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Numerical strategy for unbiased homogenization of random materials
Author(s) -
Cottereau R.
Publication year - 2013
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.4502
Subject(s) - homogenization (climate) , computation , mathematics , mathematical optimization , ergodicity , operator (biology) , boundary value problem , random matrix , statistical physics , computer science , algorithm , mathematical analysis , eigenvalues and eigenvectors , physics , biodiversity , ecology , biochemistry , statistics , chemistry , repressor , gene , transcription factor , biology , quantum mechanics
SUMMARY This paper presents a numerical strategy that allows to lower the costs associated to the prediction of the value of homogenized tensors in elliptic problems. This is performed by solving a coupled problem, in which the complex microstructure is confined to a small region and surrounded by a tentative homogenized medium. The characteristics of this homogenized medium are updated using a self‐consistent approach and are shown to converge to the actual solution. The main feature of the coupling strategy is that it really couples the random microstructure with the deterministic homogenized model, and not one (deterministic) realization of the random medium with a homogenized model. The advantages of doing so are twofold: (a) the influence of the boundary conditions is significantly mitigated, and (b) the ergodicity of the random medium can be used in full through appropriate definition of the coupling operator. Both of these advantages imply that the resulting coupled problem is less expensive to solve, for a given bias, than the computation of homogenized tensor using classical approaches. Examples of 1‐D and 2‐D problems with continuous properties, as well as a 2‐D matrix‐inclusion problem, illustrate the effectiveness and potential of the method. Copyright © 2013 John Wiley & Sons, Ltd.