Méthodes de Galerkine Discontinues pour l imagerie sismique en domaine fréquentiel

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1 Méthodes de Galerkine Discontinues pour l imagerie sismique en domaine fréquentiel Julien Diaz, Équipe Inria Magique 3D Séminaire de Mathématiques Appliquées, Laboratoire de Mathématiques Nicolas Oresme

2 Introduction Objective : imaging the subsurface Séminaire de Mathématiques Appliquées, Caen 25 avril

3 Introduction Objective : imaging the subsurface 1 Find the hidden interfaces ; kinematic information Séminaire de Mathématiques Appliquées, Caen 25 avril

4 Introduction Objective : imaging the subsurface 1 Find the hidden interfaces ; kinematic information 2 Determine the constitutive parameters ; dynamic information Séminaire de Mathématiques Appliquées, Caen 25 avril

5 Introduction Objective : imaging the subsurface 1 Find the hidden interfaces ; kinematic information 2 Determine the constitutive parameters ; dynamic information Tool: reflection of waves Séminaire de Mathématiques Appliquées, Caen 25 avril

6 Introduction Objective : imaging the subsurface 1 Find the hidden interfaces ; kinematic information 2 Determine the constitutive parameters ; dynamic information Tool: reflection of waves Means: numerical solution of wave equations Séminaire de Mathématiques Appliquées, Caen 25 avril

7 Introduction Objective : imaging the subsurface 1 Find the hidden interfaces ; kinematic information 2 Determine the constitutive parameters ; dynamic information Tool: reflection of waves Means: numerical solution of wave equations Objective (1) : time-reversibility Séminaire de Mathématiques Appliquées, Caen 25 avril

8 Introduction Objective : imaging the subsurface 1 Find the hidden interfaces ; kinematic information 2 Determine the constitutive parameters ; dynamic information Tool: reflection of waves Means: numerical solution of wave equations Objective (1) : time-reversibility Objective (2) : inverse problem Séminaire de Mathématiques Appliquées, Caen 25 avril

9 Things in common Approximation: the propagation medium is highly heterogeneous Efficiency: any algorithm must be elaborated for High Performance Computing (HPC) Séminaire de Mathématiques Appliquées, Caen 25 avril

10 The numerical method we use Finite element approximation based on discontinuous polynomials Séminaire de Mathématiques Appliquées, Caen 25 avril

11 The numerical method we use Finite element approximation based on discontinuous polynomials Séminaire de Mathématiques Appliquées, Caen 25 avril

12 The numerical method we use Finite element approximation based on discontinuous polynomials RTM requires solving plenty of full-wave equations Séminaire de Mathématiques Appliquées, Caen 25 avril

13 The numerical method we use Finite element approximation based on discontinuous polynomials RTM requires solving plenty of full-wave equations RTM is computationally intensive Séminaire de Mathématiques Appliquées, Caen 25 avril

14 Reverse Time Migration in elastic domains Séminaire de Mathématiques Appliquées, Caen 25 avril

15 Reverse Time Migration Séminaire de Mathématiques Appliquées, Caen 25 avril

16 Reverse Time Migration Let (x i ) 1 i N be a set of given points and (S i ) 1 i N be a set of point sources. Séminaire de Mathématiques Appliquées, Caen 25 avril

17 Reverse Time Migration Let (x i ) 1 i N be a set of given points and (S i ) 1 i N be a set of point sources. Basically, in time-domain:s i (x, t) = δ xi f (t). Séminaire de Mathématiques Appliquées, Caen 25 avril

18 Reverse Time Migration Let (x i ) 1 i N be a set of given points and (S i ) 1 i N be a set of point sources. Basically, in time-domain:s i (x, t) = δ xi f (t). and, in harmonic-domain: Ŝ i (x, t) = δ xi. Séminaire de Mathématiques Appliquées, Caen 25 avril

19 Reverse Time Migration Let (x i ) 1 i N be a set of given points and (S i ) 1 i N be a set of point sources. Basically, in time-domain:s i (x, t) = δ xi f (t). and, in harmonic-domain: Ŝ i (x, t) = δ xi. For each (time-domain) source S i, let g ij (t) be the recorded field at point x j and R i (x, t) = δ j g ij (T t) 1 j N Séminaire de Mathématiques Appliquées, Caen 25 avril

20 Reverse Time Migration Let (x i ) 1 i N be a set of given points and (S i ) 1 i N be a set of point sources. Basically, in time-domain:s i (x, t) = δ xi f (t). and, in harmonic-domain: Ŝ i (x, t) = δ xi. For each (time-domain) source S i, let g ij (t) be the recorded field at point x j and R i (x, t) = δ j g ij (T t) 1 j N For each (harmonic-domain) source Ŝi, let ĝ ij (ω) be the recorded field at point x j and ˆR i (x, ω) = δ xj ĝ ij 1 j N Séminaire de Mathématiques Appliquées, Caen 25 avril

21 Reverse Time Migration in time-domain Solve N forward wave equations: 1 c 2 2 u Si t 2 u Si = S i (x, t), i = 1,..., N, Séminaire de Mathématiques Appliquées, Caen 25 avril

22 Reverse Time Migration in time-domain Solve N forward wave equations: 1 c 2 2 u Si t 2 u Si = S i (x, t), i = 1,..., N, Solve N backward wave equations: 1 c 2 2 u Ri t 2 u Ri = R i (x, t), i = 1,..., N, Séminaire de Mathématiques Appliquées, Caen 25 avril

23 Reverse Time Migration in time-domain Solve N forward wave equations: 1 c 2 2 u Si t 2 u Si = S i (x, t), i = 1,..., N, Solve N backward wave equations: 1 c 2 2 u Ri t 2 u Ri = R i (x, t), i = 1,..., N, Apply the imaging condition: N T I(x) = u Si (x, t) u Ri (x, t)dt. i=1 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

24 Reverse Time Migration in harmonic-domain Solve N forward harmonic wave equations: ω 2 c 2 ûs i + û Si = Ŝ i (x, ω), i = 1,..., N, Séminaire de Mathématiques Appliquées, Caen 25 avril

25 Reverse Time Migration in harmonic-domain Solve N forward harmonic wave equations: ω 2 c 2 ûs i + û Si = Ŝ i (x, ω), i = 1,..., N, Solve N backward harmonic wave equations: ω 2 c 2 ûr i + û Ri = ˆR i (x, ω), i = 1,..., N, Séminaire de Mathématiques Appliquées, Caen 25 avril

