Comparison of solvers performance when solving the 3D Helmholtz elastic wave equations using the Hybridizable Discontinuous Galerkin method

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Comparison of solvers performance when solving the 3D

Helmholtz elastic wave equations using the Hybridizable

Discontinuous Galerkin method

Marie Bonnasse-Gahot, Henri Calandra, Julien Diaz, Stephane Lanteri

To cite this version:

Marie Bonnasse-Gahot, Henri Calandra, Julien Diaz, Stephane Lanteri. Comparison of solvers perfor-mance when solving the 3D Helmholtz elastic wave equations using the Hybridizable Discontinuous Galerkin method. Workshop DIP - Depth Imaging Partnership, Oct 2016, Houston, United States. �hal-01400656�

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October 10, 2016

Comparison of solvers performance when solving

the 3D Helmholtz elastic wave equations using

the Hybridizable Discontinuous Galerkin method

M. Bonnasse-Gahot1,2, H. Calandra3, J. Diaz1 and S. Lanteri2

1_{INRIA Bordeaux-Sud-Ouest, team-project Magique 3D} 2_{INRIA Sophia-Antipolis-Méditerranée, team-project Nachos} 3_{TOTAL Exploration-Production}

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Motivations

Imaging methods

I Reverse Time Migration (RTM) : based on the reversibility

of wave equation

I Full Wave Inversion (FWI) : inversion process requiring to

solve many forward problems

Seismic imaging : time-domain or harmonic-domain ?

I Time-domain : imaging condition complicatedbutquite low computational cost

I Harmonic-domain : imaging condition simplebuthuge computational cost

(4)

Motivations

Imaging methods

Memory usage

(5)

Motivations

Imaging methods

(6)

Motivations

Resolution of the forward problem of the inversion process

I Elastic wave propagation in the frequency domain : Helmholtz

equation

First order formulation of Helmholtz wave equations

x = (x , y , z) ∈ Ω ⊂ R3_,

(

i ωρ(x)v(x) = ∇·σ(x) +fs(x)

i ωσ(x) =C(x) ε(v(x))

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Motivations

equation

x = (x , y , z) ∈ Ω ⊂ R3_, ( i ωρ(x)v(x) = ∇·σ(x) +fs(x) i ωσ(x) =C(x) ε(v(x)) I v : velocity vector I σ : stress tensor I ε : strain tensor

(8)

Motivations

equation

x = (x , y , z) ∈ Ω ⊂ R3_,

(

i ωρ(x)v(x) = ∇·σ(x) +fs(x)

i ωσ(x) =C(x) ε(v(x))

I ρ : mass density

I C : elasticity tensor I fs : source term, fs ∈ L

2_(Ω)

(9)

Approximation methods

Discontinuous Galerkin Methods

3unstructured tetrahedral meshes

3combination between FEM and finite volume method (FVM)

3hp-adaptivity

3easily parallelizable method

(10)

Approximation methods

3hp-adaptivity

7 7large number of DOF as compared to classical FEM

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Approximation methods

3hp-adaptivity

7 7large number of DOF as compared to classical FEM

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

(12)

Approximation methods

Hybridizable Discontinuous Galerkin Methods

3same advantages as DG methods : unstructured tetrahedral

meshes, hp-adaptivity, easily parallelizable method, discontinuous basis functions

3introduction of a new variable defined only on the interfaces

3lower number of coupled DOF than classical DG methods

7time-domain increases computational costs

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Approximation methods

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b bb b b b b b b bb b b b b b b b b b b b b b b b b b b b b

(14)

Approximation methods

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb b b b b b b b bb b b b b b b bb b b b b b b b b b b b b b b b b b b b b

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Hybridizable Discontinuous Galerkin method

B. Cockburn, J. Gopalakrishnan and R. Lazarov. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems. SIAM Journal

on Numerical Analysis, Vol. 47 :1319-1365, 2009.

