Constrained optimisation in granular network flows: Games with a loaded dice

Abstract

Flows in real world networks are rarely the outcome of unconditional random allocations as, say, the roll of a dice. Think, for example, of force transmission through a contact network in a quasistatically deforming granular material. Forces ‘flow’ through this network in a highly conditional manner. How much force is transmitted between two contacting particles is always conditional not only on all the other forces acting between the particles in question but also on those acting on the other particles in the system. Broadly, we are interested in the nature and extent to which flows through a contact network favour certain pathways over others, and how the mechanisms that govern such biased flows for a given imposed loading history determine the future evolution of the contact network. Our first step is to solve a selection of fundamental combinatorial optimisation problems on the contact network from the perspective of force transmission. Here we report on solutions to the Maximum Flow Minimum Cost Problem for a weighted contact network where the weights assigned to the links of the contact network are varied according to their contact types. We found that those pathways through which the maximum flow of force is transmitted, in the direction of the maximum principal stress, at minimum cost – pass through the great majority of the force chains. Although the majority of the contacts in these pathways are elastic, the plastic contacts bear an undue influence on the minimum cost.