26 Reverse Time Migration in harmonic-domain Solve N forward harmonic wave equations: ω 2 c 2 ûs i + û Si = Ŝ i (x, ω), i = 1,..., N, Solve N backward harmonic wave equations: ω 2 c 2 ûr i + û Ri = ˆR i (x, ω), i = 1,..., N, Apply the imaging condition: N + I(x) = u Si (x, ω) u Ri (x, t)dω. i=1 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

27 So what? Is there any interest in working in the frequency domain? Indeed, Séminaire de Mathématiques Appliquées, Caen 25 avril

28 So what? Is there any interest in working in the frequency domain? Indeed, Claerbout imaging condition N T I(x) = u Sj (x, t) u Rj (x, T t) i=1 0 Require to store the wavefields at each point and each time-step: lots of computations and memory overflow. Besides, solve 2N wave equations. Séminaire de Mathématiques Appliquées, Caen 25 avril

29 So what? Is there any interest in working in the frequency domain? Indeed, Claerbout imaging condition N T I(x) = u Sj (x, t) u Rj (x, T t) i=1 0 Require to store the wavefields at each point and each time-step: lots of computations and memory overflow. Besides, solve 2N wave equations. Solve 2N Helmholtz equations? Multiple right-hand sides. Hence, invert the same linear system twice Séminaire de Mathématiques Appliquées, Caen 25 avril

30 So what? Is there any interest in working in the frequency domain? Indeed, Claerbout imaging condition N T I(x) = u Sj (x, t) u Rj (x, T t) i=1 0 Require to store the wavefields at each point and each time-step: lots of computations and memory overflow. Besides, solve 2N wave equations. Solve 2N Helmholtz equations? Multiple right-hand sides. Hence, invert the same linear system twice No free lunch: solving the Helmholtz equation in heterogeneous medium is still a tricky task. Séminaire de Mathématiques Appliquées, Caen 25 avril

31 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Séminaire de Mathématiques Appliquées, Caen 25 avril

32 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Séminaire de Mathématiques Appliquées, Caen 25 avril

33 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Adapted to strongly varying velocities; Séminaire de Mathématiques Appliquées, Caen 25 avril

34 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Adapted to strongly varying velocities; Efficient computational methods: Séminaire de Mathématiques Appliquées, Caen 25 avril

35 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Adapted to strongly varying velocities; Efficient computational methods: Volume calculi are made locally; Séminaire de Mathématiques Appliquées, Caen 25 avril

36 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Adapted to strongly varying velocities; Efficient computational methods: Volume calculi are made locally; Communications between elements thanks to conditions on the faces only ((N 1)D calculi); Séminaire de Mathématiques Appliquées, Caen 25 avril

37 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Adapted to strongly varying velocities; Efficient computational methods: Volume calculi are made locally; Communications between elements thanks to conditions on the faces only ((N 1)D calculi); High-order elements to overcome dispersion effects. Séminaire de Mathématiques Appliquées, Caen 25 avril

38 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Adapted to strongly varying velocities; Efficient computational methods: Volume calculi are made locally; Communications between elements thanks to conditions on the faces only ((N 1)D calculi); High-order elements to overcome dispersion effects. Flexible choice of interpolation orders (p adaptativity) Séminaire de Mathématiques Appliquées, Caen 25 avril

39 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Adapted to strongly varying velocities; Efficient computational methods: Volume calculi are made locally; Communications between elements thanks to conditions on the faces only ((N 1)D calculi); High-order elements to overcome dispersion effects. Flexible choice of interpolation orders (p adaptativity) Increased computational cost as compared to FEM Séminaire de Mathématiques Appliquées, Caen 25 avril

40 Discontinuous Galerkin Methods Quasi-explicit scheme: mass matrix is block-diagonal; Based on tetrahedras (easy to handle); Adapted to strongly varying velocities; Efficient computational methods: Volume calculi are made locally; Communications between elements thanks to conditions on the faces only ((N 1)D calculi); High-order elements to overcome dispersion effects. Flexible choice of interpolation orders (p adaptativity) Increased computational cost as compared to FEM DoF of FEM DoF of DGM Séminaire de Mathématiques Appliquées, Caen 25 avril

41 Outline of the talk Séminaire de Mathématiques Appliquées, Caen 25 avril

42 Outline of the talk Interior Penalty Discontinuous Galerkin (IPDG) method for elasto-acoustic problem Séminaire de Mathématiques Appliquées, Caen 25 avril

43 Outline of the talk Interior Penalty Discontinuous Galerkin (IPDG) method for elasto-acoustic problem Numerical illustrations: Reverse Time-Harmonic Migration Séminaire de Mathématiques Appliquées, Caen 25 avril

44 Outline of the talk Interior Penalty Discontinuous Galerkin (IPDG) method for elasto-acoustic problem Numerical illustrations: Reverse Time-Harmonic Migration Hybridizable Discontinuous Galerkin (HDG) method for elastodynamic Séminaire de Mathématiques Appliquées, Caen 25 avril

45 Outline of the talk Interior Penalty Discontinuous Galerkin (IPDG) method for elasto-acoustic problem Numerical illustrations: Reverse Time-Harmonic Migration Hybridizable Discontinuous Galerkin (HDG) method for elastodynamic Performance comparison between IPDG and HDG Séminaire de Mathématiques Appliquées, Caen 25 avril

46 Outline of the talk Interior Penalty Discontinuous Galerkin (IPDG) method for elasto-acoustic problem Numerical illustrations: Reverse Time-Harmonic Migration Hybridizable Discontinuous Galerkin (HDG) method for elastodynamic Performance comparison between IPDG and HDG Concluding remarks Séminaire de Mathématiques Appliquées, Caen 25 avril

47 The direct elasto-acoustic scattering problem Continuous formulation Séminaire de Mathématiques Appliquées, Caen 25 avril

48 The direct elasto-acoustic scattering problem Continuous formulation σ(u) + ω 2 ρ s u = 0 in Ω s p + k 2 p = 0 in Ω f (k = ω/c f ) Séminaire de Mathématiques Appliquées, Caen 25 avril

49 The direct elasto-acoustic scattering problem Continuous formulation σ(u) + ω 2 ρ s u = 0 in Ω s p + k 2 p = 0 in Ω f (k = ω/c f ) ω 2 ρ f u n = p n + pinc n on Γ σ(u)n = pn p inc n on Γ Séminaire de Mathématiques Appliquées, Caen 25 avril

