Asymptotics of the Solutions of the Random Schrödinger Equation

We consider solutions of the Schrödinger equation with a weak time-dependent random potential. It is shown that when the two-point correlation function of the potential is rapidly decaying, then the Fourier transform \({\hat\zeta_\epsilon(t,\xi)}\) of the appropriately scaled solution converges point-wise in ξ to a stochastic complex Gaussian limit. On the other hand, when the two-point correlation function decays slowly, we show that the limit of \({\hat\zeta_\epsilon(t,\xi)}\) has the form \({\hat\zeta_0(\xi){\rm exp}(iB_\kappa(t,\xi))}\) where B _κ (t, ξ) is a fractional Brownian motion.     

Random vibration of hysteretic systems under Poisson white noise excitations

Hysteresis widely exists in civil structures, and dissipates the mechanical energy of systems. Research on the random vibration of hysteretic systems, however,is still insufficient, particularly when the excitation is non-Gaussian. In this paper, the radial basis function(RBF) neural network(RBF-NN) method is adopted as a numerical method to investigate the random vibration of the Bouc-Wen hysteretic system under the Poisson white noise excitations. The solution to the reduced generalized Fokker...     

Homogenization of a Hele–Shaw Problem in Periodic and Random Media

We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary velocity, and we establish the uniform convergence of the free boundaries.     

Classical Limit for a System of Non-Linear Random Schrödinger Equations

This work is concerned with the semi-classical analysis of mixed state solutions to a Schrödinger–Position equation perturbed by a random potential with weak amplitude and fast oscillations in time and space. We show that the Wigner transform of the density matrix converges weakly and in probability to solutions of a Vlasov–Poisson–Boltzmann equation with a linear collision kernel.Aconsequence of this result is that a smooth non-linearity such as the Poisson potential (repulsive or attractive) does not change the statistical stability property of the Wigner transform observed in linear problems.We obtain, in addition, that the local density and current are self-averaging, which is of importance for some imaging problems in random media. The proof brings together the martingale method for stochastic equations with compactness techniques for non-linear PDEs in a semi-classical regime. It relies partly on the derivation of an energy estimate that is straightforward in a deterministic setting but requires the use of a martingale formulation and well-chosen perturbed test functions in the random context.     

Random vortex induced vibration response of suspended fexible cable to fuctuating wind

A popular dynamical model for the vortex induced vibration(VIV) of a suspended flexible cable consists of two coupled equations. The first equation is a partial differential equation governing the cable vibration. The second equation is a wake oscillator that models the lift coefficient acting on the cable. The incoming wind acting on the cable is usually assumed as the uniform wind with a constant velocity, which makes the VIV model be a deterministic one. In the real world, however, the wind v...     

A Random Field Formulation of Hooke’s Law in All Elasticity Classes

For each of the 8 symmetry classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set \(\mathsf {V}\) of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders 1 and 2 of such a field, and the field’s spectral expansion.     

Porosity Effects on Mechanical Properties of 3D Random Fibrous Materials at Elevated Temperatures

In this study,we prepare the specimens of three-dimensional random fibrous(3D RF)material along its through-the-thickness(TTT)and in-plane(IP)directions.The experimental tests of tensile and compressive properties as well as fracture toughness of 3D RF material are performed at elevated temperatures.Then,the porosity(83%,87%and 89%)and temperature dependence of the tensile and compressive strength,elastic modulus,fracture toughness and fracture surface energy of the 3D RF materials for both the TTT and IP directions are analyzed.From the results of the tensile strength and elastic modulus versus material porosities at various temperatures,we find that tensile strength and elastic modulus for the TTT direction are more sensitive to the porosity,but not for the IP direction.Fracture toughness increases firstly and then decreases at a certain critical temperature.Such critical temperature is found to be the lowest for the porosity of 83%.On the other hand,at below 1073 K,the temperature-dependent fracture surface energies with three porosities for the TTT direction show similar variation trends.     

