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

Wellander, Niklas; Guenneau, Sébastien; Cherkaev, Elena

doi:10.1002/mma.4345

Cited by 17 publications

(10 citation statements)

References 25 publications

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“…We remark that the former set corresponds also to the set of periods for 𝑓 which are close to the hyperplane identified with R d . This argument reminds the cut-and-project arguments typical of quasicrystalline structures (see previous works [3][4][5][6][7][8]). The existence of such geometric quasiperiods is not sufficient to prove the necessary translation-invariance properties for 𝑓 0 and the homogenization asymptotic formula.…”

Section: Introductionsupporting

confidence: 88%

A note on the homogenization of incommensurate thin films

Anello

Braides

Caragiulo

2023

Math Methods in App Sciences

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Dimension‐reduction homogenization results for thin films have been obtained under hypotheses of periodicity or almost periodicity of the energies in the directions of the mid‐plane of the film. In this note, we consider thin films, obtained as sections of a periodic medium with a mid‐plane that may be incommensurate, that is, not containing periods other than 0. A geometric almost periodicity argument similar to the cut‐and‐project argument used for quasicrystals allows to prove a general homogenization result.

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Section: Introductionsupporting

confidence: 88%

A note on the homogenization of incommensurate thin films

Anello

Braides

Caragiulo

2023

Math Methods in App Sciences

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“…Through out this section, we shall always assume that (ε i ) 1≤i≤n satisfies the scale-separation condition (1.2). Our proof follows closely the ideas of [21, 1, 2] and [10,31].…”

Section: Reiterated Homogenization Of Quasi-periodic Operatorsmentioning

confidence: 73%

Compactness and stable regularity in multiscale homogenization

Niu¹,

Zhuge²

2021

Preprint

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In this paper we develop some new techniques to study the multiscale elliptic equations in the form of −div Aε∇uε = 0, where Aε(x) = A(x, x/ε1, • • • , x/εn) is an n-scale oscillating periodic coefficient matrix, and (εi) 1≤i≤n are scale parameters. We show that the C α -Hölder continuity with any α ∈ (0, 1) for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary (ε1, ε2, • • • , εn) ∈ (0, 1] n and particularly is independent of the ratios between εi's. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by H-convergence. The Lipschitz estimate for arbitrary (εi) 1≤i≤n still remains open. However, for special laminate structures, i.e., Aε(x) = A(x, x1/ε1, • • • , x d /εn), we show that the Lipschitz estimate is stable for arbitrary (ε1, ε2, • • • , εn) ∈ (0, 1] n . This is proved by a technique of reperiodization.

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“…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.…”

mentioning

confidence: 87%

“…The paper is organized as follows. We provide the proofs of the statements in [27] and extend the examples to electrostatic, elastostatic, and quasistatic magnetic problems. The examples are presented in Section 2.…”

Section: Introductionmentioning

confidence: 97%

“…To do that, one has to complement existing tools with the cut-and-projection operator, this was done in [6]. In this paper, we revisit this extension, the two-scale cut-and-projection convergence method presented in [6] which in [27] was further developed to characterize the two-scale limit of differential operators. Interestingly, Braides, Riey, and Solci have independently proposed a Γ-convergence approach to homogenize Penrose tilings [7] using general theorems on almost-periodic functions in W 1,p spaces [8].…”

Section: Introductionmentioning

confidence: 99%

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

Wellander,

Guenneau,

Cherkaev

2019

Preprint

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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.

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

Abstract: International audienc

Cited by 17 publications

References 25 publications

A note on the homogenization of incommensurate thin films

A note on the homogenization of incommensurate thin films

Compactness and stable regularity in multiscale homogenization

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

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