Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

Abdulle, Assyr; Pouchon, Timothée Noé

doi:10.1142/s0218202516500627

Cited by 14 publications

(39 citation statements)

References 19 publications

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“…For uniformly periodic tensors, i.e. a(x, y) = a(y) ∀x ∈ Ω, the family E simplifies to a parametrized subset of the family defined in [8]. Indeed, in that case we verify that L 1 = 0 and…”

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confidence: 69%

“…Compared to the uniformly periodic case in [8], we observe that a supplementary term coming from the variation x → a(x, y) appears in this equation. To satisfy this equality, it is sufficient to define…”

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confidence: 76%

“…In the literature, several papers [43,32,31,36,25,26,9,11,8] investigated the research of long time effective equations in the case of a uniformly periodic tensor, i.e., a ε (x) = a x ε with a(y) Y -periodic. In particular, a recent result in [8] defines a family of effective equations whose elements are proved to approximate u ε at large timescales O(ε −2 ). The family is composed of equations of the form (we use the convention that repeated indices are summed)…”

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confidence: 99%

“…In this paper, we generalize the family of effective equations from [8] to the case of a locally periodic tensor. This analysis constitutes the first result in the study of long time wave propagation in locally periodic media.…”

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confidence: 99%

“…Let us note that the accuracy required on the adaptation in (3.2b) is dictated by the order of the timescale T ε = O(ε −2 ) (see [8] or [41,Chapter 4]). In the rest of the section, we proceed with the construction of the adaptation B εũ and of the effective equations.…”

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confidence: 99%

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Effective Models for Long Time Wave Propagation in Locally Periodic Media

Abdulle¹,

Pouchon²

2018

SIAM J. Numer. Anal.

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Abstract. A family of effective equations for the wave equation in locally periodic media over long time is derived. In particular, explicit formulas for the effective tensors are provided. To validate the derivation, an a priori error estimate between the effective solutions and the original wave is proved. As the dependence of the estimate on the domain is explicit, the result holds in arbitrarily large periodic hypercube. This constitutes the first analysis for the description of long time effects for the wave equation in locally periodic media. Thanks to this result, the long time a priori error analysis of the numerical homogenization method presented in [A. Abdulle and T. Pouchon, SIAM J. Numer. Anal., 54, 2016Anal., 54, , pp. 1507Anal., 54, -1534 is generalized to the case of a locally periodic tensor.

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confidence: 69%

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confidence: 76%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Effective Models for Long Time Wave Propagation in Locally Periodic Media

Abdulle¹,

Pouchon²

2018

SIAM J. Numer. Anal.

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Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales

Abdulle

Pouchon

2020

Found Comput Math

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A family of effective equations for wave propagation in periodic media for arbitrary timescales O(ε −α ), where ε 1 is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models [A. Benoit and A. Gloria, arXiv:1701.08600, 2017], [G. Allaire, A. Lamacz, and J. Rauch, arXiv:1803.09455, 2018]. The effective solutions in the family are proved to be ε close to the original wave in a norm equivalent to the L ∞ (0, ε −α T ; L 2 (Ω)) norm. In addition, a numerical procedure for the computation of the effective tensors of arbitrary order is provided. In particular, we present a new relation between the correctors of arbitrary order, which allows to substantially reduce the computational cost of the effective tensors of arbitrary order. This relation is not limited to the effective equations presented in this paper and can be used to compute the effective tensors of alternative effective models.

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A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures

Allaire

Briane

Vanninathan

2016

SeMA

152

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in pressInternational audienceIn this paper we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is well-known that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogenized equation [5] where the homogenized equation can be improved by adding small additional higher order differential terms. The next non-zero high order term is a fourth-order term, accounting for dispersion effects (see e.g. [23], [18], [15]). Surprisingly, this homogenized fourth-order tensor is not equal to the fourth-order tensor arising in the Taylor expansion of the first Bloch eigenvalue, which is often called Burnett tensor. Here, we establish an exact relation between the homogenized fourth-order tensor and the Burnett fourth-order tensor. It was proved in [11] that the Burnett fourth-order tensor has a sign. For the special case of a simple laminate we prove that the homogenized fourth-order tensor may change sign. In the elliptic case we explain the difference between the homogenized and Burnett fourth-order tensors by a difference in the source term which features an additional corrector term. Finally, for the wave equation, the two fourth-order tensors coincide again, so dispersion is unambiguously defined, and only the source terms differ as in the elliptic case

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Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

Cited by 14 publications

References 19 publications

Effective Models for Long Time Wave Propagation in Locally Periodic Media

Effective Models for Long Time Wave Propagation in Locally Periodic Media

Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales

A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures

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