Spectral Element Method and Discontinuous Galerkin

Home , Spectral element method

Spectral Element Method and Discontinuous Galerkin approximation for elasto-acoustic problems Hélène Barucq, Henri Calandra, Aurélien Citrain, Julien Diaz, Christian Gout

To cite this version:

Hélène Barucq, Henri Calandra, Aurélien Citrain, Julien Diaz, Christian Gout. Spectral Element Method and Discontinuous Galerkin approximation for elasto-acoustic problems. MATHIAS – TOTAL Symposium on Mathematics, Oct 2017, Paris, France. hal-01690670

HAL Id: hal-01690670 https://hal.archives-ouvertes.fr/hal-01690670 Submitted on 23 Jan 2018

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the coupling of Spectral Element Method with Discontinuous Galerkin approximation for elasto-acoustic problems.

HélèneBarucq1, Henri Calandra2, AurélienCitrain1,3, Julien Diaz1 and Christian Gout3

1 Team project Magique.3D, INRIA.UPPA.CNRS, Pau, France.

2 TOTAL SA, CSTJS, Pau, France.

3 INSA Rouen-Normandie Universit´e,LMI EA 3226, 76000, Rouen.

MATHIAS 2017 October 25-27

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 1 / 22 Why using hybrid meshes?

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

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 2 / 22 Elastodynamic system

d x ∈ Ω ⊂ R , t ∈ [0, T ], T > 0 :

 ∂v ρ(x) (x, t)= ∇ · σ(x, t)  ∂t

 ∂σ  (x, t)= C(x)(v(x, t)) ∂t

With : ρ(x) the density C(x) the elasticity tensor (x, t) the deformation tensor v(x, t), the wavespeed σ(x, t) the strain tensor

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 3 / 22 Elasticus software

Software written in Fortran 90 for wave propagation simulation in the time domain

Features Simulation: on various types of meshes (unstructured triangle, structured quadrangle, hybrid) on heterogeneous media (acoustic, elastic and elasto-acoustic)

Discontinuous Galerkin (DG) on quadrangle, triangle and hybrid mesh Spectral Element Method (SEM) only on quadrangle mesh with various time-schemes : Runge-Kutta (2 or 4), Leap-Frog with p-adaptivity, multi-order computation...

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 4 / 22 Table of contents

1 Numerical Methods

2 Comparison DG/SEM on structured quadrangle mesh

3 DG/SEM coupling

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 5 / 22 1 Numerical Methods Discontinous Galerkin Method (DG) Spectral Element Method (SEM) Advantages of each method

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 6 / 22 Discontinuous Galerkin Method

Use discontinuous functions :

Degrees of freedom necessary on each cell :

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 7 / 22 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)

N+1 Z X f (x)dx ≈ ωj f (ξj ) j=1

ϕi (ξj ) = δij

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 8 / 22 Spectral Element Method

Main change with DG DG discontinuous, SEM continuous Need to deﬁne local to global numbering Global matrices needed for SEM Basis functions computed diﬀerently

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 9 / 22 Advantages of each method

DG Element per element computation ( hp-adaptivity) Time discretization quasi explicit (block diagonal mass matrix) Simple to parallelize

SEM Couples the flexibility of FEM with the accuracy of the pseudo-spectral method Reduces the computational cost when you use structured meshes in comparison with DG Simplifies the mass and stiff matrices (mass matrix diagonal)

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 10 / 22 2 Comparison DG/SEM on structured quadrangle mesh Description of the test cases Comparative tables

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 11 / 22 Description of the test cases

Physical parameters 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 Four threads parallel execution with OpenMP

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

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 12 / 22 Comparative tables

Comparison between numerical solution and analytical solution obtained using the software Gar6more

Quadrangle mesh 10000 elements: CFL L2-error CPU-time Nb of time steps DG 1.99e-3 2.5e-2 61.96 500 SEM 4.9e-3 1.3e-1 0.73 204 SEM(DG CFL) 1.99e-3 4.8e-2 1.48 502

Quadrangle mesh 22500 elements: CFL L2-error CPU-time Nb of time steps DG 1.33e-3 1.8e-2 252.20 750 SEM 3.26e-3 7e-2 2.42 306 SEM(DG CFL) 1.33e-3 1.2e-2 4.70 751

SEM ﬁfty time much faster on a mesh with 22500 cells than DG

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 13 / 22 3 DG/SEM coupling Hybrid meshes structures Variationnal formulation Space discretization

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 14 / 22 Hybrid meshes structures

Need to couple Pk and Qk structures. Need to extend or split some of the structures (e.g. neighbour indexes) Necessity to deﬁne new face matrices

Z Z Z K,L K L K,L K L K,L K L Mij = φi φj , Mij = ψi ψj , Mij = φi ψj K∩L K∩L K∩L

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 15 / 22 Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 16 / 22 Variationnal formulation

Global context

Domain in two parts : Ωh,1 (structured quadrangle + SEM), Ωh,2 (unstructured triangle + DG)

w1,w2 the tests-function and ξ1, ξ2 the tests-tensors

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 17 / 22 Variationnal formulation

SEM variationnal formulation :

