Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, p. 1431-1449
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

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.

DOI : https://doi.org/10.1051/m2an/2014002
Classification:  35C20,  35J25,  41A60,  74F10
Keywords: asymptotic expansions, equivalent boundary conditions, elasto-acoustic coupling
     author = {P\'eron, Victor},
     title = {Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {5},
     year = {2014},
     pages = {1431-1449},
     doi = {10.1051/m2an/2014002},
     zbl = {1302.35101},
     mrnumber = {3264360},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_5_1431_0}
Péron, Victor. Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 5, pp. 1431-1449. doi : 10.1051/m2an/2014002. http://www.numdam.org/item/M2AN_2014__48_5_1431_0/

[1] T. Abboud and H. Ammari, Diffraction at a curved grating: TM and TE cases, homogenization. J. Math. Anal. Appl. 202 (1996) 995-1026. | MR 1408364 | Zbl 0865.35122

[2] H. Ammari, E. Beretta, E. Francini, H. Kang and M. Lim, Reconstruction of small interface changes of an inclusion from modal measurements ii: The elastic case. J. Math. Pures Appl. 94 (2010) 322-339. | MR 2679030 | Zbl 1197.35308

[3] H. Ammari and J.C. Nédélec, Time-harmonic electromagnetic fields in thin chiral curved layers. SIAM J. Math. Anal. 29 (1998) 395-423. | MR 1616499 | Zbl 0913.35131

[4] V. Andreev and A. Samarski, Méthode aux différences pour les équations elliptiques. Edition de Moscou, Moscou (1978). | MR 502018 | Zbl 0377.35004

[5] X. Antoine, H. Barucq and L. Vernhet, High-frequency asymptotic analysis of a dissipative transmission problem resulting in generalized impedance boundary conditions. Asymptot. Anal. 26 (2001) 257-283. | MR 1844544 | Zbl 0986.76080

[6] A. Bendali and K. Lemrabet, 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. | MR 1417476 | Zbl 0869.35068

[7] G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptot. Anal. 50 (2006) 121-173. | MR 2286939 | Zbl 1136.35021

[8] G. Caloz, M. Dauge, E. Faou and V. Péron, On the influence of the geometry on skin effect in electromagnetism. Comput. Methods Appl. Mech. Engrg. 200 (2011) 1053-1068. | MR 2796144 | Zbl 1225.78003

[9] M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains (2010) hal-00453934.

[10] 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.

[11] M. Durán and J.-C. Nédélec, Un problème spectral issu d'un couplage élasto-acoustique. ESAIM: M2AN 34 (2000) 835-857. | Zbl 0992.74028

[12] B. Engquist and J.C. Nédélec, Effective boundary condition for acoustic and electromagnetic scattering in thin layers. Technical Report of CMAP 278 (1993).

[13] H. Haddar, P. Joly and H.-M. Nguyen, 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. | MR 2463780 | Zbl 1170.35094

[14] T. Huttunen, J.P. Kaipio and P. Monk, An ultra-weak method for acoustic fluid-solid interaction. J. Comput. Appl. Math. 213 (2008) 166-185. | MR 2382712 | Zbl 1182.76949

[15] D.S. Jones, Low-frequency scattering by a body in lubricated contact. Quart. J. Mech. Appl. Math. 36 (1983) 111-138. | MR 696559 | Zbl 0552.73023

[16] O.D. Lafitte, 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. | MR 1668205 | Zbl 0959.78009

[17] O.D. Lafitte and G. Lebeau, 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. | MR 1221645 | Zbl 0780.35108

[18] K. Lemrabet, 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. | MR 880120 | Zbl 0624.73066

[19] 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.

[20] C.J. Luke and P.A. Martin, Fluid-solid interaction: acoustic scattering by a smooth elastic obstacle. SIAM J. Appl. Math. 55 (1995) 904-922. | MR 1341532 | Zbl 0832.73045

[21] P. Monk and V. Selgas, An inverse fluid-solid interaction problem. Inverse Probl. Imaging 3 (2009) 173-198. | MR 2558285 | Zbl 1183.65138

[22] 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. | MR 1726554 | Zbl 0998.74028

[23] 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). | Zbl pre06357682

[24] S.M. Rytov, Calcul du skin effect par la méthode des perturbations. J. Phys. 11 (1940) 233-242. | JFM 66.1129.02

[25] 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). | Zbl 0828.73001