Homogenization in open sets with holes

Cioranescu, Doina; Paulin, J. Saint Jean

doi:10.1016/0022-247x(79)90211-7

Cited by 380 publications

(265 citation statements)

References 2 publications

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“…In particular, we prove the continuity of the We begin by proving a Poincar6 inequality in H~ (e 2~i°, Y*). [9]) also works in the present context. We leave the details to the reader.…”

Section: 4 Completeness Of the Limit Spectrumsupporting

confidence: 51%

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Bloch-wave homogenization for a spectral problem in fluid-solid structures

Allaire

Conca

1996

Arch. Rational Mech. Anal.

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This paper is concerned with the study of the vibrations of a coupled fluid-solid periodic structure. As the period goes to zero, an asymptotic analysis of the spectrum (i.e., the set of eigenfrequencies) is performed with the help of a new method, the so-called Bloch-wave homogenization method (which is a blend of two-scale convergence and Bloch-wave decomposition). The limit spectrum is made of three parts: the macroscopic or homogenized spectrum, the microscopic or Bloch spectrum, and the boundary-layer spectrum. The two first parts are completely characterized: The homogenized and the Bloch spectra are purely essential, and have a band structure. The boundary-layer spectrum is shown to be empty in the special case of periodic boundary condition.

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“…In particular, we prove the continuity of the We begin by proving a Poincar6 inequality in H~ (e 2~i°, Y*). [9]) also works in the present context. We leave the details to the reader.…”

Section: 4 Completeness Of the Limit Spectrumsupporting

confidence: 51%

“…Since we are studying an homogenization problem in a perforated domain t)~, we use a well-known technical lemma [9] for extending the solution u~ of (20) to the whole limit domain fl. This allows us to study the convergence of the sequence u, in the fixed space H~(fl).…”

Section: U(x)v~(x)--'~y T { Uo(xy)vo(xy)dy In Ll(n) Weaklymentioning

confidence: 99%

Bloch-wave homogenization for a spectral problem in fluid-solid structures

Allaire

Conca

1996

Arch. Rational Mech. Anal.

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“…Proof of Theorem 1,5. The first step of the proof consists in proving that the séquence {P e z z } remains bounded in HQ(£1), as e -• 0.…”

Section: Proofs Of the Results For The Case Of A Homogeneoüs Neumann mentioning

confidence: 99%

Non-homogeneous Neumann problems in domains with small holes

Conca¹,

Donato²

1988

ESAIM: M2AN

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“…Fruitful results have been published with a wide range of applications, as listed in recent literature review papers [3][4][5]. Homogenization, especially asymptotic homogenization [6][7][8], provides a powerful mathematical tool for bridging different scale modeling problem, and solving micro-macro, local-global, nano-macro, etc., multiscale modeling problems. The applicable areas include so-called heterogeneous materials, cellular materials, granular materials, fiber-reinforced polymers, etc.…”

Section: Introductionmentioning

confidence: 99%

Homogenization Method for Designing Novel Architectured Cellular Materials

Ma¹

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Two types of novel architectured cellular materials have been developed [1,2]. To estimate the effective material properties and to optimally design such materials, a suitable homogenization method was needed. An existing asymptotic homogenization process has provided an effective means for the multiscale modeling of continuum solids; however, that method was not easily adapted to the aforementioned materials, which are naturally discrete systems. First, we developed a strain-based homogenization method equivalent to the existing asymptotic homogenization method, but it was developed based on an engineering approach rather than on a mathematical approach such as the one used in asymptotic homogenization. The new approach separates the strain field into a homogenized strain field and a strain variation field in the local cellular domain superposed on the homogenized strain field. The Principle of Virtual Displacements for the relationship between the strain variation field and the homogenized strain field is then used to condense the strain variation field onto the homogenized strain field, and the homogenization process becomes a coordinate reduction process comparable to the Guyan Reduction used in structural dynamics analyses. The characteristic modes and the stress recovery process are also discussed. The new method is then extended to a stress-based homogenization process based on the Principle of Virtual Forces, and it is further applied to address the discrete systems of the aforementioned architectured cellular materials.

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Homogenization in open sets with holes

Cited by 380 publications

References 2 publications

Bloch-wave homogenization for a spectral problem in fluid-solid structures

Bloch-wave homogenization for a spectral problem in fluid-solid structures

Non-homogeneous Neumann problems in domains with small holes

Homogenization Method for Designing Novel Architectured Cellular Materials

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