Multiscale Analysis of Vibrations of Streamers

Transcription

1 Multiscale Analysis of Vibrations of Streamers Leszek Demkowicz Joint work with S. Prudhomme, W. Rachowicz, W. Qiu and L. Chamoin Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin Austin, Texas 78712, U.S.A. Summer Workshop on Mutiscale Modeling and Analysis ICES, Austin, August 4-8, 2008

2 Outline of presentation Austin, August 4-8, 2008 Streamers 2 / 49

3 Outline of presentation The AP stretch - streamer system. Austin, August 4-8, 2008 Streamers 2 / 49

4 Outline of presentation The AP stretch - streamer system. Determining streamer impedance. Austin, August 4-8, 2008 Streamers 2 / 49

5 Outline of presentation The AP stretch - streamer system. Determining streamer impedance. Multiscale analysis of streamers. Austin, August 4-8, 2008 Streamers 2 / 49

6 Outline of presentation The AP stretch - streamer system. Determining streamer impedance. Multiscale analysis of streamers. Dual-Mixed formulation for elasticity. Austin, August 4-8, 2008 Streamers 2 / 49

7 Outline of presentation The AP stretch - streamer system. Determining streamer impedance. Multiscale analysis of streamers. Dual-Mixed formulation for elasticity. Additional background info: Coupled elasticity-acoustics problem, Exact sequence. Projection-Based Interpolation. Automatic hp-adaptivity and coupled multiphysics problems. Austin, August 4-8, 2008 Streamers 2 / 49

8 Surveying the Ocean Floor with Acoustical Streamers An array of 8 streamers, 6km long, pulled by a tugboat Austin, August 4-8, 2008 Streamers 3 / 49

9 The AP Stretch - Streamer System Austin, August 4-8, 2008 Streamers 4 / 49

10 Geometry of the AP Stretch Streamer System Austin, August 4-8, 2008 Streamers 5 / 49

11 Material Data for the Streamer component E(GPa) ρ(kg.m 3 ) ν Length(m) height(m) Connector Spacer Rope Skin Gel E gel (ω, T ) E gel (ω, T ) = E r (ω, T ) + ie r (ω, T ). At T = 10 C, we have: E r (ω, T ) = log 10 (ω) [P a] E i (ω, T ) = log 10 (ω) [P a] Austin, August 4-8, 2008 Streamers 6 / 49

12 2D Model: Axisymmetric Elasticity Find u such that ρω 2 u i (E ijkl u k,l ),j = 0 u i = u D,i E ijkl u k,l n j + iωρ f c 2 = 0 in Ω on Γ D on Γ C Austin, August 4-8, 2008 Streamers 7 / 49

13 2D Model: Axisymmetric Elasticity Initial mesh Austin, August 4-8, 2008 Streamers 8 / 49

14 2D Model: Axisymmetric Elasticity Optimal mesh corresponding to 2% error in energy Austin, August 4-8, 2008 Streamers 8 / 49

15 2D Model: Axisymmetric Elasticity Pressure contours for the initial mesh Austin, August 4-8, 2008 Streamers 9 / 49

16 2D Model: Axisymmetric Elasticity Pressure contours for the optimal mesh Austin, August 4-8, 2008 Streamers 9 / 49

17 2D Model: Axisymmetric Elasticity Pressure profile for the initial mesh Austin, August 4-8, 2008 Streamers 10 / 49

18 2D Model: Axisymmetric Elasticity Pressure profile for the optimal mesh Austin, August 4-8, 2008 Streamers 10 / 49

19 2D Model: Coupled Elasticity/Acoustics Find u,p such that ρω 2 u i (E ijkl u k,l ),j = 0 in Ω s p,ii ( ) ω 2 c p = 0 in Ωf u i = u D,i, p,i n i = ρ f ω 2 u D,i n i on Γ D E ijkl u k,l n j = pn i, p,i n i = ρ f ω 2 u i n i on Γ I + Sommerfeld radiation condition at The structure is axisymmetric. Austin, August 4-8, 2008 Streamers 11 / 49

20 2D Model: Coupled Elasticity/Acoustics Optimal hp mesh corresponding to 3.6 percent error Austin, August 4-8, 2008 Streamers 12 / 49

21 2D Model: Coupled Elasticity/Acoustics Pressure distribution on the optimal mesh. Range: db Austin, August 4-8, 2008 Streamers 13 / 49

22 2D Model: Coupled Elasticity/Acoustics Difference between coarse and fine grid pressures in db Austin, August 4-8, 2008 Streamers 13 / 49

