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High order discretization of seismic waves-problems based upon DG-SE methods elien Citrain 3 , 1 , Julien Diaz 1 and Christian Gout 3 ene Barucq 1 , Henri Calandra 2 , Aur H el` 1 Team project Magique.3D, INRIA, E2S UPPA, CNRS, Pau,

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

H´ el` ene Barucq1, Henri Calandra2, Aur´ elien Citrain3,1, Julien Diaz1 and Christian Gout3

1 Team project Magique.3D, INRIA, E2S UPPA, CNRS, Pau, France. 2 TOTAL SA, CSTJF, Pau, France. 3 LMI, Normandie Universit´ e, INSA ROUEN, 76000 Rouen, France.

WAVES 2019

The authors thank the M2NUM project which is co-financed by the European Union with the European regional development fund (ERDF, HN0002137) and by the Normandie Regional Council. †This work is dedicated to the memory of Dimitri Komatitsch. Aur´ elien Citrain DG-SE coupling WAVES 2019 1 / 44

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Seismic imaging

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Why using hybrid meshes?

Useful when the use of unstructured grid is non-sense (e.g. medium with a layer of water). Well suited for the coupling of numerical methods in order to reduce the computational cost and improve the accuracy.

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Why using hybrid meshes?

water water sand salt sandstone

Useful when the use of unstructured grid is non-sense (e.g. medium with a layer of water). Well suited for the coupling of numerical methods in order to reduce the computational cost and improve the accuracy.

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Elastodynamic system

x ∈ Ω ⊂ Rd, t ∈ [0, T], T > 0 :          ρ(x) ∂v ∂t (x, t) = ∇ · σ(x, t), ∂σ ∂t (x, t) = C(x)ǫ(v(x, t)). With: ρ(x) the density, C(x) the elasticity tensor, ǫ(x, t) the deformation tensor, v(x, t), the wavespeed, σ(x, t) the strain tensor.

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Elasticus software

Software written in Fortran for wave propagation simulation in the time domain

Features

Simulation:

  • n various types of meshes (unstructured triangles and tetrahedra),
  • n heterogeneous media (acoustic, elastic and elasto-acoustic).

Discontinuous Galerkin (DG) based on unstructured triangles and unstructured tetrahedra, with various time-schemes : Runge-Kutta (2 or 4), Leap-Frog, with multi-order computation(p-adaptivity)...

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Table of contents

1

DGm and SEm

2

Comparison DG/SEM on structured quadrangle mesh

3

DG/SEM coupling

4

DGSEM vs DG

5

3D extension

6

Perfectly Matched Layer(PML)

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1

DGm and SEm Discontinous Galerkin Method (DG) Spectral Element Method (SEM) Advantages of each method

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Discontinuous Galerkin Method

Use discontinuous functions : Degrees of freedom necessary on each cell :

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Spectral Element Method

General principle

Finite Element Method (FEM) discretization + Gauss-Lobatto quadrature, Gauss-Lobatto points as degrees of freedom (gives us exponential convergence on L2-norm).

  • f (x)dx ≈

N+1

  • j=1

ωjf (ξj), ϕi(ξj) = δij.

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Advantages of each method

DG

Element per element computation ( hp-adaptivity). Time discretization quasi explicit (block diagonal mass matrix). Simple to parallelize. Robust to brutal changes of physics and geometry

SEM

Couples the flexibility of FEM with the accuracy of the pseudo-spectral method. Simplifies the mass and stiffness matrices (mass matrix diagonal).

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2

Comparison DG/SEM on structured quadrangle mesh Description of the test cases Comparative tables

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Description of the test cases

Physical parameters

P wavespeed 1000 m.s−1 Density 1 kg.m−3 Second order Ricker Source in Pwave (fpeak = 10Hz)

General context

Acoustic homogeneous medium. Four different meshes : 10000 cells, 22500 cells, 90000 cells, 250000 cells. CFL computed using power iteration method. Leap-Frog time scheme. Eight threads parallel execution with OpenMP.

