An adaptive scheme for homogenised domains
D.A. Paladim, Pierre Kerfriden, J.P.M. Almeida, Stéphane Bordas · UPCommons institutional repository (Universitat Politècnica de Catalunya) · 2015
In this paper, we extend the concept of modelling error estimation for the homogenisation of ellipticPDEs. In order to do so, we fully acknowledge that the rapid spatial variation of microscopicdiffusion constants cannot be known exactly. Therefore, we represent the microscopic diffusioncoefficients as a random field. In this context, the accuracy of surrogate models, such ashomogenisation schemes, can be quantified by estimating the error in the first moments of theprobability density function of a quantity of interest.We propose a way to bound the error in the two first moments, following and extending the seminalwork of [1]. Our derivations rely on the Constitutive Relation Error[2] (CRE), which states that acertain distance between the solutions delivered by the primal and a dual surrogates of the originalstochastic problem is equal to some measure of the exact and unaffordable errors. We further assumethat these surrogates are deterministic, consistently with the theory of homogenisation. Minimisingthe CRE in this subset of homogenisation schemes leads us to an optimal surrogate that is closelyrelated to the classical Voigt and Reuss models. This result is used in a goal-oriented setting toestablish upper and lower bounds for the first two moments of the quantity of interest. We show that the method respect the numerical separation of scales, and is therefore affordable andeasy to implement, and that it produces useful results as long as the mismatch between the diffusioncoefficients of the microstructure remains small. We will propose extensions for the case of highmismatch, by allowing the surrogate solutions to fluctuate in the stochastic domain.