Analysis of modified Reynolds equation using the wavelet‐multigrid scheme

The combined study of effects of surface roughness and poroelasticity on the squeeze film behavior of bearings in general and that of synovial joints in particular are presented. The modified form of Reynolds equation, which incorporates the randomized roughness structure as well as elastic nature of articular cartilage, is derived. Christensen stochastic theory describing roughness structure of cartilage surfaces is used by assuming the roughness asperity heights to be small compared to the film thickness. A recently developed wavelet‐multigrid method is used for the solution of Reynolds equation. The method has the greatest advantage of minimizing the errors using wavelet transforms in obtaining accurate solution, as grid size tends to zero. Based on the results obtained, the influence of roughness and elasticity on bearing characteristics are discussed in some detail. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Homogenization of a Class of Integro-Differential Equations with Lévy Operators

The periodic homogenization of the integro-differential equation (PIDE) with the Lévy operator with the alpha-stable density, is studied in this paper. The formal asymptotic expansion method is employed to derive the cell problem, the ergodic problem for the Lévy operator without the second-order uniformly elliptic term. The effective equation is then obtained by using the result of the ergodic problem. Finally, the formal argument is justified rigorously by the perturbed test function method.     

Wear and fluid-solid-thermal transient coupled model for journal bearings

In this study, the transient interactions between the sliding wear behaviour and fluid–solid–thermal (FST) characteristics of journal bearings are revealed using an established mathematical model. The calculated temperature distribution is validated by a comparison with experimental results from the literature. Furthermore, a wear test for lubricated journal bearings is conducted to verify the predicted wear rate. The time-varying wear and FST performances of the journal bearing, including the wear rate, wear depth, fluid pressure, contact pressure, and maximum temperature are calculated numerically. Through numerical simulations, the effects of the boundary friction coefficient and surface roughness on the wear and FST performances are evaluated. To demonstrate the importance of considering the three-dimensional (3D) thermal effect during the wear analysis of lubricated journal bearings, the numerical results predicted by the isothermal model and the thermal model are compared systematically within a wide range of operating conditions. The numerical results reveal that the worn surface profile slightly decreases the maximum temperature. Additionally, the worn region is primarily located at both edges of the bearing, and the time-varying worn surface profile may be beneficial for improving the hydrodynamic effect. Furthermore, the effect of the 3D thermal characteristics on the wear prediction of journal bearings cannot be ignored when the external load, boundary friction coefficient, surface roughness are relatively large.     

Homogenization of a Class of Nonlinear Variational Inequalities with Applications in Fluid Film Flow

The authors consider the homogenization of a class of nonlinear variational inequalities, which include rapid oscillations with respect to a parameter. The homogenization of the corresponding class of differential equations is also studied. The results are applied to some models for the pressure in a thin fluid film fluid between two surfaces which are in relative motion. This is an important problem in the lubrication theory. In particular, the analysis includes the effects of surface roughness on both faces and the phenomenon of cavitation. Moreover, the fluid can be modeled as Newtonian or non-Newtonian by using a Rabinowitsch fluid model.     

一类小周期椭圆方程的三重尺度渐近分析

利用三重尺度方法对一类小周期椭圆方程进行了三重尺度渐近展开分析,构造了对应的三重尺度形式渐近展开式,得到了均匀化常数和均匀化方程.在形式渐近展开的基础上,构造了对应边值问题解的三重尺度渐近近似解,并分析了对应三重尺度形式渐近误差估计.     

Effets d'anisotropie par homogénéisation dans un problème à frontière libre

The Elrod–Adams model is a pressure–saturation formulation which takes into account cavitation phenomena in thin fluid films mechanics. We study the asymptotic behavior of the model, using the two-scale convergence technique, in devices such as journal bearings. Interest is highly motivated by the roughness effects of the surfaces. Anisotropic effects appear on the coefficients, which is usual, but also on the saturation function, since various saturation functions appear through the homogenization process. To cite this article: G. Bayada et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The KdV Equation in Stratified Fluid Flow

This paper describes a method for the formal derivation of the equation governing long waves on the surface of a shallow fluid which is stably and continuously stratified. By using a three variable expansion procedure involving the normal time scale together with two slow time scales suggested as a result of the formulation of the problem the governing equation is shown to be the ordinary Korteweg-de Vries equation.

AMS(MOS) CLASSIFICATION: 76D30, 35Q20, 35C20.     

