# Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition

Mario Durán; Eduardo Godoy; Jean-Claude Nédélec

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 4, page 671-692
- ISSN: 0764-583X

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topDurán, Mario, Godoy, Eduardo, and 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: Mathematical Modelling and Numerical Analysis 44.4 (2010): 671-692. <http://eudml.org/doc/250769>.

@article{Durán2010,

abstract = {
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. IIB334 (2006) 725–731].
},

author = {Durán, Mario, Godoy, Eduardo, Nédélec, Jean-Claude},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Green's function; half-plane; time-harmonic elasticity; impedance boundary condition; surface waves},

language = {eng},

month = {6},

number = {4},

pages = {671-692},

publisher = {EDP Sciences},

title = {Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition},

url = {http://eudml.org/doc/250769},

volume = {44},

year = {2010},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2010/6//

PB - EDP Sciences

VL - 44

IS - 4

SP - 671

EP - 692

AB -
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. IIB334 (2006) 725–731].

LA - eng

KW - Green's function; half-plane; time-harmonic elasticity; impedance boundary condition; surface waves

UR - http://eudml.org/doc/250769

ER -

## References

top- H. Bateman, Tables of Integral Transformations, Volume I. McGraw-Hill Book Company, Inc. (1954).
- W.W. Bell, Special Functions for Scientists and Engineers. Dover Publications, Inc., New York, USA (1968). Zbl0167.34401
- M. Bonnet, Boundary Integral Equation Methods for Solids and Fluids. John Wiley & Sons Ltd., Chichester, UK (1995).
- Z. Chen and M. Dravinski, 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. Zbl1194.74110
- Z. Chen and M. Dravinski, 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. Zbl1194.74111
- P. Colton and R. Kress, Integral Equations Methods in Scattering Theory. John Wiley, New York, USA (1983). Zbl0522.35001
- J. Dompierre, Équations Intégrales en Axisymétrie Généralisée, Application à la Sismique Entre Puits. Ph.D. Thesis, École Centrale de Paris, France (1993).
- M. Durán, E. Godoy and J.-C. Nédélec, Computing Green's function of elasticity in a half-plane with impedance boundary condition. C. R. Acad. Sci. Paris, Ser. IIB334 (2006) 725–731.
- M. Durán, I. Muga and J.-C. Nédélec, The Helmholtz equation in a locally perturbed half-plane with passive boundary. IMA J. Appl. Math.71 (2006) 853–876. Zbl1152.35338
- M. Durán, R. Hein and J.-C. Nédélec, Computing numerically the Green's function of the half-plane Helmholtz operator with impedance boundary conditions. Numer. Math.107 (2007) 295–314. Zbl1124.65113
- G.R. Franssens, 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. Zbl0527.73018
- F.B. Jensen, W.A. Kuperman, M.B. Porter and H. Schmidt, Computational Ocean Acoustics. Springer-Verlag, New York, USA (1994).
- L.R. Johnson, Green function's for Lamb's Problem. Geophys. J. Roy. Astro. Soc.37 (1974) 99–131. Zbl0298.73029
- A.M. Linkov, Boundary Integral Equations in Elasticity Theory. Kluwer Academic Publishers, Dordrecht, Boston (2002). Zbl1046.74001
- A.M. Linkov, A theory of rupture pulse on softening interface with application to the Chi-Chi earthquake. J. Geophys. Res.111 (2006) 1–14.
- J.-C. Nédélec, Acoustic and Electromagnetic Equations – Integral Representations for Harmonic Problems, Applied Mathematical Sciences144. Springer, Germany (2001). Zbl0981.35002
- C. Richter and G. Schmid, A Green's function time-domain boundary element method for the elastodynamic half-plane. Int. J. Numer. Meth. Engrg.46 (1999) 627–648. Zbl0978.74080
- M. Spies, Green's tensor function for Lamb's problem: The general anisotropic case. J. Acoust. Soc. Am.102 (1997) 2438–2441.
- T.R. Stacey and C.H. Page, Practical Handbook for Underground Rock Mechanics, Series on Rock and Soil Mechanics12. Trans Tech Publications, Germany (1986).
- C.-Y. Wang and J.D. Achenbach, Lamb's problem for solids of general anisotropy. Wave Mot.24 (1996) 227–242. Zbl0954.74523

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