A Retarded Mean-Field Approach for Interacting Fiber Structures

TL;DR

This paper develops a mean-field approach to model and analyze the interactions of fibers in a lay-down process, exploring asymptotic limits, numerical methods, and comparing solutions across multiple scales.

Contribution

It introduces a novel mean-field framework for fiber interactions and provides numerical analysis bridging microscopic, mesoscopic, and macroscopic models.

Findings

Mean-field equations accurately approximate microscopic behavior

Numerical methods effectively solve stochastic and mean-field models

Solutions across scales show consistent dynamics

Abstract

We consider an interacting system of one-dimensional structures modelling fibers with fiber-fiber interaction in a fiber lay-down process. The resulting microscopic system is investigated by looking at different asymptotic limits of the corresponding stochastic model. Equations arising from mean-field and diffusion limits are considered. Furthermore, numerical methods for the stochastic system and its mean-field counterpart are discussed. A numerical comparison of solutions corresponding to the different scales (microscopic, mesoscopic and macroscopic) is included.