10 High-order homogenization of the Poisson equation in a perforated periodic domain

Abstract

We derive high order homogenized models for the Poisson problem in a cubic domain periodically perforated with holes where Dirichlet boundary conditions are applied. These models have the potential to unify the three possible kinds of limit problems derived by the literature for various asymptotic regimes (namely the "unchanged" Poisson equation, the Poisson problem with a strange reaction term, and the zeroth order limit problem) of the ratio η ≡ aε/ε between the size aε of the holes and the size ε of the periodic cell. The derivation relies on algebraic manipulations on formal two-scale power series in terms of ε and more particularly on the existence of a "criminal" ansatz, which allows to reconstruct the oscillating solution uε as a linear combination of the derivatives of its formal average u * ε weighted by suitable corrector tensors. The formal average is itself the solution of a formal, infinite order homogenized equation. Classically, truncating the infinite order homogenized equation yields in general an ill-posed model. Inspired by a variational method introduced in [52, 23], we derive, for any K ∈ N, well-posed corrected homogenized equations of order 2K + 2 which yields approximations of the original solutions with an error of order O(ε 2K+4 ) in the L 2 norm. Finally, we find asymptotics of all homogenized tensors in the low volume fraction regime η → 0 and in dimension d ≥ 3. This allows us to show that our higher order effective equations converge coefficient-wise to either of the classical homogenized regimes of the literature which arise when η is respectively equivalent, or greater than the critical scaling η crit ∼ ε 2/(d−2) .

Layer potentials for Lamé systems and homogenization of perforated elastic medium with clamped holes

“…We extend the approach to Lamé system in this paper. Some recent related works on homogenization in perforated domains with Dirichlet conditions on the holes can be found in [20,16,15,13,14]. We remark that when other boundary conditions such as Neumann, Robin or transmission conditions are imposed at the boundary of the holes/inclusions, the asymptotic behavior could be very different; see e.g.…”

High Order Homogenized Stokes Models Capture all Three Regimes

“…Notation conventions. In the whole paper, we use the same notation conventions for tensor related operations as in our previous works [12,13]. These are summarized in the nomenclature below.…”

“…which we considered in [12]. In fact, it turns out that in scalar context of (1.4), free of the divergence constraint, the approximation error on the solution u ε committed by the homogenized model of order 2K + 2 improves rather surprisingly up to the order…”

Learning high-order spatial discretisations of PDEs with symmetry-preserving iterative algorithms

“…To determine a macroscale description of a multiscale system with microscale heterogeneity we seek an homogenisation, or 'average', over the microscale which accounts for the heterogeneity without retaining unnecessary fine-scale details (Geers et al 2017). For example, for composite materials with periodic microstructures, which are common in nature and are increasingly synthesised for novel industrial applications (e.g., Nemat-Nasser et al 2011;Bargmann et al 2018), asymptotic homogenisation constructs a power series in scale separation with coefficients dependent on a periodic 'representative volume element' (e.g., a unit cell) which describes the microscale heterogeneity (Dutra et al 2020;Feppon 2020). However, homogenisation often requires substantial user input in the form of an algebraic analysis of the given microscale system prior to numerical implementation, and theoretical support is often only guaranteed in unphysical circumstances; for example, typically one needs to explicitly identify 'fast' and 'slow' variables, and also assume an unphysical infinite scale separation between the two (e.g., Engquist and Souganidis 2008).…”