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time Nb of time steps DG(α = 0) 3.18e-3 2e-1 5.13 629 SEM 4.9e-3 5e-2 0.80 409

Table: DG not penalized and SEM comparison on the 10000 cells case

CFL L2-error CPU-time(s) Nb of time steps DG(α = 0) 2.12e-3 7e-1 18.11 943 SEM 3.26e-3 4e-2 3.54 613

Table: DG not penalized and SEM comparison on the 20000 cells case

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time Nb of time steps DG(α = 0) 3.18e-3 2e-1 5.13 629 SEM 4.9e-3 5e-2 0.80 409

Table: DG not penalized and SEM comparison on the 10000 cells case

CFL L2-error CPU-time(s) Nb of time steps DG(α = 0) 2.12e-3 7e-1 18.11 943 SEM 3.26e-3 4e-2 3.54 613

Table: DG not penalized and SEM comparison on the 20000 cells case

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time Nb of time steps DG(α = 0) 3.18e-3 2e-1 5.13 629 SEM 4.9e-3 5e-2 0.80 409

Table: DG not penalized and SEM comparison on the 10000 cells case

CFL L2-error CPU-time(s) Nb of time steps DG(α = 0) 2.12e-3 7e-1 18.11 943 SEM 3.26e-3 4e-2 3.54 613

Table: DG not penalized and SEM comparison on the 20000 cells case

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time Nb of time steps DG(α = 0.5) 2e-3 3e-2 7.93 1000 SEM 4.9e-3 5e-2 0.80 409

Table: DG penalized and SEM comparison on the 10000 cells case

CFL L2-error CPU-time(s) Nb of time steps DG(α = 0.5) 1.33e-3 2e-2 32.98 1502 SEM 3.26e-3 4e-2 3.54 613

Table: DG penalized and SEM comparison on the 20000 cells case

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time Nb of time steps DG(α = 0.5) 2e-3 3e-2 7.93 1000 SEM 4.9e-3 5e-2 0.80 409

Table: DG penalized and SEM comparison on the 10000 cells case

CFL L2-error CPU-time(s) Nb of time steps DG(α = 0.5) 1.33e-3 2e-2 32.98 1502 SEM 3.26e-3 4e-2 3.54 613

Table: DG penalized and SEM comparison on the 20000 cells case

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time Nb of time steps DG(α = 0.5) 2e-3 3e-2 7.93 1000 SEM 4.9e-3 5e-2 0.80 409

Table: DG penalized and SEM comparison on the 10000 cells case

CFL L2-error CPU-time(s) Nb of time steps DG(α = 0.5) 1.33e-3 2e-2 32.98 1502 SEM 3.26e-3 4e-2 3.54 613

Table: DG penalized and SEM comparison on the 20000 cells case

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time Nb of time steps DG(α = 0.5) 2e-3 3e-2 7.93 1000 SEM 4.9e-3 5e-2 0.80 409

Table: DG penalized and SEM comparison on the 10000 cells case

CFL L2-error CPU-time(s) Nb of time steps DG(α = 0.5) 1.33e-3 2e-2 32.98 1502 SEM 3.26e-3 4e-2 3.54 613

Table: DG penalized and SEM comparison on the 20000 cells case

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time(s) Nb of time steps DG(α = 0.5) 2e-3 3e-2 7.93 1000 SEM 2e-3 3e-2 2.12 1000

Table: DG penalized and SEM comparison using the same CFL on a 10000 thousands cells mesh

CFL L2-error CPU-time(s) Nb of time steps DG (α = 0.5) 1.33e-3 2e-2 32.98 1502 SEM 1.33e-3 2e-2 8.67 1502

Table: DG penalized and SEM comparison using the same CFL on a 20000 thousands cells mesh

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time(s) Nb of time steps DG(α = 0.5) 2e-3 3e-2 7.93 1000 SEM 2e-3 3e-2 2.12 1000