S. Lanteri, L. Li and R. Perrussel. Numerical investigation of a high order hybridizable discontinuous Galerkin method for 2d

time-harmonic Maxwell’s equations. COMPEL, 32(3)1112-1138, 2013.

N.C. Nguyen, J. Peraire and B. Cockburn. High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. Journal of Computational Physics, 230 :7151-7175, 2011

N.C. Nguyen and B. Cockburn. Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics.

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HDG method

HDG formulation of the equations

Local HDG formulation

(

i ωρv− ∇ ·σ = 0

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HDG formulation of the equations

Local HDG formulation        Z K i ωρKvK · w + Z K σK : ∇w − Z ∂K b σ∂K · n · w = 0 Z K i ωσK : ξ + Z K vK · ∇ ·CKξ− Z ∂Kb v∂K ·CKξ · n = 0 b σK andbv

K _{are numerical traces of}_σK _and_vK _{respectively on ∂K}

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HDG formulation of the equations

We define :

b

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HDG formulation of the equations

We define : bv ∂K ₌ _λF_, _{∀F ∈ F} h, b σ∂K · n = σK· n − τ I vK −λF , on ∂K

where τ is the stabilization parameter (τ > 0)

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HDG formulation of the equations

Local HDG formulation        Z K i ωρKvK· w − Z K ∇ ·σK · w + Z ∂K τ I vK−λF · w = 0 Z K i ωσK: ξ + Z K vK· ∇ ·CKξ− Z ∂K λF ·CKξ· n = 0 We define : WK =VxK, VyK, VzK, σxxK, σyyK, σzzK, σxyK, σxzK, σyzK T Λ= ΛF1_, _ΛF2_{, ...,} _ΛFnfT , where nf = card(Fh)

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HDG formulation of the equations

Local HDG formulation        Z K i ωρKvK· w − Z K ∇ ·σK · w + Z ∂K τ I vK−λF · w = 0 Z K i ωσK: ξ + Z K vK· ∇ ·CKξ− Z ∂K λF ·CKξ · n = 0 We define : WK =VxK, VyK, VzK, σxxK, σyyK, σzzK, σxyK, σxzK, σyzK T Λ= ΛF1_, _ΛF2_{, ...,} _ΛFnfT , where nf = card(Fh)

Discretization of the local HDG formulation

AKWK+ X

F ∈∂K

CK ,FΛ= 0

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HDG formulation of the equations

Local HDG formulation        Z K i ωρKvK· w − Z K ∇ ·σK · w + Z ∂K τ I vK−λF · w = 0 Z K i ωσK: ξ + Z K vK· ∇ ·CKξ− Z ∂K λF ·CKξ · n = 0 We define : WK =VxK, VyK, VzK, σxxK, σyyK, σzzK, σxyK, σxzK, σyzK T Λ= ΛF1_, _ΛF2_{, ...,} _ΛFnfT_{, where n}_f _{= card(F}_h₎ Discretization of the local HDG formulation

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HDG formulation of the equations

Transmission condition

In order to determineλF_{, the continuity of the normal component}

ofσ_b∂K is weakly enforced, rendering this numerical trace

conservative : _Z

F

[[_bσ∂K · n]] · η = 0

Replacing σ_b∂K · n and summing over all faces, the transmission

condition becomes : X K ∈Th Z ∂K σK · n· η − X K ∈Th Z ∂K τ IvK−λF· η = 0

Discretization of the transmission condition

X

K ∈Th

BKWK+LKΛ = 0

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HDG formulation of the equations

conservative : _Z

F

[[_bσ∂K · n]] · η = 0

X

K ∈Th

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HDG formulation of the equations

conservative : _Z

F

[[_bσ∂K · n]] · η = 0

X

K ∈Th

BKWK+LKΛ = 0

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HDG formulation of the equations