23 2D Model: Coupled Elasticity/Acoustics Pressure profile for the initial mesh Austin, August 4-8, 2008 Streamers 14 / 49

24 2D Model: Coupled Elasticity/Acoustics Pressure profile for the optimal mesh Austin, August 4-8, 2008 Streamers 14 / 49

25 Determining Streamer Impedance Austin, August 4-8, 2008 Streamers 15 / 49

26 Streamer Impedance Define streamer impedance as β = N where N is the force across the u stretch/streamer interface, and u is the displacement of the interface. N = βu provides a Cauchy B.C. for a 1D Stretch model. Task: Compute impedance β as function of frequency ω and water temperature. Austin, August 4-8, 2008 Streamers 16 / 49

27 Two D.O.F. Model of the Streamer β = F u 0 = ω2 (k + iωc) ω 2 + iωc + k Impedance is singular at resonant frequencies Austin, August 4-8, 2008 Streamers 17 / 49

28 Iterative Procedure to Compute the Streamer Impedance Austin, August 4-8, 2008 Streamers 18 / 49

29 Computation of Impedance for a Single Section Integrate by parts: b(u, v) = Ω (σ ijv i,j ρω 2 u i v i ) dx + Γ C β ij u j v i ds = Ω (σ ij,j ρω 2 u i ) v i dx + Γ D t i v i ds + Γ C β ij u j v i ds Choose a special test function v = (v z, v r ) where v z = 1, v r = 0 on Γ D and v = 0 on the rest of the boundary. It follows that: Γ D t z ds = b(u, v) b(u hp, v) Austin, August 4-8, 2008 Streamers 19 / 49

30 Numerical Results: 2D Elasticity Model Streamer impedance as a function of frequency. T = 10 o C Austin, August 4-8, 2008 Streamers 20 / 49

31 Numerical Results: 2D Coupled Model Streamer impedance as a function of frequency. T = 10 o C Austin, August 4-8, 2008 Streamers 21 / 49

32 Comparison of Elastic and Coupled Models Streamer impedance as a function of frequency. T = 10 o C Austin, August 4-8, 2008 Streamers 22 / 49

33 Local Analysis of Hydrophones A Multiscale Approach Austin, August 4-8, 2008 Streamers 23 / 49

34 3D Model: Local Analysis of Hydrophones Task: Analyze pressure distribution around the microphones Issue: How to define appropriate boundary conditions? (Periodic?) Austin, August 4-8, 2008 Streamers 24 / 49

35 Axi-symmetric Streamer Section In order to determine the BC on one periodic cell, we studied the response on a 75m long streamer section. We used the axisymmetric model for detailed studies, but results were confirmed on the 3D simplified model. In particular we measured the displacement along the streamer and accross the two-end sections of one cell in the streamer. Results are shown in following slides. Austin, August 4-8, 2008 Streamers 25 / 49

36 Study of 75m Streamer Section Displacement u z (real part, imaginary part, amplitude) along streamer. Austin, August 4-8, 2008 Streamers 26 / 49

37 Study of 75m Streamer Section Displacement u r (real part, imaginary part, amplitude) across mid-section in the gel cavities before and after one spacer. Austin, August 4-8, 2008 Streamers 27 / 49

38 Study of 75m Streamer Section Displacement u z (real part, imaginary part, amplitude) across mid-section in the gel cavities before and after one spacer. Austin, August 4-8, 2008 Streamers 28 / 49

39 Study of 75m Streamer Section Strain ɛ zz (real part, imaginary part, amplitude) across mid-section in the gel cavities before and after one spacer. Austin, August 4-8, 2008 Streamers 29 / 49

40 Observations from Numerical Study 1. This is clearly a two-scale problem: u(x) = U(x) + u s (x) 2. The small scales u s are proportional to the large scales and not to their derivatives. u R r = u L r u 3. R z = u L z + C Re(ɛ zz ) R = Re(ɛ zz ) L + C Im(ɛ zz ) R Im(ɛ zz ) L + C 4. The displacement of the small scales in one cell is symmetric with respect to the cross-section that passes through the middle of the spacer (see zoom below). Austin, August 4-8, 2008 Streamers 30 / 49

