performance analysis of dg and hdg methods for the simulation … · 2015. 1. 6. ·...
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September 22-26, 2014
Performance analysis of DG and HDG methodsfor the simulation of seismic wave propagationin harmonic domainM. Bonnasse-Gahot1,2, H. Calandra3, J. Diaz1 and S. Lanteri21 INRIA Bordeaux-Sud-Ouest, team-project Magique 3D2 INRIA Sophia-Antipolis-Méditerranée, team-project Nachos
3 TOTAL Exploration-Production
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Motivation
Examples of seismic applications
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MotivationImaging method : the full wave inversion
I Iterative procedure using the wavefield in order to obtainquantitative high resolution images of the subsurface physicalparameters
Seismic imaging : time-domain or harmonic-domain ?I Time-domain : imaging condition complicated but low
computational costI Harmonic-domain : imaging condition simple but huge
computational cost
Forward problem of the inversion processI Elastic wave propagation in harmonic domain : Helmholtz
equationI Reduction of the size of the linear system
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MotivationImaging method : the full wave inversion
I Iterative procedure using the wavefield in order to obtainquantitative high resolution images of the subsurface physicalparameters
Seismic imaging : time-domain or harmonic-domain ?I Time-domain : imaging condition complicated but low
computational costI Harmonic-domain : imaging condition simple but huge
computational cost
Forward problem of the inversion processI Elastic wave propagation in harmonic domain : Helmholtz
equationI Reduction of the size of the linear system
M. Bonnasse-Gahot - DG and HDG methods for Helmholtz wave equations September 8, 2014 - 3/30
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MotivationImaging method : the full wave inversion
I Iterative procedure using the wavefield in order to obtainquantitative high resolution images of the subsurface physicalparameters
Seismic imaging : time-domain or harmonic-domain ?I Time-domain : imaging condition complicated but low
computational costI Harmonic-domain : imaging condition simple but huge
computational cost
Forward problem of the inversion processI Elastic wave propagation in harmonic domain : Helmholtz
equationI Reduction of the size of the linear system
M. Bonnasse-Gahot - DG and HDG methods for Helmholtz wave equations September 8, 2014 - 3/30
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MotivationSeismic imaging in heterogeneous complex media
I Complex topographyI High heterogeneities
Use of unstructured meshes with FE methods
DG methodI Flexible choice of interpolation orders (p − adaptativity)I Highly parallelizable methodI Increased computational cost as compared to classical FEM
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DOF of classical FEM
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DOF of DGM
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MotivationSeismic imaging in heterogeneous complex media
I Complex topographyI High heterogeneities
Use of unstructured meshes with FE methods
DG methodI Flexible choice of interpolation orders (p − adaptativity)I Highly parallelizable methodI Increased computational cost as compared to classical FEM
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DOF of classical FEM
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DOF of DGM
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MotivationSeismic imaging in heterogeneous complex media
I Complex topographyI High heterogeneities
Use of unstructured meshes with FE methods
DG methodI Flexible choice of interpolation orders (p − adaptativity)I Highly parallelizable methodI Increased computational cost as compared to classical FEM
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DOF of classical FEM