50 The direct elasto-acoustic scattering problem Continuous formulation σ(u) + ω 2 ρ s u = 0 in Ω s p + k 2 p = 0 in Ω f (k = ω/c f ) ω 2 ρ f u n = p n + pinc n on Γ σ(u)n = pn p inc n on Γ ( ) p lim r r + r ikp = 0 (r = x 2 ) Séminaire de Mathématiques Appliquées, Caen 25 avril

51 A finite element solver Main features Séminaire de Mathématiques Appliquées, Caen 25 avril

52 A finite element solver Main features 1. A BVP reduced to Ω f b Ω s via an absorbing boundary condition on an artificial boundary Σ; Séminaire de Mathématiques Appliquées, Caen 25 avril

53 A finite element solver Main features 1. A BVP reduced to Ω f b Ω s via an absorbing boundary condition on an artificial boundary Σ; 2. A Discontinuous Galerkin method with interior penalty; Séminaire de Mathématiques Appliquées, Caen 25 avril

54 A finite element solver Main features 1. A BVP reduced to Ω f b Ω s via an absorbing boundary condition on an artificial boundary Σ; 2. A Discontinuous Galerkin method with interior penalty; 3. Curved elements on the boundaries Γ et Σ. Séminaire de Mathématiques Appliquées, Caen 25 avril

55 The reduced direct fluid-solid problem An absorbing boundary condition σ(u) + ω 2 ρ s u = 0 in Ω s p + ω2 cf 2 p = 0 in Ω f b ω 2 ρ f u n = p n + pinc n on Γ σ(u)n = pn p inc n on Γ p + Λ(p) = 0 on Σ. n Séminaire de Mathématiques Appliquées, Caen 25 avril

56 The reduced direct fluid-solid problem An absorbing boundary condition σ(u) + ω 2 ρ s u = 0 in Ω s p + ω2 cf 2 p = 0 in Ω f b ω 2 ρ f u n = p n + pinc n on Γ σ(u)n = pn p inc n on Γ First-order boundary condition: p ikp = 0 on Σ. n. Séminaire de Mathématiques Appliquées, Caen 25 avril

57 The Interior Penalty Discontinuous Galerkin Method (IPDG method) Notations f h and s h: mesh of Ω f b and Ω s of dim. N s h and N f h; F f h,σ and F f,s h,γ: sets of faces on Σ and Γ; F f h,int, F s h,int: sets of internal faces; [q] := q + q and {q} := q+ + q. 2 Séminaire de Mathématiques Appliquées, Caen 25 avril

58 The IPDG method Approximation spaces Vh f := {q L 2 (Ω f b) : q P m (), h}; f Vh s := {v (L 2 (Ω s )) 2 : v (P m ()) 2, h}; s (ϕ k ) k=1,,n : Lagrange functions of degree m in P m (); p h = N f h n j=1 k=1 p j h,kϕ j k = Nf i=1 p h,i ϕ i V f h ; Séminaire de Mathématiques Appliquées, Caen 25 avril

59 The IPDG method Weak continuity constraints through each interior element [p] = 0 and [u] = 0; [ 1 ρ f p n] = 0 and [σ(u)n] = 0. Séminaire de Mathématiques Appliquées, Caen 25 avril

60 The IPDG method IPDG matricial formulation Af B t B A s P U = F 1 F 2. where P and U are the unknown vectors: P = (p h,i ) t 1 i N f and U = (U x, U y ) t = (u h,i ) t 1 i N s. Séminaire de Mathématiques Appliquées, Caen 25 avril

61 The IPDG method A f = + + h f IPDG matricial formulation 1 ω 2 F Fh,int f F Fh,int f F F f h,σ ( 1 ω 2 F 1 ω 2 F 1 ω 2 F 1 ϕ i ϕ j dx 1 k 2 ρ f ρ f ϕ i ϕ j dx) ({ 1 ϕ i }n[ϕ j ] + { 1 ϕ j }n[ϕ i ]) ds ρ f ρ f α f c f max h F [ϕ i ][ϕ j ] ds 1 Λ(ϕ i )ϕ j ds ρ f 1 i,j N f 1 i,j N f 1 i,j N f 1 i,j N f Séminaire de Mathématiques Appliquées, Caen 25 avril

62 The IPDG method A f = + + IPDG matricial formulation ( 1 ω 2 h f 1 ω 2 F Fh,int f F F Fh,int f F F f h,σ 1 ω 2 F 1 ω 2 F 1 ϕ i ϕ j dx 1 k 2 ρ f ρ f ϕ i ϕ j dx) ({ 1 ϕ i }n[ϕ j ] + { 1 ϕ j }n[ϕ i ]) ds ρ f ρ f α f c f max h F [ϕ i ][ϕ j ] ds 1 Λ(ϕ i )ϕ j ds ρ f 1 i,j N f 1 i,j N f 1 i,j N f 1 i,j N f Séminaire de Mathématiques Appliquées, Caen 25 avril

63 The IPDG method A f = + + IPDG matricial formulation ( 1 ω 2 h f 1 ω 2 F Fh,int f F F Fh,int f F F f h,σ 1 ω 2 F 1 ω 2 F 1 ϕ i ϕ j dx 1 k 2 ρ f ρ f ϕ i ϕ j dx) ({ 1 ϕ i }n[ϕ j ] + { 1 ϕ j }n[ϕ i ]) ds ρ f ρ f α f c f max h F [ϕ i ][ϕ j ] ds 1 Λ(ϕ i )ϕ j ds ρ f 1 i,j N f 1 i,j N f 1 i,j N f 1 i,j N f Séminaire de Mathématiques Appliquées, Caen 25 avril

64 The IPDG method IPDG matricial formulation A s = + ( h s F Fh,int s F F Fh,int s F σ(ψ i ) : ψ j dx ω 2 ρ s ψ i ψ j dx ) ({σ(ψ i )}n[ψ j ] + {σ(ψ j )}n[ψ i ]) ds α f c s max h F [ψ i ][ψ j ] ds 1 i,j N s 1 i,j N s 1 i,j N s Séminaire de Mathématiques Appliquées, Caen 25 avril

65 The IPDG method IPDG matricial formulation B = F 1 = F 2 = F F f,s h,γ F F f,s h,γ F F F f,s h,γ ψ i nϕ j ds 1 ω 2 F F 1 i N s,1 j N f 1 g nϕ j ds ρ f gnψ j ds 1 i,j Ns. 1 i,j Nf, Séminaire de Mathématiques Appliquées, Caen 25 avril

