Bloch wave homogenization in a medium perforated by critical holes

Jaime Ortega; Jorge San Martín; Loredana Smaranda

doi:10.1016/j.crme.2007.01.001

Bloch wave homogenization in a medium perforated by critical holes

Presented by: Évariste Sanchez-Palencia

Jaime Ortega ¹; Jorge San Martín ²; Loredana Smaranda ^3,⁴

¹ Departamento de Ciencias Básicas, Universidad del Bío-Bío, Casilla 447, Campus Fernando May, Chillán, Chile
² Departemento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile
³ Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile
⁴ Department of Applied Mathematics, University of Piteşti, Romania

Comptes Rendus. Mécanique, Volume 335 (2007) no. 2, pp. 75-80.

Abstract
Résumé

In this Note, we use the Bloch wave method to study the asymptotic behavior of the solution of the Laplace equation in a periodically perforated domain, under a non-homogeneous Neumann condition on the boundary of the holes, as the hole size goes to zero more rapidly than the domain period. We prove that for a critical size, the non-homogeneous boundary condition generates an additional term in the homogenized problem, commonly referred to as ‘the strange term’ in the literature.

Dans cette Note, nous utilisons la méthode des ondes de Bloch dans l'étude du comportement asymptotique de la solution de l'équation de Laplace dans un domaine périodiquement perforé sous une condition Neumann non homogène sur la frontière des perforations quand la taille des trous converge vers zéro plus rapidement que la période du domaine. On prouve que pour une taille critique, la condition non homogène génère un terme additionnel dans le problème homogénéisé, lequel est connue dans la littérature comme « le terme étrange ».

Received: 2006-12-13
Accepted: 2006-12-20
Published online: 2007-02-01

DOI: 10.1016/j.crme.2007.01.001

Keywords: Computational solid mechanics, Bloch wave method
Mot clés : Mécanique des solides numérique, Méthode des ondes de Bloch

Author's affiliations:

Jaime Ortega ¹; Jorge San Martín ²; Loredana Smaranda ^{3, 4}

¹ Departamento de Ciencias Básicas, Universidad del Bío-Bío, Casilla 447, Campus Fernando May, Chillán, Chile
² Departemento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile
³ Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile
⁴ Department of Applied Mathematics, University of Piteşti, Romania

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     author = {Jaime Ortega and Jorge San Mart{\'\i}n and Loredana Smaranda},
     title = {Bloch wave homogenization in a medium perforated by critical holes},
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     language = {en},
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Jaime Ortega; Jorge San Martín; Loredana Smaranda. Bloch wave homogenization in a medium perforated by critical holes. Comptes Rendus. Mécanique, Volume 335 (2007) no. 2, pp. 75-80. doi : 10.1016/j.crme.2007.01.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.01.001/

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