Two-Scale Convergence for Locally Periodic Microstructures and Homogenization of Plywood Structures

Abstract

The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in H 1 (Ω) are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.

Multiscale Modelling and Analysis of Signalling Processes in Tissues with Non-Periodic Distribution of Cells

“…To define the locally periodic microstructure related to the original non-periodic one, we consider, similarly to [8,29], the partition covering of by a family of open nonintersecting cubes { ε n } 1≤n≤N ε of side ε r , with 0 < r < 1, such that For each x ∈ R 3 , we consider a transformation matrix D(x) ∈ R 3×3 and assume that D, D −1 ∈ Lip(R 3 ; R 3×3 ) and 0…”

“…Then, the convergence results for the l-p unfolding operator and l-t-s convergence, see [29,30] or Appendix, imply that there exist subsequences (denoted again by c ε , r ε f and r ε b ) and the functions c ∈ L 2 (0, T ;…”

“…See Appendix or [29] for the definition of the locally periodic approximation L ε . The regularity assumptions on α, β, K, and γ ensure that α, β ∈ C( ; C per ( Y x )).…”

“…First, assuming the C 2 -regularity for the rotation angle γ , we define the locally periodic microstructure which approximates the original non-periodic plywood-like structure. Similar approximation of non-periodic plywood-like microstructure by locally periodic one was considered in [7,29]. Then, applying techniques of locally periodic homogenization (locally periodic two-scale convergence (l-t-s) and l-p unfolding operator), we derive macroscopic equations for the original microscopic model.…”

“…Previous results on homogenization in locally periodic media constitute the multiscale analysis of a heat-conductivity problem defined in domains with non-periodically distributed spherical balls [3,8,31], and elliptic and Stokes equations in non-periodic fibrous materials [4,6,7,29]. Formal asymptotic expansion and two-scale convergence defined for periodic test functions, [27], were used to derive macroscopic equations for models posed in domains with locally periodic perforations, i.e., domains consisting of periodic cells with smoothly changing perforations [5,10,11,22,23,33].…”