Z Z Z  ρ∂t v1 · w1 = − σ1 · ∇w1 + (σ1n1) · w1   Ωh,1 Ωh,1 Γout Z Z Z   ∂t σ1 : ξ1 = − (∇(Cξ1)) · v1 + (Cξ1n1) · v1 Ωh,1 Ωh,1 Γout

DG variationnal formulation: Z Z Z Z  ρ∂t v2 · w2 = − σ2 · ∇w2 + (σ2n2) · w2+ {{σ2}}[[w2]] · n2   Ωh,2 Ωh,2 Γout Γint Z Z Z Z   ∂t σ2 : ξ2 =− (∇(Cξ2)) · v2 + (Cξ2n2) · v2+ {{v2}}[[Cξ2]] · n2 Ωh,2 Ωh,2 Γout Γint

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 17 / 22 Variationnal formulation

SEM variationnal formulation :

Z Z Z Z  ρ∂t v1 · w1 = − σ1 · ∇w1 + (σ1n1) · w1 + (σ1n1) · w1  Ω Ω Γ Γ  h,1 h,1 out DG/SEM

Z Z Z Z  ∂t σ1 : ξ1 = − (∇(Cξ1)) · v1 + (Cξ1n1) · v1+ (Cξ1n1) · v1  Ωh,1 Ωh,1 Γout ΓDG/SEM

DG variationnal formulation: Z Z Z Z Z  ρ∂t v2 · w2 = − σ2 · ∇w2 + (σ2n2) · w2+ {{σ2}}[[w2]] · n2+ (σ2n2) · w2  Ω Ω Γ Γ Γ  h,2 h,2 out int DG/SEM

Z Z Z Z Z  ∂t σ2 : ξ2 =− (∇(Cξ2)) · v2 + (Cξ2n2) · v2+ {{v2}}[[Cξ2]] · n2 + (Cξ2n2) · v2  Ωh,2 Ωh,2 Γout Γint ΓDG/SEM

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 17 / 22 Variationnal formulation

Computation steps

1 Simplify the coupling terms and separates the two parts + put σ · n = 0

Z Z 1 Z  ρ∂t v1 · w1 = − σ1 · ∇w1 + (σ1 + σ2)n1 · w1  Ω Ω 2 Γ  h,1 h,1 DG/SEM

Z Z 1 Z  ∂t σ1 : ξ1 = − (∇(Cξ1)) · v1 + (Cξ1n1) · (v1 + v2)  Ωh,1 Ωh,1 2 ΓDG/SEM

Z Z Z Z 1  ρ∂t v2 · w2 = − σ2 · ∇w2 + {{σ2}}[[w2]] · n2− 2 (σ1 + σ2)n1 · w2  Ω Ω Γ Γ  h,2 h,2 int DG/SEM

Z Z Z 1 Z  ∂t σ2 : ξ2 = − (∇(Cξ2)) · v2 + {{v2}}[[Cξ2]] · n2− (Cξ2n1) · (v1 + v2)  Ωh,2 Ωh,2 Γint 2 ΓDG/SEM

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 18 / 22 Space discretization : SEM part

ϕi : SEM basis functions

ψi : DG basis functions

 M ∂ v + R σ + R2,1σ = 0  v 1 t h,1 σ1 h,1 σ2 h,2

 2,1 Mσ1 ∂t σh,1 + Rv 1 v h,1 + Rv 2 v h,2 = 0

(r+1)d Z X X X Mij = ϕi ϕj ≈ ωk ϕi (ξk )ϕj (ξk ) = ωi δi,j the mass matrix Ω e∈supp(ϕi )∩supp(ϕj ) k=1 e∈supp(ϕi )∩supp(ϕj )

Z ∂ϕj Rpij = ϕi stiﬀness matrix Ω ∂p

Matrix of DG/SEM coupling : Z R2,1 = ψ ϕ σ2,ij i j ∂Ω1∩∂Ω2

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 19 / 22 Space discretization : DG part

ρM ∂ v + R σ − R1,2σ = 0  v 2 t h,2 σ2 h,2 σ1 h,1  M ∂ σ + R v − R1,2v = 0  σ2 t h,2 v 2 h,2 v 1 h,1

Z K K K Mij = ψi ψj mass matrix, K Z ∂ψK RK = ψK j stiﬀness matrix, pij i K ∂p Z K,L K L Rpij = ψi ψj nK · ep the mass-face matrices ∂K∩∂L 1,2 Two new matrices which come from the DG/SEM hybridation R? . Block composed :

1 Z R1,2 = R1,2 = − ψK2 ϕ ∀i, j = 1..N (1) v1 σ1 j i m 2 ∂Ω2∩∂K1

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 20 / 22 Conclusion and perspectives

Conclusion 1 As expected, SEM is more eﬃcient on structured quadrangle mesh than DG 2 Show the utility on the use of hybrid meshes and method coupling (reduce computational cost,...) 3 Build a variationnal formulation for DG/SEM coupling and ﬁnd a CFL condition that ensures stability

Perspectives Implement DG/SEM coupling on the code (2D) Develop h-adaptivity for the structured part Develop DG/SEM coupling in 3D

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 21 / 22 Thank you for your attention !

Questions?

Aur´elienCitrain Coupling DG/SEM MATHIAS 2017 October 25-27 22 / 22