Global HDG formulation                    Z K i ωρKvK· w − Z K ∇ ·σK · w + Z ∂K τ I vK−λF · w = 0 Z K i ωσK: ξ + Z K vK· ∇ ·CKξ− Z ∂K λF ·CKξ · n = 0 X K ∈Th Z ∂K σK· n · η − X K ∈Th Z ∂K τ I vK−λF · η = 0

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HDG formulation of the equations

Global HDG discretization        AKWK+CKΛ= 0 X K ∈Th BKWK+LKΛ = 0

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HDG formulation of the equations

Global HDG discretization        WK = −(AK)−1CKΛ X K ∈Th BKWK+LKΛ = 0

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HDG formulation of the equations

Global HDG discretization

X

K ∈Th

−BK(AK)−1CK+LKΛ= 0

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HDG method Symmetric HDG formulation

Symmetric HDG formulation

Local HDG formulation ( i ωρv− ∇ ·σ = 0 i ωσ−Cε (v) = 0 ξ = −DKξ0

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Symmetric HDG formulation

Local HDG formulation ( i ωρv− ∇ ·σ = 0 i ωσ−Cε (v) = 0

C invertible and symmetric tensor, i.e for a symmetric σ :

σ =Cε (u) and ε (u) =Dσ

withD =C−1 andu= i ωv

ξ = −DKξ0

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Symmetric HDG formulation

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Symmetric HDG formulation

ξ = −DKξ0

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Symmetric HDG formulation

Local HDG formulation                Z K i ωρKvK · w + Z K σK : ∇w − Z ∂K b σ∂K · n · w = 0 − Z K i ωσK :DKξ0− Z K vK· ∇ ·CKDKξ0 + Z ∂K b v∂K ·CKDKξ0· n = 0 ξ = −DKξ0

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Symmetric HDG formulation

Local HDG formulation        Z K i ωρKvK · w + Z K σK : ∇w − Z ∂K b σ∂K · n · w = 0 − Z K i ωDKσK : ξ0− Z K vK · ∇ · ξ0+ Z ∂K bv ∂K _{· ξ}0_{· n = 0}

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Symmetric HDG formulation

Local HDG formulation        Z K i ωρKvK · w + Z K σK : ∇w − Z ∂K b σ∂K · n · w = 0 − Z K i ωDKσK : ξ0− Z K vK · ∇ · ξ0+ Z ∂K bv ∂K _{· ξ}0_{· n = 0} bv ∂K ₌ _λF_, _{∀F ∈ F} h, b σ∂K · n = σK· n − τ I vK −λF , on ∂K

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Symmetric HDG formulation

Local HDG formulation        Z K i ωρKvK· w − Z K ∇ ·σK · w + Z ∂K τ I vK−λF · w = 0 − Z K i ωDKσK : ξ0− Z K vK· ∇ · ξ0+ Z ∂K λF · ξ0· n = 0

A2K symmetric matrix

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Symmetric HDG formulation

A2KWK+

X

F ∈∂K

C2K ,FΛ= 0

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Symmetric HDG formulation

A2KWK+C2KΛ= 0

A2K symmetric matrix

(41)

Symmetric HDG formulation

A2KWK+C2KΛ= 0

(42)

Symmetric HDG formulation

Transmission condition Z F [[_bσ∂K · n]] · η = 0 X K ∈Th Z ∂K σK · n· η − X K ∈Th Z ∂K τ IvK−λF· η = 0

X

K ∈Th

WK+LKΛ = 0

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Symmetric HDG formulation

X

K ∈Th

(44)

Symmetric HDG formulation

X

K ∈Th

BKWK+LKΛ = 0

(45)

Symmetric HDG formulation

Transmission condition Z F [[_bσ∂K · n]] · η = 0 X K ∈Th Z ∂K σK · n· η − X K ∈Th Z ∂K τ I vK−λF · η = 0

X

K ∈Th

BKWK+LKΛ = 0

(46)

Symmetric HDG formulation

X

K ∈Th

(C2K)TWK+LKΛ = 0

(47)