Table: DG penalized and SEM comparison using the same CFL on a 10000 thousands cells mesh

CFL L2-error CPU-time(s) Nb of time steps DG (α = 0.5) 1.33e-3 2e-2 32.98 1502 SEM 1.33e-3 2e-2 8.67 1502

Table: DG penalized and SEM comparison using the same CFL on a 20000 thousands cells mesh

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Comparative tables

Error computed as the difference between an analytical and a numerical solution for each method. Three cases considered : DG without penalization terms, DG with penalization terms and SEM. CFL L2-error CPU-time(s) Nb of time steps DG(α = 0.5) 2e-3 3e-2 7.93 1000 SEM 2e-3 3e-2 2.12 1000

Table: DG penalized and SEM comparison using the same CFL on a 10000 thousands cells mesh

CFL L2-error CPU-time(s) Nb of time steps DG (α = 0.5) 1.33e-3 2e-2 32.98 1502 SEM 1.33e-3 2e-2 8.67 1502

Table: DG penalized and SEM comparison using the same CFL on a 20000 thousands cells mesh

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Advantages of each method

DG

Element per element computation ( hp-adaptivity). Time discretization quasi explicit (block diagonal mass matrix). Simple to parallelize. Robust to brutal changes of physics and geometry

SEM

Couples the flexibility of FEM with the accuracy of the pseudo-spectral method. Simplifies the mass and stiffness matrices (mass matrix diagonal). Reduces the computational costs on structured quadrangle cells in comparison with DG

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3

DG/SEM coupling Hybrid meshes structures Variational formulation

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Hybrid meshes structures

Aim at coupling Pk and Qk structures. Need to extend or split some structures (e.g. neighbour indices). Define new face matrices: MK,L

ij

=

  • K∩L

φK

i φL j ,

MK,L

ij

=

  • K∩L

ψK

i ψL j ,

MK,L

ij

=

  • K∩L

φK

i ψL j .

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Variational formulation

Global context

Domain in two parts : Ωh,1 (structured quadrangles + SEM), Ωh,2 (unstructured triangles + DG).

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Variational formulation

Γint K1 K2 Fluxes

Definitions

Jump and average. [[u]] = (uK1nK1 + uK2nK2) {{u}} = 1 2 (uK2 + uK1)

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Variational formulation

SEM variational formulation :           

  • Ωh,1

ρ∂tv1 · w1 = −

  • Ωh,1

σ1 : ∇w1 +

  • Γout,1

(σ1n1) · w1,

  • Ωh,1

∂tσ1 : ξ1 =

  • Ωh,1

(Cξ1) : ∇v1. DG variational formulation :           

  • Ωh,2

ρ∂tv2 · w2 = −

  • Ωh,2

σ2 : ∇w2 +

  • Γout,2

(σ2n2) · w2+

  • Γint

{{σ2}} : [[w2]],

  • Ωh,2

∂tσ2 : ξ2 =−

  • Ωh,2

(∇ · (Cξ2)) · v2 +

  • Γout,2

(Cξ2n2) · v2+

  • Γint

{{v2}} · [[Cξ2]].

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Variational formulation

                                                                

  • Ωh,1

ρ∂tv1 · w1 +

  • Ωh,2

ρ∂tv2 · w2 = −

  • Ωh,1

σ1 : ∇w1 −

  • Ωh,2

σ2 : ∇w2 +

  • Γout,1

(σ1n1) · w1 +

  • Γout,2

(σ2n2) · w2 +

  • Γint

{{σ2}} : [[w2]] +

  • Γ1/2

[[σw]],

  • Ωh,1

∂tσ1 : ξ1 +

  • Ωh,2

∂tσ2 : ξ2 =

  • Ωh,1

(Cξ1) : ∇v1 −

  • Ωh,2

(∇ · (Cξ2)) · v2 +

  • Γout,2

(Cξ2n2) · v2 +

  • Γint

{{v2}} · [[Cξ2]] +

  • Γ1/2

[[(Cξ)v]].