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DOF of DGM
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MotivationSeismic imaging in heterogeneous complex media
I Complex topographyI High heterogeneities
Use of unstructured meshes with FE methods
DG methodI Flexible choice of interpolation orders (p − adaptativity)I Highly parallelizable methodI Increased computational cost as compared to classical FEM
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DOF of classical FEM
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DOF of DGM
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Motivation
Objective of this workI Development of an hybridizable DG (HDG) methodI Comparison with a reference method : a standard nodal DG
method
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Figure : Degrees offreedom of DGM
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Figure : Degrees offreedom of HDGM
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HDG methods
HDG methodsI B. Cockburn, J. Gopalakrishnan, R. Lazarov Unified
hybridization of discontinuous Galerkin, mixed and continuousGalerkin methods for second order elliptic problems, SIAMJournal on Numerical Analysis, Vol. 47 (2009)
I S. Lanteri, L. Li, R. Perrussel, Numerical investigation of ahigh order hybridizable discontinuous Galerkin method for 2dtime-harmonic Maxwell’s equations, COMPEL, Vol. 32 (2013)(time-harmonic domain)
I N.C. Nguyen, J. Peraire, B. Cockburn, High-order implicithybridizable discontinuous Galerkin methods for acoustics andelastodynamics, J. of Comput. Physics, Vol. 230 (2011) (timedomain for seismic applications)
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2D Helmholtz equations
Contents
2D Helmholtz elastic equations
Notations and definitions
Hybridizable Discontinuous Galerkin method
Numerical results
Conclusions-Perspectives
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2D Helmholtz equations
2D Helmholtz elastic equations
First order formulation of Helmholtz wave equationsx = (x , y) ∈ Ω ⊂ R2,
iωρ(x)v(x) = ∇·σ(x) + fs(x)
iωσ(x) = C(x) ε(v(x))
I Free surface condition : σn = 0 on Γl
I Absorbing boundary condition : σn = vp(v · n)n + vs(v · t)t on Γa
I v : velocity vectorI σ : stress tensorI ε : strain tensor
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2D Helmholtz equations
2D Helmholtz elastic equations
First order formulation of Helmholtz wave equationsx = (x , y) ∈ Ω ⊂ R2,
iωρ(x)v(x) = ∇·σ(x) + fs(x)
iωσ(x) = C(x) ε(v(x))
I Free surface condition : σn = 0 on Γl
I Absorbing boundary condition : σn = vp(v · n)n + vs(v · t)t on Γa
I ρ : mass densityI C : tensor of elasticity
coefficients
I vp : P-wave velocityI vs : S-wave velocityI fs : source term, fs ∈ L2(Ω)
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Definitions
Contents
2D Helmholtz elastic equations
Notations and definitions
Hybridizable Discontinuous Galerkin method
Numerical results
Conclusions-Perspectives
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Definitions
Notations and definitions
NotationsI Th mesh of Ω composed of triangles K
I Fh set of all faces F of ThI n the normal outward vector of an element K
K
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Definitions
Notations and definitions
NotationsI Th mesh of Ω composed of triangles KI Fh set of all faces F of Th
I n the normal outward vector of an element K
F
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Definitions
Notations and definitions
NotationsI Th mesh of Ω composed of triangles KI Fh set of all faces F of ThI n the normal outward vector of an element K
nK
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Definitions
Notations and definitions
Approximations spacesI Pp(K ) set of polynomials of degree at most p on K
I Vph = v ∈
(L2(Ω)
)2: v|K ∈ Vp(K ) = (Pp(K ))2 , ∀K ∈ Th
I Σph = σ ∈
(L2(Ω)