Symmetric HDG formulation of the equations

Global HDG formulation                    Z K i ωρKvK· w − Z K ∇ ·σK· w + Z ∂K τ IvK−λF· w = 0 − Z K i ωDKσK: ξ0− Z K vK· ∇ · ξ0+ Z ∂K λF· ξ0· n = 0 X K ∈Th Z ∂K σK· n· η − X K ∈Th Z ∂K τ IvK−λF· η = 0

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Symmetric HDG formulation of the equations

Global HDG discretization        A2KWK+C2KΛ= 0 X K ∈Th (C2K)TWK+LKΛ = 0

⇒ Symmetric linear system

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Symmetric HDG formulation of the equations

Global HDG discretization        WK = −(A2K)−1C2KΛ X K ∈Th (C2K)TWK+LKΛ = 0

(50)

Symmetric HDG formulation of the equations

X

K ∈Th

−(C2K)T(A2K)−1C2K+LKΛ= 0

⇒ Symmetric linear system

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Symmetric HDG formulation of the equations

X

K ∈Th

−(C2K)T(A2K)−1C2K+LKΛ= 0

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HDG method Algorithm

Main steps of the HDG algorithm

1. Construction of the global matrixM

withM= X K ∈Th h −_BK(AK)−1CK +LK i for K = 1 to Nbtri do

Computation of matrices BK, (AK)−1,CK andLK

Construction of the corresponding section ofM

end for

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Main steps of the HDG algorithm

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Main steps of the HDG algorithm

2. Construction of the right hand sideS

3. Resolution MΛ = S, with a direct solver (MUMPS) or hybrid

solver (MaPhys)

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Main steps of the HDG algorithm

solver (MaPhys)

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Main steps of the HDG algorithm

solver (MaPhys)

4. Computation of the solutions of the initial problem

for K = 1 to Nbtri do

ComputeWK = −(AK)−1CKΛ

end for

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2D Numerical results : comparison of the two HDG formulations

Plane wave

10000 m 10000 m Computational domain Ω setting I Physical parameters : I _ρ_{= 2000kg .m}−3 I λ= 16GPa I µ= 8GPa I Plane wave :

u = ∇ei (k cos θx +k sin θy )

where k = ω vp I θ = 0, vp= 4000 m.s−1, ω = 4π I Three meshes : I 3000 elements I 10000 elements I 45000 elements

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Plane wave : Convergence order

5 5.5 6 6.5 −5 0 5 10 15 20 h max ||W a − W e || 1 2.3 1 3.4 1 4.0 1 5.4 P 1 P 2 P 3 P 4

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Plane wave : Memory consumption

P1 P2 P3 P4 1 2 3 4 5 Interpolation order Memo ry (GB)

Finest mesh (45000 elements)

HDGm 1 HDGm 2

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Plane wave : Memory consumption

P1 P2 P3 P4 0 10 20 30 Interpolation order Memo ry (GB)

HDGm 1 HDGm 2 IPDGm

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Plane wave : CPU time

P2 P3 P4 0 50 100 150 200 Interpolation order CPU time (s)

HDGm const. 1 HDGm res. 1 P2 P3 P4 0 50 100 150 200 HDGm const. 2 HDGm res. 2

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Plane wave : CPU time

P2 P3 P4 0 500 1,000 1,500 CPU time (s)

HDGm const. 1 HDGm res. 1 P2 P3 P4 0 500 1,000 1,500 HDGm const. 2 HDGm res. 2 P2 P3 P4 0 500 1,000 1,500 IPDGm const. IPDGm res.

