A comparison between the finite element method and a kinematic model derived from robot swarms for first and second gradient continua

Abstract

AbstractIn this paper, we consider a deformable continuous medium and its discrete representation realized by a lattice of points. The former is solved using the classical variational formulation with the finite element method. The latter, a 2D discrete “kinematic” model, instead is conceived to determine the displacements of the lattice points depending on interaction rules among them and thus provides the final configuration of the system. The kinematic model assigns the displacements of some points, so-called leaders, by solving Newton’s law; the other points, namely followers, are left to rearrange themselves according to the lattice structure and the flocking rules. These rules are derived from the effort to describe the behaviour of a robot swarm as a single whole organism. The advantage of the kinematic model lies in reducing computational cost and the easiness of managing complicated structures and fracture phenomena. In addition, generalizing the discrete model to non-local interactions, such as for second gradient materials, is easier than solving partial differential equations. This paper aims to compare and discuss the deformed configurations obtained by these two approaches. The comparison between FEM and the kinematic model shows a reasonable agreement even in the case of large deformations for the standard case of the first gradient continuum.