We investigate the case of a medium with two inclusions or inhomogeneities with nearly touching corner singularities. We present two different asymptotic models to describe the phenomenon under specific geometrical assumptions. These asymptotic expansions are analysed and compared in a common framework. We conclude by a representation formula to characterise the detachment of the corners and we provide the possible extensions of the geometrical hypotheses.
Nous considérons le problème de Laplace pour un matériau contenant deux inclusions ou inhomogénéités avec coins proches. Nous présentons deux approches asymptotiques différentes pour décrire le phénomène sous certaines conditions géométriques. Ces développements asymptotiques sont étudiés et comparés dans un même cadre. Nous aboutissons à une formule de représentation qui caractérise l’éloignement des deux coins et proposons une relaxation des hypothèses géométriques.
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Keywords: Asymptotic expansion, corner singularity
@article{AHL_2020__3__381_0,
author = {Bonnaillie-No\"el, Virginie and Poignard, Clair and Vial, Gr\'egory},
title = {Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner},
journal = {Annales Henri Lebesgue},
pages = {381--406},
publisher = {\'ENS Rennes},
volume = {3},
year = {2020},
doi = {10.5802/ahl.36},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/ahl.36/}
}
TY - JOUR AU - Bonnaillie-Noël, Virginie AU - Poignard, Clair AU - Vial, Grégory TI - Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner JO - Annales Henri Lebesgue PY - 2020 SP - 381 EP - 406 VL - 3 PB - ÉNS Rennes UR - http://archive.numdam.org/articles/10.5802/ahl.36/ DO - 10.5802/ahl.36 LA - en ID - AHL_2020__3__381_0 ER -
%0 Journal Article %A Bonnaillie-Noël, Virginie %A Poignard, Clair %A Vial, Grégory %T Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner %J Annales Henri Lebesgue %D 2020 %P 381-406 %V 3 %I ÉNS Rennes %U http://archive.numdam.org/articles/10.5802/ahl.36/ %R 10.5802/ahl.36 %G en %F AHL_2020__3__381_0
Bonnaillie-Noël, Virginie; Poignard, Clair; Vial, Grégory. Asymptotic expansions for the conductivity problem with nearly touching inclusions with corner. Annales Henri Lebesgue, Volume 3 (2020), pp. 381-406. doi : 10.5802/ahl.36. http://archive.numdam.org/articles/10.5802/ahl.36/
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