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2D Numerical results : comparison of the two HDG formulations Anisotropic test case

Anisotropic test case

I Three meshes :

I 600 elements

I 3000 elements

I 28000 elements

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Anisotropic case : Memory consumption

M1 M2 M3 0 500 1,000 1,500 2,000 Mesh Memo ry (MB) P3 interpolation order HDGm 1 HDGm 2

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Anisotropic case : Memory consumption

M1 M2 M3 0 2,000 4,000 6,000 8,000 Mesh Memo ry (MB) P3 interpolation order HDGm 1 HDGm 2 IPDGm

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Anisotropic case : CPU time (s)

M1 M2 M3 0 20 40 60 CPU time (s) P3 interpolation order HDGm const. 1 HDGm res. 1 M1 M2 M3 0 20 40 60 HDGm const. 2 HDGm res. 2

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Anisotropic case : CPU time (s)

M1 M2 M3 0 200 400 600 Mesh CPU time (s) P3 interpolation order HDGm const. 1 HDGm res. 1 M1 M2 M3 0 200 400 600 HDGm const. 2 HDGm res. 2 M1 M2 M3 0 200 400 600 IPDGm const. IPDGm res.

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Conclusion

I HDG method more efficient than classical DG methods for a

same accuracy

I Memory

I Computational time

2D specific study of HDG formulation

I Anisotropic HDG algorithm without any additional

computational cost

I Computational gain without loss of accuracy using

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3D numerical results

Main steps of the HDG algorithm

solver (MaPhys)

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Main steps of the HDG algorithm

3. Resolution MΛ = S, with a direct solver (MUMPS) or

hybrid solver (MaPhys)

4. Computation of the solutions of the initial problem

(73)

MaPhys Vs MUMPS

Pattern of the HDG global matrix for P1 interpolation and for a 3D

(74)

MaPhys Vs MUMPS

Software packages for solving systems of linear equations Ax = b, where A is a sparse matrix

I MUMPS (MUltifrontal Massively Parallel sparse direct

Solver) :

I Direct factorization A = LU or A = LDLT

I Multifrontal approach

I MaPhys (Massively Parallel Hybrid Solver) :

I Direct and iterative methods

I non-overlapping algebraic domain decomposition method (Schur complement method)

I resolution of each local problem thanks to direct solver such as MUMPS or PaStiX.

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3D numerical results 3D plane wave in an homogeneous medium

3D plane wave in an homogeneous medium

1000 m 1000 m 1000 m Configuration of the computational domain Ω . I Physical parameters : I ρ= 1 kg.m−3 I λ= 16 GPa I µ= 8 GPa I Plane wave : u = ∇ei (kxx +kyy +kzz) where kx = ω vp cos θ0cos θ1, ky = ω vp

sin θ0cos θ1, and kz = ω vp sin θ1 I ω = 2πf , f = 8 Hz I θ0 = 30◦, θ1 = 0◦ I Mesh composed of 21 000 elements

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Cluster configuration

Features of the nodes :

I 2 Dodeca-core Haswell Intel Xeon E5-2680

I Frequency : 2,5 GHz

I RAM : 128 Go

I Storage : 500 Go

I Infiniband QDR TrueScale : 40Gb/s

I Ethernet : 1Gb/s

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3D Plane wave : Memory consumption

48 96 192 384 576 1 10 # cores Memo ry (GB)

Maximum local memory for HDG-P3 method

MaPhys 8 MPI, 3 threads 4 MPI, 6 threads 2 MPI, 12 threads MUMPS 8 MPI, 3 threads 4 MPI, 6 threads 2 MPI, 12 threads (matrix order = 1 287 360, # nz=298 598 400 )

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3D Plane wave : Memory consumption

2 4 8 16 10 100 # nodes Memo ry (GB)

Average memory for one node (8 MPI by node and 3 threads by MPI)

MaPhys MUMPS Slope = 2 Slope = 2.5

(matrix order = 1 287 360, # nz=298 598 400 )

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3D Plane wave : Execution time

48 96 192 384 576

100

# cores

Time

(s)

Execution time for the resolution of the HDG-P3 system

MaPhys 8 MPI, 3 threads 4 MPI, 6 threads 2 MPI, 12 threads

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3D Plane wave : Execution time

