Fokker-Planck modeling of many-agent systems in swarm manufacturing: asymptotic analysis and numerical results

TL;DR

This paper introduces a Fokker-Planck model for large-agent systems in swarm manufacturing, analyzing its long-term behavior and providing numerical methods for 2D cases.

Contribution

It presents a novel mean-field Fokker-Planck model with discontinuous flux for swarm manufacturing, including theoretical analysis and numerical methods for 2D extension.

Findings

Existence and uniqueness of a diffusion coefficient ensuring target domain coverage.

Convergence to equilibrium in one dimension.

Numerical extension to two dimensions using structure-preserving methods.

Abstract

In this paper we study a novel Fokker-Planck-type model that is designed to mimic manufacturing processes through the dynamics characterizing a large set of agents. In particular, we describe a many-agent system interacting with a target domain in such a way that each agent/particle is attracted by the center of mass of the target domain with the aim to uniformly cover this zone. To this end, we first introduce a mean-field model with discontinuous flux whose large time behavior is such that the steady state is globally continuous and uniform over a connected portion of the domain. We prove that a diffusion coefficient that guarantees that a given portion of mass enters in the target domain exists and that it is unique. Furthermore, convergence to equilibrium in 1D is provided through a reformulation of the initial problem involving a nonconstant diffusion function. The extension to 2D is explored numerically by means of recently introduced structure preserving methods for Fokker-Planck equations.