An algorithm for adaptive introduction and arrangement of velocity discontinuities within 3D finite-element-based upper bound limit analysis approaches

Abstract

This paper presents a new adaptive strategy to efficiently exploit velocity discontinuities in 3D finite-element-based upper bound limit analysis formulations. Based on an initial upper bound result, obtained with a conventional approach without velocity discontinuities, possible planes of plastic flow localisation are determined at each strain-rate evaluation node and, subsequently, this information is used to sequentially introduce discontinuities into the considered discretised structure. During a few iterations, by means of introducing new and adjusting existing discontinuities, an optimal velocity discontinuity layout is obtained. For the general 3D case, the geometry of this layout is defined by the well-known level set method, standardly used to define the geometry of cracks in the extended finite element method.To make this method also applicable for orthotropic strength behaviours, traction-based yield functions defining the plastic flow across discontinuities are derived from their stress-based counterparts. This procedure is outlined in detail and the obtained traction-based yield functions are verified numerically, to guarantee a consistent strength behaviour throughout the whole discretised structure.By means of three different examples, including isotropic as well as orthotropic yield functions, the performance of the proposed strategy is investigated and upper bound results as well as failure modes are compared to reference solutions. The proposed approach delivers reliable upper bounds for each example and the majority of plastic flow takes place across the sensibly-arranged discontinuities. For this reason, very good upper bounds can be obtained with a quite coarse finite element mesh and only few introduced velocity discontinuities. This represents an attractive alternative to commonly-used adaptive mesh refinement strategies, where often a huge number of degrees of freedom need to be added to capture localised failure.