Guaranteed error bounds for the homogenisation of random materials
Daniel Paladim  1@  , Jose Paulo Moitinho De Almeida  2@  , Stephane Bordas  3@  , Pierre Kerfriden  4@  
1 : Institute of Mechanics and Advanced Materials, Cardiff University
Cardiff School of Engineering, Cardiff University, Wales -  United Kingdom
2 : Instituto Superior Tecnico [Lisboa]  (IST)  -  Website
Av. Rovisco Pais, 1 1049-001 Lisboa -  Portugal
3 : Faculté des Sciences, de la Technologie et de la Communication
Université du Luxembourg, L-1359 -  Luxembourg
4 : Institute of Mechanics Materials and Advanced Manufacturing
Cardiff University, CF24 3AA -  United Kingdom

The present paper aims to quantify the error due to homogenisation of highly heterogeneous diffusion fields in the solution of linear elliptic PDEs which are common in the modelling of packed particulate composites. This work takes as starting point the pioneering work Oden and Zohdi (1997) and extends it to bound the error in the expectation and second moment of quantities of interest without solving the intractable stochastic fine-scale problem. All the computations involved are deterministic, macroscopic and indepedent of the scale ratio. In the present work, the guaranteed error bounds are re-derived using the Prager-Synge hypercircle theorem. This enabled us to optimise and fully characterise the effectivity of the presented estimates. We also interpret our results in terms of the Reuss and Voigt approaches for the homogenisation of composites. Finally, an efficient procedure is presented to tighten the estimates through the local approximation of the fine-scale model.

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