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Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer∗

Published online by Cambridge University Press:  13 August 2014

Victor Péron
Victor Péron*
Affiliation:
LMAP CNRS UMR 5142 & Team MAGIQUE 3D INRIA Bordeaux Sud-Ouest, Université de Pau et des Pays de l’Adour, Avenue de l’Université, BP 1155, 64013 Pau Cedex, France.. victor.peron@univ-pau.fr
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Abstract

We present equivalent conditions and asymptotic models for the diffraction problem of elastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. We derive and validate equivalent conditions up to the fourth order for the elastic displacement. These conditions approximate the acoustic waves which propagate in the fluid region. This approach leads to solve only elastic equations. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem.

Keywords

Asymptotic expansionsequivalent boundary conditionselasto-acoustic coupling
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1431 - 1449
Copyright
© EDP Sciences, SMAI 2014

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References

Abboud, T. and Ammari, H., Diffraction at a curved grating: TM and TE cases, homogenization. J. Math. Anal. Appl. 202 (1996) 995–1026. Google Scholar
Ammari, H., Beretta, E., Francini, E., Kang, H. and Lim, M., Reconstruction of small interface changes of an inclusion from modal measurements ii: The elastic case. J. Math. Pures Appl. 94 (2010) 322–339. Google Scholar
Ammari, H. and Nédélec, J.C., Time-harmonic electromagnetic fields in thin chiral curved layers. SIAM J. Math. Anal. 29 (1998) 395–423. Google Scholar
V. Andreev and A. Samarski, Méthode aux différences pour les équations elliptiques. Edition de Moscou, Moscou (1978).
Antoine, X., Barucq, H. and Vernhet, L., High-frequency asymptotic analysis of a dissipative transmission problem resulting in generalized impedance boundary conditions. Asymptot. Anal. 26 (2001) 257–283. Google Scholar
Bendali, A. and Lemrabet, K., The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation. SIAM J. Appl. Math. 56 (1996) 1664–1693. Google Scholar
Caloz, G., Costabel, M., Dauge, M. and Vial, G., Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptot. Anal. 50 (2006) 121–173. Google Scholar
Caloz, G., Dauge, M., Faou, E. and Péron, V., On the influence of the geometry on skin effect in electromagnetism. Comput. Methods Appl. Mech. Engrg. 200 (2011) 1053–1068. Google Scholar
M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains (2010) hal-00453934.
J. Diaz and V. Péron, Equivalent Conditions for Elasto-Acoustics, in Waves 2013: The 11th International Conference on Math. Numer. Aspects of Waves. Gammarth, Tunisie (2013) 345–346.
Durán, M. and Nédélec, J.-C., Un problème spectral issu d’un couplage élasto-acoustique. ESAIM: M2AN 34 (2000) 835–857. Google Scholar
B. Engquist and J.C. Nédélec, Effective boundary condition for acoustic and electromagnetic scattering in thin layers. Technical Report of CMAP 278 (1993).
Haddar, H., Joly, P. and Nguyen, H.-M., Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell’s equations. Math. Models Methods Appl. Sci. 18 (2008) 1787–1827. Google Scholar
Huttunen, T., Kaipio, J.P. and Monk, P., An ultra-weak method for acoustic fluid-solid interaction. J. Comput. Appl. Math. 213 (2008) 166–185. Google Scholar
Jones, D.S., Low-frequency scattering by a body in lubricated contact. Quart. J. Mech. Appl. Math. 36 (1983) 111–138. Google Scholar
Lafitte, O.D., Diffraction in the high frequency regime by a thin layer of dielectric material. I. The equivalent impedance boundary condition. SIAM J. Appl. Math. 59 (1999) 1028–1052. Google Scholar
Lafitte, O.D. and Lebeau, G., Équations de Maxwell et opérateur d’impédance sur le bord d’un obstacle convexe absorbant. C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 1177–1182. Google Scholar
Lemrabet, K., Le problème de Ventcel pour le système de l’élasticité dans un domaine de R3. C. R. Acad. Sci. Paris Sér. I Math. 304 (1987) 151–154. Google Scholar
M.A. Leontovich, Approximate boundary conditions for the electromagnetic field on the surface of a good conductor, in Investigations on radiowave propagation, vol. 2. Printing House of the USSR Academy of Sciences, Moscow (1948) 5–12.
Luke, C.J. and Martin, P.A., Fluid-solid interaction: acoustic scattering by a smooth elastic obstacle. SIAM J. Appl. Math. 55 (1995) 904–922. Google Scholar
Monk, P. and Selgas, V., An inverse fluid-solid interaction problem. Inverse Probl. Imaging 3 (2009) 173–198. Google Scholar
D. Natroshvili, A.-M. Sändig and W.L. Wendland, Fluid-structure interaction problems, in Mathematical aspects of boundary element methods (Palaiseau, 1998), vol. 414. Research Notes Math. Chapman & Hall/CRC, Boca Raton, FL (2000) 252–262.
V. Péron, Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer. Rapport de recherche RR-8163, INRIA (2013).
Rytov, S.M., Calcul du skin effect par la méthode des perturbations. J. Phys. 11 (1940) 233–242. Google Scholar
T.B.A. Senior and J.L. Volakis, and Institution of Electrical Engineers. Approximate Boundary Conditions in Electromagnetics. IEE Electromagnetic Waves Series. Inst of Engineering & Technology (1995).