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Variational formulation

                                                                

  • Ωh,1

ρ∂tv1 · w1 +

  • Ωh,2

ρ∂tv2 · w2 = −

  • Ωh,1

σ1 : ∇w1 −

  • Ωh,2

σ2 : ∇w2 +

  • Γout,1

(σ1n1) · w1 +

  • Γout,2

(σ2n2) · w2 +

  • Γint

{{σ2}} : [[w2]] +

  • Γ1/2

{{σ}} : [[w]] + [[σ]] · {{w}},

  • Ωh,1

∂tσ1 : ξ1 +

  • Ωh,2

∂tσ2 : ξ2 =

  • Ωh,1

(Cξ1) : ∇v1 −

  • Ωh,2

(∇ · (Cξ2)) · v2 +

  • Γout,2

(Cξ2n2) · v2 +

  • Γint

{{v2}} · [[Cξ2]] +

  • Γ1/2

{{v}} · [[Cξ]] + {{Cξ}} : [[v]].

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Variational formulation

                                                                

  • Ωh,1

ρ∂tv1 · w1 +

  • Ωh,2

ρ∂tv2 · w2 = −

  • Ωh,1

σ1 : ∇w1 −

  • Ωh,2

σ2 : ∇w2 +

  • Γout,1

(σ1n1) · w1 +

  • Γout,2

(σ2n2) · w2 +

  • Γint

{{σ2}} : [[w2]] +

  • Γ1/2

{{σ}} : [[w]] ✭✭✭✭✭

✭

+[[σ]] · {{w}},

  • Ωh,1

∂tσ1 : ξ1 +

  • Ωh,2

∂tσ2 : ξ2 =

  • Ωh,1

(Cξ1) : ∇v1 −

  • Ωh,2

(∇ · (Cξ2)) · v2 +

  • Γout,2

(Cξ2n2) · v2 +

  • Γint

{{v2}} · [[Cξ2]] +

  • Γ1/2

{{v}} · [[Cξ]] ✭✭✭✭✭

✭

+{{Cξ}} : [[v]].

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Energy study

Goal : Show that our coupling preserves the energy We set ξ1 = σ1, ξ2 = σ2, w1 = v1, w2 = v2 d dt E = 0 with E =

  • Ωh,1

ρ∂tv1 · v1 +

  • Ωh,2

ρ∂tv2 · v2 +

  • Ωh,1

∂tσ1 : σ1 +

  • Ωh,2

∂tσ2 : σ2

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4

DGSEM vs DG

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DGSEM vs DG

Two dimensional Mesh with 74969 cells : 53969 triangles and 21000 quadrangles.

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DGSEM vs DG

Two dimensional Mesh with 74969 cells : 53969 triangles and 21000 quadrangles. Elastic isotropic case with order 2 on triangles (DG) and order 8 on the quadrangles (DG or SEM), point source at the center of the 3000x3000 domain. CFL CPU-time Nb of time steps DG(α = 0.5) 1e-4 4148 10000 DG/SEM 1e-4 2645 10000

Figure: DG and DG/SEM comparison on an elastic isotropic case

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DGSEM vs DG

Two dimensional Mesh with 74969 cells : 53969 triangles and 21000 quadrangles. Elastic isotropic case with order 2 on triangles (DG) and order 8 on the quadrangles (DG or SEM), point source at the center of the 3000x3000 domain. ”Salt dome” case with order 2 on the triangles (DG) and order 8 on the quadrangles (DG or SEM), point source on the top layer of water. CFL CPU-time Nb of time steps DG(α = 0.5) 1e-4 4148 10000 DG/SEM 1e-4 2645 10000

Figure: DG and DG/SEM comparison on an elastic isotropic case

CFL CPU-time Nb of time steps DG(α = 0.5) 3e-5 38378 33333 DG/SEM 3e-5 22260 33333

Figure: DG and DG/SEM comparison on a ”salt dome” case

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5

3D extension

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General settings

Figure: Hexa/Tet boundary configuration

Only deal with a simple case of 3D hybrid meshes : one hexahedron has only two tetrahedra as neighbour. Extend SEM in 3D (basis functions...). Require introducing a new matrix which handles the rotation cases between two elements.

