Multiscale convergence and reiterated homogenisation

1996 ◽  
Vol 126 (2) ◽  
pp. 297-342 ◽  
Author(s):  
G. Allaire ◽  
M. Briane
Keyword(s):  
Composite Material ◽  
Heat Equation ◽  
Model Problem ◽  
Perforated Domain ◽  
Periodic Oscillations ◽  
Multi Scale ◽  
Multiscale Convergence ◽  
Convergence Method ◽  
Oscillating Coefficients ◽  
The Neumann Problem

This paper generalises the notion of two-scale convergence to the case of multiple separated scales of periodic oscillations. It allows us to introduce a multi-scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients. This new method is applied to a model problem with a finite or infinite number of microscopic scales, namely the homogenisation of the heat equation in a composite material. Finally, it is generalised to handle the homogenisation of the Neumann problem in a perforated domain.

Stochastic homogenization and macroscopic modelling of composites and flow through porous media

2002 ◽  
pp. 337-378 ◽  
Author(s):  
Jozef Telega ◽  
Wlodzimierz Bielski
Keyword(s):  
Porous Medium ◽  
Random Distribution ◽  
Flow Through Porous Media ◽  
Macroscopic Models ◽  
Linear Elastic ◽  
Multi Scale ◽  
Convergence Method ◽  
Flow Through ◽  
The Mean ◽  
Random Porous Medium

The aim of this contribution is mainly twofold. First, the stochastic two-scale convergence in the mean developed by Bourgeat et al. [13] is used to derive the macroscopic models of: (i) diffusion in random porous medium, (ii) nonstationary flow of Stokesian fluid through random linear elastic porous medium. Second, the multi-scale convergence method developed by Allaire and Briane [7] for the case of several microperiodic scales is extended to random distribution of heterogeneities characterized by separated scales (stochastic reiterated homogenization). .

scholarly journals Moderate deviations for a fractional stochastic heat equation with spatially correlated noise

2017 ◽  
Vol 17 (04) ◽  
pp. 1750025 ◽  
Author(s):  
Yumeng Li ◽  
Ran Wang ◽  
Nian Yao ◽  
Shuguang Zhang
Keyword(s):  
Heat Equation ◽  
Moderate Deviation ◽  
Stochastic Heat Equation ◽  
Correlated Noise ◽  
Derivative Operator ◽  
Spatially Correlated ◽  
Whole Space ◽  
Convergence Method ◽  
Spatially Correlated Noise ◽  
Weak Convergence Method

In this paper, we study the Moderate Deviation Principle for a perturbed stochastic heat equation in the whole space [Formula: see text]. This equation is driven by a Gaussian noise, white in time and correlated in space, and the differential operator is a fractional derivative operator. The weak convergence method plays an important role.

ASYMPTOTIC EXPANSION AND CONVERGENCE THEOREM OF CONTROL AND OBSERVATION ON THE BOUNDARY FOR SECOND-ORDER ELLIPTIC EQUATION WITH HIGHLY OSCILLATORY COEFFICIENTS

2004 ◽  
Vol 14 (03) ◽  
pp. 417-437 ◽  
Author(s):  
LI-QUN CAO
Keyword(s):  
Asymptotic Expansion ◽  
Elliptic Equation ◽  
Asymptotic Expansions ◽  
Convergence Theorem ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Oscillating Coefficients ◽  
Highly Oscillatory ◽  
Second Order Elliptic Equation ◽  
The Neumann Problem

In this paper, we shall study systems governed by the Neumann problem of second-order elliptic equation with rapidly oscillating coefficients and with control and observations on the boundary. The multiscale asymptotic expansions of the solution for considering problem in the case without any constraints, and homogenized equation in the case with constraints will be given, their rigorous proofs will also be proposed.

Multi-Scale and Finite Element Analysis of the Mixed Boundary Value Problem in a Perforated Domain under Coupled Thermoelasticity

2005 ◽  
Vol 9 ◽  
pp. 153-162
Author(s):  
Yongping Feng ◽  
Junzhi Cui
Keyword(s):  
Finite Element ◽  
Mixed Boundary ◽  
Asymptotic Expression ◽  
Perforated Domain ◽  
Scale Analysis ◽  
Element Analysis ◽  
Coupled Thermoelasticity ◽  
Multi Scale ◽  
Error Estimations ◽  
Small Periodic

The two-scale asymptotic expression and error estimations based on two-scale analysis (TSA) are presented for the solution of the increment of temperature and the displacement of a composite structure with small periodic configurations under coupled thermoelasticity condition in a perforated domain. The two-scale coupled relation between the increment of temperature and displacement is established.The multi-scale finite element algorithms corresponding to TSA are described and numerical results are presented.

scholarly journals Homogenization in Sobolev Spaces with Nonstandard Growth: Brief Review of Methods and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-16
Author(s):  
Brahim Amaziane ◽  
Leonid Pankratov
Keyword(s):  
Sobolev Spaces ◽  
Variational Problems ◽  
Double Porosity ◽  
Energy Characteristics ◽  
Perforated Domains ◽  
Growth Functions ◽  
Homogenization Approach ◽  
Convergence Method ◽  
Oscillating Coefficients ◽  
Γ Convergence

We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed.

Multi-Scale Strain Measurements of a Particulate Composite Material

2002 ◽  
Author(s):  
C. T. Liu ◽  
C. W. Smith ◽  
G. Ravichandran
Keyword(s):  
Composite Material ◽  
Particulate Composite ◽  
Experimental Results ◽  
Strain Fields ◽  
Length Scales ◽  
Strain Measurements ◽  
Multi Scale ◽  
Hard Particles ◽  
Particulate Composite Material

In this study, the strain fields on two different length scales in a particulate composite material containing hard particles embedded in a rubbery matrix were investigated, using two different techniques. The experimental results were analyzed and are discussed.

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