41 Some Mathematics Hypothesis: The small scale u s satisfies periodic boundary conditions. Case 1: Periodic kinematic BC: u R s = u L s + δu, ɛ R s,zz = ɛ L s,zz Case 2: Periodic strain BC: u R s = u L s, ɛ R s,zz = ɛ L s,zz + δɛ For a symmetric geometry of the spacer, it can be shown that Case 1 necessarily yields an antisymmetric solution and Case 2 a symmetric solution. So we choose Case 2. Global/local ansatz: u s and u are now approximated by ū s and ū such that so ū s CU, with U = (0, 0, U z ) and C = constant ū = U + ū s u and δɛ U Austin, August 4-8, 2008 Streamers 31 / 49

42 Solution of the Local Problem The small scale (solution of the local problem) is driven by the large scale represented by complex constant C in the BC. It changes with the location of the spacer/microphone. Austin, August 4-8, 2008 Streamers 32 / 49

43 Solution of the Local Problem The small scale (solution of the local problem) is driven by the large scale represented by complex constant C in the BC. It changes with the location of the spacer/microphone. Quantities like ratio of pressures at different locations or phase difference are independent of constant C and can be evaluated w/o its knowledge. Austin, August 4-8, 2008 Streamers 32 / 49

44 Solution of the Local Problem The small scale (solution of the local problem) is driven by the large scale represented by complex constant C in the BC. It changes with the location of the spacer/microphone. Quantities like ratio of pressures at different locations or phase difference are independent of constant C and can be evaluated w/o its knowledge. Pressure (derivatives) depends mainly upon the small scale. Austin, August 4-8, 2008 Streamers 32 / 49

45 Comsol Problem Setting Geometry of Hydrophone Austin, August 4-8, 2008 Streamers 33 / 49

46 Solution with Comsol Pressure phase on the slice which passes on top of the hydrophone. Austin, August 4-8, 2008 Streamers 34 / 49

47 Verification with hp3d Goal-driven h-adaptivity: 8th fine mesh, 31k elements, 730k dofs Goal: Average pressure over the microfone. Austin, August 4-8, 2008 Streamers 35 / 49

48 Verification with hp3d Pressure Over the Microfone: min/max=67/137 [db] Austin, August 4-8, 2008 Streamers 36 / 49

49 Dual-Mixed Formulation for Elasticity Austin, August 4-8, 2008 Streamers 37 / 49

50 Dual-Mixed Formulation for Elasticity σ ij = E ijkl ɛ kl = E ijkl (u k,l ω kl ) = C klij σ ij = u k,l ω kl σ ij,j + ρω 2 u i = f i σ ij n j = g i on Γ t Ω C klijσ ij τ kl + Ω u kτ kl,l + Ω u kω kl τ kl = Γ u ū k τ kl n l τ kl : τ kl n l = 0 on Γ t Ω σ ij,jv i + ρω 2 Ω u iv i = Ω f iv i v i Ω σ klq kl = 0 q kl = q lk Austin, August 4-8, 2008 Streamers 38 / 49

51 Elasticity Complex (Arnold at al. decoded ) Φ(W ) Λ 0 (W ) id id A 0 Λ 1 (W ) A 1 Λ 2 (W ) A 2 Λ 3 (W ) 0 π 1 π 2 id Φ(W ) Λ 0 (W ) A 0 Γ 1 A 1 Γ 2 A 2 Λ 3 (W ) 0 W = K V, K = {ω ij = ɛ ijk Ψ k }, V = {φ i } Λ 0 (W ) = {(Φ m, φ i )} Λ 1 (W ) = {(Ek m, ei k )} Γ1 = {(Ek m, ei k ) : ɛ nlkek,l m (en m e k k δn m) = 0} Λ 2 (W ) = {(Vn m, vm)} i Γ 2 = {(0, vm)} i Λ 3 (W ) = {(Ψ m, ψ i )} A 0 (Ψ m, φ i ) = (Ek m, ei k ) = (Φm,k + ɛmkα φ α, φ i,k ) A 1 (Ek m, ei k ) = (V n m, vm) i = (ɛ nlk Ek,l m (en m e k k δn m), ɛ mlk e i k,l ) A 2 (Vn m, vm) i = (Ψ m, ψ i ) = (Vn,n m 1 2 ɛmin vn, i vm,m) i Austin, August 4-8, 2008 Streamers 39 / 49

52 p-convergence Test Geometry and boundary conditions. Austin, August 4-8, 2008 Streamers 40 / 49

53 Regular Solution Manufactured solution: u 1 = cos(x + 2y) u 2 = sin(3x + y) Convergence rates (ln error vs. number of d.o.f.). Austin, August 4-8, 2008 Streamers 41 / 49

