homogenisation; error; RVE; upper bound; lower bound; model selection
Abstract :
[en] This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion coefficients. We typically consider linear elliptic diffusion problems in randomly packed particulate composites. Our work extends the pioneering work presented in [26,32] in order to bound the error in the expectation and second moment of quantities of interest, without ever solving the fine-scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine-scale features. Our computations are purely macroscopic, deterministic, and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager-Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss-Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation.
Disciplines :
Materials science & engineering
Author, co-author :
Paladim, Daniel-Alves
de Almeida, José Paulo Baptista
BORDAS, Stéphane ; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Engineering Research Unit
Kerfriden, Pierre
External co-authors :
yes
Language :
English
Title :
Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales
Publication date :
2017
Journal title :
International Journal for Numerical Methods in Engineering
ISSN :
0029-5981
eISSN :
1097-0207
Publisher :
Wiley, Chichester, United Kingdom
Volume :
110
Issue :
2
Pages :
103–132
Peer reviewed :
Peer Reviewed verified by ORBi
Focus Area :
Computational Sciences Physics and Materials Science