A simple numerical absorbing layer method in elastodynamics

Jean-François Semblat; Ali Gandomzadeh; Luca Lenti

doi:10.1016/j.crme.2009.12.004

A simple numerical absorbing layer method in elastodynamics
[Une méthode simple de couche absorbante pour les problèmes élastodynamiques]

Jean-François Semblat ¹ ; Ali Gandomzadeh ¹ ; Luca Lenti ¹

¹ Université Paris-Est, LCPC, 58 boulevard Lefebvre, 75015 Paris, France

Comptes Rendus. Mécanique, Volume 338 (2010) no. 1, pp. 24-32.

Résumé
Abstract

La simulation numérique de la propagation d'ondes élastiques en milieux infinis peut s'avérer délicate du fait des réflexions d'ondes parasites sur les frontières du modèle discrétisé. Plusieurs méthodes sophistiquées, telles que les frontières absorbantes, les éléments infinis ou les couches absorbantes (e.g. « Perfectly Matched Layers ») permettent une réduction importante des réflexions parasites. Dans cette Note, une méthode de couche absorbante simple et efficace est proposée dans le cadre de la méthode des éléments finis. Cette méthode s'appuie sur une formulation de l'amortissement dans la couche de type Rayleigh/Caughey et ses principes sont d'abord détaillés. L'efficacité de la méthode est alors démontrée grâce à des simulations unidimensionnelles en considérant un amortissement homogène ou variable dans la couche absorbante. Des modèles bidimensionnels permettent ensuite d'apprécier l'efficacité de la méthode de couche absorbante proposée pour différents types d'ondes (ondes de surface, ondes de volume) et des incidences variées (normale à rasante). La méthode s'avère ainsi efficace pour différents types d'ondes élastiques et pourrait être utilisée pour traiter divers problèmes élastodynamiques en milieux non bornés.

The numerical analysis of elastic wave propagation in unbounded media may be difficult to handle due to spurious waves reflected at the model artificial boundaries. Several sophisticated techniques, such as nonreflecting boundary conditions, infinite elements or absorbing layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this Note, a simple and efficient absorbing layer method is proposed in the framework of the Finite Element Method. This method considers Rayleigh/Caughey damping in the absorbing layer and its principle is presented first. The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types (surface waves, body waves) and incidences (normal to grazing). The method is shown to be efficient for different types of elastic waves and may thus be used for various elastodynamic problems in unbounded domains.

Reçu le : 2009-06-26
Accepté le : 2009-11-29
Publié le : 2009-12-22

DOI : 10.1016/j.crme.2009.12.004

Keywords: Waves, Continuum mechanics, Finite Element Method, Elastodynamics
Mot clés : Ondes, Milieux continus, Méthode des éléments finis, Élastodynamiques

Affiliations des auteurs :

Jean-François Semblat ¹ ; Ali Gandomzadeh ¹ ; Luca Lenti ¹

¹ Université Paris-Est, LCPC, 58 boulevard Lefebvre, 75015 Paris, France

@article{CRMECA_2010__338_1_24_0,
     author = {Jean-Fran\c{c}ois Semblat and Ali Gandomzadeh and Luca Lenti},
     title = {A simple numerical absorbing layer method in elastodynamics},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {24--32},
     publisher = {Elsevier},
     volume = {338},
     number = {1},
     year = {2010},
     doi = {10.1016/j.crme.2009.12.004},
     language = {en},
}

TY  - JOUR
AU  - Jean-François Semblat
AU  - Ali Gandomzadeh
AU  - Luca Lenti
TI  - A simple numerical absorbing layer method in elastodynamics
JO  - Comptes Rendus. Mécanique
PY  - 2010
SP  - 24
EP  - 32
VL  - 338
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crme.2009.12.004
LA  - en
ID  - CRMECA_2010__338_1_24_0
ER  -

