ON CAUCHY PROBLEMS FOR THE RLW EQUATION IN TWO SPACE DIMENSIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｓｙｓｔｅｍｏｆｔｈｅｎｏｎｌｉｎｅａｒｄｉｓｐｅｒｓｉｖｅ ,ｔｈｅｐａｐｅｒｓ [1 ]ａｎｄ [2 ]ｈａｖｅｓｔｕｄｉｅｄｔｈｅｍｏｄｅｌｅｑｕａｔｉｏｎｏｆｓｉｍｐｌｅｗａｖｅｓｐｒｏｐａｇａｔｉｏｎｏｆｌｏｎｇｗａｖｅ ,ｕｔ+αｕｘ+ βｕｕｘ-γｕｘｘｔ=0　　ｘ∈Ｒ ,ｔ≥ 0ｉｓｃａｌｌｅｄｔｈｅＲＬＷｅｑｕａｔｉｏｎ .ＳＨＡＮＧ[3]ｏｂｔａｉｎｅｄｔｈｅｅｘａｃｔｓｏｌｉｔａｒｙｗａｖｅｓｏｌｕｔｉｏｎｓｏｆｔｈｅｔｗｏｄｉｍｅｎｓｉｏｎａｌＲＬＷｅｑｕａｔｉｏｎ .Ｉｎ…     

Solitary wave solutions of the nonlinear generalized Pochhammer–Chree and regularized long wave equations

In this paper, we implement some fast and high accuracy numerical algorithms to obtain the solitary wave solutions of generalized Pochhammer?CChree (PC) and regularized long wave (RLW) equations. We employ the discrete Fourier transform to discretize the original partial differential equations (PDEs) in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. The proposed methods are fast and accurate due to the use of the fast Fourier transform in combination with explicit fourth-order time stepping methods. For RLW equation we investigate the propagation of a single solitary and interaction of two and three solitary waves. Moreover, three invariants of motion (mass, energy, and momentum) are evaluated to determine the conservation properties of the problem, and the numerical schemes lead to accurate results. The numerical results are compared with analytical solutions and with those of other recently published methods to confirm the accuracy and efficiency of the presented schemes.     

On cauchy problems for the RLW equation in two space dimensions

A cnoidal wave solution of the two dimensional RLW equation of are obtained by elliptic integral method, and the some estimations the uniqueness and the stability of the periodic solution with both x, y to the Cauchy problem are proved by the priori estimations. Biography: Huang Zheng-hong (1956-)     

Numerical solution of the regularized long wave equation using nonpolynomial splines

In this paper, we employ nonpolynomial spline (NPS) basis functions to obtain approximate solutions of the regularized long wave (RLW) equation. By considering suitable relevant parameters, it is shown that the local truncation error behaves O(k ²+h ²) with respect to the time and space discretization. Numerical stability of the method is investigated by using a linearized stability analysis. To illustrate the applicability and efficiency of the aforementioned basis, we compare obtained numerical results with other existing recent methods. Motion of single solitary wave and double and triple solitary waves, wave undulation, generation of solitary waves using the Maxwellian initial condition and conservation properties of mass, energy, and momentum of numerical solutions of the equation are dealt with.     

三维爆炸与冲击问题仿真软件研究

针对三维复杂爆炸流场的数值模拟,开发了欧拉型三维爆炸与冲击问题数值模拟软件EXPLOSION-3D,对该软件的三个组成模块即计算程序MMIC-3D、有限差分网格自动生成前处理程序MESH-3D和可视化后处理程序VISC-3D的基本理论和算法进行了讨论。以楼群中爆炸流场的三维数值模拟为例,采用该软件对大型爆炸发生后形成的爆炸场进行了数值模拟研究,得到一些有意义的数值结果,这些结果对于爆炸灾害的预防和评估具有重要的参考价值。     

Modelings and mechanism analysis underlying both the 4D Euler equations and Hamiltonian conservative chaotic systems

Nonlinear Dynamics - Four sub-Euler equations for four sub-rigid bodies are generalized by extending the dimension of the state space from 3D to 4D. Six integrated 4D Euler equations are proposed...     