Network Approximation in the Limit of¶Small Interparticle Distance of the¶Effective Properties of a High-Contrast¶Random Dispersed Composite

We consider a high-contrast two-phase composite such as a ceramic/polymer composite or a fiberglass composite. Our objective is to determine the dependence of the effective conductivity (or the effective dielectric constant or the effective shear modulus) of the composite on the random locations of the inclusions (ceramic particles or fibers) when the concentration of the inclusions is high. We consider a two-dimensional model and show that the continuum problem can be approximated by a discrete random network (graph). We use variational techniques to provide rigorous mathematical justification for this approximation. In particular, we have shown asymptotic equivalence of the effective constant for the discrete and continuum models in the limit when the relative interparticle distance goes to zero. We introduce the geometrical interparticle distance parameter using Voronoi tessellation, and emphasize the relevance of this parameter due to the fact that for irregular (non-periodic) geometries it is not uniquely determined by the volume fraction of the inclusions. We use the discrete network to compute numerically. For this purpose we employ a computer program which generates a random distribution of disks on the plane. Using this distribution we obtain the corresponding discrete network. Furthermore, the computer program provides the distribution of fluxes in the network which is based on Keller's formula for two closely spaced disks. We compute the dependence of on the volume fraction of the inclusions V for monodispersed composites and obtaine results which are consistent with the percolation theory predictions. For polydispersed composites (random inclusions of two different sizes) the dependence is not simple and is determined by the relative volume fraction V _r of large and small particles. We found some special values of V _r for which the effective coefficient is significantly decreased. The computer program which is based on our network model is very efficient and it allows us to collect the statistical data for a large number of random configurations.     

Three-Dimensional Lattice Boltzmann Simulations of Single-Phase Permeability in Random Fractal Porous Media with Rough Pore–Solid Interface

Single-phase permeability k has intensively been investigated over the past several decades by means of experiments, theories and simulations. Although the effect of surface roughness on fluid flow and permeability in single pores and fractures as well as networks of fractures was studied previously, its influence on permeability in a random mass fractal porous medium constructed of pores of different sizes remained as an open question. In this study, we, therefore, address the effect of pore–solid interface roughness on single-phase flow in random fractal porous media. For this purpose, we apply a mass fractal model to construct porous media with a priori known mass fractal dimensions \(2.579 \le D_{\mathrm{m}} \le 2.893\) characterizing both solid matrix and pore space. The pore–solid interface of the media is accordingly roughened using the Weierstrass–Mandelbrot approach and two parameters, i.e., surface fractal dimension \(D_{\mathrm{s}}\) and root-mean-square (rms) roughness height. A single-relaxation-time lattice Boltzmann method is applied to simulate single-phase permeability in the corresponding porous media. Results indicate that permeability decreases sharply with increasing \(D_{\mathrm{s}}\) from 1 to 1.1 regardless of \(D_{\mathrm{m}}\) value, while k may slightly increase or decrease, depending on \(D_{\mathrm{m}}\), as \(D_{\mathrm{s}}\) increases from 1.1 to 1.6.     

Slip–Brinkman Flow Through Corrugated Microannulus with Stationary Random Roughness

This paper presents a boundary perturbation method of the Brinkman-extended Darcy model to investigate the flow in corrugated microannuli cylindrical tubes with slip surfaces. The stationary random model is used to mimic the surface roughness of the cylindrical walls. The tube is filled with a porous medium. We shall consider the two cases where corrugations are either perpendicular or parallel to the flow, and particular attention is given to the effect of the phase shift. The effects of the corrugations on the flow rate and pressure gradient are investigated as functions of wavelength, the permeability of the medium, the radius ratio and the slip parameter. Particular surface roughnesses are examined as special cases of stationary random surface. It is found that the effect of the partial slip is significant on the corrugation functions. The limiting cases of Stokes and Darcy’s flows and no-slip case are discussed.     

Random Data Cauchy Theory for the Generalized Incompressible Navier–Stokes Equations

In this paper, we consider the generalized Navier?CStokes equations where the space domain is ${\mathbb{T}^N}$ or ${\mathbb{R}^N, N\geq3}$ . The generalized Navier?CStokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier?CStokes equations by the more general operator (???) ^?? with ${\alpha\in (\frac{1}{2},\frac{N+2}{4})}$ . After a suitable randomization, we obtain the existence and uniqueness of the local mild solution for a large set of the initial data in ${H^s, s\in[-\alpha,0]}$ , if ${1 < \alpha < \frac{N+2}{4}, s\in(1-2\alpha,0]}$ , if ${\frac{1}{2} < \alpha\leq 1}$ . Furthermore, we obtain the probability for the global existence and uniqueness of the solution. Specially, our result shows that, in some sense, the Cauchy problem of the classical Navier?CStokes equation is local well-posed for a large set of the initial data in H ^?1+, exhibiting a gain of ${\frac{N}{2}-}$ derivatives with respect to the critical Hilbert space ${H^{\frac{N}{2}-1}}$ .     