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General settings

Figure: Hexa/Tet boundary configuration

Only deal with a simple case of 3D hybrid meshes : one hexahedron has only two tetrahedra as neighbour. Extend SEM in 3D (basis functions...). Require introducing a new matrix which handles the rotation cases between two elements.

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General settings

Figure: Hexa/Tet boundary configuration

Only deal with a simple case of 3D hybrid meshes : one hexahedron has only two tetrahedra as neighbour. Extend SEM in 3D (basis functions...). Require introducing a new matrix which handles the rotation cases between two elements.

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General settings

Figure: Hexa/Tet boundary configuration

Only deal with a simple case of 3D hybrid meshes : one hexahedron has only two tetrahedra as neighbour. Require introducing a new matrix which handles the rotation cases between two elements.

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6

Perfectly Matched Layer(PML) Mathematical formulation ADE-PML (Auxiliary Differential Equation) DGM vs DG/SEM simulation

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What is a PML and why using it?

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What is a PML and why using it?

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What is a PML and why using it?

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What is a PML and why using it?

Easy to implement

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What is a PML and why using it?

Easy to implement Not really adapted to the simulation of ”infinite” domain

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What is a PML and why using it?

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What is a PML and why using it?

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What is a PML and why using it?

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What is a PML and why using it?

Possibility to simulate an infinite domain

Aur´ elien Citrain DG-SE coupling WAVES 2019 30 / 44

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SLIDE 67

What is a PML and why using it?

Possibility to simulate an infinite domain Complicated to implement when the order of the condition increase

Aur´ elien Citrain DG-SE coupling WAVES 2019 30 / 44

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SLIDE 68

What is a PML and why using it?

Possibility to simulate an infinite domain Complicated to implement when the order of the condition increase Apparition of reflection when it comes to waves with grazing incidence

Aur´ elien Citrain DG-SE coupling WAVES 2019 30 / 44

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SLIDE 69

What is a PML and why using it?

Aur´ elien Citrain DG-SE coupling WAVES 2019 31 / 44

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SLIDE 70

What is a PML and why using it?

Aur´ elien Citrain DG-SE coupling WAVES 2019 31 / 44

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SLIDE 71

What is a PML and why using it?

Possibility to simulate an infinite domain

Aur´ elien Citrain DG-SE coupling WAVES 2019 31 / 44

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SLIDE 72

What is a PML and why using it?

Possibility to simulate an infinite domain

Aur´ elien Citrain DG-SE coupling WAVES 2019 32 / 44

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SLIDE 73

What is a PML and why using it?

Possibility to simulate an infinite domain Some stability problems on elastic case using DGM

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SLIDE 74

Classic PML formulation

Step 1: Rewrite the system in the frequency domain      iωρv = ∇ · σ iωσ = C : ∇v

Aur´ elien Citrain DG-SE coupling WAVES 2019 33 / 44

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SLIDE 75

Classic PML formulation

Step 1: Rewrite the system in the frequency domain      iωρv = ∇ · σ iωσ = C : ∇v Step 2: Introduce a new system of complex coordinates Define dz the damping :

  • dz > 0 if z ∈ ΩPML

dz = 0 if z / ∈ ΩPML

Aur´ elien Citrain DG-SE coupling WAVES 2019 33 / 44

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SLIDE 76

Classic PML formulation

Step 2: Introduce a new system of complex coordinates Define dz the damping :

  • dz > 0 if z ∈ ΩPML

dz = 0 if z / ∈ ΩPML

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SLIDE 77

Classic PML formulation

Step 2: Introduce a new system of complex coordinates Define dz the damping :

