A Three-Dimensional Model for Fiber Lay-Down Processes

We present an improved stochastic model concerning the lay-down of fibers on a conveyor belt in the production of nonwovens. The model is based on stochastic differential equations describing the resulting position of the fiber on the belt having regard to its motion in the deposition region under influence of turbulent air flow. Our aim is to generalize an existing model to 3D. By introducing a parameter we have an alternative to consider both isotropic and anisotropic orientation of the fibers generating the nonwoven. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Asymptotic Behaviour of a Stochastic 3D LANS-α Model

The long-time behaviour of a stochastic 3D LANS-α model on a bounded domain is analysed. First, we reformulate the model as an abstract problem. Next, we establish sufficient conditions ensuring the existence of stationary (steady state) solutions of this abstract nonlinear stochastic evolution equation, and study the stability properties of the model. Finally, we analyse the effects produced by stochastic perturbations in the deterministic version of the system (persistence of exponential stability as well as possible stabilisation effects produced by the noise). The general results are applied to our stochastic LANS-α system throughout the paper.     

Weak solutions to stochastic 3D Navier-Stokes-α model of turbulence: α-Asymptotic behavior

We study the asymptotic behavior of weak solutions to the stochastic 3D Navier-Stokes-α model as α approaches zero. The main result provides a new construction of the weak solutions of stochastic 3D Navier-Stokes equations as approximations by sequences of solutions of the stochastic 3D Navier-Stokes-α model.     

A mixed R&D projects and securities portfolio selection model

The business environment is full of uncertainty. Allocating the wealth among various asset classes may lower the risk of overall portfolio and increase the potential for more benefit over the long term. In this paper, we propose a mixed single-stage R&D projects and multi-stage securities portfolio selection model. Specifically, we present a bi-objective mixed-integer stochastic programming model. Moreover, we use semi-absolute deviation risk functions to measure the risk of mixed asset portfolio. Based on the idea of moments approximation method via linear programming, we propose a scenario generation approach for the mixed single-stage R&D projects and multi-stage securities portfolio selection problem. The bi-objective mixed-integer stochastic programming problem can be solved by transforming it into a single objective mixed-integer stochastic programming problem. A numerical example is given to illustrate the behavior of the proposed mixed single stage R&D projects and multi-stage securities portfolio selection model.     

Backward uniqueness of 3D MHD‐α model driven by linear multiplicative Gaussian noise

In this paper, we devote to proving the backward uniqueness property of the solution to three‐dimensional stochastic magnetohydrodynamic‐α model (3D stochastic MHD‐α model) driven by linear multiplicative Gaussian noise, which involves not only the study of multiplicative noise but also the challenging nonlinear drift terms.     

Entropy-based Inhomogeneity Detection in Fiber Materials

We study a change-point problem for random fields based on a univariate detection of outliers via the 3σ-rule in order to recognize inhomogeneities in glass fiber reinforced polymers (GFRP). In particular, we focus on GFRP modeled by stochastic fiber processes with high fiber intensity and search for abrupt changes in the direction of the fibers. As a measure of change, the entropy of the directional distribution is locally estimated within a window that scans the region to be analyzed.     

The Analysis of Stochastic Fiber Lay-Down Models: Geometry and Convergence to Equilibrium of the Basic Model

The so-called fiber lay-down models arise in the production process of nonwovens. We introduce the generalized version of the basic fiber lay-down model which can precisely be formulated in abstract form as some manifold-valued stochastic differential equation. An important criterion for the quality of the nonwoven material is how the solution to the associated Fokker-Planck equation converges towards its stationary state. Especially, one is interested in determining the speed of convergence. Here we present some results concerning the long-time behavior by using classical stochastic methods as well as modern analytic methods from the theory of hypocoercivity. Demanding mathematical difficulties arising since the equation is degenerate. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smooth solutions of non-linear stochastic partial differential equations driven by multiplicative noises

In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.     

Fiber waviness in textile composites and its stochastic modeling

A generic stochastic theory of composite materials with continuous, randomly curved (imperfect) fiber reinforcements, recently developed by the present authors, enables one to quantify the effect of fiber deviations from the assumed perfect paths. The theory of random functions and stochastic extension of the orientational averaging approach are utilized to evaluate the mean values and standard deviations of the full set of anisotropic stiffness characteristics. The major advantage of this novel stochastic approach is its applicability to practically any fiber reinforcement architecture, from unidirectional to multidirectional, 3-D woven, and braided composites. Importantly, the approach does not ask for exact quantification of the reinforcement imperfections, but needs only a limited knowledge of the mean path of the reinforcement and standard deviation of the local tangent. Numerical examples illustrating applications of the stochastic theory developed consider three types of composites having (i) unidirectional, (ii) biaxial, 2-D braided, and (iii) 3-D orthogonally woven reinforcements. The first example concerns validation of the model. The second example is selected due to the commonly observed significant randomness of the fiber architecture in biaxially braided composite shell elements. The third example illustrates the effect of Z-yarn waviness (illustrated by optical microscopy) in orthogonally woven composites on their elastic characteristics.     

