Stochastic Fluid Research Articles

Tangential fluid flow within 3D narrow fissures: Conservative velocity fields on associated triangulations and transport processes

For a fissured medium with uncertainty in the knowledge of fractures' geometry, a conservative tangential flow field is constructed, which is consistent with the physics of stationary fluid flow in porous media and an interpolated geometry of the cracks. The flow field permits computing preferential fluid flow directions of the medium, rates of mechanical energy dissipations, and a stochastic matrix modeling stream lines and fluid mass transportation, for the analysis of solute/contaminant mass advection‐diffusion as well as drainage times.

Gibbsian Dynamics and Ergodicity of Stochastic Micropolar Fluid System

The theory of micropolar fluids emphasizes the micro-structure of fluids by coupling the Navier–Stokes equations with micro-rotational velocity, and is widely viewed to be well fit, better than the Navier–Stokes equations, to describe fluids consisting of bar-like elements such as liquid crystals made up of dumbbell molecules or animal blood. Following the work of Weinan et al. (Commun Math Phys 224:83–106, 2001), we prove the existence of a unique stationary measure for the stochastic micropolar fluid system with periodic boundary condition, forced by only the determining modes of the noise and therefore a type of finite-dimensionality of micropolar fluid flow. The novelty of the manuscript is a series of energy estimates that is reminiscent from analysis in the deterministic case.

Well-posedness of stochastic second grade fluids

The theory of turbulent Newtonian fluids shows that the choice of the boundary condition is a relevant issue because it can modify the behavior of a fluid by creating or avoiding a strong boundary layer. In this study, we consider stochastic second grade fluids filling a two-dimensional bounded domain with the Navier-slip boundary condition (with friction). We prove the well-posedness of this problem and establish a stability result. Our stochastic model involves a multiplicative white noise and a convective term with third order derivatives, which significantly complicate the analysis.

Optimization of Manufacturing Supply Chain with Stochastic Demand and Planned Delivery Time

In this paper, we study a manufacturing system composed by a single-product machine, a buffer and a constant demand. A stochastic fluid model is adopted to describe the system and to take into account delivery times and returned products from the customers (withdraw right). The objective of this paper is to determine the optimal buffer level taken into account machine failures, delivery times and withdrawal right. This optimal buffer allows minimizing the sum of inventory, transportation, lost sales and returned products costs. Infinitesimal perturbation analysis method is used for optimizing the proposed system. Using the stochastic fluid model, the trajectories of buffer level are studied and the infinitesimal perturbation analysis estimators are evaluated. These estimators are shown to be unbiased and then they are implanted in an optimization algorithm which determines the optimal buffer level in the presence of delivery time and withdrawal right.

Almost sure exponential stability of stochastic fluid networks with nonlinear control

ABSTRACTDue to the objective of controlling fluid flows, fluid networks are high-order nonlinear stochastic systems in essential. Taking the nonlinearities in pressure drop of fan branch, measurement error of inertia coefficients, and random factors into consideration, this paper presents a stochastic fluid networks model described by Itô stochastic differential equations. As drift term of the stochastic fluid networks does not satisfy the linear growth condition, the almost sure exponential stability cannot get from the moment exponential stability. We directly investigate the almost sure exponential stability of the tracking and regulation problems for fluid networks under decentralised and adaptive control. It shows that the fluid flows in all branches can reach their reference values if we set controller only in links of the networks. We illustrate the results with two examples.

Feedback control: Two-sided Markov-modulated Brownian motion with instantaneous change of phase at boundaries

We consider a Markov-modulated Brownian motion {Y(t),ρ(t)} with two boundaries at 0 and b>0, and allow for the controlling Markov chain {ρ(t)} to instantaneously undergo a change of phase upon hitting either of the two boundaries at semi-regenerative epochs defined to be the first time the process reaches a boundary since it last hits the other boundary. We call this process a flexible Markov-modulated Brownian motion.Using the recently-established links between stochastic fluid models and Markov-modulated Brownian motions, we determine important characteristics of first exit times of a Markov-modulated Brownian motion from an interval with a regulated boundary. These results allow us to follow a Markov-regenerative approach and obtain the stationary distribution of the flexible process. This highlights the effectiveness of the regenerative approach in analyzing Markov-modulated Brownian motions subject to more general boundary behaviors than the classic regulated boundaries.

Modeling Dynamics of Online Video Popularity

Large Internet video delivery systems serve millions of videos to tens of millions of users on daily basis, via Video-on-Demand and live streaming. Video popularity evolves over time. It represents the workload, as welll as business value, of the video to the overall system. The ability to predict video popularity is very helpful for improving service quality and operating efficiency. Previous studies adopted simple models for video popularity, or directly adopted patterns from measurement studies. In this paper, we develop a stochastic fluid model that tries to capture two hidden processes that give rise to different patterns of a given video's popularity evolution: the information spreading process, and the user reaction process. Specifically, these processes model how the video is recommended to the user, the videos inherent attractiveness, and users reaction rate, and yield specific popularity evolution patterns. We then validate our model by matching the predictions of the model with observed patterns from our collaborator, a large content provider in China. This model thus gives us the insight to explain the common and different video popularity evolution patterns and why.

