This work presents an effective and accurate method for determining, from a theoretical and computational point of view, the time-harmonic Green's function of an isotropic elastic half-plane where an impedance boundary condition is considered. This method, based on the previous work done by Durán et al. (cf. [Numer. Math. 107 (2007) 295-314; IMA J. Appl. Math. 71 (2006) 853-876]) for the Helmholtz equation in a half-plane, combines appropriately analytical and numerical techniques, which has an important advantage because the obtention of explicit expressions for the surface waves. We show, in addition to the usual Rayleigh wave, another surface wave appearing in some special cases. Numerical results are given to illustrate that. This is an extended and detailed version of the previous article by Durán et al. [C. R. Acad. Sci. Paris, Ser. IIB 334 (2006) 725-731].
Mots-clés : Green's function, half-plane, time-harmonic elasticity, impedance boundary condition, surface waves
@article{M2AN_2010__44_4_671_0,
author = {Dur\'an, Mario and Godoy, Eduardo and N\'ed\'elec, Jean-Claude},
title = {Theoretical aspects and numerical computation of the time-harmonic {Green's} function for an isotropic elastic half-plane with an impedance boundary condition},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {671--692},
publisher = {EDP-Sciences},
volume = {44},
number = {4},
year = {2010},
doi = {10.1051/m2an/2010020},
mrnumber = {2683578},
zbl = {1194.31002},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2010020/}
}
TY - JOUR AU - Durán, Mario AU - Godoy, Eduardo AU - Nédélec, Jean-Claude TI - Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2010 SP - 671 EP - 692 VL - 44 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010020/ DO - 10.1051/m2an/2010020 LA - en ID - M2AN_2010__44_4_671_0 ER -
%0 Journal Article %A Durán, Mario %A Godoy, Eduardo %A Nédélec, Jean-Claude %T Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition %J ESAIM: Modélisation mathématique et analyse numérique %D 2010 %P 671-692 %V 44 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010020/ %R 10.1051/m2an/2010020 %G en %F M2AN_2010__44_4_671_0
Durán, Mario; Godoy, Eduardo; Nédélec, Jean-Claude. Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 4, pp. 671-692. doi : 10.1051/m2an/2010020. http://www.numdam.org/articles/10.1051/m2an/2010020/
[1] , Tables of Integral Transformations, Volume I. McGraw-Hill Book Company, Inc. (1954).
[2] , Special Functions for Scientists and Engineers. Dover Publications, Inc., New York, USA (1968). | Zbl
[3] , Boundary Integral Equation Methods for Solids and Fluids. John Wiley & Sons Ltd., Chichester, UK (1995). | Zbl
[4] and , Numerical evaluation of harmonic Green's functions for triclinic half-space with embedded sources - Part I: A 2D model. Int. J. Numer. Meth. Engrg. 69 (2007) 347-366. | Zbl
[5] and , Numerical evaluation of harmonic Green's functions for triclinic half-space with embedded sources - Part II: A 3D model. Int. J. Numer. Meth. Engrg. 69 (2007) 367-389. | Zbl
[6] and , Integral Equations Methods in Scattering Theory. John Wiley, New York, USA (1983). | Zbl
[7] , Équations Intégrales en Axisymétrie Généralisée, Application à la Sismique Entre Puits. Ph.D. Thesis, École Centrale de Paris, France (1993).
[8] , and , Computing Green's function of elasticity in a half-plane with impedance boundary condition. C. R. Acad. Sci. Paris, Ser. IIB 334 (2006) 725-731.
[9] , and , The Helmholtz equation in a locally perturbed half-plane with passive boundary. IMA J. Appl. Math. 71 (2006) 853-876. | Zbl
[10] , and , Computing numerically the Green's function of the half-plane Helmholtz operator with impedance boundary conditions. Numer. Math. 107 (2007) 295-314. | Zbl
[11] , Calculation of the elasto-dynamics Green's function in layered media by means of a modified propagator matrix method. Geophys. J. Roy. Astro. Soc. 75 (1983) 669-691. | Zbl
[12] , , and , Computational Ocean Acoustics. Springer-Verlag, New York, USA (1994). | Zbl
[13] , Green function's for Lamb's Problem. Geophys. J. Roy. Astro. Soc. 37 (1974) 99-131. | Zbl
[14] , Boundary Integral Equations in Elasticity Theory. Kluwer Academic Publishers, Dordrecht, Boston (2002). | Zbl
[15] , A theory of rupture pulse on softening interface with application to the Chi-Chi earthquake. J. Geophys. Res. 111 (2006) 1-14.
[16] , Acoustic and Electromagnetic Equations - Integral Representations for Harmonic Problems, Applied Mathematical Sciences 144. Springer, Germany (2001). | Zbl
[17] and , A Green's function time-domain boundary element method for the elastodynamic half-plane. Int. J. Numer. Meth. Engrg. 46 (1999) 627-648. | Zbl
[18] , Green's tensor function for Lamb's problem: The general anisotropic case. J. Acoust. Soc. Am. 102 (1997) 2438-2441.
[19] and , Practical Handbook for Underground Rock Mechanics, Series on Rock and Soil Mechanics 12. Trans Tech Publications, Germany (1986).
[20] and , Lamb's problem for solids of general anisotropy. Wave Mot. 24 (1996) 227-242. | Zbl
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