A pseudospectral model of linear elastic wave propagation is described based on the first order stress-velocity equations of elastodynamics. k-space adjustments to the spectral gradient calculations are derived from the dyadic Green’s function solution to the second-order elastic wave equation and used to (a) ensure the solution is exact for homogeneous wave propagation for timesteps of arbitrarily large size, and (b) also allows larger time steps without loss of accuracy in heterogeneous media. The formulation in k-space allows the wavefield to be split easily into compressional and shear parts. A perfectly matched layer (PML) absorbing boundary condition was developed to effectively impose a radiation condition on the wavefield. The staggered grid, which is essential for accurate simulations, is described, along with other practical details of the implementation. The model is verified through comparison with exact solutions for canonical examples and further examples are given to show the efficiency of the method for practical problems. The efficiency of the model is by virtue of the reduced point-per-wavelength requirement, the use of the fast Fourier transform (FFT) to calculate the gradients in k space, and larger time steps made possible by the k-space adjustments.

1.
H.
Sato
and
M. C.
Fehler
,
Seismic Wave Propagation and Scattering in the Heterogeneous Earth
, 1st ed. (
Springer
,
Berlin
,
2009
),
308
pp.
2.
J.
Carcione
,
Waves and Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, and Porous Media
(Handbook of Geophysical Exploration, Seismic Exploration Series) (
Pergamon
,
Amsterdam
,
2001
), Vol. 31,
538
pp.
3.
A.
Leger
and
M.
Deschamps
(Eds.),
Ultrasonic Wave Propagation in Non Homogeneous Media
(
Springer
Proceedings in Physics) (Springer,
Berlin
,
2009
), Vol.
128
,
435
pp.
4.
B.
Luthi
,
Physical Acoustics in the Solid State
(
Springer
,
Berlin
,
2005
),
420
pp.
5.
J. D.
Cheeke
,
Fundamentals and Applications of Ultrasonic Waves
(
CRC Press
,
London
,
2002
),
480
pp.
6.
A.
Arnau Vives
,
Piezoelectric Transducers and Applications
, 2nd ed. (
Springer
,
Berlin
,
2008
),
532
pp.
7.
E.
Bossy
,
M.
Talmant
, and
P.
Laugier
, “
Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models
,”
J. Acoust. Soc. Am.
115
(
5
),
2314
24
(
2004
).
8.
W. D.
Smith
, “
The application of finite element analysis to body wave propagation problems
,”
Geophys. J. R. Astron. Soc.
42
(
2
),
747
768
(
1975
).
9.
F. J.
Rizzo
,
D. J.
Shippy
, and
M.
Rezayat
, “
A boundary integral equation method for radiation and scattering of elastic waves in three dimensions
,”
Int. J. Numer. Meth. Eng.
21
(
1
),
115
129
(
1985
).
10.
E.
Dormy
and
A.
Tarantola
, “
Numerical simulation of elastic wave propagation using a finite volume method
,”
J. Geophys. Res.
100
(
B2
),
2123
2133
(
1995
).
11.
R. J.
Leveque
,
Finite Volume Methods for Hyperbolic Problems
(Cambridge Texts in Applied Mathematics Series, No. 31) (
Cambridge University Press
,
Cambridge, UK
,
2002
),
578
pp.
12.
W. C.
Chew
,
M. S.
Tong
, and
B.
Hu
,
“Integral equation methods for electromagnetic and elastic waves,”
Synthesis Lectures on Computational Electromagnetic
, No. 12 (
Morgan and Claypool Publishers
,
San Rafael, CA
,
2009
),
241
pp.
13.
J.
Virieux
, “
P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method
,”
Geophysics
51
(
4
),
889
901
(
1986
).
14.
A.
Levander
, “
Fourth-order finite-difference P-SV seismograms
,”
Geophysics
53
(
11
),
1425
1436
(
1988
).
15.
P.
Moczo
,
J. O. A.
Robertsson
, and
L.
Eisner
, “
The finite-difference time-domain method for modeling of seismic wave propagation
,”
Adv. Geophys.
48
,
421
516
(
2007
).
16.
D. D.
Kosloff
and
E.
Baysal
, “
Forward modeling by a Fourier method
,”
Geophysics
47
,
1402
1412
(
1982
).
17.
B.
Fornberg
, “
The pseudospectral method: Comparisons with finite differences for the elastic wave equation
,”
Geophysics
52
(
4
),
483
501
(
1987
).
18.
B.
Fornberg
, “
The pseudospectral method: Accurate representation of interfaces in elastic wave calculations
,”
Geophysics
53
(
5
),
625
637
(
1988
).
19.
D.
Gottlieb
and
J. S.
Hesthaven
, “
Spectral methods for hyperbolic problems
,”
J. Comp. Appl. Math.
128
(
1–2
),
83
131
(
2001
).
20.
J. S.
Hesthaven
,
S.
Gottlieb
, and
D.
Gottlieb
,
Spectral Methods for Time-Dependent Problems
(Cambridge Monographs on Applied and Computational Mathematics Series, No. 21) (
Cambridge University Press
,
New York
,
2007
),
284
pp.
21.
C. G.
Canuto
,
M. Y.
Hussaini
,
A.
Quarteroni
, and
T. A.
Zang
,
Spectral Methods: Fundamentals in Single Domains
(
Springer
,
Berlin
,
2006
),
581
pp.
22.
B.
Fornberg
,
A Practical Guide to Pseudospectral Methods
(Cambridge Monographs on Applied and Computational Mathematics Series, No. 1) (
Cambridge University Press
,
New York
,
1998
),
244
pp.