Reliability analysis of pipe conveying fluid with stochastic structural and fluid parameters

In this article, Monte Carlo simulation method is used in conjunction with finite elements for probabilistic self-excited vibration and stability analyses of pipes conveying fluid. For fluid–structure interaction, Euler–Bernoulli beam model is used for analyzing pipe structure and plug flow model is used for representing internal fluid flow in the pipe. By considering structural and fluid parameters of the system as random fields, the governing deterministic partial differential equation (PDE) of continuous system is transformed into a stochastic PDE. The continuous random fields are discretized by mid-point and local average discretization methods. For self-excited vibration analysis, the complex-valued eigenvalue problem is solved for investigating the eigenvalues and critical eigenfrequencies. Consequently, using complex eigenfrequencies and divergence velocities for every realization, the statistical responses of stochastic problem are obtained as expected values, standard deviations, probability density functions, and the probability of divergence occurrence. Moreover, the randomness effects of fluid parameters on the system are compared to those of structural parameters.

The velocity tracking problem for Wick-stochastic Navier–Stokes flows using Weiner chaos expansion

In this paper, we numerically investigate a distributed control problem of stochastic incompressible Newtonian fluid flows modeled by Navier–Stokes equations with additive white noise. Wick product is applied to define the nonlinear convective term. The objective of optimization is to track a given deterministic or stochastic velocity field. An optimization algorithm based on Weiner chaos expansion is proposed. Numerical tests are performed to validate our methodology.

Stochastic fluid structure interaction of three-dimensional plates facing a uniform flow

An experiment on a flat rectangular plate facing a uniform flow at $Re=264\,000$ shows the importance of the base pressure loading on the asymmetric static modes of the turbulent wake. The plate is free to rotate around its short symmetry axis. For plates with aspect ratio ${\it\kappa}<6$, the angular position exhibits strong random discontinuities between steady states of non-zero angles. The steady states have long time durations, more than one order of magnitude greater than the convective time scale. The discontinuities, comparable to rare and violent events, are due to strong fluid forces associated with a drastic global change of the three-dimensional wake – mainly the switching between the static asymmetric modes. A clear transition occurs at ${\it\kappa}=6$, for which the angular fluctuations are minimum, leading for ${\it\kappa}>6$ to a classical fluid structure interaction with periodic fluctuations. The transition is supported by a recent global stability analysis of rectangular fixed plates in the laminar regime.

On blocking time in multi‐class optical burst switching nodes

SummaryRecently, there has been an increasing research interest being directed to optical burst switching (OBS) networks because of its mature electronic processing capability and high‐capacity optical transport capability. The blocking time for an OBS node, representing the time duration in which a certain channel is occupied by some class of bursts, plays an important role in characterizing the performance of an OBS network. In this paper, we analyze this important metric for an OBS node with a horizonal and a single channel theoretically and numerically. On the basis of that, the burst arrivals are assumed to follow the Poisson process and the burst lengths have a phase‐type distribution, we are able to formulate a working mechanism of an OBS node into a multilayer stochastic fluid model. Using a level crossing arguments method, we are able to obtain the Laplace–Stieltjes transforms of the blocking time, from which n‐order moments and the probability density functions of the blocking time can be further derived. We provide a numerical example to illustrate the theoretical analyses, and we further discuss the effects of the offset time and the arrival density on the blocking time through this numerical example. Copyright © 2016 John Wiley & Sons, Ltd.

On the overflow time of a fluid model

We consider a continuous time stochastic fluid model with alternating on- and off-periods. During on-periods the process increases linearly, during off periods there is an additional negative decrease rate, proportional to the content level. While the durations of the on-periods have an exponential distribution we allow for general distributions for the durations of the off-periods. We study the overflow time of the system and its behavior as the overflow level tends to infinity. Such systems can be related to queuing systems which switch between a one-server mode and phases with infinitely many available servers.

An (s, k, S) fluid inventory model with exponential leadtimes and order cancellations

We consider a stochastic fluid inventory model based on a (s, k, S) policy. The content level W = {W(t): t ≥ 0} increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). W starts at W(0) = S; whenever W(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches k, the order is suppressed. We obtain explicit formulas for the expected discounted costs. The derivations are based on the optional sampling theorem (OST) to the multidimensional martingale and on fluid flow techniques.

Characterization of the law for 3D stochastic hyperviscous fluids

We consider the 3D hyperviscous Navier-Stokes equations in vorticity form, where the dissipative term $-\Delta \vec \xi $ of the Navier-Stokes equations is substituted by $(-\Delta )^{1+c} \vec \xi $. We investigate how big the correction term $c$ has to be in order to prove, by means of Girsanov transform, that the vorticity equations are equivalent (in law) to easier reference equations obtained by neglecting the stretching term. This holds as soon as $c>\frac 12$, improving previous results obtained with $c>\frac 32$ in a different setting in [5, 14].

