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Coupling of the Discontinuous Galerkin and Finite Difference techniques to simulate seismic waves in the presence of sharp
interfaces
J. Diaz, D. Kolukhin, J. Diaz, D. Kolukhin, V. LisitsaV. Lisitsa, V. , V. TcheverdaTcheverda
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Motivation for Motivation for mathematiciansmathematicians
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σ = 1.38, I = 44.9 м
Free-surface perturbation
Motivation for Motivation for mathematiciansmathematicians
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σ = 1.38, I = 44.9 м
N
jjXN
XRMS1
21)(
Free-surface perturbation
30%
Motivation for Motivation for geophysicistsgeophysicists
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x (m)
z (m
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vs (m/s)
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x (m)
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Motivation for geophysicistsMotivation for geophysicists
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x (m)
z (m
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ux, t = 0.55s
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Original source
Motivation for Motivation for geophysicistsgeophysicists
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x (m)
z (m
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ux, t = 1.75s
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x 104
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Diffraction of Rayleigh wave, secondary sources
MotivationMotivation
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x (m)
time
(s)
uz
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x 104
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Standard staggered grid Standard staggered grid schemescheme
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1,
tC
tuu
tt
u T
zz
yy
xx
66
55
44
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c
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Standard staggered grid Standard staggered grid schemescheme
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1,
tC
tuu
tt
u T
• Easy to implement• Able to handle complex models• High computational efficiency• Suitable accuracy• Poor approximation of sharp
interfaces
Discontinuous Galerkin Discontinuous Galerkin methodmethod
Elastic wave equation in Cartesian coordinates:
dim
1
*
0
0
0
0
j
u
jj
j
f
fu
xB
Bu
tS
I
FVGt
VA
)(
Discontinuous Galerkin Discontinuous Galerkin methodmethod
FVGt
VA
)(
kkkk DDDD
dxWFdsWnVGdxVWGdxWt
VA
])([)]([
Discontinuous Galerkin Discontinuous Galerkin methodmethod
kkkk DDDD
dxWFdsWnVGdxVWGdxWt
VA
])([)]([
dim
1
)(j
inoutjj VVnBnVG
Discontinuous Galerkin Discontinuous Galerkin methodmethod
kkkk DDDD
dxWFdsWnVGdxVWGdxWt
VA
])([)]([
• Use of polyhedral meshes• Accurate description of sharp
interfaces• Hard to implement for complex
models• Computationally intense• Strong stability restrictions (low
Courant numbers)
Dispersion analysis Dispersion analysis (P1)(P1)
Courant ratio 0.25
Dispersion analysis Dispersion analysis (P2)(P2)
Courant ratio 0.144
Dispersion analysis Dispersion analysis (P3)(P3)
Courant ratio 0.09
DG + FDDG + FD
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Finite differences:•Easy to implement•Able to handle complex models•High computational efficiency•Suitable accuracy•Poor approximation of sharp interfaces
Discontinuous Galerkin method:•Use of polyhedral meshes•Accurate description of sharp interfaces•Hard to implement for complex models•Computationally intense•Strong stability restrictions (low Courant numbers)
A sketchA sketch
P1-P3 DG on irregular triangular grid to match free-surface topography
P0 DG on regular rectangular grid = conventional (non-staggered grid scheme) – transition zone
Standard staggered grid scheme
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ExperimentsExperiments
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PPW Reflection
15 ~3 %
30 ~0.5 %
60 ~0.1 %
120 ??? %
DG
FD+DG on rectangular FD+DG on rectangular gridgrid
P0 DG on regular rectangular grid
Standard staggered grid scheme
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Spurious ModesSpurious Modes2D example in Cartesian coordinates
Spurious ModesSpurious Modes2D example in Cartesian coordinates
InterfaceInterfaceIncident waves Reflected waves
Transmitted artificial waves
Transmitted true waves
Conjugation Conjugation conditionsconditions
Incident waves Reflected waves
Transmitted artificial waves
Transmitted true waves
ExperimentsExperiments
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PPW Reflection
15 1.6 %
30 0.5 %
60 0.1 %
120 0.03 %
Numerical Numerical experimentsexperiments
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P
S
Surface Xs=4000, Zs=110 (10
meters below free surface), volumetric source, freq=30HzZr=5 meters below free surfaceVertical component is presented
Source
Comparison with FDComparison with FD
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DG P1 h=2.5 m.FD h=2.5 m.
The same amplitude normalizationNumerical diffraction
Comparison with FDComparison with FD
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DG P1 h=2.5 m.FD h=1m.
The same amplitude normalization Numerical diffraction
Numerical Numerical experimentsexperiments
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Xs=4500, Zs= 5 meters below free surface, volumetric source, freq=20HzZr=5 meters below free surface
Numerical Numerical ExperimentsExperiments
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Numerical Experiment – Sea Numerical Experiment – Sea
BedBed
Source position x=12,500 m, z=5 m Ricker pulse with central frequency of 10 Hz Receivers were placed at the seabed.
Numerical ExperimentsNumerical Experiments
ConclusionsConclusions
• Discontinuous Galerkin method allows properly handling wave interaction with sharp interfaces, but it is computationally intense
• Finite differences are computationally efficient but cause high diffractions because of stair-step approximation of the interfaces.
• The algorithm based on the use of the DG in the upper part of the model and FD in the deeper part allows properly treating the free surface topography but preserves the efficiency of FD simulation.
Thank you Thank you for attentionfor attention
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