Homogenization of quasiperiodic structures and two-scale cut-and-projection convergence

Abstract

Quasiperiodic arrangements of the constitutive materials in composites result in effective properties with very unusual electromagnetic and elastic properties. The paper discusses the cut-and-projection method that is used to characterize effective properties of quasiperiodic materials. Characterization of cut-and-projection convergence limits of partial differential operators is presented, and correctors are established. We provide the proofs of the results announced in (Wellander et al., 2018) and give further examples. Applications to problems of interest in physics include electrostatic, elastostatic and quasistatic magnetic cases.

Two‐scale cut‐and‐projection convergence; homogenization of quasiperiodic structures

“…However, in the decomposition in (47), the gradient of the potential in the "direction " of the hyperplane can be obtained from the gradient of the potential via a rotation of coordinate system. Further note that unlike in [WGC18,WGC19], in the proof below we use the notion of 2-scale cut-and-projection convergence in distributional sense (Definition 1.1), and not in weak sense (Definition 1.2).…”

“…In order to homogenize nonlinear PDEs with a monotone partial differential operator as in (3), we need to identify the differential relationship between χ and u 0 , given a bounded sequence (u η ) in W 1,p (Ω) (such that u η R ⇀ ⇀ u 0 and ∇u η R ⇀ ⇀ χ). This problem was solved by Allaire in the case of periodic functions [All92] and extended by Bouchitté et al for quasiperiodic functions [BGZ10] in W 1,2 (Ω) and revisited in [WGC18,WGC19].…”

“…We have the following integration by parts type generalization to the L p case of Lemma 6 given in the L 2 setting in [WGC19] Lemma 2.1 (Green's Identity). It holds that…”

“…To do that one has to complement existing tools with the cut-and-projection operator, this was done in [BGZ10] in the framework of W 1,2 , making use of Fourier representation for two-scale limits of gradients. Importantly, this was revisited in [WGC18,WGC19].…”

“…In this paper we extend the two-scale cut-and-projection convergence method to Sobolev spaces W 1,p , 1 < p < ∞. We build upon [WGC19] to characterize the limits of nonlinear partial differential operators in this setting. We illustrate the method on a nonlinear electrostatic problem that was previously homogenized using the tool of G-convergence for a larger class of almost periodic functions in [BCPD92].…”