66 The IPDG method IPDG matricial formulation Af B t B A s P U = F 1 F 2. Séminaire de Mathématiques Appliquées, Caen 25 avril

67 The IPDG method IPDG matricial formulation Af B t B A s P U = F 1 F 2. Solver MUMPS. Séminaire de Mathématiques Appliquées, Caen 25 avril

68 Curved elements on the boundaries Idea Modelling of the curved boundaries. (a) P 3 straight elements (b) P 3 curved elements Séminaire de Mathématiques Appliquées, Caen 25 avril

69 A circular elastic inclusion problem with analytical solution Configuration Physical parameters: c f = 1500 m/s, ρ f = 1000 kg/m 3, c p = 6198 m/s, c s = 3122 m/s, ρ s = 2700 kg/m 3. p inc = e ikx d with d = (1, 0). Séminaire de Mathématiques Appliquées, Caen 25 avril

70 A circular elastic inclusion problem with analytical solution Configuration 1 Physical parameters: c f = 1500 m/s, ρ f = 1000 kg/m 3, c p = 6198 m/s, c s = 3122 m/s, ρ s = 2700 kg/m 3. p inc = e ikx d with d = (1, 0). 1 T. Huttunen, J. aipio and P. Monk, An ultra-weak method for acoustic fluid-solid interaction, J. Comput. Appl. Math., 213, pp , Séminaire de Mathématiques Appliquées, Caen 25 avril

71 A circular elastic inclusion problem with analytical solution Configuration 2 Physical parameters: c f = 1500 m/s, ρ f = 1000 kg/m 3, c p = 6198 m/s, c s = 3122 m/s, ρ s = 2700 kg/m 3. p inc = e ikx d with d = (1, 0). 2 H. Barucq, E. Éstécahandy and R. Djellouli, Efficient DG-like formulation equipped with curved boundary edges for solving elasto-acoustic scattering problems, IJNME, 98, pp , Séminaire de Mathématiques Appliquées, Caen 25 avril

72 A circular elastic inclusion problem Fourier series of the analytical solution p(r, θ) = + n=0 [ An H (1) n (kr) + B n H n (2) u(r, θ) = φ + ( e z ) ψ where (kr) ] cos(nθ) φ(r, θ) = + C n J n (k p r) cos(nθ), n=0 ψ(r, θ) = + D n J n (k s r) sin(nθ). n=0 Séminaire de Mathématiques Appliquées, Caen 25 avril

73 A circular elastic inclusion problem Fourier series of the analytical solution p(r, θ) = + n=0 [ An H (1) n (kr) + B n H n (2) u(r, θ) = φ + ( e z ) ψ where (kr) ] cos(nθ) φ(r, θ) = + C n J n (k p r) cos(nθ), n=0 ψ(r, θ) = + D n J n (k s r) sin(nθ). n=0 Séminaire de Mathématiques Appliquées, Caen 25 avril

74 A circular elastic inclusion problem Fourier series of the analytical solution X n = (A n, B n, C n, D n ) t solution of a system arising from the boundary conditions: E n X n = b n. Séminaire de Mathématiques Appliquées, Caen 25 avril

75 A circular elastic inclusion problem Fourier series of the analytical solution X n = (A n, B n, C n, D n ) t solution of a system arising from the boundary conditions: Issue: E n X n = b n. Location of the Jones resonance frequencies? Séminaire de Mathématiques Appliquées, Caen 25 avril

76 A circular elastic inclusion problem Fourier series of the analytical solution X n = (A n, B n, C n, D n ) t solution of a system arising from the boundary conditions: Issue: E n X n = b n. Location of the Jones resonance frequencies? Idea: Study of the invertibility of the matrix E n for each mode n. Séminaire de Mathématiques Appliquées, Caen 25 avril

77 A circular elastic inclusion problem On the identification of the Jones frequencies Mesh 1: 8 points per wavelength relatively to ka 10, composed of 6442 elements. Figure : Circular elastic inclusion problem - Mesh 1. Séminaire de Mathématiques Appliquées, Caen 25 avril

78 A circular elastic inclusion problem On the identification of the Jones frequencies Figure : Relative error as a function of ka on the range [4, 21] with P 3 straight finite elements. Séminaire de Mathématiques Appliquées, Caen 25 avril

79 A circular elastic inclusion problem On the identification of the Jones frequencies Figure : Relative error as a function of ka on the range [4, 21] with P 3 straight finite elements. Maximal error at ka = : f R = 221 khz. Séminaire de Mathématiques Appliquées, Caen 25 avril

80 A circular elastic inclusion problem On the identification of the Jones frequencies Figure : Normalized determinant dete n as a function of ka for the third mode number n = 2. Séminaire de Mathématiques Appliquées, Caen 25 avril

81 A circular elastic inclusion problem On the identification of the Jones frequencies Séminaire de Mathématiques Appliquées, Caen 25 avril

82 A circular elastic inclusion problem On the identification of the Jones frequencies Conclusion: The peaks in the error coincide with the Jones frequencies. Séminaire de Mathématiques Appliquées, Caen 25 avril

83 A circular elastic inclusion problem Objective A study of the IPDG method combined with curved finite elements on three frequencies: The resonance: f R = 221 khz; Two closed values: f 1 = 219 khz and f 2 = 223 khz. Séminaire de Mathématiques Appliquées, Caen 25 avril

84 A circular elastic inclusion problem IPDG method combined with curved finite elements Mesh 2: 645 elements. Figure : Circular elastic inclusion problem - Mesh 2. Séminaire de Mathématiques Appliquées, Caen 25 avril

85 A circular elastic inclusion problem Curved finite elements on Mesh 2 f (khz) p u x u y P P E E-002 P E Table : Relative error (%) for P 1 to P 3 approximations with curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

86 A circular elastic inclusion problem Curved finite elements on Mesh 2 f (khz) p u x u y P P E E-002 P E Table : Relative error (%) for P 1 to P 3 approximations with curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

87 A circular elastic inclusion problem Curved finite elements on Mesh 2 f (khz) p u x u y P P E E-002 P E Table : Relative error (%) for P 1 to P 3 approximations with curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

88 A circular elastic inclusion problem Curved finite elements on Mesh 2 f (khz) p u x u y P P E E-002 P E Table : Relative error (%) for P 1 to P 3 approximations with curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