  • dz > 0 if z ∈ ΩPML

dz = 0 if z / ∈ ΩPML ˜ z(z) = z − i ω z dz(s)ds

Aur´ elien Citrain DG-SE coupling WAVES 2019 34 / 44

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SLIDE 78

Classic PML formulation

Step 2: Introduce a new system of complex coordinates Define dz the damping :

  • dz > 0 if z ∈ ΩPML

dz = 0 if z / ∈ ΩPML ˜ z(z) = z − i ω z dz(s)ds Define the associate differential operator : ∂˜

z =

iω iω + dz ∂z = 1 − dz iω + dz ∂z

Aur´ elien Citrain DG-SE coupling WAVES 2019 34 / 44

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SLIDE 79

Classic PML formulation

Step 3: Rewrite the system iωρvx = ∂xσxx + ∂˜

zσxz,

iωρvz = ∂xσxz + ∂˜

zσzz,

iωσxx = (λ + 2µ)∂xvx + λ∂˜

zvz,

iωσzz = λ∂xvx + (λ + 2µ)∂˜

zvz,

iωσxz = µ(∂xvz + ∂˜

zvx).

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SLIDE 80

ADE-PML (Auxiliary Differential Equation)

Martin, Roland and Komatitsch, Dimitri and Gedney, Stephen D and Bruthiaux, Emilien and

  • thers

A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML) Computer Modeling in Engineering and Sciences (CMES),2010 Introduction of 2d + 1 new variables ψ⋆ ∈ H1 → Simplify the implementation BUT add new equation to the linear system

Aur´ elien Citrain DG-SE coupling WAVES 2019 36 / 44

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SLIDE 81

ADE-PML (Auxiliary Differential Equation)

Martin, Roland and Komatitsch, Dimitri and Gedney, Stephen D and Bruthiaux, Emilien and

  • thers

A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML) Computer Modeling in Engineering and Sciences (CMES),2010 Introduction of 2d + 1 new variables ψ⋆ ∈ H1 → Simplify the implementation BUT add new equation to the linear system iωρvz = ∂xσxz + ∂˜

zσzz

Aur´ elien Citrain DG-SE coupling WAVES 2019 36 / 44

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SLIDE 82

ADE-PML (Auxiliary Differential Equation)

Martin, Roland and Komatitsch, Dimitri and Gedney, Stephen D and Bruthiaux, Emilien and

  • thers

A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML) Computer Modeling in Engineering and Sciences (CMES),2010 Introduction of 2d + 1 new variables ψ⋆ ∈ H1 → Simplify the implementation BUT add new equation to the linear system iωρvz = ∂xσxz + ∂˜

zσzz

↓ Back to original coordinates

Aur´ elien Citrain DG-SE coupling WAVES 2019 36 / 44

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SLIDE 83

ADE-PML (Auxiliary Differential Equation)

Martin, Roland and Komatitsch, Dimitri and Gedney, Stephen D and Bruthiaux, Emilien and

  • thers

A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML) Computer Modeling in Engineering and Sciences (CMES),2010 Introduction of 2d + 1 new variables ψ⋆ ∈ H1 → Simplify the implementation BUT add new equation to the linear system iωρvz = ∂xσxz + ∂˜

zσzz

↓ Back to original coordinates iωρvz = ∂xσxz + ∂zσzz − dz dz + iω ∂zσzz

Aur´ elien Citrain DG-SE coupling WAVES 2019 36 / 44

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SLIDE 84

ADE-PML (Auxiliary Differential Equation)

Martin, Roland and Komatitsch, Dimitri and Gedney, Stephen D and Bruthiaux, Emilien and

  • thers

A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML) Computer Modeling in Engineering and Sciences (CMES),2010 Introduction of 2d + 1 new variables ψ⋆ ∈ H1 → Simplify the implementation BUT add new equation to the linear system iωρvz = ∂xσxz + ∂˜

zσzz

↓ Back to original coordinates iωρvz = ∂xσxz + ∂zσzz − dz dz + iω ∂zσzz ↓ Back to time domain

