Coupling DG/SEM Hybrid space discretization to solve elasto-acoustic coupling elien Citrain 1 , 3 , Julien Diaz 1 and Christian ene Barucq 1 , Henri Calandra 2 , Aur´ H´ el` Gout 3 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. ( m 2 op ) V Congress - Bilbao 24/05/2018
Coupling DG/SEM Why using hybrid meshes? water water sand sandstone salt 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
Coupling DG/SEM Elastodynamic system 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
Coupling DG/SEM Elasticus software Written in Fortran 90 for wave propagation simulation in the time domain Features Using various types of meshes ( unstructured triangle, structured quadrilaterals, hybrid ) Modelling of various physics ( acoustic, elastic and elasto-acoustic ) Discontinuous Galerkin Method (DG) based on structured quadrilaterals, triangle and hybrid meshes Spectral Element Method (SEM) only on structured quadrilaterals mesh DG/SEM coupling on 2D hybrid meshes with various time-schemes : Runge-Kutta (2 or 4), Leap-Frog with p-adaptivity , multi-order computation...
Coupling DG/SEM Table of contents 1 Numerical Methods 2 Comparison DG/SEM on structured quadrangle mesh 3 DG/SEM coupling 4 Comparison between DG/SEM an DG on hybrid meshes 5 3D extension
Coupling DG/SEM Numerical Methods 1 Numerical Methods Discontinuous Galerkin Method (DG) Spectral Element Method (SEM) Advantages of each method
Coupling DG/SEM Numerical Methods Discontinuous Galerkin Method (DG) Discontinuous Galerkin Method Use discontinuous functions : Degrees of freedom necessary on each cell :
Coupling DG/SEM Numerical Methods Spectral Element Method (SEM) Spectral Element Method General principle Finite Element Method (FEM) discretization + Gauss-Lobatto quadrature Gauss-Lobatto points as degrees of freedom ( exponential convergence on L 2 -norm) N +1 � � f ( x ) dx ≈ ω j f ( ξ j ) j =1 ϕ i ( ξ j ) = δ ij
Coupling DG/SEM Numerical Methods Spectral Element Method (SEM) Spectral Element Method Main change with DG DG discontinuous, SEM continuous Need of defining local to global numbering Global matrices required by SEM Basis functions computed differently
Coupling DG/SEM Numerical Methods Advantages of each method 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 using structured meshes in comparison with DG
Coupling DG/SEM DG/SEM Comparison 2 Comparison DG/SEM on structured quadrangle mesh Description of the test cases Comparative tables
Coupling DG/SEM DG/SEM Comparison Description of the test cases Description of the test cases Physical parameters General context Acoustic homogeneous medium Four differents 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 1000 m . s − 1 P wavespeed 1 kg . m − 3 Density Second order Ricker Source in Pwave ( f peak = 10 Hz )
Coupling DG/SEM DG/SEM Comparison Comparative tables Comparative tables Error computed as the difference between an analytical and a numerical solution for each method Quadrangle mesh 10000 elements: CFL L2-error CPU-time Nb of time steps DG 1.99e-3 2.5e-2 19.30 500 SEM 4.9e-3 1.3e-1 0.36 204 SEM(DG CFL) 1.99e-3 4.8e-2 0.73 502 Quadrangle mesh 22500 elements: CFL L2-error CPU-time Nb of time steps DG 1.33e-3 1.8e-2 100.48 750 SEM 3.26e-3 7e-2 1.19 306 SEM(DG CFL) 1.33e-3 1.2e-2 2.82 751 SEM fifty time faster than DG on a mesh with 22500 cells
Coupling DG/SEM DG/SEM coupling 3 DG/SEM coupling Hybrid meshes structures Variational formulation Space discretization
Coupling DG/SEM DG/SEM coupling Hybrid meshes structures Hybrid meshes structures Aim at coupling P k and Q k structures. Need to extend or split some of the structures (e.g. neighbour indexes) Necessity to define new face matrices � � � M K , L M K , L M K , L φ K i φ L ψ K i ψ L φ K i ψ L = j , = j , = j ij ij ij K ∩ L K ∩ L K ∩ L
