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Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales. (English) Zbl 1365.80011

Summary: 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. More precisely, in the context of linear elliptic diffusion problems in randomly packed particulate composites, we develop an approach to strictly 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.

MSC:

80M40 Homogenization for problems in thermodynamics and heat transfer
65N15 Error bounds for boundary value problems involving PDEs
49K45 Optimality conditions for problems involving randomness
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
80A20 Heat and mass transfer, heat flow (MSC2010)

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References:

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