Fuzzy model of a joint maintenance and production control under sustainability constraints

This paper considers a single-stage single-product failure-prone manufacturing system on which we try to minimize economical costs in a sustainable context. By sustainability, we mean to act on the three spheres of sustainable development, i.e. economy, environment and society. More precisely, the aim of this study is to minimize inventory and maintenance costs while minimizing environmental impacts such as pollutants and greenhouse gases emission and improving the well being of direct labor. The machine of this manufacturing system is subject to random breakdowns and a stochastic fluid model is defined. Only one customer with a constant demand per time unit is considered and a buffer of final products is defined between the machine and the customer to limit losses. Then, an integrated production and maintenance strategy is defined to minimize the total discounted cost. The chosen production policy is the well-known hedging point policy and a systematic preventive maintenance policy is defined and adapted by using fuzzy logic in order to ensure sustainability.

Optimal production control of a manufacturing-remanufacturing system with delivery activity

This work considers a manufacturing-remanufacturing system composed of two parallel machines, a manufacturing inventory, a purchase warehouse, a recovery inventory and customers who demand a constant quantity of product. Stochastic fluid model is adopted to describe the system and to take into account machine failure, returned products, remanufacturing products and delivery activity. The objective of this paper is to establish an optimal production control which allows minimizing the sum of inventory, transportation and lost sales costs. Perturbation analysis is applied to the stochastic fluid model to determine the optimal manufacturing inventory level that describes the optimal production control. The trajectories of the manufacturing inventory level are studied and the perturbation analysis estimates are evaluated. The unbiasedness of these estimates is proved and then they are implemented in an optimization algorithm which determines the optimal manufacturing inventory in the presence of returned products and delivery time.

Formal Methods in Manufacturing [Book News

This book presents a current, state-of-the-art solution to common problems in manufacturing systems, and all chapters are written by leading experts in their respective fields. The book consists of 23 chapters grouped into four parts as follows: modeling and simulation of manufacturing systems; supervisory control of manufacturing systems; performance evaluation of manufacturing systems and supply chains, and fault diagnosis of manufacturing sytems. In the 23 chapters, all problems are studied from different perspectives, using following formal models: Petri nets, finite-state automata, stochastic fluid models, and max-plus algebra. Also, many examples are given, allowing easy understanding of the topics and makingthe book more attractive for the readers.

The analysis of cyclic stochastic fluid flows with time-varying transition rates

We consider a stochastic fluid model (SFM) $$\{(\widehat{X}(t),J(t)),t\ge 0\}$${(X^(t),J(t)),t?0} driven by a continuous-time Markov chain $$\{J(t),t\ge 0 \}$${J(t),t?0} with a time-varying generator $$T(t)$$T(t) and cycle of length 1 such that $$T(t)=T(t+1)$$T(t)=T(t+1) for all $$t\ge 0$$t?0. We derive theoretical expressions for the key periodic measures for the analysis of the model, and develop efficient methods for their numerical computation. We illustrate the theory with numerical examples. This work is an extension of the results in Bean et al. (Stoch. Models 21(1):149---184, 2005) for a standard SFM with time-homogeneous generator, and suggests a possible alternative approach to that developed by Yunan and Whitt (Queueing Syst. 71(4):405---444, 2012).

Intrinsic fault tolerance of multilevel Monte Carlo methods

Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods applied to solvers for Partial Differential Equations with random input data are proved to exhibit intrinsic failure resilience. Sufficient conditions are provided for non-recoverable loss of a random fraction of MC samples not to fatally damage the asymptotic accuracy versus work of a MC simulation. Specifically, the convergence behavior of MLMC methods on massively parallel hardware with runtime faults is analyzed mathematically and investigated computationally. Our mathematical model assumes node failures which occur uncorrelated of MC sampling and with general sample failure statistics on the different levels and which also assume absence of checkpointing, i.e., we assume irrecoverable sample failures with complete loss of data. Modifications of the MLMC with enhanced resilience are proposed. The theoretical results are obtained under general statistical models of CPU failure at runtime. Particular attention is paid to node failures with so-called Weibull failure models on massively parallel stochastic finite volume computational fluid dynamics simulations are discussed. We discuss the resilience of massively parallel stochastic Finite Volume computational fluid dynamics simulations.

Erlangian approximation to finite time probability of blocking time of multi-class OBS nodes

In an optical burst switching (OBS) network, the blocking time, representing the time interval during which the channel is occupied for a given class of incoming burst, is a key metric for performance evaluation and traffic shaping. In this paper, we study a horizon-based single-channel multi-class OBS node, for which the multiple traffic classes are differentiated using different offset time of each class. By assuming Poisson burst arrivals and phase-type distributed burst lengths and using the theory of Multi-layer stochastic fluid model, we obtain the Erlangian approximation for the finite time probability of the blocking time for a given class of burst in an OBS node. We further propose an explicit algorithm and procedure to calculate the Erlangian approximation. Numerical results are provided to illustrate the accuracy and the speed of convergence of the proposed method.