Random Kick-Forced 3D Navier–Stokes Equations in a Thin Domain

We consider the Navier–Stokes equations in the thin 3D domain , where is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ε → 0) to a unique stationary measure for the Navier–Stokes equation on . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains.     

Pesin’s Formula for Random Dynamical Systems on \mathbf {R}^d

Pesin’s formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on $\mathbf {R}^d$ which have an invariant probability measure absolutely continuous to the Lebesgue measure on $\mathbf {R}^d$ . Finally we will show that a broad class of stochastic flows on $\mathbf {R}^{d}$ of a Kunita type satisfies Pesin’s formula.     

Investigation of the Stability Problems of Elastic Bodies Using the Method of Mathematical Theory of Elasticity

In this paper,using the equilibrium equations and boundary conditionsof elastic stability problem of Новожилов and the method of mathematicaltheory of elasticity,we solve some elastic stability problems,which werestudied byищлинскииandвоицеховская,and obtained more reason-able results than theirs.     

Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity

Denoting by the stress tensor, by the linearized strain tensor, by A the elasticity tensor, and assuming that is a convex potential, the inclusion accounts for nonlinear viscoelasticity, and encompasses both the linear Kelvin–Voigt model of solid-type viscoelasticity and the Prager model of rigid plasticity with linear kinematic strain-hardening. This relation is assumed to represent the constitutive behavior of a space-distributed system, and is here coupled with the dynamical equation. An initial- and boundary-value problem is formulated, and the existence and uniqueness of the solution are proved via classical techniques based on compactness and monotonicity. A composite material is then considered, in which the function and the tensor A rapidly oscillate in space. A two-scale model is derived via Nguetseng’s notion of two-scale convergence. This provides a detailed account of the mesoscopic state of the system. Any dependence on the fine-scale variable is then eliminated, and the existence of a solution of a new single-scale macroscopic model is proved. The final outcome is at variance with the nonlinear extension of the generalized Kelvin–Voigt model, which is based on an apparently unjustified mean-field-type hypothesis.     

Model of Self‐Organization of an Ensemble of Cracks Radiating Sound

A model of selforganization of cracks arising in a rock specimen (granite) compressed by a press is proposed. The model is based on the assumption of acoustic wave interaction between the cracks. To construct the model of selforganization of cracks, solutions of the Fokker–Planck equation are used. The experimentally observed spontaneous increase in the activity of acoustic emission, spatial and temporal clusterization, and formation of a fractal structure in rock specimens under constant and slowly varying loads are explained.     

Estimation of overall properties of random polycrystals with the use of invariant decompositions of Hooke’s tensor

In the paper the theoretical analysis of bounds and self-consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented. In the study two invariant decompositions of Hooke’s tensors are used. The applied method enables derivation of novel expressions for estimates of the bulk and shear moduli, which depend on invariants of local stiffness tensor. With use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained.     

Modeling of the Front of Melting and Destruction of a Melt Film During Gas‐Laser Cutting of Metals

The products of laser cutting of metals on an automated laser setup are investigated. Results of model experiments are presented, where soft wax was used instead of metal transforming into the melt; soft wax filled a narrow flat slot between two glass plates and was removed by a heated air stream. The physical processes of melting of the liquidwax film, its destruction, and entrainment by the gas jet being assumed to be analogous to the processes of metalmelt spraying inside the cut in fullscale experiments, the characteristic size of drops formed thereby is evaluated. The modeling results are in qualitative agreement with the results of fullscale experiments. It is shown that the quality of laser cutting of metals directly depends on the character of spraying of the liquid melt and the process of its removal.     

Table of Contents of Volume 35 – 2000

Volume Contents

Table of Contents of Volume 35 – 2000     

Table of Contents of Volume 36 – 2001

Volume Contents

Table of Contents of Volume 36 – 2001