2016
DOI: 10.3233/asy-151350
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On the homogenization of thin perforated walls of finite length

Abstract: The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of peri… Show more

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Cited by 8 publications
(11 citation statements)
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“…As figure 8 shows, the analysis appears to capture the resonance shift correctly, as well as the 1 variation of the peak amplitude. To gain more accuracy in the resonance shift, we expect it would be necessary to consider local approximations in the vicinity of the corners (which were neglected in our analysis) and match these to the boundary layer and outer expansions, following the procedure outlined in [ 13 15 ].…”
Section: Discussionmentioning
confidence: 99%
“…As figure 8 shows, the analysis appears to capture the resonance shift correctly, as well as the 1 variation of the peak amplitude. To gain more accuracy in the resonance shift, we expect it would be necessary to consider local approximations in the vicinity of the corners (which were neglected in our analysis) and match these to the boundary layer and outer expansions, following the procedure outlined in [ 13 15 ].…”
Section: Discussionmentioning
confidence: 99%
“…If the macroscopic part of the solution is extended in a smooth way to the endpoints, the extension is not necessarily regular, e.g. it may tend to infinity at the endpoints of the interface Γ [20,21]. A similar behaviour has been observed for the macroscopic solution for problems with oscillating boundaries with corners [28] or a domain with rounded corners [29].…”
Section: (D) Solution Representationmentioning
confidence: 52%
“…The surface homogenization leads to an asymptotic solution representation, which can be used to construct effective boundary or transmission conditions [12][13][14][15][16][17][18], if their endpoints are ignored. At the endpoints the asymptotic solution representation has to incorporate the interaction of the microstructure and the singularities correctly (this has been done for the Poisson problem [19,20] and for the Helmholtz problem [21]). In particular, the interaction with the singular behaviour, which is macroscopically measurable, is mathematically involved.…”
Section: Introductionmentioning
confidence: 99%
“…The asymptotic (3.7) then follows from the application of the results of Nazarov [32] (see also Section 4 of Ref. [21] for a detailed description of this decomposition). A rigorous estimation of the remainder o (R ± ) −k can be done through the introduction of non-uniform weigthed Sobolev spaces [32].…”
Section: The Families S ± Nmentioning
confidence: 89%