinuous displacement field. It is well known (and confirmed here) that the accuracy of the latter approach is highly dependent on the alignment of the discontinuities, such that it can perform poorly if an unstructured mesh is employed. Second, the optimization of the displacement field is formulated as a standard second-order cone programming (SOCP) problem. Using a state-of-the-art SOCP code developed by researchers in mathematical programming, very large example problems are solved with outstanding speed. The examples concern plane strain and the Mohr-Coulomb criterion, but the same approach can be used in 3D with the Drucker-Prager criterion, and can readily be extended to other yield criteria having a similar conic quadratic form.

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