International audienceHybrid meshes comprised of hexahedras and te-trahedras are particularly interesting for representing media with local complex geometrical features like the seabed in offshore applications. We develop a coupled finite element method for solving elasto-acoustic wave equations. It combines Discontinuous Galerkin (DG) finite elements for solving elastodynamics with spectral finite elements (SE) for solving the acoustic wave equation. SE method has demonstrated very good performances in 3D with hexahedral meshes and contributes to reduce the computational burden by having less discrete unknowns than DG. The implementation of the method is performed both in 2D and 3D and it turns out that the coupling contributes to reduce the computational costs significantly: for the same time step and the same elementary mesh size, the CPU time of the coupled method is almost halved when compared to the one of a full DG method

Barucq, Hélène

Calandra, Henri

Citrain, Aurélien

Diaz, Julien

Gout, Christian

English

Hal-Diderot

High order discretization of seismic waves-problems based upon DG-SE methods

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High order discretization of seismic waves-problems
based upon DG-SE methods
Hélène Barucq, Henri Calandra, Aurélien Citrain, Julien Diaz, Christian Gout
To cite this version:
Hélène Barucq, Henri Calandra, Aurélien Citrain, Julien Diaz, Christian Gout. High order discretiza-
tion of seismic waves-problems based upon DG-SE methods. WAVES 2019 - 14th International Con-
ference on Mathematical and Numerical Aspects of Wave Propagation, Aug 2019, Vienne, Austria.
￿hal-02277988￿
WAVES 2019, Vienna, Austria 1
High order discretization of seismic waves-problems based upon DG-SE methods†
Hélène Barucq 1, Henri Calandra2, Aurélien Citrain3,1,∗, Julien Diaz1, Christian Gout3
1Team project Magique.3D, INRIA, E2S UPPA, CNRS, Pau, France
2TOTAL SA, CSTJF, Pau, France
3INSA Rouen-Normandie Université, LMI EA 3226, 76000, Rouen
∗Email: aurelien.citrain@insa-rouen.fr
†This work is dedicated to the memory of Dimitri Komatitsch.
Abstract
Hybrid meshes comprised of hexahedras and te-
trahedras are particularly interesting for repre-
senting media with local complex geometrical
features like the seabed in offshore applications.
We develop a coupled finite element method for
solving elasto-acoustic wave equations. It com-
bines Discontinuous Galerkin (DG) finite ele-
ments for solving elastodynamics with spectral
finite elements (SE) for solving the acoustic wave
equation. SE method has demonstrated very
good performances in 3D with hexahedral me-
shes and contributes to reduce the computa-
tional burden by having less discrete unknowns
than DG. The implementation of the method
is performed both in 2D and 3D and it turns
out that the coupling contributes to reduce the
computational costs significantly: for the same
time step and the same elementary mesh size,
the CPU time of the coupled method is almost
halved when compared to the one of a full DG
method.
Keywords: Hybrid meshes, Discontinuous Ga-
lerkin method, Spectral Element method, cou-
pling
1 Introduction
We focus on the first-order elasto-dynamic sys-
tem due to space constraint. We denote by Ω a
rectangular domain in 2D or a parallelepiped in
3D. We consider the system of wave equations:
ρ(x)
∂v
∂t
(x, t) = ∇ · σ(x, t),
∂σ
∂t
(x, t) = C(x)(v(x, t)),
(1)
where ρ is the density, C the elasticity tensor
and  the deformation tensor. The space vari-
able is x ∈ Rd (with d = 2, 3) and t ≥ 0 is
the time variable. The two unknowns are v the
wavespeed and σ the strain tensor. This system
can be solved by using DGm (see [1, 2]) or a
SEm (see [3–5]). Our objective is to construct a
variational formulation resulting from the com-
bination of both approximations. The difficulty
of such a coupling is the communication between
the two different schemes.
2 Variational Formulation
Offshore geophysical exploration can be repre-
sented by a reference domain composed of a
layer of water over the ocean bottom (see Fig
1). The computational domain is covered by a
hybrid grid composed of hexahedra on the top
and tetrahedra within the bottom. Basically,
we define two areas: Ωh,1 composed of cartesian
cells and Ωh,2 paved with unstructured tetrahe-
dra capable of following the topography of the