)3: σ|K ∈ Σp(K ) = (Pp(K ))3 ,∀K ∈ Th
I Mh = η ∈(L2(Fh)
)2: η|F ∈ (Pp(F ))2 ,∀F ∈ Fh
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Definitions
Notations and definitions
Approximations spacesI Pp(K ) set of polynomials of degree at most p on KI Vp
h = v ∈(L2(Ω)
)2: v|K ∈ Vp(K ) = (Pp(K ))2 , ∀K ∈ Th
I Σph = σ ∈
(L2(Ω)
)3: σ|K ∈ Σp(K ) = (Pp(K ))3 ,∀K ∈ Th
I Mh = η ∈(L2(Fh)
)2: η|F ∈ (Pp(F ))2 ,∀F ∈ Fh
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Definitions
Notations and definitions
Approximations spacesI Pp(K ) set of polynomials of degree at most p on KI Vp
h = v ∈(L2(Ω)
)2: v|K ∈ Vp(K ) = (Pp(K ))2 , ∀K ∈ Th
I Σph = σ ∈
(L2(Ω)
)3: σ|K ∈ Σp(K ) = (Pp(K ))3 ,∀K ∈ Th
I Mh = η ∈(L2(Fh)
)2: η|F ∈ (Pp(F ))2 ,∀F ∈ Fh
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Definitions
Notations and definitions
Approximations spacesI Pp(K ) set of polynomials of degree at most p on KI Vp
h = v ∈(L2(Ω)
)2: v|K ∈ Vp(K ) = (Pp(K ))2 , ∀K ∈ Th
I Σph = σ ∈
(L2(Ω)
)3: σ|K ∈ Σp(K ) = (Pp(K ))3 ,∀K ∈ Th
I Mh = η ∈(L2(Fh)
)2: η|F ∈ (Pp(F ))2 ,∀F ∈ Fh
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Definitions
Notations and definitions
DefinitionsI Jump [[·]] of a vector v through F :
[[v]] = v+ ·n+ +v− ·n− = v+ ·n+−v− ·n+
I Jump of a tensor σ through F :
[[σ]] = σ+n+ + σ−n− = σ+n+ − σ−n+
K+
K−n+
n−
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HDG method
Contents
2D Helmholtz elastic equations
Notations and definitions
Hybridizable Discontinuous Galerkin methodFormulationDiscretization
Numerical results
Conclusions-Perspectives
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HDG method Formulation
HDG formulation of the equations
Local HDG formulationiωρv−∇ · σ = 0
iωσ − Cε (v) = 0
∫
KiωρK vK ·w +
∫KσK : ∇w−
∫∂Kσ∂K · n ·w = 0∫
KiωσK : ξ +
∫K
vK · ∇ ·(CKξ
)−∫∂K
v∂K · CKξ · n = 0
We define :
vF = λF , ∀F ∈ Fh,σ∂K · n = σK · n− τ I
(vK − λ∂K) , on ∂K
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HDG method Formulation
HDG formulation of the equations
Local HDG formulation
∫
KiωρK vK ·w +
∫KσK : ∇w−
∫∂Kσ∂K · n ·w = 0∫
KiωσK : ξ +
∫K
vK · ∇ ·(CKξ
)−∫∂K
v∂K · CKξ · n = 0
We define :
vF = λF , ∀F ∈ Fh,σ∂K · n = σK · n− τ I
(vK − λ∂K) , on ∂K
σK and vK are numerical traces of σK and vK respectively on ∂K
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HDG method Formulation
HDG formulation of the equations
Local HDG formulation
∫
KiωρK vK ·w +
∫KσK : ∇w−
∫∂Kσ∂K · n ·w = 0∫
KiωσK : ξ +
∫K
vK · ∇ ·(CKξ
)−∫∂K
v∂K · CKξ · n = 0
We define :
vF = λF , ∀F ∈ Fh,σ∂K · n = σK · n− τ I
(vK − λ∂K) , on ∂K
where τ is the stabilization parameter (τ > 0)
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HDG method Formulation
HDG formulation of the equations
Local HDG formulationWe replace vK and
(σK · n
)by their definitions into the local
equations
∫KiωρK vK ·w +
∫KσK : ∇w−
∫∂KσK · n ·w
+
∫∂Kτ I(
vK − λ∂K)·w = 0∫
KiωσK : ξ +
∫K
vK · ∇ ·(CKξ
)−∫∂Kλ∂K · CKξ · n = 0
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HDG method Formulation
HDG formulation of the equations
Local HDG formulation∫
KiωρK vK ·w−
∫K
(∇ · σK) ·w +
∫∂Kτ I(vK − λ∂K) ·w = 0∫
KiωσK : ξ +
∫K
vK · ∇ ·(CKξ
)−∫∂Kλ∂K · CKξ · n = 0
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HDG method Formulation
HDG formulation of the equations
Transmission conditionIn order to determine λK , the continuity of the normal componentof σK is weakly enforced, rendering this numerical traceconservative : ∫
F[[σK · n]] · η = 0
Replacing(σK · n
)and summing over all faces, the transmission
condition becomes :∑K∈Th
∫∂K
(σK · n
)· η −
∑K∈Th
∫∂Kτ I(
vK − λ∂K)· η = 0
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HDG method Formulation
HDG formulation of the equations
Transmission conditionIn order to determine λK , the continuity of the normal componentof σK is weakly enforced, rendering this numerical traceconservative : ∫
F[[σK · n]] · η = 0
Replacing(σK · n