89 A circular elastic inclusion problem Curved finite elements on Mesh 2 f (khz) p u x u y P P E E-002 P E Table : Relative error (%) for P 1 to P 3 approximations with curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

90 A circular elastic inclusion problem Curved finite elements on Mesh 2 f (khz) p u x u y P P E E-002 P E Table : Relative error (%) for P 1 to P 3 approximations with curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

91 A circular elastic inclusion problem Curved finite elements on Mesh 2 (a) Approximate solution (b) Exact solution Figure : Pressure modulus and displacement amplitude fields for the frequency f 1 = 219kHz with P 3 finite elements and curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

92 A circular elastic inclusion problem Curved finite elements on Mesh 2 Figure : Absolute error between both solutions - f 1 = 219kHz. Séminaire de Mathématiques Appliquées, Caen 25 avril

93 A circular elastic inclusion problem Curved finite elements on Mesh 2 (a) Approximate solution (b) Exact solution Figure : Pressure modulus and displacement amplitude fields for the frequency f 2 = 223kHz with P 3 finite elements and curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

94 A circular elastic inclusion problem Curved finite elements on Mesh 2 Figure : Absolute error between both solutions - f 2 = 223kHz. Séminaire de Mathématiques Appliquées, Caen 25 avril

95 A circular elastic inclusion problem Curved finite elements on Mesh 2 (a) Approximate solution (b) Exact solution Figure : Pressure modulus and displacement amplitude fields for the frequency f R = 221kHz with P 3 finite elements and curved elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

96 A circular elastic inclusion problem Curved finite elements on Mesh 2 Figure : Absolute error between both solutions - f R = 221kHz. Séminaire de Mathématiques Appliquées, Caen 25 avril

97 A circular elastic inclusion problem Remark on the Jones frequency (a) Displacement error (b) Fourier mode n = 2 Figure : Comparison between the error in displacement and the third Fourier mode for the resonance frequency f R = 221kHz. Séminaire de Mathématiques Appliquées, Caen 25 avril

98 A circular elastic inclusion problem Curved finite elements on Mesh 2 Figure : Relative error as a function of ka on the range [4, 21] with P 3 finite elements combined with curved finite elements on the boundaries. Séminaire de Mathématiques Appliquées, Caen 25 avril

99 A circular elastic inclusion problem Curved finite elements on Mesh 2 Straight finite elements on Mesh 1 Séminaire de Mathématiques Appliquées, Caen 25 avril

100 A circular elastic inclusion problem IPDG method combined with curved finite elements Conclusion: Accuracy; Robustness in the Jones frequency regions; No additional numerical resonance; A large range of frequencies covered by a single coarse mesh. Séminaire de Mathématiques Appliquées, Caen 25 avril

101 IPDG method combined with curved finite elements Convergence study Figure : Convergence of the relative error as a function of kh in the resonance region with P 3 finite elements combined with curved finite elements on the boundaries for ka = (f 1 = 219 khz). Séminaire de Mathématiques Appliquées, Caen 25 avril

102 IPDG method combined with curved finite elements Convergence study Figure : Convergence of the relative error as a function of kh in the resonance region with P 3 finite elements combined with curved finite elements on the boundaries for ka = (f 2 = 223 khz). Séminaire de Mathématiques Appliquées, Caen 25 avril

103 IPDG method combined with curved finite elements Convergence study Figure : Convergence of the relative error as a function of kh in the resonance region with P 3 finite elements combined with curved finite elements on the boundaries for ka = (f R = 221 khz). Séminaire de Mathématiques Appliquées, Caen 25 avril

104 IPDG method combined with curved finite elements Convergence study Conclusion: Order 4. Séminaire de Mathématiques Appliquées, Caen 25 avril

105 IPDG method combined with curved finite elements Convergence study Conclusion: Order 4. Comparison with straight finite elements: Non-resonant cases: order 2; Jones frequency case: order 3/2. Séminaire de Mathématiques Appliquées, Caen 25 avril

106 IPDG method combined with curved finite elements Stability study in the low and mid-frequency regimes Figure : Relative error as a function of ka on range [5, 200] with P 3 finite elements combined with curved finite elements on the boundaries (Mesh 1 refined twice). Séminaire de Mathématiques Appliquées, Caen 25 avril

107 IPDG method combined with curved finite elements Stability study in the high-frequency regime ka p u x u y E E E E E E E E E E Table : Relative error (%) in the high-frequency regime with P 3 finite elements combined with curved finite elements on the boundaries (Mesh 1 refined three times). Séminaire de Mathématiques Appliquées, Caen 25 avril

108 IPDG method combined with curved finite elements Stability study in the high-frequency regime ka p u x u y E E E E E E E E E E Table : Relative error (%) in the high-frequency regime with P 3 finite elements combined with curved finite elements on the boundaries (Mesh 1 refined three times). Séminaire de Mathématiques Appliquées, Caen 25 avril

109 IPDG method combined with curved finite elements Stability study Conclusion: Stability in the mid and high-frequency regimes. Séminaire de Mathématiques Appliquées, Caen 25 avril

110 Numerical applications : Reverse Time-Harmonic Migration Two cases: The object contains corner The object has a curved interface Séminaire de Mathématiques Appliquées, Caen 25 avril

111 Object with corners Ω 2 Ω 1 Séminaire de Mathématiques Appliquées, Caen 25 avril

112 Object with corners Ω 1 : fluid medium with c = 1500 m/s Ω 2 : solid medium with V p = 6200 m/s and V s = 4300 m/s We use 100 sources placed on an ellipse with major axis of 2 km and minor axis of 1 km. The data are obtained using PML. The forward and backward problems are coupled with ABC on an elliptic artificial boundary of major axis of 2.2 km and minor axis of 2.1 km. Ω 2 Ω 1 Séminaire de Mathématiques Appliquées, Caen 25 avril

113 Object with corners RTM image for f = 10 Hz Séminaire de Mathématiques Appliquées, Caen 25 avril

114 Object with corners RTM image for f = 20 Hz Séminaire de Mathématiques Appliquées, Caen 25 avril

115 Object with corners RTM image for f = 40 Hz Séminaire de Mathématiques Appliquées, Caen 25 avril

116 Object with curved interfaces Ω 1 Ω 2 Séminaire de Mathématiques Appliquées, Caen 25 avril

117 Object with curved interfaces Ω 1 : fluid medium with c = 1500 m/s Ω 2 : solid medium with V p = 6200 m/s and V s = 4300 m/s We use 100 sources placed on an ellipse with major axis of 2 km and minor axis of 1 km. The data are obtained using PML. The forward and backward problems are coupled with either PML or ABC on an elliptic artificial boundary of major axis of 2.2 km and minor axis of 2.1 km. Ω 2 Ω 1 Séminaire de Mathématiques Appliquées, Caen 25 avril