Convergence for a Splitting-Up Scheme for the 3D Stochastic Navier-Stokes-α Model

We propose and analyze a splitting-up scheme for the numerical approximation of the 3D stochastic Navier-Stokes-α model. We prove the convergence of the scheme to the unique variational solution of the 3D stochastic Navier-Stokes-α model when the time step tends to zero.     

STOCHASTIC SIMPLIFIED BARDINA TURBULENT MODEL： EXISTENCE OF WEAK SOLUTION

In this paper, we consider the stochastic version of the 3D Bardina model arising from the turbulent flows of fluids. We obtain the existence of probabilistie weak solution for the model with the non-Lipschitz condition.     

Solution Properties of a 3D Stochastic Euler Fluid Equation

We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.

     

Random point fields with Markovian refinements and the geometry of fractally disordered media

We give a general construction of the probability measure for describing stochastic fractals that model fractally disordered media. For these stochastic fractals, we introduce the notion of a metrically homogeneous fractal Hansdorff-Karathéodory measure of a nonrandom type. We select a classF[q] of random point fields with Markovian refinements for which we explicitly construct the probability distribution. We prove that under rather weak conditions, the fractal dimension D for random fields of this class is a self-averaging quantity and a fractal measure of a nonrandom type (the Hausdorff D-measure) can be defined on these fractals with probability 1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 490–505, September, 2000.     

Stochastic flows of diffeomorphisms on an open set in rn

In this paper, we will give sufficient conditions for the solution to a stochastic differential equation (SDE) on an open set D in R" to define a stochastic flow of diffeomorphisms of D onto itself. Since a necessary and sufficient condition for the solution to determine a stochastic flow of diffeomorphisms is that the original SDE and its adjoint SDE are both strictly conservative, we will concentrate our attention on finding sufficient conditions for the SDE to be strictly conservative. It will be etablished that the strict conservativeness follows if the vector fields governing the SDE decay suitably near the boundary dD in the direction transversal to 3D and some additional assumptions are satisfied.     

Stochastic stability and bifurcation for the chronic state in Marchuk’s model with noise

A stochastic differential equation modelling a Marchuk’s model is investigated. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. Firstly, the stochastic Marchuk’s model has been simplified by applying stochastic center manifold and stochastic average theory. Secondly, by using Lyapunov exponent and singular boundary theory, we analyze the local stochastic stability and global stochastic stability for stochastic Marchuk’s model, respectively. Thirdly, we explore the stochastic bifurcation of the stochastic Marchuk’s model according to invariant measure and stationary probability density. Some new criteria ensuring stochastic pitchfork bifurcation and P-bifurcation for stochastic Marchuk’s model are obtained, respectively.     

Theoretical Justification and Error Analysis for Slender Body Theory

Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three-dimensional fiber, in part due to the fact that simply prescribing data along a 1D curve does not result in a well-posed boundary value problem for the Stokes equations in ℝ³ . Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well-posedness framework, allows us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry. © 2019 Wiley Periodicals, Inc.     

Stochastic cohomology of Chen–Souriau and line bundle over the Brownian bridge

We define a stochastic cohomology theory related to a stochastic diffeology for the Hoelder loop space. We show that the stochastic de Rham cohomology groups are equal to the deterministic de Rham cohomology groups of the Hoelder loop space. As an application, we show that a stochastic line bundle over the Brownian bridge (with fiber almost surely defined) is isomorphic to a true line bundle over the Hoelder loop space. Received: 9 November 1998 / Revised version: 14 July 2000 / Published online: 26 April 2001     

A scenario‐based stochastic programming model for the control or dummy wafers downgrading problem

The subject of this paper is to study a realistic planning environment in wafer fabrication for the control or dummy (C/D) wafers problem with uncertain demand. The demand of each product is assumed with a geometric Brownian motion and approximated by a finite discrete set of scenarios. A two‐stage stochastic programming model is developed based on scenarios and solved by a deterministic equivalent large linear programming model. The model explicitly considers the objective to minimize the total cost of C/D wafers. A real‐world example is given to illustrate the practicality of a stochastic approach. The results are better in comparison with deterministic linear programming by using expectation instead of stochastic demands. The model improved the performance of control and dummy wafers management and the flexibility of determining the downgrading policy. Copyright © 2008 John Wiley & Sons, Ltd.