54 Singular Solution Manufactured solution: u 1 = u 2 = r α sin(α(θ + π )), α = Convergence rates (ln error vs. ln d.o.f.). Austin, August 4-8, 2008 Streamers 42 / 49

55 Additional Background Info Austin, August 4-8, 2008 Streamers 43 / 49

56 Coupled Acoustics/Elasticity acoustics in Ω a : elasticity in Ω e : { c 2 iωp + ρ f w i,i = 0 ρ f iωw i + p,i = 0 { ρs ω 2 u i σ ij,j = 0 σ ij = µ(u i,j + u j,i ) + λu k,k δ ij iωu i n i = w i n i σ ij n j = pn i on interface Γ I + standard boundary conditions on the boundary Austin, August 4-8, 2008 Streamers 44 / 49

57 Weak Coupling Step 1: Formulate conservation of mass (acoustics) and balance of momentum (elasticity) in a weak form: Ω a ( ω c ) 2 pq + iωρf v i q,i Γ I iωρ f v i n i q = B.T. q Ω e ω 2 ρu i v i + σ ij v i,j Γ I σ ij n j v i = B.T. v i Austin, August 4-8, 2008 Streamers 45 / 49

58 Weak Coupling Step 1: Formulate conservation of mass (acoustics) and balance of momentum (elasticity) in a weak form: Ω a ( ω c ) 2 pq + iωρf v i q,i Γ I iωρ f v i n i q = B.T. q Ω e ω 2 ρu i v i + σ ij v i,j Γ I σ ij n j v i = B.T. v i Step 2: Use the remaining equations in the strong form to eliminate fluid velocity and elastic stresses: ( ω ) 2 Ω a c pq + p,i q,i Γ I iωρ f v i n i q = B.T. q Ω e ω 2 ρu i v i + µ(u i,j + u j,i )v i,j + λu k,k v k,k Γ I σ ij n j v i = B.T. v i Austin, August 4-8, 2008 Streamers 45 / 49

59 Weak Coupling Step 1: Formulate conservation of mass (acoustics) and balance of momentum (elasticity) in a weak form: Ω a ( ω c ) 2 pq + iωρf v i q,i Γ I iωρ f v i n i q = B.T. q Ω e ω 2 ρu i v i + σ ij v i,j Γ I σ ij n j v i = B.T. v i Step 2: Use the remaining equations in the strong form to eliminate fluid velocity and elastic stresses: ( ω ) 2 Ω a c pq + p,i q,i Γ I iωρ f v i n i q = B.T. q Ω e ω 2 ρu i v i + µ(u i,j + u j,i )v i,j + λu k,k v k,k Γ I σ ij n j v i = B.T. v i Step 3: Use the interface conditions to couple the two variational formulations: Ω a ( ω c ) 2 pq + p,i q,i + ω 2 Γ I ρ f u i n i q = B.T. q Ω e ω 2 ρu i v i + µ(u i,j + u j,i )v i,j + λu k,k v k,k + Γ I pv i n i = B.T. v i Austin, August 4-8, 2008 Streamers 45 / 49

60 Abstract Variational Formulation u ũ D + V e, p p + V a b ee (u, v) + b ae (p, v) = l e (v) v V e b ea (u, q) + b aa (p, q) = l a (q) q V a where V e = {v H 1 (Ω e ) : v = 0 on Γ De } V a = {q H 1 (Ω a ) : q = 0 on Γ Da } b ee (u, v) = Ω e (E ijkl u k,l v i,j ρ s ω 2 u i v i ) dx + Γ Ce β ij u i v j ds b ae (p, v) = Γ I pv n ds b ea (u, q) = Γ I u n q ds b aa (p, q) = 1 ω 2 ρ f Ω a ( p q k 2 pq) dx l e (v) = Ω e f i v i dx + Γ Ne Γ Ce g i v i ds l a (q) = Ω a fq dx + Γ Na Γ Ca gv ds For scattering problems: l e (v) = Γ I p inc v n ds, l a (q) = 1 ω 2 ρ f Γ I p inc q ds n Austin, August 4-8, 2008 Streamers 46 / 49

61 de Rham Diagram IR H 1 H(curl) id L 2 0 H(div) Π grad Π curl Π div P IR W hp Qhp V hp Y hp 0 where the Projection-Based Interpolation Operators Π grad, Π curl, Π div, and L 2 -projection P make the diagram commute. Austin, August 4-8, 2008 Streamers 47 / 49

62 A Two-Grid Paradigm Our approach is to determine an optimal refinement strategy for a given coarse grid by examining the solution on a corresponding fine grid obtained by a global hp-refinement. Global hp-refinement p = 8 Coarse grid 0 Fine grid 0 p = 1 Austin, August 4-8, 2008 Streamers 48 / 49