On the optimization of surface textures for lubricated contacts

The pressure field that develops inside a lubricated contact obeys an elliptic equation known as Reynolds equation, with coefficients that depend on the shape of the contacting surfaces. The load-carrying capacity of a contact, defined as the integral of the pressure field, is an important performance indicator that should be as high as possible to avoid wear and damage of the surfaces. In this article, the effect of arbitrary uniform periodic textures on the load-carrying capacity of lubricated devices known as thrust bearings is investigated theoretically by means of homogenization techniques and first-order perturbation analysis. It is shown that the untextured shape is a local optimum for the load-carrying capacity of the homogenized pressure field. This is proved for bearings of general shape and considering both incompressible and compressible models for the lubricant. The homogenization technique however implies an error. Suitable bounds for the effect of this error are provided in a simplified case.     

Derivation and analysis of an effective model for biofilm growth in evolving porous media

The present article deals with the growth of biofilms produced by bacteria within a saturated porous medium. Starting from the pore‐scale, the process is essentially described by attachment/detachment of mobile microorganisms to a solid surface and their ability to build biomass. The increase in biomass on the surface of the solid matrix changes the porosity and impedes flow through the pores. Using formal periodic homogenization, we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients provided by the evolving microstructure at the pore‐scale. Assuming, that the underlying pore geometry may be described by a single parameter, for example, porosity, the level set equation locating the biofilm‐liquid interface transforms into an ordinary differential equation (ODE) for the parameter. For such a setting, we state significant analytical and algebraic properties of these effective parameters. A further objective of this article is the analytical investigation of the resulting coupled PDE–ODE model. In a weak sense, unique solvability either global in time or at least up to a possible clogging phenomenon is shown. Copyright © 2016 John Wiley & Sons, Ltd.     

Étude asymptotique des phénomènes acoustiques dans un mélange fluide-solide thermoélastique

Asymptotic study of the macroscopie behaviour of a fluid-thermoelastic solid mixture. The homogenization theory is applied to the study of small perturbations in a mixture of a thermoelastic solid and a viscous compressible fluid. There appear memory phenomena in the bulk behaviour as well as a modification in the initial data regarding the original problem. The convergence of the formal asymptotic expansion to the solution of the homogenized equations is proved.     

Homogenization of Boundary-Value Problem in a Locally Periodic Perforated Domain

We consider a model homogenization problem for the Poisson equation in a locally periodic perforated domain with the smooth exterior boundary, the Fourier boundary condition being posed on the boundary of the holes. In the paper we construct the leading terms of formal asymptotic expansion. Then, we justify the asymptotics obtained and estimate the residual.     

Nonlinear dynamic analysis of a complex dual rotor-bearing system based on a novel model reduction method

In this paper, a dynamic model of a complex dual rotor-bearing system of an aero-engine is established based on the finite element method with three types of beam elements (rigid disc, cylindrical beam element and conical beam element), as well as taking into account the nonlinearities of all of the supporting rolling element bearings. To rapidly and accurately analyze dynamic behaviors of the complex dual rotor-bearing system, a two-level model order reduction (MOR) method is proposed by combining component mode synthesis (CMS) method and proper orthogonal decomposition (POD) technique. The first-level reduced-order model (ROM) of the dual rotors is obtained by CMS method with a high precision for the original system. Then, the POD method is applied to second-level model order reduction to further decrease the degrees of freedom (DOFs) of first-level ROM. Second-level ROM with mode expansion and direct second-level ROM are obtained, and the nonlinear displacement responses of the two ROMs are compared with the first-level ROM. The numerical results demonstrate that the proposed method has a higher computational efficiency and accuracy in terms of mode expansion than the direct model reduction by using POD method. In addition, the nonlinear vibration responses of the dual rotor-bearing system are studied by this second-level ROM in the case of different clearances of the inter-shaft bearing. The results indicate that the dynamic characteristics of the dual rotor-bearing system are very complicated for a large clearance.     