Aur´ elien Citrain DG-SE coupling WAVES 2019 36 / 44

slide-85
SLIDE 85

ADE-PML (Auxiliary Differential Equation)

Martin, Roland and Komatitsch, Dimitri and Gedney, Stephen D and Bruthiaux, Emilien and

  • thers

A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using Auxiliary Differential Equations (ADE-PML) Computer Modeling in Engineering and Sciences (CMES),2010 Introduction of 2d + 1 new variables ψ⋆ ∈ H1 → Simplify the implementation BUT add new equation to the linear system iωρvz = ∂xσxz + ∂˜

zσzz

↓ Back to original coordinates iωρvz = ∂xσxz + ∂zσzz − dz dz + iω ∂zσzz ↓ Back to time domain ρ∂tvx = ∂xσxz + ∂zσzz − dz dz + ∂t ∂zσzz

Aur´ elien Citrain DG-SE coupling WAVES 2019 36 / 44

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SLIDE 86

ADE-PML (Auxiliary Differential Equation)

Definition of the memory variable ψσzz : ψσzz = − dz dz + ∂t ∂zσzz

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SLIDE 87

ADE-PML (Auxiliary Differential Equation)

Definition of the memory variable ψσzz : ψσzz = − dz dz + ∂t ∂zσzz ρ∂tvz = ∂xσxz + ∂zσzz + ψσzz

Aur´ elien Citrain DG-SE coupling WAVES 2019 37 / 44

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SLIDE 88

ADE-PML (Auxiliary Differential Equation)

Definition of the memory variable ψσzz : ψσzz = − dz dz + ∂t ∂zσzz ρ∂tvz = ∂xσxz + ∂zσzz + ψσzz ∂tψσzz = −dz∂zσzz − dzψσzz

Aur´ elien Citrain DG-SE coupling WAVES 2019 37 / 44

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SLIDE 89

ADE-PML (Auxiliary Differential Equation)

Definition of the memory variable ψσzz : ψσzz = − dz dz + ∂t ∂zσzz ρ∂tvz = ∂xσxz + ∂zσzz + ψσzz ∂tψσzz = −dz∂zσzz − dzψσzz

ADE with C-PML (Convolutional PML)

∂˜

z =

  • 1 −

dz αz + iω + dz

  • ∂z

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SLIDE 90

ADE-PML (Auxiliary Differential Equation)

Definition of the memory variable ψσzz : ψσzz = − dz dz + ∂t ∂zσzz ρ∂tvz = ∂xσxz + ∂zσzz + ψσzz ∂tψσzz = −dz∂zσzz − dzψσzz

ADE with C-PML (Convolutional PML)

∂˜

z =

  • 1 −

dz αz + iω + dz

  • ∂z

∂tψσzz = −dz∂zσzz − (dz + αz)ψσzz

Aur´ elien Citrain DG-SE coupling WAVES 2019 37 / 44

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SLIDE 91

DG and DGSEM comparison

(a) t=1.5s (b) t=1.5s

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SLIDE 92

DG and DGSEM comparison

(c) t=3s (d) t=3s

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SLIDE 93

DG and DGSEM comparison

(e) t=5s (f) t=5s

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SLIDE 94

DG and DGSEM comparison

(g) t=8s (h) t=8s

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SLIDE 95

DG and DGSEM comparison

(i) t=10s (j) t=10s

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SLIDE 96

Conclusion and perspectives

Conclusion

1

SEM is more efficient on structured quadrangle mesh than DG

2

Build a variational formulation for DG/SEM coupling and find a CFL condition that ensures stability

3

Show the utility of using hybrid meshes and method coupling (reduce computational cost,...)

Achievements

Implement DG/SEM coupling on the code (2D) Develop DG/SEM coupling in 3D Develop PML in the hexahedral part

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SLIDE 97

Thank you for your attention ! Questions?

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