13 research outputs found
Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes
This paper is devoted to the adaptation of the method developed in [4,3] to a
Fokker-Planck equation for fiber lay-down which has been studied in [1,5].
Exponential convergence towards a unique stationary state is proved in a norm
which is equivalent to a weighted norm. The method is based on a micro /
macro decomposition which is well adapted to the diffusion limit regime.Comment: 8 page
Exponential decay to equilibrium for a fibre lay-down process on a moving conveyor belt
We show existence and uniqueness of a stationary state for a kinetic
Fokker-Planck equation modelling the fibre lay-down process in the production
of non-woven textiles. Following a micro-macro decomposition, we use
hypocoercivity techniques to show exponential convergence to equilibrium with
an explicit rate assuming the conveyor belt moves slow enough. This work is an
extension of (Dolbeault et al., 2013), where the authors consider the case of a
stationary conveyor belt. Adding the movement of the belt, the global Gibbs
state is not known explicitly. We thus derive a more general hypocoercivity
estimate from which existence, uniqueness and exponential convergence can be
derived. To treat the same class of potentials as in (Dolbeault et al., 2013),
we make use of an additional weight function following the Lyapunov functional
approach in (Kolb et al., 2013)
Single to double mill small noise transition via semi-Lagrangian finite volume methods
We show that double mills are more stable than single mills under stochastic perturbations in swarming dynamic models with basic attraction-repulsion mechanisms. In order to analyse this fact accurately, we will present a numerical technique for solving kinetic mean field equations for swarming dynamics. Numerical solutions of these equations for different sets of parameters will be presented and compared to microscopic and macroscopic results. As a consequence, we numerically observe a phase transition diagram in terms of the stochastic noise going from single to double mill for small stochasticity fading gradually to disordered states when the noise strength gets larger. This bifurcation diagram at the inhomogeneous kinetic level is shown by carefully computing the distribution function in velocity space
Geometric Langevin equations on submanifolds and applications to the stochastic melt-spinning process of nonwovens and biology
In this article we develop geometric versions of the classical Langevin
equation on regular submanifolds in euclidean space in an easy, natural way and
combine them with a bunch of applications. The equations are formulated as
Stratonovich stochastic differential equations on manifolds. The first version
of the geometric Langevin equation has already been detected before by
Leli\`evre, Rousset and Stoltz with a different derivation. We propose an
additional extension of the models, the geometric Langevin equations with
velocity of constant absolute value. The latters are seemingly new and provide
a galaxy of new, beautiful and powerful mathematical models. Up to the authors
best knowledge there are not many mathematical papers available dealing with
geometric Langevin processes. We connect the first version of the geometric
Langevin equation via proving that its generator coincides with the generalized
Langevin operator proposed by Soloveitchik, Jorgensen and Kolokoltsov. All our
studies are strongly motivated by industrial applications in modeling the fiber
lay-down dynamics in the production process of nonwovens. We light up the
geometry occuring in these models and show up the connection with the spherical
velocity version of the geometric Langevin process. Moreover, as a main point,
we construct new smooth industrial relevant three-dimensional fiber lay-down
models involving the spherical Langevin process. Finally, relations to a class
of self-propelled interacting particle systems with roosting force are
presented and further applications of the geometric Langevin equations are
given
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Modelling fibre laydown and web uniformity in nonwoven fabric
The mechanical and functional performance of nonwoven fabric critically depends on the fibre architecture. The fibre laydown process plays a key role in controlling this architecture. The fibre dynamic behaviour during laydown is studied through a finite element model which describes the role of the parameters in defining the area covered by a single fibre when deposited on the conveyor belt. The path taken by a fibre is described in terms of the radius of gyration, which characterises the area covered by the fibre in the textile, and the spectrum of curvature, which describes the degree of fibre looping as a function of the arc length. Starting from deterministic and idealised fibre curvature spectra, stochastic Monte Carlo simulations are undertaken to generate full nonwoven web samples and reproduce the uniformity of fibre density. A novel image analysis technique that allows measurement of the uniformity of real spunbonded nonwoven samples from images of textiles is used to confirm the validity of the model. It is shown that the main parameter that governs the fibre density uniformity is the ratio of the fibre spinning velocity to the velocity of conveyor belt, while fibre oscillations prior to deposition play a secondary role.Fitesa Germany gmb