Coupling DG/SEM DG/SEM coupling Variational formulation Variational formulation Global context Domain in two parts : Ω h , 1 ( structured quadrangle + SEM ), Ω h , 2 ( unstructured triangle + DG )
Coupling DG/SEM DG/SEM coupling Variational formulation Variational formulation SEM variational formulation : � � � ρ∂ t v 1 · w 1 = − σ 1 · ∇ w 1 + ( σ 1 n 1 ) · w 1 Ω h , 1 Ω h , 1 Γ out , 1 � � � ∂ t σ 1 : ξ 1 = − ( ∇ ( C ξ 1 )) · v 1 + ( C ξ 1 n 1 ) · v 1 Ω h , 1 Ω h , 1 Γ out , 1 DG variational formulation : � � � � ρ∂ t v 2 · w 2 = − σ 2 · ∇ w 2 + ( σ 2 n 2 ) · w 2 + {{ σ 2 }} [[ w 2 ]] · n 2 Ω h , 2 Ω h , 2 Γ out , 2 Γ int � � � � ∂ t σ 2 : ξ 2 = − ( ∇ ( C ξ 2 )) · v 2 + ( C ξ 2 n 2 ) · v 2 + {{ v 2 }} [[ C ξ 2 ]] · n 2 Ω h , 2 Ω h , 2 Γ out , 2 Γ int
Coupling DG/SEM DG/SEM coupling Variational formulation Variational formulation Add the average of the solution of each part at the interface + put σ ⋆ n ⋆ = 0 σ 1 · ∇ w 1 + 1 � � � ρ∂ t v 1 · w 1 = − ( σ 1 + σ 2 ) n 1 · w 1 2 Ω h , 1 Ω h , 1 Γ 1 / 2 ( ∇ ( C ξ 1 )) · v 1 + 1 � � � ∂ t σ 1 : ξ 1 = − ( C ξ 1 n 1 ) · ( v 1 + v 2 ) 2 Ω h , 1 Ω h , 1 Γ 1 / 2 {{ σ 2 }} [[ w 2 ]] · n 2 − 1 � � � � ρ∂ t v 2 · w 2 = − σ 2 · ∇ w 2 + ( σ 1 + σ 2 ) n 1 · w 2 2 Ω h , 2 Ω h , 2 Γ int Γ 1 / 2 � � � ∂ t σ 2 : ξ 2 = − ( ∇ ( C ξ 2 )) · v 2 + {{ v 2 }} [[ C ξ 2 ]] · n 2 Ω h , 2 Ω h , 2 Γ int − 1 � ( C ξ 2 n 1 ) · ( v 1 + v 2 ) 2 Γ 1 / 2
Coupling DG/SEM DG/SEM coupling Variational formulation Continuous energy study Goal : Show that our coupling preserve the energy We set ξ 1 = σ 1 , ξ 2 = σ 2 , w 1 = v 1 , w 2 = v 2 We add the equations of the two parts variational formulation d dt E = 0
Coupling DG/SEM DG/SEM coupling Space discretization Space discretization : SEM part ϕ i : SEM basis functions ψ i : DG basis functions M v 1 ∂ t v h , 1 + R σ 1 σ h , 1 + R 2 , 1 σ 2 σ h , 2 = 0 M σ 1 ∂ t σ h , 1 + R v 1 v h , 1 + R 2 , 1 v 2 v h , 2 = 0 ( r +1) d � � � � M ij = ϕ i ϕ j ≈ ω k ϕ i ( ξ k ) ϕ j ( ξ k ) = the ω i δ i , j Ω e ∈ supp ( ϕ i ) ∩ supp ( ϕ j ) k =1 e ∈ supp ( ϕ i ) ∩ supp ( ϕ j ) mass matrix � ∂ϕ j R p ij = ϕ i stiffness matrix ∂ p Ω Matrix of DG/SEM coupling : σ 2 , ij = 1 � R 2 , 1 ψ i ϕ j 2 ∂ Ω 1 ∩ ∂ Ω 2
Coupling DG/SEM DG/SEM coupling Space discretization Space discretization : DG part ρ M v 2 ∂ t v h , 2 + R σ 2 σ h , 2 − R 1 , 2 σ 1 σ h , 1 = 0 M σ 2 ∂ t σ h , 2 + R v 2 v h , 2 − R 1 , 2 v 1 v h , 1 = 0 � M K ψ K i ψ K ij = mass matrix, j K ∂ψ K � j ψ K R K p ij = stiffness matrix, i ∂ p K � R K , L ψ K i ψ L = j n K · e p the mass-face matrix p ij ∂ K ∩ ∂ L Two new matrices which come from the DG/SEM coupling R 1 , 2 ⋆ . Block composed : σ 1 = − 1 � ψ K 2 R 1 , 2 = R 1 , 2 ϕ i (1) v 1 j 2 ∂ Ω 2 ∩ ∂ K 1
Coupling DG/SEM DG/SEM coupling Space discretization
Coupling DG/SEM Comparison DG/SEM and DG 4 Comparison between DG/SEM an DG on hybrid meshes Experimentation context Comparative tables
Coupling DG/SEM Comparison DG/SEM and DG Experimentation context Context Acoustic homogeneous medium 54000 triangles 21000 quadrangles Using Leap-Frog time scheme Parallel computation using OpenMP Done with different orders of discretization
Coupling DG/SEM Comparison DG/SEM and DG Comparative tables Comparative tables P 1 − Q 1 computation : P 2 − Q 1 computation : CFL L 2 -error CPU-time CFL L 2 -error CPU-time DG 2e-4 0.05 57.39 DG 1e-4 0.009 279 DG/SEM 2e-4 0.05 17.74 DG/SEM 1e-4 0.01 247 P 1 − Q 2 computation: P 2 − Q 2 computation : CFL L 2 -error CPU-time CFL L 2 -error CPU-time DG 4e-5 0.04 780 DG 3e-5 0.003 1437.05 DG/SEM 4e-5 0.03 114.44 DG/SEM 3e-5 0.008 490
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