Multiscale coupling of compliant and rigid walls blood flow models

Numerical methods based on geometrical multiscale models of blood flows solve for averaged flow statistics on a network of vessels while providing more detailed information about fluid dynamics in a specific region of interest. In such an approach, a 3D model based on the Navier–Stokes equations posed in a domain with rigid walls is often used to describe blood flow dynamics in the refined region. While ignoring elasticity effects in 3D models is plausible in certain applications and saves computational time significantly, coupling such models with 1D flow models may result in non‐physiological phenomena in the computed solutions. Thus, the immediate coupling conditions based on continuity of normal stresses, flow rate, pressure, or a combination of thereof do not account for the inconsistency between elasticity effects in the 1D model and the non‐compliance of the 3D model. In this paper, we introduce and study an auxiliary absorbing 0D model, which is placed at the interface between 1D and 3D models. A virtual device mimics the effect of the 3D model compliance and hence reduces pressure wave reflection and instabilities caused by the inconsistency. The absorbing model is developed from basic mechanical principles. As a result, parameters of the 0D model can be designed based on hemodynamic data. We analyze the stability of the geometrical multiscale model and perform several numerical experiments to assess its computational efficiency. Copyright © 2016 John Wiley & Sons, Ltd.     

A method to discretize non‐planar fractures for 3D subsurface flow and transport simulations

A method is presented to discretize inclined non‐planar 2D fractures within a 3D finite element grid for subsurface flow and transport simulations. Each 2D fracture is represented as a triangulated surface. Each triangle is then discretized by 2D fracture elements that can be horizontal, vertical or inclined and that can be triangular or rectangular. The 3D grid representing a porous rock formation consists of hexahedra and can be irregular to allow grid refinement. An inclined fracture was discretized by (a) inclined triangles and (b) orthogonal rectangles and flow/transport simulations were run to compare the results. The comparison showed that (i) inclined fracture elements must be used to simulate 2D transient flow, (ii) results of 2D/3D steady‐state and 3D transient flow simulations are identical for both discretization methods, (iii) inclined fracture elements must be used to simulate 2D/3D transport because orthogonal fracture elements significantly underestimate concentrations, and (iv) orthogonal elements can be used to simulate 2D/3D transport if fracture permeability is corrected and multiplied by the ratio of fracture surface areas (orthogonal to inclined). Groundwater flow at a potential site for long‐term disposal of spent nuclear fuel was simulated where a complex 3D fracture network was discretized with this technique. The large‐scale simulation demonstrates that the proposed discretization procedure offers new possibilities to simulate flow and transport in complex 3D fracture networks. The new procedure has the further advantage that the same grid can be used for different realizations of a fracture network model with no need to regenerate the grid. Copyright © 2007 John Wiley & Sons, Ltd.     

二维材料纳米尺度摩擦行为及其机制

二维材料是指厚度仅有单层或数层原子的晶体或非晶材料,其优异的物理、力学和化学性能给纳米尺度超薄固体润滑材料的设计和发展带来了新的契机。同时,二维材料独特而简单的拓扑结构也为深入了解摩擦的微观机制提供了一个理想的对象。本文综述了以石墨烯为主的二维材料纳米尺度摩擦和磨损研究的进展。根据相对运动形式的不同,我们分别介绍了二维材料的层间滑动和表面摩擦行为,并详细阐述了这些独特行为背后的微观物理机制;同时我们还重点介绍了若干种影响和调控二维材料表面摩擦性能的典型方法和策略,以及二维材料纳米尺度的磨损行为及其失效模式。最后,我们还对纳米尺度二维材料摩擦研究进行了小结,并展望了该领域尚待探索的若干研究方向。     

One- and Three-Dimensional Soil Transportation Models for Volatiles Migrating from Soils to House Interiors

Two models are presented for the transient migration of a volatile organic compound (VOC) from soil to the interor of a house with a crawl space. The migration in the house is taken as one-dimensional (1D) and coupled to a soil transportation model with diffusion, leaching and VOC degradation. The diffusion is either vertical, providing a 1D model, or three-dimensional (3D) axisymmetric, providing a 3D model. The initial subsurface VOC deposition is assumed to be a finite layer extending over the footprint of the house in the 1D case or as a cylinder of arbitrary radius in the 3D case. Using data for benzene as the VOC, comparisons are made for results from the two models for VOC concentrations in soil, crawl space and dwelling space as well as the cumulative dwelling space concentration used in human health assessment, for a wide range of soil parameters and subsurface sizes of the initial VOC cylinder. In all cases the values from the 1D transport model were slightly higher than those for the 3D model. The analysis uses Laplace transforms with numerical inversions. For the 3D soil transport model, different soil surface flux boundary conditions underneath and outside the house give rise to novel dual integral equations.     

