Discrete Random Model Research Articles

On a Non-local Sobolev–Galpern-Type Equation Associated with Random Noise

This paper aims to retrieve the initial value for a non-local fractional Sobolev–Galpern problem. The given data are subject to noise by the discrete random model. We show that the solution to the problem is ill-posed in the sense of Hadamard. In this paper, we applied the Fourier truncation method to construct the regularized solution. We estimate the convergence between the solution and the regularized solution. In addition, the numerical example is also proposed to assess the efficiency of the theory.

A frontal air intake may improve the natural ventilation in urban buses.

In this report we analyze the air flow across the open windows (natural ventilation) of an urban bus model and the consequent dispersion of aerosols emitted in the passengers area. The methods include computational fluid dynamics simulations and three ways to characterize the dispersion of passive tracers: a continuous concentration-based model, a discrete random model and a parametric scalar based on the so-called mean age of air. We also conducted experiments using a 1:10 scale bus model and [Formula: see text] as a passive tracer to assess the ventilation characteristics. We found that dispersion and expulsion of aerosols is driven by a negative pressure in the standard bus design equipped with lateral windows. Also, the average age of air is 6 minutes while the air flow promotes aerosol accumulation to the front (driver's area). To speed up the expulsion of aerosols and reduce their in-cabin accumulation, we propose a bus bodywork prototype having a frontal air intake. All the numerical models and experiments conducted in this work agreed that the expulsion of aerosols in this novel configuration is significantly increased while the average age of air is reduced to 50 seconds. The average air flow also changes with the presence of frontal air intakes and, as a consequence, the expulsion of aerosols is now driven by a frontal velocity field.

Study of Pulse Wave Propagation and Attenuation Mechanism in Shale Reservoirs During Pulse Hydraulic Fracturing

Hydraulic fracturing (HF) is indispensable in shale gas, but there are also problems existed in high ratio of energy consumption and low ratio of energy utilization. Based on hydraulic fracturing and pulsating jet, pulse hydraulic fracturing (PHF) has been widely used and achieved good results in the coal-bed methane. This paper attempts to study the mechanism of pulse wave propagation and attenuation during PHF in shale reservoirs, which provides theoretical support for application. On the basis of the motion equation and continuity equation, the two-dimensional propagation and attenuation equation with damping and fissure width during PHF is established and fracture network is simulated based on DFN discrete random model. The mathematical and physical model is verified by laboratory experiment The study of PHF superiority, PHF waveform effect, PHF propagation and attenuation characters are also performed. Results indicate that PHF propagation has two stages: The first is the pressure fluctuation is sharp in the process of the initial fracturing. The second is a stable state is appeared after a period of time. Compared with HF, PHF can pressurize obviously the pressure gradient of pulse waveform, which is an important parameter of affecting pulse pressure: the greater pressure gradient, the more obvious water hammer and supercharging effect. In the process of PHF propagation, pressure distribution complicates under the supercharging. Pressure wave has different attenuation characteristics in two steps. These research results will provide guidance for optimizing technical parameters for using PHF technology in shale reservoirs.

Countinuous and discrete models for abrasion processes

The origin of the shapes of stones and other particles formed by water or wind has always attracted the attention of geologists and mathematicians. A classical model of abrasion due to W. J. Firey leads to a geometric partial differential equation representing the continuum limit of the process. This model predicts convergence to spheres from an arbitrary initial form; analogously, the two-dimensional version of the model predicts convergence to circles. The shapes of real stones are, however, not always round. Most notably, coastal pebbles tend to be smooth but somewhat flat, and ventifacts (e.g. pyramidal dreikanters) often have completely different shapes with sharp edges. Inspired by Firey´s results, a new PDE is derived in this paper, which not only appears to be a natural mathematical generalization of Firey´s PDE, but also represents the continuum limit of a genezalized abrasion model based on recurrent loss of material due to collisions of nearby pebbles. We also introduce a related, mezo-scale discre te random model which is ideally suited for analyzing wear processes in specific geometric scenarios. Preliminary results suggest that our model is capable to predict a broad range of limit shapes: polygonal shapes with sharp edges develop due to sand blasting (big stone surrounded by infinitesimally small particles), round stones emerge due to collisions with relatively big stones, and flat shapes are the typical outcome in the intermediate case. The results show nice agreement with real data despite the model´s simplicity.

Discrete and continuous models for in plane loaded random elastic brickwork

Abstract The aim of this paper is to study non-periodic masonries – typical of historical buildings – by means of a perturbation approach and to evaluate the effect of a random perturbation on the elastic response of a periodic masonry wall. The random masonry is obtained starting from a periodic running bond pattern. A random perturbation on the horizontal positions of the vertical interfaces between the blocks which form the masonry wall is introduced. In this way, the height of the blocks is uniform, while their width in the horizontal direction is random. The perturbation is limited such as each block has still exactly 6 neighboring blocks. In a first discrete model, the blocks are modeled as rigid bodies connected by elastic interfaces (mortar thin joints). In other words, masonry is seen as a “skeleton” in which the interactions between the rigid blocks are represented by forces and moments which depend on their relative displacements and rotations. A second continuous model is based on the homogenization of the discrete model. Explicit upper and lower bounds on the effective elastic moduli of the homogenized continuous model are obtained and compared to the well-known effective elastic moduli of the regular periodic masonry. It is found that the effective moduli are not very sensitive to the random perturbation (less than 10%). At the end, the Monte Carlo simulation method is used to compare the discrete random model and the continuous model at the structural level (a panel undergoing in plane actions). The randomness of the geometry requires the generation of several samples of size L of the discrete masonry. For a sample of size L , the structural discrete problem is solved using the same numerical procedure adopted in [Cecchi, A., Sab, K., 2004. A comparison between a 3D discrete model and two homogenized plate models for periodic elastic brickwork, International Journal of Solids Structures 41 (9–10), 2259–2276] and the average solution over the samples gives an estimation which depends on L . As L increases, an asymptotic limit is reached. One issue is to find the minimum size for L and to compare the asymptotic average solution to the one obtained from the continuous homogenized model.