)and summing over all faces, the transmission
condition becomes :∑K∈Th
∫∂K
(σK · n
)· η −
∑K∈Th
∫∂Kτ I(
vK − λ∂K)· η = 0
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HDG method Formulation
HDG formulation of the equations
Global HDG formulation
∫KiωρK vK ·w−
∫K
(∇ · σK) ·w +
∫∂Kτ I(vK − λ∂K) ·w = 0∫
KiωσK : ξ +
∫K
vK · ∇ ·(CKξ
)−∫∂Kλ∂K · CKξ · n = 0
∑K∈Th
∫∂K
(σK · n
)· η −
∑K∈Th
∫∂Kτ I(vK − λ∂K) · η = 0
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HDG method Discretization
Discretization of the HDG formulation
Local HDG formulation∫
KiωρK vK ·w−
∫K
(∇ · σK) ·w +
∫∂Kτ I(vK − λ∂K) ·w = 0∫
KiωσK : ξ +
∫K
vK · ∇ ·(CKξ
)−∫∂Kλ∂K · CKξ · n = 0
We define :W K =
(vx
K , vzK , σxx
K , σzzK , σxz
K)T
Λ =(ΛF1 , ΛF2 , ..., ΛFnf
)T, where nf = card(Fh)
Discretization of the local HDG formulationAKW K + CK Λ = 0
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HDG method Discretization
Discretization of the HDG formulation
Transmission condition∑K∈Th
∫∂K
(σK · n
)· η −
∑K∈Th
∫∂K
S(
vK − λ∂K)· η = 0
Discretization of the transmission condition∑K∈Th
[BKWK + LK Λ
]= 0
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HDG method Discretization
Discretization of the HDG formulationTransmission condition∑
K∈Th
[BKWK + LK Λ
]= 0
Local HDG scheme
AKWK + CK Λ = 0
Global HDG system
KΛ = 0
with K =∑
K∈Th
[−BK (AK )−1CK + LK
]
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HDG method Discretization
Discretization of the HDG formulationTransmission condition∑
K∈Th
[BKWK + LK Λ
]= 0
Expression of W K in terms of Λ
WK = −(AK )−1CK Λ
Global HDG system
KΛ = 0
with K =∑
K∈Th
[−BK (AK )−1CK + LK
]
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HDG method Discretization
Discretization of the HDG formulationTransmission condition∑
K∈Th
[BKWK + LK Λ
]= 0
Expression of W K in terms of Λ
WK = −(AK )−1CK Λ
Global HDG system
KΛ = 0
with K =∑
K∈Th
[−BK (AK )−1CK + LK
]M. Bonnasse-Gahot - DG and HDG methods for Helmholtz wave equations September 8, 2014 - 18/30
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Numerical results
Contents
2D Helmholtz elastic equations
Notations and definitions
Hybridizable Discontinuous Galerkin method
Numerical resultsPlane wave in an homogeneous mediumDisk-shaped scatterer problemMarmousi test-case
Conclusions-Perspectives
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Numerical results Plane wave in an homogeneous medium
Plane wave
10000 m
10000 m
Computational domain Ωsetting
I Physical parameters :I ρ = 2000kg .m−3I λ = 16GPaI µ = 8GPa
I Plane wave :
u = ∇ei(k cos θx+k sin θy)
where k =ω
vpI θ = 0I Three meshes :
I 3000 elementsI 10000 elementsI 45000 elements
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Numerical results Plane wave in an homogeneous medium
Plane wave
5 5.5 6 6.5−5
0
5
10
15
20
hmax
||Wa −
We ||
12.3
13.4
1
4.0
1
5.4
P1
P2
P3
P4
Convergence order of the HDG scheme
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Numerical results Disk-shaped scatterer problem
Disk-shaped scatterer problem
ΓlΓa
Ω
a
b
Computational domain Ωsetting
I a = 2000.0m and b = 8000.0mI Physical parameters in Ω :
I ρ = 1kg .m−3I λ = 8GPaI µ = 4GPa
I Γl free surface boundary :σn = 0
I Γa absorbing boundary :σn = vp(v · n)n + vs(v · t)t
I Three meshes :I 1200 elementsI 5400 elementsI 22000 elements
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Numerical results Disk-shaped scatterer problem
Disk-shaped scatterer problem
5.5 6 6.5 711
12
13
14
15
16
17
hmax
||Wa −
We ||
P2
h2.5
P3
h3.22
P4
h2.27
Convergence order of the HDG schemeM. Bonnasse-Gahot - DG and HDG methods for Helmholtz wave equations September 8, 2014 - 22/30
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Numerical results Disk-shaped scatterer problem
Disk-shaped scatterer problem
Elements Order CPU Time (s) Memory (MB)HDG UDG IPDG HDG UDG IPDG
1200 2 0.7
2.6 2.4
32
269 70
5100 2 3.0
15.0 11.9
161
1360 369
21000 2 14.0