Hyperchaos,chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points

Based on three-dimensional (3D) Lü chaotic system, we introduce a four-dimensional (4D) nonlinear system with infinitely many equilibrium points. The Lyapunov-exponent spectrum is obtained for the 4D chaotic system. A hyperchaotic attractor and a chaotic attractor are emerged in this 4D nonlinear system. Furthermore, to verify the existence of hyperchaos, the chaotic dynamics of this 4D nonlinear system is also studied by means of topological horseshoe theory and numerical computation.     

Natural Convection Heat Transfer in 2D and 3D Trapezoidal Enclosures Filled with Nanofluid

The purpose of the present study is to investigate the heat transfer performance due to free convection of nanofluids with variable properties inside 2D and 3D channels with trapezoidal cross sections. The governing equations are solved numerically using the finite volume method and the SIMPLER algorithm. In this study, the effect of the nanoparticle volume fraction, Rayleigh number, side wall angles of the trapezoidal section, and axial slope of the 3D channel are examined. The presented results include the average Nusselt number, flow circulation streamlines, and isothermal contours. The heat transfer rate (i.e., Nusselt number) is seen to increase in both 2D and 3D channels with an increase in the Rayleigh number. In 2D trapezoidal enclosures, the Nusselt number decreases with an increase in the nanoparticle volume fraction from zero to 2% and increases if the nanoparticle volume fraction is greater than 2%. In 3D channels, an increase in the axial slope of the channel leads to an increase in the Nusselt number.     

Effect of 2D Image Resolution on 3D Stochastic Reconstruction and Developing Petrophysical Trend

Multi-resolution digital rock physics (DRP) makes it possible to up-scale petrophysical properties from micron size to core sample size using two-dimensional (2D) thin section images. Resolution of 3D images and sample size are challenging problems in DRP where high-resolution images are acquired from small samples using inefficient and expensive micro-CT facilities. Three-dimensional stochastic reconstruction is an alternative approach to overcome these challenges. In this paper, we use multi-resolution images and investigate effect of 2D image resolution on 3D stochastic reconstruction and development of petrophysical trends for our two sandstone and carbonate original representative volume elements (RVEs). The proposed method includes three steps. In the first step, the spatial resolution of our original RVEs is decreased synthetically. In the second step, stochastic RVEs are realized for each resolution using two perpendicular images, correlation functions, and phase recovery algorithm. In the reconstruction method, a full set of two-point correlation functions (TPCFs) is extracted from two perpendicular 2D images. Then TPCF vectors are decomposed and averaged to realize 3D stochastic RVEs. In the third step, petrophysical properties like relative and absolute permeability as well as porosity and formation factor are computed. The output is used to develop trends for petrophysical properties in different resolutions. Experimental results illustrate that the proposed method can be used to predict petrophysical properties and reconstruct 3D RVEs for resolutions unavailable in the acquired 2D or 3D data.     

Families of gap solitons and their complexes in media with saturable nonlinearity and fractional diffraction

Nonlinear Dynamics - We demonstrate the existence of various types of gap localized modes, including one- and two-dimensional (1D and 2D) single solitons and soliton clusters, as well as the...     

Two and three dimensional modeling of fluidized bed with multiple jets in a DEM-CFD framework

Fluidized beds with multiple jets have widespread industrial applications. The objective of this paper is to investigate the jet interactions and hydrodynamics of a fluidized bed with multiple jets. Discrete element modeling coupled with in-house CFD code GenlDLEST has been used to simulate a bed with nine jets. The results are compared with published experiments. Mono dispersed particles of size 550 ~m are used with 1.4 times the minimum fluidization velocity of the particles. Both two and three dimensional computations have been performed. To the best of our knowledge, the results presented in this paper are the first full 3D simulations of a fluidized bed performed with multiple jets. Discrepancies between the experiment and simulations are discussed in the context of the dimensionality of the simulations. The 2D solid fraction profile compares well with the experiment close to the distributor plate. At higher heights, the 2D simulation over-predicts the solid fraction profiles near the walls. The 3D simulation on the other hand is better able to capture the solid fraction profile higher up in the bed compared to that near the distributor plate. Similarly, the normalized particle velocities and the particle fluxes compare well with the experiment closer to the distributor plate for the 2D simulation and the freeboard for the 3D simulation, respectively. A lower expanded bed height is predicted in the 2D simulation compared to the 3D simulation and the experiment. The results obtained from DEM computations show that a 2D simulation can be used to capture essential jetting trends near the distributor plate regions, whereas a full scale 3D simulation is needed to capture the bubbles near the freeboard regions. These serve as validations for the experiment and help us understand the complex jet interaction and solid circulation patterns in a multiple jet fluidized bed system.     