118 Object with curved interfaces RTM image for f = 10 Hz Séminaire de Mathématiques Appliquées, Caen 25 avril

119 Conclusion IPDG method combined with curved finite elements Accuracy and robustness; Optimal convergence order recovered even in the Jones frequency case; Well-suited to address the pollution effects. Séminaire de Mathématiques Appliquées, Caen 25 avril

120 Conclusion for preliminary computations RTHM is efficient to recover hidden objects. And what about limited aperture? ABCs can be used to reduce the size of the linear system. But because of the difficulty of solving huge linear systems, RTHM should be kept for 2D seismic imaging. What do we intend to investigate now? The influence of ABCs is unclear: more numerical experiments are needed In particular, is it relevant to enrich the modelling related to the exterior boundary? Séminaire de Mathématiques Appliquées, Caen 25 avril

121 How can we improve the efficiency? Hybridizable Discontinuous Galerkin Methods same advantages as DG methods: unstructured triangular meshes, hp-adaptivity, easily parallelizable method, discontinuous basis functions introduction of a new variable defined only on the interfaces lower number of coupled DOF than classical DG methods Séminaire de Mathématiques Appliquées, Caen 25 avril

122 How can we improve the efficiency? Hybridizable Discontinuous Galerkin Methods same advantages as DG methods: unstructured triangular meshes, hp-adaptivity, easily parallelizable method, discontinuous basis functions introduction of a new variable defined only on the interfaces lower number of coupled DOF than classical DG methods Séminaire de Mathématiques Appliquées, Caen 25 avril

123 How can we improve the efficiency? Hybridizable Discontinuous Galerkin Methods same advantages as DG methods: unstructured triangular meshes, hp-adaptivity, easily parallelizable method, discontinuous basis functions introduction of a new variable defined only on the interfaces lower number of coupled DOF than classical DG methods time-domain, implicit scheme Séminaire de Mathématiques Appliquées, Caen 25 avril

124 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: , 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) , N.C. Nguyen, J. Peraire and B. Cockburn. High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. Journal of Computational Physics, 230: , 2011 N.C. Nguyen and B. Cockburn. Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics. Journal of Computational Physics 231: , 2012 Séminaire de Mathématiques Appliquées, Caen 25 avril

125 HDG formulation of the equations Local HDG formulation { iωρv σ = 0 iωσ Cε (v) = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

126 HDG formulation of the equations Local HDG formulation iωρ v w + iωσ : ξ + v σ : w ( C ξ ) σ n w = 0 v C ξ n = 0 σ and v are numerical traces of σ and v respectively on Séminaire de Mathématiques Appliquées, Caen 25 avril

127 HDG formulation of the equations We define: v = λ F, F F h, Séminaire de Mathématiques Appliquées, Caen 25 avril

128 HDG formulation of the equations We define: v = λ F, ( F F h, σ ) n = σ n τi v λ F, on where τ is the stabilization parameter (τ > 0) Séminaire de Mathématiques Appliquées, Caen 25 avril

129 HDG formulation of the equations Local HDG formulation ( ) We replace v and σ n by their definitions into the local equations iωρ v w + σ : w σ n w ( + τi v λ F w = 0 ( iωσ : ξ + v C ξ ) λ F C ξ n = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

130 HDG formulation of the equations Local HDG formulation iωρ v w iωσ : ξ + v ( σ ) w + ( ) C ξ τi ( v λ F ) w = 0 λ F C ξ n = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

131 HDG formulation of the equations Local HDG formulation iωρ v w iωσ : ξ + v ( σ ) w + ( ) C ξ τi ( v λ F ) w = 0 λ F C ξ n = 0 We define: W = (V ) x, V y, V z, σ xx, σ yy, σ zz, σ xy, σ T xz, σ yz Λ = ( ) T Λ F1, Λ F2,..., Λ Fn f, where nf = card(f h ) Discretization of the local HDG formulation A W + C,F Λ = 0 F Séminaire de Mathématiques Appliquées, Caen 25 avril

132 HDG formulation of the equations Local HDG formulation iωρ v w iωσ : ξ + v ( σ ) w + ( ) C ξ τi ( v λ F ) w = 0 λ F C ξ n = 0 We define: W = (V ) x, V y, V z, σ xx, σ yy, σ zz, σ xy, σ T xz, σ yz Λ = ( ) T Λ F1, Λ F2,..., Λ Fn f, where nf = card(f h ) Discretization of the local HDG formulation A W + C Λ = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

133 HDG formulation of the equations Transmission condition In order to determine λ F, the continuity of the normal component of σ is weakly enforced, rendering this numerical trace conservative : [[ σ n]] η = 0 F Séminaire de Mathématiques Appliquées, Caen 25 avril

134 HDG formulation of the equations Transmission condition In order to determine λ F, the continuity of the normal component of σ is weakly enforced, rendering this numerical trace conservative : [[ σ n]] η = 0 T h F ( ) Replacing σ n and summing over all faces, the transmission condition becomes : ( ) σ n η ( ) τi v λ F η = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

135 HDG formulation of the equations Transmission condition In order to determine λ F, the continuity of the normal component of σ is weakly enforced, rendering this numerical trace conservative : [[ σ n]] η = 0 T h F ( ) Replacing σ n and summing over all faces, the transmission condition becomes : ( ) σ n η ( ) τi v λ F η = 0 T h Discretization of the transmission condition [ B W + L Λ ] = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

136 HDG formulation of the equations Global HDG formulation iωρ v w iωσ : ξ + T h ( σ ) w + v ( σ n ) η T h ( ) C ξ τi ( v λ F ) w = 0 λ F C ξ n = 0 τi ( v λ F ) η = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

137 HDG formulation of the equations Global HDG formulation iωρ v w iωσ : ξ + T h ( σ ) w + v ( σ n ) η T h ( ) C ξ τi ( v λ F ) w = 0 λ F C ξ n = 0 τi ( v λ F ) η = 0 Global HDG discretization A W + C Λ = 0 [ B W + L Λ ] = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

