A simple numerical absorbing layer method in elastodynamics

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.

[1]  J. Bielak,et al.  Domain Reduction Method for Three-Dimensional Earthquake Modeling in Localized Regions, Part II: Verification and Applications , 2001 .

[2]  R. Clough,et al.  Dynamics Of Structures , 1975 .

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  J. Virieux.  P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method , 1986 .

[5]  Olivier Coussy,et al.  Acoustics of Porous Media , 1988 .

[6]  T. Caughey,et al.  Classical Normal Modes in Damped Linear Dynamic Systems , 1960 .

[7]  James S. Sochacki,et al.  Absorbing boundary conditions and surface waves , 1987 .

[8]  Moshe Reshef,et al.  A nonreflecting boundary condition for discrete acoustic and elastic wave equations , 1985 .

[9]  Patrick Dangla,et al.  A plane strain soil‐structure interaction model , 1988 .

[10]  Dimitri E. Beskos,et al.  Boundary Element Methods in Dynamic Analysis: Part II (1986-1996) , 1997 .

[11]  I. Babuska,et al.  Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation , 1995 .

[12]  J. Semblat,et al.  EFFICIENCY OF HIGHER ORDER FINITE ELEMENTS FOR THE ANALYSIS OF SEISMIC WAVE PROPAGATION , 2000, 0901.3715.

[13]  Anil K. Chopra,et al.  Dynamics of Structures: Theory and Applications to Earthquake Engineering , 1995 .

[14]  D. Komatitsch,et al.  An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation , 2007 .

[15]  D Komatitsch,et al.  CASTILLO-COVARRUBIAS JM, SANCHEZ-SESMA FJ. THE SPECTRAL ELEMENT METHOD FOR ELASTIC WAVE EQUATIONS-APPLICATION TO 2-D AND 3-D SEISMIC PROBLEMS , 1999 .

[16]  D. Givoli.  Non-reflecting boundary conditions , 1991 .

[17]  Jeroen Tromp,et al.  The spectral element method for elastic wave equations: Application seismic problems to 2D and 3D , 1998 .

[18]  Peter Moczo,et al.  On the rheological models used for time‐domain methods of seismic wave propagation , 2005 .

[19]  J. Vilotte,et al.  The Newmark scheme as velocity–stress time-staggering: an efficient PML implementation for spectral element simulations of elastodynamics , 2005 .

[20]  Guy Bonnet,et al.  Nonlinear Viscoelastic Wave Propagation: An Extension of Nearly Constant Attenuation Models , 2009 .

[21]  José M. Carcione,et al.  Time-domain Modeling of Constant-Q Seismic Waves Using Fractional Derivatives , 2002 .

[22]  J. Bielak,et al.  Domain Reduction Method for Three-Dimensional Earthquake Modeling in Localized Regions, Part I: Theory , 2003 .

[23]  K L Lee,et al.  Seismic Response by Variable Damping Finite Elements , 1974 .

[24]  R. Christensen,et al.  Theory of Viscoelasticity , 1971 .

[25]  Gaetano Festa,et al.  PML Absorbing Boundaries , 2003 .

[26]  P. Bettess,et al.  Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements , 1999 .

[27]  Dan Givoli,et al.  High-order nonreflecting boundary conditions without high-order derivatives , 2001 .

[28]  Alain Pecker,et al.  Waves and Vibrations in Soils: Earthquakes, Traffic, Shocks, Construction works , 2009 .

[29]  George P. Mavroeidis,et al.  A Mathematical Representation of Near-Fault Ground Motions , 2003 .

[30]  A. Chopra.  Dynamics of Structures: A Primer , 1981 .

[31]  J. Kristek,et al.  3D Heterogeneous Staggered-grid Finite-difference Modeling of Seismic Motion with Volume Harmonic and Arithmetic Averaging of Elastic Moduli and Densities , 2002 .

[32]  R. Christensen.  Theory of viscoelasticity : an introduction , 1971 .

[33]  Jean-Franccois Semblat.  Rheological Interpretation of Rayleigh Damping , 1997 .

[34]  Ezio Faccioli,et al.  2d and 3D elastic wave propagation by a pseudo-spectral domain decomposition method , 1997 .

[35]  M. Bonnet,et al.  A new fast multi‐domain BEM to model seismic wave propagation and amplification in 3‐D geological structures , 2009 .

[36]  Antonio Munjiza,et al.  An M(M−1K)m proportional damping in explicit integration of dynamic structural systems , 1998 .

[37]  J. Bernard Minster,et al.  Numerical simulation of attenuated wavefields using a Padé approximant method , 1984 .

[38]  Kristel C. Meza-Fajardo,et al.  A Nonconvolutional, Split-Field, Perfectly Matched Layer for Wave Propagation in Isotropic and Anisotropic Elastic Media: Stability Analysis , 2008 .

[39]  M. Bonnet.  Boundary Integral Equation Methods for Solids and Fluids , 1999 .

[40]  A. Chopra,et al.  Perfectly matched layers for time-harmonic elastodynamics of unbounded domains : Theory and finite-element implementation , 2003 .

[41]  M. Korn,et al.  Incorporation of attenuation into time-domain computations of seismic wave fields , 1987 .