63 A Two-Grid Paradigm Our approach is to determine an optimal refinement strategy for a given coarse grid by examining the solution on a corresponding fine grid obtained by a global hp-refinement. p = 8 Error estimate and optimal Coarse grid 1 hp-refinement Fine grid 0 p = 1 Austin, August 4-8, 2008 Streamers 48 / 49

64 A Two-Grid Paradigm Our approach is to determine an optimal refinement strategy for a given coarse grid by examining the solution on a corresponding fine grid obtained by a global hp-refinement. Global hp-refinement p = 8 Coarse grid 1 Fine grid 1 p = 1 Austin, August 4-8, 2008 Streamers 48 / 49

65 A Two-Grid Paradigm Our approach is to determine an optimal refinement strategy for a given coarse grid by examining the solution on a corresponding fine grid obtained by a global hp-refinement. p = 8 Error estimate and optimal Coarse grid 2 hp-refinement Fine grid 1 p = 1 Austin, August 4-8, 2008 Streamers 48 / 49

66 A Two-Grid Paradigm Our approach is to determine an optimal refinement strategy for a given coarse grid by examining the solution on a corresponding fine grid obtained by a global hp-refinement. Global hp-refinement p = 8 Coarse grid 2 Fine grid 2 p = 1 Austin, August 4-8, 2008 Streamers 48 / 49

67 A Two-Grid Paradigm Our approach is to determine an optimal refinement strategy for a given coarse grid by examining the solution on a corresponding fine grid obtained by a global hp-refinement. p = 8 Error estimate and optimal Coarse grid 3 hp-refinement Fine grid 2 p = 1 Austin, August 4-8, 2008 Streamers 48 / 49

68 A Two-Grid Paradigm Our approach is to determine an optimal refinement strategy for a given coarse grid by examining the solution on a corresponding fine grid obtained by a global hp-refinement. Global hp-refinement p = 8 Coarse grid 3 Fine grid 3 p = 1 Austin, August 4-8, 2008 Streamers 48 / 49

69 The energy-driven mesh optimization algorithm Find optimal hp-refinements of the current coarse grid hp yielding the next coarse grid hp next such that (u = u h/2,p+1 ), u Π hp u u Π hp nextu N hp next N hp max Austin, August 4-8, 2008 Streamers 49 / 49

70 The energy-driven mesh optimization algorithm Find optimal hp-refinements of the current coarse grid hp yielding the next coarse grid hp next such that (u = u h/2,p+1 ), u Π hp u u Π hp nextu N hp next N hp max The algorithm reflects the logic of the projection-based interpolation and consists of three steps: Determining optimal refinement of edges Determining optimal refinement of faces Determining optimal refinement of element interiors Each of the steps sets up initial conditions for the next step, limiting the number of cases to be considered. Austin, August 4-8, 2008 Streamers 49 / 49

71 Coupled Multiphysics Problems Simultaneous use of H 1, H(curl), H(div), L 2 - conforming elements, C-preprocessing is used only to differentiate between the real and complex versions of the code. Austin, August 4-8, 2008 Streamers 50 / 49

72 Coupled Multiphysics Problems Simultaneous use of H 1, H(curl), H(div), L 2 - conforming elements, C-preprocessing is used only to differentiate between the real and complex versions of the code. The domain is decomposed into geometrical blocks. Each block may involve only a subset of all variables. Austin, August 4-8, 2008 Streamers 50 / 49

73 Coupled Multiphysics Problems Simultaneous use of H 1, H(curl), H(div), L 2 - conforming elements, C-preprocessing is used only to differentiate between the real and complex versions of the code. The domain is decomposed into geometrical blocks. Each block may involve only a subset of all variables. Problem dependent code: computation of (unconstrained) element matrices, choice of norm. Austin, August 4-8, 2008 Streamers 50 / 49

74 Coupled Multiphysics Problems Simultaneous use of H 1, H(curl), H(div), L 2 - conforming elements, C-preprocessing is used only to differentiate between the real and complex versions of the code. The domain is decomposed into geometrical blocks. Each block may involve only a subset of all variables. Problem dependent code: computation of (unconstrained) element matrices, choice of norm. Problem independent code: description of geometry and multiphysics, constrained approximation, graphics, linear solvers, adaptivity, automatic h-,p- and hp-adaptivity. Austin, August 4-8, 2008 Streamers 50 / 49