138 HDG formulation of the equations Global HDG formulation iωρ v w iωσ : ξ + T h ( σ ) w + v ( σ n ) η T h ( ) C ξ τi ( v λ F ) w = 0 λ F C ξ n = 0 τi ( v λ F ) η = 0 Global HDG discretization W = (A ) 1 C Λ [ B W + L Λ ] = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

139 HDG formulation of the equations Global HDG formulation iωρ v w iωσ : ξ + T h ( σ ) w + v ( σ n ) η T h ( ) C ξ τi ( v λ F ) w = 0 λ F C ξ n = 0 τi ( v λ F ) η = 0 Global HDG discretization [ B (A ) 1 C + L ] Λ = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

140 Symmetric HDG formulation Local HDG formulation { iωρv σ = 0 iωσ Cε (v) = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

141 Symmetric HDG formulation Local HDG formulation { iωρv σ = 0 iωσ Cε (v) = 0 C invertible and symmetric tensor, i.e for a symmetric σ : with D = C 1 and u = iωv σ = Cε (u) and ε (u) = Dσ Séminaire de Mathématiques Appliquées, Caen 25 avril

142 Symmetric HDG formulation Local HDG formulation iωρ v w + iωσ : ξ + v σ : w ( C ξ ) σ n w = 0 v C ξ n = 0 σ and v are numerical traces of σ and v respectively on Séminaire de Mathématiques Appliquées, Caen 25 avril

143 Symmetric HDG formulation Local HDG formulation iωρ v w + iωσ : ξ + v σ : w ( C ξ ) σ n w = 0 v C ξ n = 0 σ and v are numerical traces of σ and v respectively on ξ = D ξ Séminaire de Mathématiques Appliquées, Caen 25 avril

144 Symmetric HDG formulation Local HDG formulation iωρ v w + σ : w σ n w = 0 iωσ : D ξ v (C D ξ ) + v C D ξ n = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

145 Symmetric HDG formulation Local HDG formulation iωρ v w + iωd σ : ξ σ : w v ξ + σ n w = 0 v ξ n = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

146 Symmetric HDG formulation Local HDG formulation iωρ v w + iωd σ : ξ σ : w v ξ + σ n w = 0 v ξ n = 0 v = λ F, ( F F h, σ ) n = σ n τi v λ F, on Séminaire de Mathématiques Appliquées, Caen 25 avril

147 Symmetric HDG formulation Local HDG formulation iωρ v w iωd σ : ξ ( σ ) w + v ξ + τi ( v λ F ) w = 0 λ F ξ n = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

148 Symmetric HDG formulation Local HDG formulation iωρ v w iωd σ : ξ ( σ ) w + v ξ + τi ( v λ F ) w = 0 λ F ξ n = 0 Discretization of the local HDG formulation A 2 W + C,F 2 Λ = 0 F Séminaire de Mathématiques Appliquées, Caen 25 avril

149 Symmetric HDG formulation Local HDG formulation iωρ v w iωd σ : ξ ( σ ) w + v ξ + τi ( v λ F ) w = 0 λ F ξ n = 0 Discretization of the local HDG formulation A 2 W + C 2 Λ = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

150 Symmetric HDG formulation Local HDG formulation iωρ v w iωd σ : ξ ( σ ) w + v ξ + τi ( v λ F ) w = 0 λ F ξ n = 0 Discretization of the local HDG formulation A 2 W + C 2 Λ = 0 A 2 symmetric matrix Séminaire de Mathématiques Appliquées, Caen 25 avril

151 Symmetric HDG formulation Transmission condition [[ σ n]] η = 0 F Séminaire de Mathématiques Appliquées, Caen 25 avril

152 Symmetric HDG formulation Transmission condition [[ σ n]] η = 0 F T h ( ) σ n η T h ( ) τi v λ F η = 0 Séminaire de Mathématiques Appliquées, Caen 25 avril

153 Symmetric HDG formulation Transmission condition [[ σ n]] η = 0 F T h ( ) σ n η T h ( ) τi v λ F η = 0 Discretization of the transmission condition [ B W + L Λ ] = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

154 Symmetric HDG formulation Transmission condition [[ σ n]] η = 0 F T h ( ) σ n η T h ( ) τi v λ F η = 0 Discretization of the transmission condition [ B W + L Λ ] = 0 T h B = (C 2 ) T Séminaire de Mathématiques Appliquées, Caen 25 avril

155 Symmetric HDG formulation Transmission condition [[ σ n]] η = 0 F T h ( ) σ n η T h ( ) τi v λ F η = 0 Discretization of the transmission condition [ (C2 ) T W + L Λ ] = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

156 Symmetric HDG formulation of the equations Global HDG formulation ( iωρ v w ) σ w + τi ( v λ ) F w = 0 iωd σ : ξ v ξ + λ F ξ n = 0 ( σ n ) η τi ( v λ ) F η = 0 T h T h Séminaire de Mathématiques Appliquées, Caen 25 avril

157 Symmetric HDG formulation of the equations Global HDG formulation ( iωρ v w ) σ w + τi ( v λ ) F w = 0 iωd σ : ξ v ξ + λ F ξ n = 0 ( σ n ) η τi ( v λ ) F η = 0 T h T h Global HDG discretization A 2 W + C 2 Λ = 0 [ (C2 ) T W + L Λ ] = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

158 Symmetric HDG formulation of the equations Global HDG formulation ( iωρ v w ) σ w + τi ( v λ ) F w = 0 iωd σ : ξ v ξ + λ F ξ n = 0 ( σ n ) η τi ( v λ ) F η = 0 T h T h Global HDG discretization W = (A 2 ) 1 C 2 Λ [ (C2 ) T W + L Λ ] = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

159 Symmetric HDG formulation of the equations Global HDG formulation ( iωρ v w ) σ w + τi ( v λ ) F w = 0 iωd σ : ξ v ξ + λ F ξ n = 0 ( σ n ) η τi ( v λ ) F η = 0 T h T h Global HDG discretization [ (C2 ) T (A 2 ) 1 C 2 + L ] Λ = 0 T h Séminaire de Mathématiques Appliquées, Caen 25 avril

160 Symmetric HDG formulation of the equations Global HDG formulation ( iωρ v w ) σ w + τi ( v λ ) F w = 0 iωd σ : ξ v ξ + λ F ξ n = 0 ( σ n ) η τi ( v λ ) F η = 0 T h T h Global HDG discretization [ (C2 ) T (A 2 ) 1 C 2 + L ] Λ = 0 T h Symmetric linear system Séminaire de Mathématiques Appliquées, Caen 25 avril

