Peter Wall

REITERATED HOMOGENIZATION OF NONLINEAR MONOTONE OPERATORS

In this paper, the authors study reiterated homogenization of nonlinear equations of the form -div(a(x,x/e,x/e2,Due))=f, where a is periodic in the first two arguments and monotone in the third. It is proved that ue converges weakly in W1,p(Ω) (and even in some multiscale sense), as e→0 to the solution u0 of a limit problem. Moreover, an explicit expression for the limit problem is given. The main results were also stated in [15]. This article presents the complete proofs of these results.

Reiterated homogenization applied in hydrodynamic lubrication

This work is devoted to studying the combined effect that arises due to surface texture and surface roughness in hydrodynamic lubrication. An effective approach in tackling this problem is by using the theory of reiterated homogenization with three scales. In the numerical analysis of such problems, a very fine mesh is needed, suggesting some type of averaging. To this end, a general class of problems is studied that, e.g. includes the incompressible Reynolds problem in both Cartesian and cylindrical coordinate forms. To demonstrate the effectiveness of the method several numerical results are presented that clearly show the convergence of the deterministic solutions towards the homogenized solution. Moreover, the convergence of the friction force and the load-carrying capacity of the lubricant film is also addressed in this paper. In conclusion, reiterated homogenization is a feasible mathematical tool that facilitates the analysis of this type of problem.

Similarities and Differences Between the Flow Factor Method by Patir and Cheng and Homogenization

Different averaging techniques have proved to be useful for analyzing the effects of surface roughness in hydrodynamic lubrication. This paper compares two of these averaging techniques, namely the ...

Commutators of Hardy operators in vanishing Morrey spaces

In this paper we study boundedness of commutators of the multi-dimensional Hardy type operators with BMO coefficients, in weighted global and/or local generalized Morrey spaces LΠp,φ(Rn,w) and vanishing local Morrey spaces VLlocp,φ(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(|x|). This study is made in the perspective of posterior applications of the weighted results to some problems in the theory of PDE. We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, and also in terms of the Matuszewska-Orlicz indices of φ and w, for such a boundedness.

A New Approach for Studying Cavitation in Lubrication

The underlying theory, in this paper, is based on clear physical arguments related to conservation of mass flow and considers both incompressible and compressible fluids. The result of the mathematical modeling is a system of equations with two unknowns, which are related to the hydrodynamic pressure and the degree of saturation of the fluid. Discretization of the system leads to a linear complementarity problem (LCP), which easily can be solved numerically with readily available standard methods and an implementation of a model problem in matlab code is made available for the reader of the paper. The model and the associated numerical solution method have significant advantages over todays most frequently used cavitation algorithms, which are based on Elrod–Adams pioneering work.

Two-scale convergence with respect to measures and homogenization of monotone operators

In 1989 Nguetseng introduced two-scale convergence, which now is a frequently used tool in homogenization of partial differential operators. In this paper we discuss the notion of two-scale convergence with respect to measures. We make an exposition of the basic facts of this theory and develope it in various ways. In particular, we consider both variable Lp spaces and variable Sobolev spaces. Moreover, we apply the results to a homogenization problem connected to a class of monotone operators.

A Comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai Equations

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.

On some sharp bounds for the homogenized p-poisson equation

We consider homogenization of a scale of p-Poisson equations in RN. Some new bounds of the effective energy are proved and compared with the non-linear Wiener -and Hashin-Shtrikman bounds. Moreover, we point out concrete nontrivid examples where these bounds even coincide. Some new examples of “optimal” microstructures are presented.

Reiterated homogenization of nonlinear monotone operators in a general deterministic setting

We study reiterated homogenization of a nonlinear non-periodic elliptic differential operator in a general deterministic setting as opposed to the usual stochastic setting. Our approach proceeds from an appropriate notion of convergence termed reiterated Σ -convergence. A general deterministic homogenization theorem is proved and several concrete examples are studied under various structure hypotheses ranging from the classical periodicity hypothesis to more complicated, but realistic, structure hypotheses.

Bounds on the effective behavior of a homogenized generalized Reynolds equation

We study upper and lower bounds for estimating the effective behavior described by homogenizing a problem which is a generalization of the Reynold equation. All cases when these bounds coincide are also found.