Time‐domain hybrid formulations for wave simulations in three‐dimensional PML‐truncated heterogeneous media

SUMMARY We are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly-matched-layer-(PML)-truncated media. We extend in three dimensions a recently developed twodimensional formulation, by treating the PML via an unsplit-field, but mixed-field, displacement-stress formulation, which is then coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient hybrid scheme. The hybrid treatment leads to, at most, third-order in time semi-discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi-axial PML. We discuss several time-marching schemes, which can be used a la carte, depending on the application: (a) an extended Newmark scheme for third-order in time, either unsymmetric or fully symmetric semi-discrete forms; (b) a standard implicit Newmark for the second-order, unsymmetric semi-discrete forms; and (c) an explicit Runge–Kutta scheme for a first-order in time unsymmetric system. The latter is well-suited for large-scale problems on parallel architectures, while the second-order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains. We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd.

[1]  O. Ghattas,et al.  A Newton-CG method for large-scale three-dimensional elastic full-waveform seismic inversion , 2008 .

[2]  Matthew G. Knepley,et al.  PETSc Users Manual (Rev. 3.4) , 2014 .

[3]  Weng Cho Chew,et al.  A 3D perfectly matched medium from modified maxwell's equations with stretched coordinates , 1994 .

[4]  H. Kreiss,et al.  Time-Dependent Problems and Difference Methods , 1996 .

[5]  Gary Cohen,et al.  Mixed Spectral Finite Elements for the Linear Elasticity System in Unbounded Domains , 2005, SIAM J. Sci. Comput..

[6]  Maxim Dmitriev,et al.  Application of M-PML absorbing boundary conditions to the numerical simulation of wave propagation in anisotropic media. Part II: Stability , 2012 .

[7]  Alfio Quarteroni,et al.  Numerical Mathematics (Texts in Applied Mathematics) , 2006 .

[8]  David Gottlieb,et al.  On the construction and analysis of absorbing layers in CEM , 1998 .

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

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

[11]  Loukas F. Kallivokas,et al.  Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media , 2011 .

[12]  Patrick Joly,et al.  Stability of perfectly matched layers, group velocities and anisotropic waves , 2003 .

[13]  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 .

[14]  Peter G. Petropoulos,et al.  On the Termination of the Perfectly Matched Layer with Local Absorbing Boundary Conditions , 1998 .

[15]  Jan S. Hesthaven,et al.  Long Time Behavior of the Perfectly Matched Layer Equations in Computational Electromagnetics , 2002, J. Sci. Comput..

[16]  Yu Zhang,et al.  A multiaxial perfectly matched layer (M-PML) for the long-time simulation of elastic wave propagation in the second-order equations , 2014 .

[17]  S. Kucukcoban.  The inverse medium problem in PML-truncated elastic media , 2010 .

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

[19]  Kenneth Duru,et al.  A Well-Posed and Discretely Stable Perfectly Matched Layer for Elastic Wave Equations in Second Order Formulation , 2012 .

[20]  David A. Kopriva,et al.  Implementing Spectral Methods for Partial Differential Equations , 2009 .

[21]  C. Tsogka,et al.  Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media , 2001 .

[22]  S. Gedney.  An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices , 1996 .

[23]  Jeroen Tromp,et al.  A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation , 2003 .

[24]  F. Hu.  On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer , 1995 .

[25]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[26]  W. Chew,et al.  Complex space approach to perfectly matched layers: a review and some new developments , 2000 .

[27]  Patrick Joly,et al.  FICTITIOUS DOMAINS, MIXED FINITE ELEMENTS AND PERFECTLY MATCHED LAYERS FOR 2-D ELASTIC WAVE PROPAGATION , 2001 .

[28]  F. Brezzi.  A Survey of Mixed Finite Element Methods , 1988 .

[29]  Davi Correia,et al.  Performance of regular PML, CFS‐PML, and second‐order PML for waveguide problems , 2006 .

[30]  Maxim Dmitriev,et al.  Application of M-PML reflectionless boundary conditions to the numerical simulation of wave propagation in anisotropic media. Part I: Reflectivity , 2011 .

[31]  Tsili Wang,et al.  Finite-difference modeling of elastic wave propagation: A nonsplitting perfectly matched layer approach , 2003 .

[32]  René Matzen.  An efficient finite element time‐domain formulation for the elastic second‐order wave equation: A non‐split complex frequency shifted convolutional PML , 2011 .

[33]  Roland Martin,et al.  A Variational Formulation of a Stabilized Unsplit Convolutional Perfectly Matched Layer for The Isotropic or Anisotropic Seismic Wave Equation , 2008 .

[34]  Loukas F. Kallivokas,et al.  A symmetric hybrid formulation for transient wave simulations in PML-truncated heterogeneous media , 2013 .

[35]  David Gottlieb,et al.  A Mathematical Analysis of the PML Method , 1997 .

[36]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[37]  A. Chopra,et al.  Perfectly matched layers for transient elastodynamics of unbounded domains , 2004 .

[38]  Qing Huo Liu,et al.  The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media , 2001 .

[39]  Kristel C. Meza-Fajardo,et al.  Study of the Accuracy of the Multiaxial Perfectly Matched Layer for the Elastic‐Wave Equation , 2012 .

[40]  Jacobo Bielak,et al.  A simple impedance-infinite element for the finite element solution of the three-dimensional wave equation in unbounded domains , 1997 .

[41]  Dan Givoli,et al.  Comparison of high‐order absorbing boundary conditions and perfectly matched layers in the frequency domain , 2010 .

[42]  John B. Schneider,et al.  Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation , 1996 .

[43]  E. Michielssen,et al.  Complex coordinate system as a generalized absorbing boundary condition , 1997, IEEE Antennas and Propagation Society International Symposium 1997. Digest.

[44]  P. Monk,et al.  Optimizing the Perfectly Matched Layer , 1998 .

[45]  Qing Huo Liu,et al.  Perfectly matched layers for elastic waves in cylindrical and spherical coordinates , 1999 .

[46]  Qing Huo Liu,et al.  PERFECTLY MATCHED LAYERS FOR ELASTODYNAMICS: A NEW ABSORBING BOUNDARY CONDITION , 1996 .

[47]  Sabine Fenstermacher,et al.  Numerical Approximation Of Partial Differential Equations , 2016 .

[48]  A. Giannopoulos,et al.  A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves , 2007 .

[49]  Ushnish Basu,et al.  Explicit finite element perfectly matched layer for transient three‐dimensional elastic waves , 2009 .

[50]  Gideon Juve,et al.  The ShakeOut earthquake scenario: Verification of three simulation sets , 2010 .

[51]  E. Turkel,et al.  ANALYTICAL AND NUMERICAL STUDIES OF A FINITE ELEMENT PML FOR THE HELMHOLTZ EQUATION , 2000 .

[52]  Loukas F. Kallivokas,et al.  Time-domain forward and inverse modeling of lossy soils with frequency-independent Q for near-surface applications , 2012 .

[53]  Omar Ghattas,et al.  Site characterization using full waveform inversion , 2013 .

[54]  Georg Stadler,et al.  A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media , 2010, J. Comput. Phys..

[55]  David R. O'Hallaron,et al.  Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers , 1998 .