161 Main steps of the HDG algorithm 1. Construction of the global matrix M with M = [ B ] (A ) 1 C + L T h for = 1 to Nb tri do Computation of matrices B, (A ) 1, C and L Construction of the corresponding section of M end for Séminaire de Mathématiques Appliquées, Caen 25 avril

162 Main steps of the HDG algorithm 1. Construction of the global matrix M with M = [ (C ] 2 ) T (A 2 ) 1 C 2 + L T h for = 1 to Nb tri do Computation of matrices (A 2 ) 1, C 2 and L Construction of the corresponding section of M (use of the symetry of the system, only the upper or lower corresponding section) end for Séminaire de Mathématiques Appliquées, Caen 25 avril

163 Main steps of the HDG algorithm 1. Construction of the global matrix M 2. Construction of the right hand side S Séminaire de Mathématiques Appliquées, Caen 25 avril

164 Main steps of the HDG algorithm 1. Construction of the global matrix M 2. Construction of the right hand side S 3. Resolution MΛ = S, with a direct solver (MUMPS) Séminaire de Mathématiques Appliquées, Caen 25 avril

165 Main steps of the HDG algorithm 1. Construction of the global matrix M 2. Construction of the right hand side S 3. Resolution MΛ = S, with a direct solver (MUMPS) 4. Computation of the solutions of the initial problem Séminaire de Mathématiques Appliquées, Caen 25 avril

166 Main steps of the HDG algorithm 1. Construction of the global matrix M 2. Construction of the right hand side S 3. Resolution MΛ = S, with a direct solver (MUMPS) 4. Computation of the solutions of the initial problem for = 1 to Nb tri do Compute W = (A ) 1 C Λ end for Séminaire de Mathématiques Appliquées, Caen 25 avril

167 Main steps of the HDG algorithm 1. Construction of the global matrix M 2. Construction of the right hand side S 3. Resolution MΛ = S, with a direct solver (MUMPS) 4. Computation of the solutions of the initial problem for = 1 to Nb tri do Compute W = (A 2 ) 1 C 2 Λ end for Séminaire de Mathématiques Appliquées, Caen 25 avril

168 Plane wave m Physical parameters : ρ = 2000kg.m 3 λ = 16GPa µ = 8GPa Plane wave: i(k cos θx+k sin θy) u = e m Computational domain Ω setting where k = ω v p θ = 0, v p = 4000 m.s 1, ω = 4π Three meshes: 3000 elements elements elements Séminaire de Mathématiques Appliquées, Caen 25 avril

169 Plane wave Exact solution Séminaire de Mathématiques Appliquées, Caen 25 avril

170 Plane wave HDG-P 2 solution, computed on the coarsest mesh Séminaire de Mathématiques Appliquées, Caen 25 avril

171 Plane wave: Convergence order W a W e P 1 P 2 P 3 P h max Séminaire de Mathématiques Appliquées, Caen 25 avril

172 Plane wave: Memory consumption 5 Finest mesh (45000 elements) 4 Memory (GB) HDGm 1 HDGm 2 P 1 P 2 P 3 P 4 Interpolation order Séminaire de Mathématiques Appliquées, Caen 25 avril

173 Plane wave: Memory consumption Finest mesh (45000 elements) Memory (GB) HDGm 1 HDGm 2 UDGm IPDGm P 1 P 2 P 3 P 4 Interpolation order Séminaire de Mathématiques Appliquées, Caen 25 avril

174 Plane wave: CPU time 200 Finest mesh (45000 elements) CPU time (s) HDGm const. 1 HDGm res. 1 HDGm const. 2 HDGm res. 2 0 P 2 P 3 P 4 Interpolation order Séminaire de Mathématiques Appliquées, Caen 25 avril

175 Disk shaped scatterer Computational domain Ω setting a = 2000 m and b = 8000 m Physical parameters in Ω: ρ = 1 kg.m 3 λ = 8 GPa µ = 4 GPa Γ a free surface boundary: σn = 0 Γ b absorbing boundary: σn = v p (v n)n + v s (v t)t Three meshes: 1200 elements 5400 elements elements Séminaire de Mathématiques Appliquées, Caen 25 avril

176 Disk shaped scatterer: Memory consumption Finest mesh (21000 elements) 2 Memory (GB) HDGm 1 HDGm P 2 P 3 P 4 Interpolation order Séminaire de Mathématiques Appliquées, Caen 25 avril

177 Anisotropic test case Three meshes: 600 elements 3000 elements elements Séminaire de Mathématiques Appliquées, Caen 25 avril

178 Anisotropic test case Three meshes: 600 elements 3000 elements elements Symmetric HDG formulation does not work with acoustic media Séminaire de Mathématiques Appliquées, Caen 25 avril

179 Anisotropic test case: Comparison between anisotropic HDG-P 3 formulations 1 and 2 and IPDG-P 3 performances. # elements Memory (MB) HDG 1 HDG 2 IPDG Séminaire de Mathématiques Appliquées, Caen 25 avril

180 Anisotropic test case: Comparison between anisotropic HDG-P 3 formulations 1 and 2 and IPDG-P 3 performances. # elements Const. time (s) Res. time (s) HDG 1 HDG 2 IPDG HDG 1 HDG 2 IPDG Séminaire de Mathématiques Appliquées, Caen 25 avril

181 3D plane wave in an homogeneous medium m 1000 m Configuration of the computational domain Ω. Physical parameters : ρ = 2000kg.m 3 λ = 16GPa µ = 8GPa Plane wave: i(k cos θx+k sin θy) u = e where k = ω v p θ = 0 Two meshes: 2250 elements elements Séminaire de Mathématiques Appliquées, Caen 25 avril

182 Plane wave: Memory consumption Coarsest mesh (2250 elements) 6 Memory (GB) 4 2 HDGm 1 HDGm 2 0 P 1 P 2 P 3 P 4 Interpolation order Séminaire de Mathématiques Appliquées, Caen 25 avril

183 Plane wave: CPU time 100 Coarsest mesh (2250 elements) 80 CPU time (s) HDGm const. 1 HDGm res. 1 HDGm const. 2 HDGm res. 2 0 P 1 P 2 P 3 Interpolation order Séminaire de Mathématiques Appliquées, Caen 25 avril