in-depth site. The transition between both areas
is located inside the water. Hence, the interface
Γ1/2 = Ωh,1 ∩ Ωh,2 is flat and the two regions
communicate with each other through suitable
fluxes. The portion Ωh,1 of the mesh can thus
be seen as a macro-element of the DG partition.
In the following, we use the subscript 1 to des-
ignate the fields computed over Ωh,1 while any
field with subscript 2 corresponds to a quan-
tity computed over Ωh,2. We introduce the pair
(w, ξ) to test the continuous problem (1) and
to get a variational formulation set in the whole
domain Ω. For the sake of simplicity, we denote
by aj , bj , cj and dj the bilinear forms defined
by
aj(v, w) =
∫
Ωh,j
ρ∂tv · w,
bj(σ,w) = −
∫
Ωh,j
σ · ∇w
cj(σ, ξ) =
∫
Ωh,j
∂tσ : ξ,
dj(v, ξ) = −
∫
Ωh,j
(∇(Cξ)) · v
Suggested members of the Scientific Committee:
Géza Seriani | Peter Monk
WAVES 2019, Vienna, Austria 2
Then the global variational formulation reads
as:
a1(v1, w1) + a2(v2, w2) = b1(σ1, w1) + b2(σ2, w2)
+
∑
Γ∈Γint
∫
Γ
{{σ
2
}}[[w2]] · n +∫
Γ1/2
{{σ}}[[w]] · n
c1(σ1, ξ1
) + c2(σ2, ξ2
) = d1(v1, ξ
1
) + d2(v2, ξ
2
)
+
∑
Γ∈Γint
∫
Γ
[[Cξ
2
]]{{v2}} · n +∫
Γ1/2
[[Cξ]]{{v}} · n
where Γint stands for the set of internal bound-
aries limiting DG-elements. The shortage of the
paper challenges us to omit to speak about ex-
ternal boundary conditions.
At the interface Γ1/2, we define n as the unitary
normal vector oriented from Ωh,1 to Ωh,2. We
can see how the two areas communicate through
the different fluxes involving the jump and the
mean value respectively defined by:
[[w]] = (wK2 − wK1) · n, {{ξ}} =
1
2
(ξ
K2
+ ξ
K1
)
K1 and K2 are two connected DG-cells and wKj
(resp. ξ
Kj
) is the value of w (resp. ξ) in Kj .
In comparison with the implementation of a full
SEm or a full DGm, we have to create new terms
corresponding to the handling of Γ1/2. They are
written in terms of integrals mixing DG basis
functions with a SE-one.
3 Numerical tests
We consider an exemple of anisotropic elasto-
acoustic domain depicted in Figure 1
water
water sand
salt
sandstone
Figure 1: Propagation domain
It is a square 3000 meters paved with 74969
cells composed of 53969 unstructured triangles
and 21000 structured quadrangles. The source
is a second-order Ricker point source located on
the top of the layer of water. Both DGm and
SEm have been validated separately in stratified
media for which we dispose of analytical solu-
tions. To assess the accuracy of the coupling,
we have compared the full DG solution with the
DG-SE one at order three and the results are
displayed in Table 1. We compare the CPU-
time at equal time-step and the relative error
between these two solutions and a reference so-
lution computed using DGm at order five. The
second column shows that the DG-SEm solution
has the same accuracy as the DG one. Then the
third column certifies that the coupling allows
to reduce the CPU time by a factor of 2. It is
worth noting that we have used a global time-
step and in the near future, we hope to improve
our results by using local-time stepping.
Relative error(%) CPU-time(s)
DGm 5e-4 16317
DG_SEm 1e-3 8918
Table 1: DGm vs DG_SEm comparison.
Acknowledgments
The authors acknowledge the support of the European
Union’s research FEDER program under the M2NUM
agreement HN0002137 and of the DIP Inria-TOTAL re-
search project.
References
[1] Reed, W. H. and Hill, TR., Triangular mesh
methods for the neutron transport equation,
1973
[2] Cockburn, B., Karniadakis, G. E. and, Shu,
C., The development of discontinuous Galerkin
methods, Springer, 3-50, 2000
[3] Patera, A. T., A spectral element method for
fluid dynamics: laminar flow in a channel ex-
pansion Journal of computational Physics, 54,
468-488, 1984
[4] Seriani, G., Priolo, E., Carcione, J and,
Padovani, E., High-order spectral element
method for elastic wave modeling Society of Ex-
ploration Geophysicists, 1285-1288, 1992
[5] Komatitsch, D., Méthodes spectrales et éléments
spectraux pour l’équation de l’élastodynamique
2 D et 3 D en milieu hétérogène, Thèse de doc-
torat, 1997
Suggested members of the Scientific Committee:
Géza Seriani | Peter Monk


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High order discretization of seismic waves-problems based upon DG-SE methods

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