23.
B.
Fornberg
and
G. B.
Whitham
,
“A numerical and theoretical study of certain nonlinear wave phenomena”
Philos. Trans. R. Soc. London A
289
(
1361
),
373
404
(
1978
).
24.
N. N.
Bojarski
, “
The k-space formulation of the scattering problem in the time domain
,”
J. Acoust. Soc. Am.
72
(
2
),
570
584
(
1982
).
25.
N. N.
Bojarski
, “
The k-space formulation of the scattering problem in the time domain: An improved single propagator formulation
,”
J. Acoust. Soc. Am.
77
(
3
),
826
831
(
1985
).
26.
B.
Compani-Tabrizi
, “
k-space scattering formulation of the absorptive full fluid elastic scalar wave equation in the time domain
,”
J. Acoust. Soc. Am.
79
(
4
),
901
905
(
1986
).
27.
S.
Finette
, “
Computational methods for simulating ultrasonic scattering in soft tissue
,”
IEEE Trans. Ultrason., Ferroelectr., Freq. Control
34
(
3
),
283
292
(
1987
).
28.
S.
Finette
, “
A computer model of acoustic wave scattering in soft tissue
,”
IEEE Trans. Biomed. Eng.
34
(
5
),
336
344
(
1987
).
29.
S.
Pourjavid
and
O. J.
Tretiak
, “
Numerical solution of the direct scattering problem through the transformed acoustical wave equation
,”
J. Acoust. Soc. Am.
91
(
2
),
639
645
(
1992
).
30.
Q. H.
Liu
, “
Generalisation of the k-space formulation to elastodynamic scattering problems
,”
J. Acoust. Soc. Am.
97
(
3
),
1373
1379
(
1995
).
31.
M.
Tabei
,
T. D.
Mast
, and
R. C.
Waag
, “
A k-space method for coupled first-order acoustic propagation equations
,”
J. Acoust. Soc. Am.
111
(
1
),
53
63
(
2002
).
32.
B. T.
Cox
,
S.
Kara
,
S. R.
Arridge
, and
P. C.
Beard
, “
k-space propagation models for acoustically heterogeneous media: Application to biomedical photoacoustics
,”
J. Acoust. Soc. Am.
121
(
6
),
3453
3464
(
2007
).
33.
B. E.
Treeby
and
B. T.
Cox
, “
k-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields
,”
J. Biomed. Optics
15
(
2
),
021314
(
2010
).
34.
B.
Fornberg
, “
High-order finite differences and the pseudospectral method on staggered grids
,”
SIAM J. Numer. Anal.
27
(
4
),
904
918
(
1990
).
35.
F. D.
Hastings
,
J. B.
Schneider
, and
S. L.
Broschat
, “
Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation
,”
J. Acoust. Soc. Am.
100
(
5
),
3061
3069
(
1996
).
36.
W. C.
Chew
and
Q. H.
Liu
, “
Perfectly matched layers for elastodynamics: A new absorbing boundary condition
,”
J. Comp. Acoust.
4
,
341
359
(
1996
).
37.
Z.
Alterman
and
F. C.
Karal
, “
Propagation of elastic waves in layered media by finite-difference methods
,”
Bull. Seism. Soc. Am.
58
(
1
),
367
398
(
1968
).
38.
Y. Q.
Zeng
and
Q.
Liu
, “
A multidomain PSTD method for 3D elastic wave equations
,”
Bull. Seismol. Soc. Am.
94
(
3
),
1002
1015
(
2004
).
39.
E.
Kausel
,
Fundamental Solutions in Elastodynamics: A Compendium
(
Cambridge University Press
,
New York
,
2006
),
260
pp.
40.
A. T.
de Hoop
, “
A modification of Cagniard’s method for solving seismic pulse problems
,”
Appl. Sci. Res. B
8
,
349
356
(
1960
).
41.
http://www.spice-rtn.org/library/software/EX2DELEL.html (Last viewed April 10,
2012
).
42.
https://www.geoazur.net/PERSO/operto/HTML/fwm2dpsv.html (Last viewed April 10,
2012
).
43.
C. C.
Mow
and
Y. H.
Pao
, “
The diffraction of elastic waves and dynamic stress concentrations
” (RAND Corporation, Santa Monica, CA,
1971
),
696
pp.
44.
M.
Ghrist
,
B.
Fornberg
, and
T. A.
Driscoll
, “
Staggered time integrators for wave equations
,”
SIAM J. Numer. Anal.
38
(
3
),
718
741
(
2000
).
45.
O.
Bou Matar
,
V.
Preobrazhensky
, and
P.
Pernod
, “
Two-dimensional axisymmetric numerical simulation of supercritical phase conjugation of ultrasound in active solid media
,”
J. Acoust. Soc. Am.
118
(
5
),
2880
2890
(
2005
).
46.
G.
Wojcik
,
B.
Fornberg
,
R.
Waag
,
L.
Carcione
,
J.
Mould
,
L.
Nikodym
, and
T.
Driscoll
, “
Pseudospectral methods for large-scale bioacoustic models
,”
Proc. IEEE Ultrason. Symp.
2
,
1501
1506
(
1997
).
47.
J. C.
Mould
,
G. L.
Wojcik
,
L. M.
Carcione
,
M.
Tabei
,
T. D.
Mast
, and
R. C.
Waag
, “
Validation of FFT-based algorithms for large-scale modeling of wave propagation in tissue
,”
Proc. IEEE Ultrason. Symp.
2
,
1551
1556
(
1999
).
You do not currently have access to this content.