%0 Journal Article
%A Jean-François Semblat
%A Ali Gandomzadeh
%A Luca Lenti
%T A simple numerical absorbing layer method in elastodynamics
%J Comptes Rendus. Mécanique
%D 2010
%P 24-32
%V 338
%N 1
%I Elsevier
%R 10.1016/j.crme.2009.12.004
%G en
%F CRMECA_2010__338_1_24_0

Jean-François Semblat; Ali Gandomzadeh; Luca Lenti. A simple numerical absorbing layer method in elastodynamics. Comptes Rendus. Mécanique, Volume 338 (2010) no. 1, pp. 24-32. doi : 10.1016/j.crme.2009.12.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.12.004/

Bibliographie
Cité par

[1] J. Virieux P-SV wave propagation in heterogeneous media: velocity–stress finite-difference method, Geophysics, Volume 51 (1986), pp. 889-901

[2] P. Moczo; J. Kristek; V. Vavrycuk; R.J. Archuleta; L. Halada 3D heterogeneous staggered-grid finite-difference modeling of seismic motion with volume harmonic and arithmetic averaging of elastic moduli and densities, Bulletin of the Seismological Society of America, Volume 92 (2002) no. 8, pp. 3042-3066

[3] T.J.R. Hughes Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987

[4] J.F. Semblat; A. Pecker Waves and Vibrations in Soils, IUSS Press, Pavia, Italy, 2009

[5] D.E. Beskos Boundary elements methods in dynamic analysis: Part II (1986–1996), Applied Mechanics Reviews (ASME), Volume 50 (1997) no. 3, pp. 149-197

[6] M. Bonnet Boundary Integral Equation Methods for Solids and Fluids, Wiley, Chichester, United Kingdom, 1999

[7] S. Chaillat; M. Bonnet; J.F. Semblat A new fast multi-domain BEM to model seismic wave propagation and amplification in 3D geological structures, Geophys. J. Int., Volume 177 (2009) no. 2, pp. 509-531

[8] E. Faccioli; F. Maggio; R. Paolucci; A. Quarteroni 2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method, Journal of Seismology, Volume 1 (1997), pp. 237-251

[9] D. Komatitsch; J.P. Vilotte; R. Vai; J.M. Castillo-Covarrubias; F.J. Sanchez-Sesma The spectral element method for elastic wave equations – Application to 2D and 3D seismic problems, Int. J. Numer. Meth. Eng., Volume 45 (1999), pp. 1139-1164

[10] N. Delépine; G. Bonnet; L. Lenti; J.F. Semblat Nonlinear viscoelastic wave propagation: an extension of Nearly Constant Attenuation models, Journal of Engineering Mechanics (ASCE), Volume 135 (2009) no. 11, pp. 1305-1314

[11] F. Ihlenburg; I. Babuška Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, Int. J. Numer. Meth. Eng., Volume 38 (1995), pp. 3745-3774

[12] J.F. Semblat; J.J. Brioist Efficiency of higher order finite elements for the analysis of seismic wave propagation, Journal of Sound and Vibration, Volume 231 (2000) no. 2, pp. 460-467

[13] H. Modaressi, I. Benzenati, An absorbing boundary element for dynamic analysis of two-phase media, in: 10th World Conf. on Earthquake Engineering, Madrid, 1992, pp. 1157–1163

[14] P. Dangla A plane strain soil–structure interaction model, Earthquake Engineering and Structural Dynamics, Volume 16 (1988), pp. 1115-1128

[15] J. Bielak; K. Loukakis; Y. Hisada; C. Yoshimura Domain reduction method for three-dimensional earthquake modeling in localized regions, Part I: Theory, Bulletin of the Seismological Society of America, Volume 93 (2003) no. 2, pp. 817-824

[16] C. Cerjan; D. Kosloff; R. Kosloff; M. Reshef A non-reflecting boundary condition for discrete acoustic and elastic waves, Geophysics, Volume 50 (1985), pp. 705-708

[17] J. Sochacki; R. Kubichek; G. George; W.R. Fletcher; S. Smithson Absorbing boundary conditions and surface waves, Geophysics, Volume 52 (1987), pp. 65-71