Design and analysis of a Schwarz coupling method for a dimensionally heterogeneous problem

In the present work, we propose and analyse an efficient iterative coupling method for a dimensionally heterogeneous problem. We consider the case of a 2D Laplace equation with non‐symmetric boundary conditions coupled with a corresponding 1D Laplace equation. We first show how to obtain the 1D model from the 2D one by integration along one direction, by analogy with the link between shallow water equations and the Navier–Stokes system. Then we focus on the design of a Schwarz‐like iterative coupling method. We discuss the choice of boundary conditions at coupling interfaces. We prove the convergence of such algorithms and give some theoretical results related to the choice of the location of the coupling interface, and to the control of the difference between a global 2D reference solution and the 2D coupled solution. These theoretical results are illustrated numerically. Copyright © 2014 John Wiley & Sons, Ltd.     

An Incompressible 2D Didactic Model with Singularity and Explicit Solutions of the 2D Boussinesq Equations

We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time for all time. In addition, we introduce a variant of the 2D Boussinesq equations which is perhaps a more faithful companion of the 3D axisymmetric Euler equations than the usual 2D Boussinesq equations.     

A frame-indifferent model for a thermo-elastic material beyond the three-dimensional Eulerian and Lagrangian descriptions

The covariance principle of differential geometry within a four-dimensional (4D) space-time ensures the validity of any equations and physical relations through any changes of frame of reference, due to the definition of the 4D space-time and the use of 4D tensors, operations and operators. This enables to separate covariance (i.e. frame-indifference) and material objectivity (i.e. material-indifference). We propose here a method to build a constitutive relation for thermo-elastic materials using such a 4D formalism. A 4D generalization of the classical variational approach is assumed leading to a model for a general thermo-elastic material. The isotropy of the relation can be ensured by the use of the invariants of variables, which offers new possibilities for the construction of constitutive relations. It is then possible to build a general frame-indifferent but not necessarily material-indifferent constitutive relation. It encompasses both the 3D Eulerian and Lagrangian thermo-elastic isotropic relations for finite transformations.     

Construction of hybrid 1D-0D networks for efficient and accurate blood flow simulations

The one-dimensional (1D) modeling of blood flow in complex networks of vessels and cardiovascular models can result in computationally expensive simulations. The complexity of such networks has significantly increased in the last years, in terms of both enhanced anatomical detail and modeling of physiological mechanisms and mechanical characteristics. To address such issue, the main goal of this work is to present a novel methodology to construct hybrid networks of coupled 1D and 0D vessels and to perform computationally efficient and accurate blood flow simulations in such networks. Departing from both the 1D and lumped-parameter (0D) nonlinear models for blood flow, we propose high-order numerical coupling strategies to solve the 1D, 0D, and hybrid coupling of vessels at junctions. To effectively construct hybrid networks, we explore different a-priori model selection criteria focusing in obtaining the best possible trade-off between computational cost of the simulations and accuracy of the computed solutions for the hybrid network with respect to the 1D network. The achievement of the expected order of accuracy is verified in several test cases. The novel methodology is applied to two different arterial networks, the 37-artery network and the reduced ADAN56 model, where, in order to identify the best performing a-priori model selection criteria, the quantitative assessment of CPU times and errors and the qualitative comparison between results are carried out and discussed.     

Experimental and numerical study of pseudo-2D circulating fluidized beds

We present experimental investigations and numerical simulations of a pseudo-2D riser. Experiments were performed for various airflow rates, particle types/diameters, and particle size distributions. Pressure distributions along the wall of the riser were measured. Additional measurements from a smaller pseudo-2D riser (Kallio et al., 2009; Shah et al., 2012) were used to analyze horizontal solids volume fraction profiles. The experimental data were compared with simulation results carried out using an Euler–Euler approach. A mesh sensitivity study was conducted for numerical simulations and effects associated with simplifying real 3D geometry to a 2D model were examined. In addition, the effect of using an algebraic equation to represent the granular temperature versus a full partial differential equation also was examined for numerical simulations. Results showed small but significant near-wall sensitivity of the flow variables to mesh size. Substantial differences in mean pressure, solids distribution, and solid velocities were obtained, when 2D and 3D simulation results were compared. Finally, applying the simplified granular temperature equation for turbulent fluidization and for dilute-phase transport can lead to incorrect predictions in models.