2020/02/04 by Yahya Farah, Farah, Yahya, Daniel Loghin +5
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Classical Physics (physics.class-ph) #Composite Material Mechanics #Computational Engineering #Computational Physics (physics.comp-ph) #FOS: Computer and information sciences #FOS: Physical sciences #Finance #Fluid Dynamics (physics.flu-dyn) #Numerical methods in inverse problems #and Science (cs.CE)
paper · doi:10.48550/arxiv.2002.04526
openalex publication_date 2020/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.