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Diffusion in arrays of obstacles: beyond homogenisation
arXiv - CS - Computational Engineering, Finance, and Science Pub Date : 2020-02-04 , DOI: arxiv-2002.04526
Yahya Farah; Daniel Loghin; Alexandra Tzella; Jacques Vanneste

We revisit the classical problem of diffusion of a scalar (or heat) released in a two-dimensional medium with an embedded periodic array of impermeable obstacles such as perforations. Homogenisation theory provides a coarse-grained description of the scalar at large times and predicts that it diffuses with a certain effective diffusivity, so the concentration is approximately Gaussian. We improve on this by developing a large-deviation approximation which also captures the non-Gaussian tails of the concentration through a rate function obtained by solving a family of eigenvalue problems. We focus on cylindrical obstacles and on the dense limit, when the obstacles occupy a large area fraction and non-Gaussianity is most marked. We derive an asymptotic approximation for the rate function in this limit, valid uniformly over a wide range of distances. We use finite-element implementations to solve the eigenvalue problems yielding the rate function for arbitrary obstacle area fractions and an elliptic boundary-value problem arising in the asymptotics calculation. Comparison between numerical results and asymptotic predictions confirm the validity of the latter.
更新日期:2020-02-12

 

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