48 96 192 384 576 100 400 # cores Time (s)

MUMPS 8 MPI, 3 threads 4 MPI, 6 threads 2 MPI, 12 threads

(matrix order = 1 287 360, # nz=298 598 400 )

(81)

3D Plane wave : Execution time

48 96 192 384 576

100

# cores

Time

(s)

MaPhys 8 MPI, 3 threads 4 MPI, 6 threads 2 MPI, 12 threads MUMPS 8 MPI, 3 threads 4 MPI, 6 threads 2 MPI, 12 threads (matrix order = 1 287 360, # nz=298 598 400 )

(82)

3D numerical results 3D geophysic test-case : Epati test-case

Epati test-case

Vp-velocity model (m.s−1), vertical section at y = 700 m

Mesh composed of 25 000 tetrahedrons

(83)

Epati test-case : Memory consumption

96 192 384 1 10 # cores Memo ry (GB)

Maximum local memory for HDG-P3 method

MaPhys 24 MPI, 1 thread 12 MPI, 2 threads 6 MPI, 4 threads MUMPS 24 MPI, 1 thread 12 MPI, 2 threads 6 MPI, 4 threads (matrix order = 1 600 740, # nz=365 385 600)

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Epati test-case : Memory consumption

4 8 16 3 10 30 # nodes Memo ry (GB)

Average memory for one node (24 MPI by node and 1 thread by MPI)

MaPhys MUMPS Slope = 2 Slope = 2.5

(matrix order = 1 287 360, # nz=365 385 600 )

(85)

Epati test-case : Execution time

96 192 384 10 100 1000 # cores Time (s)

MaPhys 24 MPI, 1 thread 12 MPI, 2 threads 6 MPI, 4 threads MUMPS 24 MPI, 1 thread 12 MPI, 2 threads 6 MPI, 4 threads (matrix order = 1 287 360, # nz=365 385 600 )

(86)

Conclusions-Perspectives

Conclusion-Perspectives

I more detailled analysis of the comparison between MUMPS

and MaPhys

I comparison for the symetric HDG formulation

I comparison to PaStiX solver

I extension to elasto-acoustic case

I study of the stabilization parameter τ for the 3D case

I call for projects PRACE to test bigger test-cases

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Conclusions-Perspectives

Comparison of solvers performance when solving the 3D Helmholtz elastic wave equations using the Hybridizable Discontinuous Galerkin method

HAL Id: hal-01400656

https://hal.inria.fr/hal-01400656

Comparison of solvers performance when solving the 3D

Helmholtz elastic wave equations using the Hybridizable

Discontinuous Galerkin method

Marie Bonnasse-Gahot, Henri Calandra, Julien Diaz, Stephane Lanteri

To cite this version:

Comparison of solvers performance when solving

the 3D Helmholtz elastic wave equations using

the Hybridizable Discontinuous Galerkin method

Motivations

Motivations

Motivations

Motivations

Motivations

Motivations

Approximation methods

Approximation methods

Approximation methods

Approximation methods

Approximation methods

Approximation methods

Hybridizable Discontinuous Galerkin method

Contents

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

HDG formulation of the equations

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation

Symmetric HDG formulation of the equations

Symmetric HDG formulation of the equations

Symmetric HDG formulation of the equations

Symmetric HDG formulation of the equations

Symmetric HDG formulation of the equations

Main steps of the HDG algorithm

Main steps of the HDG algorithm

Main steps of the HDG algorithm

Main steps of the HDG algorithm

Main steps of the HDG algorithm

Contents

Plane wave

Plane wave : Convergence order

Plane wave : Memory consumption

Plane wave : Memory consumption

Plane wave : CPU time

Plane wave : CPU time

Anisotropic test case

Anisotropic case : Memory consumption

Anisotropic case : Memory consumption

Anisotropic case : CPU time (s)

Anisotropic case : CPU time (s)

Conclusion

Contents

Main steps of the HDG algorithm