[18] D. Givoli Non-reflecting boundary conditions, J. Comput. Phys., Volume 94 (1991) no. 1, pp. 1-29

[19] D. Givoli High-order nonreflecting boundary conditions without high-order derivatives, J. Comput. Phys., Volume 170 (2001), pp. 849-870

[20] E. Chadwick; P. Bettess; O. Laghrouche Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements, Int. J. Numer. Meth. Eng., Volume 45 (1999), pp. 335-354

[21] U. Basu; A.K. Chopra Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation, Comput. Meth. Appl. Mech. Eng., Volume 192 (2003), pp. 1337-1375

[22] G. Festa; S. Nielsen PML absorbing boundaries, Bulletin of the Seismological Society of America, Volume 93 (2003) no. 2, pp. 891-903

[23] G. Festa; J.P. Vilotte The Newmark scheme as velocity–stress time-staggering: an efficient PML implementation for spectral element simulations of elastodynamics, Geophys. J. Int., Volume 161 (2005), pp. 789-812

[24] D. Komatitsch; R. Martin An unsplit convolutional Perfectly Matched Layer improved at grazing incidence for the seismic wave equation, Geophysics, Volume 72 (2007) no. 5, pp. 155-167

[25] K.C. Meza-Fajardo; A.S. Papageorgiou A nonconvolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: stability analysis, Bulletin of the Seismological Society of America, Volume 98 (2008) no. 4, pp. 1811-1836

[26] J.M. Carcione; F. Cavallini; F. Mainardi; A. Hanyga Time-domain modeling of constant-Q seismic waves using fractional derivatives, Pure Appl. Geophys., Volume 159 (2002) no. 7–8, pp. 1719-1736

[27] S.M. Day; J.B. Minster Numerical simulation of wavefields using a Padé approximant method, Geophys. J. Roy. Astron. Soc., Volume 78 (1984), pp. 105-118

[28] H. Emmerich; M. Korn Incorporation of attenuation into time-domain computations of seismic wave fields, Geophysics, Volume 52 (1987) no. 9, pp. 1252-1264

[29] P. Moczo; J. Kristek On the rheological models used for time-domain methods of seismic wave propagation, Geophys. Res. Lett., Volume 32 (2005), p. L01306

[30] T. Caughey Classical normal modes in damped linear systems, Journal of Applied Mechanics, Volume 27 (1960), pp. 269-271

[31] A. Munjiza; D.R.J. Owen; A.J.L. Crook An $M {(M^{- 1} K)}^{m}$ proportional damping in explicit integration of dynamic structural systems, Int. J. Numer. Meth. Eng., Volume 41 (1998), pp. 1277-1296

[32] A.K. Chopra Dynamics of Structures, Theory and Applications to Earthquake Engineering, Pearson Prentice Hall, 2007 (876 pp)

[33] J.F. Semblat Rheological interpretation of Rayleigh damping, Journal of Sound and Vibration, Volume 206 (1997) no. 5, pp. 741-744

[34] R. Christensen Theory of Viscoelasticity. An Introduction, Academic Press, New York, USA, 1982

[35] I.M. Idriss; H.B. Seed; N. Serff Seismic response by variable damping finite elements, Journal of the Geotechnical Engineering Division (ASCE), Volume 100 (1974) no. 1, pp. 1-13

[36] J.F. Semblat, J.J. Brioist, Wave propagation in non-homogeneously damped medium: numerical vs experimental approach, in: 11th European Conf. on Earthquake Eng., Paris, France, 1998

[37] G.P. Mavroeidis; A.S. Papageorgiou A mathematical representation of near-fault ground motions, Bulletin of the Seismological Society of America, Volume 93 (2003) no. 3, pp. 1099-1131

[38] T. Bourbié, O. Coussy, B. Zinzsner, Acoustics of porous media, Technip, Paris, France, 1987

Cité par Sources :

Commentaires - Politique