Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings

Published in "Networks and Heterogeneous Media", 2010, vol. 5, no 1, p. 1-29 http://hal-enpc.archives-ouvertes.fr/hal-00625537/fr/In quasi-periodic or nonlinear periodic homogenization, the corrector problem must be in general set on the whole space. Numerically computing the homogenization coefficient therefore implies a truncation error, due to the fact that the problem is approximated on a bounded, large domain. We present here an approach that improves the rate of convergence of this approximation

Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell-problems, result in an exponential decay of the resonance error

Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients sufer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell-problems, result in an exponential decay of the resonance error

MATHICSE Technical Report : Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell-problems, result in an exponential decay of the resonance error

Macroscopically consistent non-local modelling of heterogeneous media

International audienceWithin the framework of the homogenization of heterogeneous media, a non local model is proposed. A field of non-local filtered stiffness tensor is introduced by filtering the solution to the homogenization problem. The filtered stiffness tensor, depending on the filter to heterogeneity size ratio, provides a continuous transition from the actual micro-scale heterogeneous stiffness field to the macro-scale homogenized stiffness tensor. For any intermediate filter size, the homogenization of the filtered stiffness yields exactly the homogenized stiffness, therefore it is called macroscopically consistent. The non-local stiffness tensor is intrinsically non symmetric, but its spatial fluctuations are smoothed, allowing for a less refined discretization in numerical methods. As a by-product, a two step heterogeneous multiscale method is proposed to reduce memory and computational time requirements of existing direct schemes while controlling the accuracy of the result. The first step is the estimation of the filtered stiffness at sampling points by means of an oversampling strategy to reduce boundary effects. The second step is the numerical homogenization of the obtained sampled filtered stiffness