A smooth model for fiber lay-down processes and its diffusion approximations

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.Keywords interacting stochastic particles, fibers, mean-field equations, retarded potential, delay equations.2010 AMS Subject Classification: 92D50, 35B40, 82C22, 92C151. Introduction. One-dimensional structures appear in various context of industrial applications. They are used, for example, in the modelling of polymers in suspensions, composite materials, nanostructures, fiber dynamics in turbulent flows and, in particular, fiber lay-down in technical processes of non-woven materials. Furthermore, such structures have been modelled on different levels of description involving different scales. Besides microscopic models, mesoscopic kinetic or Fokker-Planck equations have been widely used for a statistical description of the fiber or polymer distributions. We refer to [1,2] and [18,21] for concrete examples in the industry. In this article, we consider non-woven materials, which are webs of long flexible fibers. Production processes and models corresponding to the lay-down of such fibers have been intensively investigated. See [4] and the above cited references.In the above mentioned investigations, the one-dimensional structures (fibers) under consideration were assumed to be mutually independent, which clearly does not represent reality. Therefore, the present work aims at including fiber-fiber interaction, thereby describing the size of each fiber and, simultaneously, the absence of intersection among fibers. This is achieved by simply including the interaction of structures into a well investigated model described in non-woven production processes. Taking into account the interaction of the structures on the microscopic level leads to coupled systems of stochastic differential equations. Its statistical description should also take into account the interactions and will consequently no longer be based on the classical Fokker-Planck model.The new model makes use of a microscopic systems of retarded stochastic differential equations, and its mesoscopic description is obtained via formal mean-field procedures. The mean-field limit is described by a McKean-Vlasov type equation with a delay term. We perform an analytical investigation of the mean field limit, as well as a numerical comparison of microscopic, mean field and macroscopic equations. The analysis of the limit is based on the work in [5,12,15,16,17,25,31]. For numerical methods for mean-field type equations we refer to [2,26,27,30].The paper is organized as follows: starting from a...

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“…This equation is similar to an equation derived in [4,19,24] for the case without an interaction potential. The stationary equation reads…”

Section: Large Diffusion Scalingsupporting

confidence: 60%

A Retarded Mean-Field Approach for Interacting Fiber Structures

Borsche¹,

Klar²,

Nessler³

et al. 2017

Multiscale Model. Simul.

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“…Indeed, the term is markedly relevant to capture the curved nature of the new state space, cf. Klar et al [6,[8][9][10] As a final note, we recover the basic model from our refinement as white noise limit as → ∞ while is kept constant, see Herty et al [6,8,11] Compare the trajectories of the two-dimensional smooth model (2) in Figure 4 to Figure 2. As before a Gaussian potential Φ = |x| 2 is supposed and we vary the diffusion parameter.…”

Section: Smooth Modelmentioning

confidence: 99%

Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

2018

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In this article we review a powerful and fairly simple hypocoercivity method and its application to qualitative analysis of industrially relevant fibre lay‐down models. The hypocoercivity strategy is formulated in an abstract Hilbert space setting with all necessary conditions; we provide some interpretation of these conditions and briefly explain why they are needed to make the machinery working. Once the conditions are fulfilled we gain exponential decay of the strongly continuous semigroup associated to the fibre lay‐down process at an explicitly known rate. We interpret this result as describing for instance homogeneity of nonwoven fibre webs. Primarily, this article portrays the concepts, however it is based on original results by Grothaus and Stilgenbauer, which can be found in the previous studies.

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“…In the 2D case the parameter A and the shape of the potential V have been estimated in [10] for different production processes. There, full simulations of a single representative fiber have been performed with the software tool FIDYST 1 , see [12,8] for a description of the algorithms. The noise amplitude A and the potential V in the 2D model (2.4) have been identified on the basis of the full simulation.…”

Section: The Large Coiling Forcementioning

confidence: 99%

“…See [5] for the 2-D case. The drawback of non-differentiable fiber path can be overcome by replacing the Wiener Process by an Ornstein-Uhlenback process leading to a more realistic smooth model [11] analogous to the 2-D case, see [8]. (15 pages, 1998) 12.…”

Section: The Large Coiling Forcementioning

confidence: 99%

A 3d Model for Fiber Lay-Down in Nonwoven Production Processes

Klar

Maringer

Wegener

2012

Math. Models Methods Appl. Sci.

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In this paper a three-dimensional stochastic model for the lay-down of fibers on a moving conveyor belt in the production process of nonwoven materials is derived. The model is based on stochastic differential equations describing the resulting position of the fiber on the belt under the influence of turbulent air flows. The model presented here is an extension of an existing surrogate model, see Refs. 6 and 3.

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A smooth model for fiber lay-down processes and its diffusion approximations

Cited by 16 publications

References 4 publications

A Retarded Mean-Field Approach for Interacting Fiber Structures

A Retarded Mean-Field Approach for Interacting Fiber Structures

Hypocoercivity for geometric Langevin equations motivated by fibre lay‐down models arising in industrial application

A 3d Model for Fiber Lay-Down in Nonwoven Production Processes

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