Explosion en temps fini de la solution d'un problème d'évolution de surface de film mince

In this Note, we are interested in the evolution of a surface of a crystal structure, constituted by an elastic substrate and a thin film. If the crystal is constrained, some morphological instabilities may appear. To study these instabilities, we made use of the model developped in Phys. Rev. B 47 (1993) 9760–9777. There, the map f of the free surface of the film satisfies a parabolic partial differential equation, depending on the elastic displacement of the substrate. For simplicity, the substrate is assumed to be linearly elastic and the structure to be infinite in one direction. Then, under some formal asymptotic assumptions, a formal expansion of the displacement can be determined after some appropriate scalings, allowing to derive a simplified parabolic nonlinear equation as in Lods et al. (Asymptotic Anal. 33 (2003) 67–91). We give here some results about the finite-time blow-up and the existence and uniqueness of the solution in an appropriate space. To validate the theoretical results, we also performed some numerical simulations using a pseudo-spectral method and also compute the initial-profile dependent critical value of the parameter θ involved in the nonlinear equation. To cite this article: M. Boutat et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Nonlinear reynolds equation for lubrication of a rapidly rotating shaft

Using the homogenization theory, we derive the nonlinear Reynolds equation governing the process of lubrication of a slipper bearing with rapidly rotating shaft. We prove that this nonliner lubrication law is an approximation of the full Navier-Stokes equations in a thin cylinder with periodic roughness. The analyticity of the nonlinear function giving the relation between the velocity and the pressure drop is proved. The first term in its Taylor's expansion is the classical linear Reynolds law. Boundary layer correctors are computed.     

Determination of wear coefficient in mixed lubrication using FEM

In a line or point contact with an elastohydrodynamic lubricant oil film, solid-to-solid contacts are common and wear will occur at these places. Given that there is only a portion of the load is supported by the direct interaction of roughness asperities, the wear coefficient should be less than that for dry contacts and account for the effect of surface roughness and oil film. Since it is difficult to obtain the wear coefficient value at different oil film thickness by experiments, this paper presents the methodology of determination of wear coefficient in mixed lubrication using finite element method (FEM). In this method, the roughness of contact surfaces is characterized as fractal surfaces by the Weierstrass–Mandelbrot (W–M) function, the sliding wear in mixed lubrication is simulated by the Coupled-Eulerian–Lagrangian (CEL) method and the wear volume is calculated according to the solid–solid contact load. Then the wear coefficient can be determined and the simulation example shows that the wear coefficient decreases nonlinearly with the increasing of oil film thickness and dynamic viscosity in mixed lubrication.     

奇摄动分数阶微分方程的渐近解

本文研究了一类奇摄动非线性分数阶微分方程初值问题.利用伸长变量构造出解的形式展开式,并利用微分不等式理论,证明了解的一致有效的渐近式.所得的结果具有较好精度的近似解.     

应力和位移约束下连续体结构拓扑优化

同时考滤应力和位移约束的连续体结构拓扑优化问题,很难用现有的均匀方法或变密度方法等求解。主要困难在于难以建立应力和位移约束与拓扑设计变量间显式关系式;即使建立了这种关系,也由于优化问题规模过大,利用常规的数学规划方法难以求解。隋允康、杨德庆曾提出了基于独立连续拓扑变量及映射变换（ＩＣＭ）的桁架结构拓扑优化模型。本文在此基础上,建立了以重量为目标,考虑应力和位移约束的连续体结构拓扑优化模型,并推导出     

一类高阶椭圆型方程Dirichlet问题

本文讨论了一类奇摄动高阶椭圆型方程Dirichlet问题，利用伸长变量和变界层校正法，得到了问题解的形式渐近展开式．再用微分不等式理论，证明了解的一致有效性．     

Homogenized elliptic equations and variational inequalities with oscillating parameters. Application to the study of thin flow behavior with rough surfaces

Double scale homogenization is used to average stationary equation or inequalities in which both highly oscillating variables and parameters appear. We demonstrate how the limit is obtained using a two-stage procedure, firstly by carrying out a classical homogenization process by freezing the oscillating parameter, then by averaging the result with respect to this parameter. These results allow us to average the pressure for a newtonian fluid in a narrow gap between two rough unstationary surfaces. Numerical results are given.     

A multiscale method for semilinear elliptic equations

At present there are many papers, based on multiscale expansion and homogenization theory, to deal with nonlinear problems with microstructure. But there is no systematic method to deal with all of the possible nonlinear partial differential equations since different nonlinear problems gives rise to different multiscale expansions parameters classes. This introduces changes in the consequent process of homogenization. In this paper, a method based on the theory of upper and lower solution is provided. It deals with nonlinear problems by reducing them to a series of linear problems. In addition numerical computations are also presented in the last part of the paper to support our theoretical analysis.