94.8 58.0
728
6578 1857
1200 3 1.7
5.4 6.8
57
525 190
5100 3 7.6
38.8 35.9
283
2921 1017
21000 3 34.8
252.0 197.8
1284
14131 5126
1200 4 3.9
10.5 15.7
86
895 428
5100 4 17.7
67.0 87.9
430
4537 2279
21000 4 79.1
452.8 520.7
1953
21186 11503
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Numerical results Disk-shaped scatterer problem
Disk-shaped scatterer problem
Elements Order CPU Time (s) Memory (MB)HDG UDG IPDG HDG UDG IPDG
1200 2 0.7 2.6 2.4 32 269 705100 2 3.0 15.0 11.9 161 1360 36921000 2 14.0 94.8 58.0 728 6578 18571200 3 1.7 5.4 6.8 57 525 1905100 3 7.6 38.8 35.9 283 2921 101721000 3 34.8 252.0 197.8 1284 14131 51261200 4 3.9 10.5 15.7 86 895 4285100 4 17.7 67.0 87.9 430 4537 227921000 4 79.1 452.8 520.7 1953 21186 11503
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Numerical results Disk-shaped scatterer problem
Disk-shaped scatterer problem
Elements Order CPU Time MemoryHDG UDG IPDG HDG UDG IPDG
1200 2 1 3.7 3.4 1 8.4 2.25100 2 1 5.0 4.0 1 8.4 2.321000 2 1 6.8 4.1 1 9.0 2.61200 3 1 3.1 4.0 1 9.2 3.35100 3 1 5.1 4.7 1 10.3 3.621000 3 1 7.2 5.7 1 11.0 4.01200 4 1 2.7 4.0 1 10.4 5.05100 4 1 3.8 5.0 1 10.5 5.321000 4 1 5.7 6.6 1 10.8 5.9
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Numerical results Disk-shaped scatterer problem
Disk-shaped scatterer problem
Elements Order CPU Time MemoryHDG UDG IPDG HDG UDG IPDG
1200 2 1 3.7 3.4 1 8.4 2.25100 2 1 5.0 4.0 1 8.4 2.321000 2 1 6.8 4.1 1 9.0 2.61200 3 1 3.1 4.0 1 9.2 3.35100 3 1 5.1 4.7 1 10.3 3.621000 3 1 7.2 5.7 1 11.0 4.01200 4 1 2.7 4.0 1 10.4 5.05100 4 1 3.8 5.0 1 10.5 5.321000 4 1 5.7 6.6 1 10.8 5.9
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Numerical results Marmousi test-case
Marmousi test-case
Computational domain Ω composed of 235000 triangles
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Numerical results Marmousi test-case
Parallel results for the Marmousi test-case with theHDG-P2 scheme
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Numerical results Marmousi test-case
Parallel results for the Marmousi test-case with theHDG-P2 scheme
CPU Time CPU Time Maximumconstruction (s) resolution. (s) Memory (MB)
sequential 67 133 99272 proc. (2/1) 32 93 58924 proc. (2/2) 15 56 33408 proc. (4/2) 8 38 209216 proc. (4/4) 4 39 369532 proc. (4/8) 2 21 131264 proc. (8/8) 1 19 893
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Numerical results Marmousi test-case
Parallel results for the Marmousi test-case with theHDG-P3 scheme
CPU Time CPU Time Maximumconstruction (s) resolution (s) Memory (MB)
sequential 207 321 176312 proc.(2/1) 96 196 100484 proc. (2/2) 47 116 58498 proc. (4/2) 23 75 520516 proc. (4/4) 12 72 462832 proc. (4/8) 6 42 168964 proc. (8/8) 3 33 1229
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Conclusion
Contents
2D Helmholtz elastic equations
Notations and definitions
Hybridizable Discontinuous Galerkin method
Numerical results
Conclusions-Perspectives
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Conclusion
Conclusions-Perspectives
ConclusionsI The HDG scheme has the correct convergence order (p + 1)I On a same mesh the HDG formulation is more competitive in
terms of memory and computational time than the upwindflux DG formulation and the IPDG method
PerspectivesI Develop 3D Upwind flux DG and HDG formulations for
Helmholtz equationsI Solution strategy for the HDG linear system
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Conclusion
Conclusions-Perspectives
ConclusionsI The HDG scheme has the correct convergence order (p + 1)I On a same mesh the HDG formulation is more competitive in
terms of memory and computational time than the upwindflux DG formulation and the IPDG method
PerspectivesI Develop 3D Upwind flux DG and HDG formulations for
Helmholtz equationsI Solution strategy for the HDG linear system
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Conclusion
Thank you !