A substructuring FE model for reduction of structural acoustic problems with dissipative interfaces

Transcription

1 A substructuring FE model for reduction of structural acoustic problems with dissipative interfaces PhD started October 2008 Romain RUMPLER Conservatoire National des Arts et Métiers - Cnam Future ESR Fellow at KTH (July 2010) Reduction methods for structural acoustic problems 1 / 28

2 Motivations Noise reduction in the context of structural acoustic applications Continue W. Larbi s work at LMSSC on dissipative interfaces modelisation Extend to applications including 3D modelisation (porous media) Develop computation-efficient methods to treat large structural acoustic problems (reduction methods) Implement models into efficient / open source FE software: FEAP (R.L. Taylor), Fortran Reduction methods for structural acoustic problems 2 / 28

3 Outline Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 3 / 28

4 (Us p F ) coupling element Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 4 / 28

5 (Us p F ) coupling element Elasto acoustic coupling: [U s,p F ] Problem equations:»» Kss K sθ ω 2 Mss K θs K θθ `ˆKF M θs M sθ M θθ Us θ s ω 2 ˆM F ˆpF Coupling on Γ sf through Neumann BC: f bs = f Fs = C sf p F f bθ = f Fθ = 0 f bf = f sf = ω 2 ρ 0 c 2 0 CT sf Us with: C sf = R Γ sf (N s) T nn F ds =» fbs f bθ = ˆf bf Implementation - Frequency domain: K ss K sθ C sf M ss M sθ 4K θs K θθ 0 5 ω 2 4 5A K F M θs M θθ 0 CsF T 0 K F U s θ s p F = 4 f bs f bθ f bf 3 5 Reduction methods for structural acoustic problems 5 / 28

6 (Us p F ) coupling element Elasto acoustic coupling: [U s,p F ] Problem equations:»» Kss K sθ ω 2 Mss K θs K θθ `ˆKF M θs M sθ M θθ Us θ s ω 2 ˆM F ˆpF Coupling on Γ sf through Neumann BC: f bs = f Fs = C sf p F f bθ = f Fθ = 0 f bf = f sf = ω 2 ρ 0 c 2 0 CT sf Us with: C sf = R Γ sf (N s) T nn F ds =» fbs f bθ = ˆf bf Implementation - Frequency domain: C sf U s ω A 4θ s 5 = CsF T 0 0 p F 0 Reduction methods for structural acoustic problems 5 / 28

7 (Us p F ) coupling element Frequency response of a cavity coupled with a plate Elasto-acoustic coupled cavity: (1m 1m 1m) cavity ( ) acoustic elements (1m 1m) plate on one face (10 10) shell elements (10 10) quad coupling elts Normal unit excitation force Coupled cavity mesh: F Mean quadratic velocity of plate (db) Mean quadratic velocity of plate coupled with cavity (Panneton) Frequency (Hz) [R. Panneton, Modélisation numérique tridimensionnelle par éléments finis des milieux poroélastiques, Ph.D Thesis, 1996] Reduction methods for structural acoustic problems 6 / 28

8 (us u f ) porous element Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 7 / 28

9 (us u f ) porous element Us-Uf porous element implementation with: Implementation - Frequency domain:»» Kss K sf ω 2 Mss M sf us = K fs K ff M ss = ρ ss RΩ P N T s NsdV M ff = ρ ff RΩ P N T f N f dv M sf = ρ sf R Ω P N T s N f dv = M T fs K ss = R Ω P ( N s) T D s N sdv K ff = R Ω P ( N f ) T D f N f dv M fs M ff u f» fbs f bf Material parameters: In vacuo frame: Es, ν, ρ s Fluid: ρ 0, γ, Pr, η, P 0 Biot porous: α, φ, σ, Λ, Λ K sf = R Ω P ( N s) T Q N f dv = K T sf f bs Neumann BC on frame f bf Neumann BC on fluid phase [J.- F. Allard, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, Elsevier Science Publishers Ltd, London, 1993] Reduction methods for structural acoustic problems 8 / 28

10 (us u f ) porous element Us-Uf porous element implementation with: Implementation - Frequency domain:»» Kss K sf ω 2 Mss M sf us = K fs K ff M ss = ρ ss RΩ P N T s NsdV M ff = ρ ff RΩ P N T f N f dv M sf = ρ sf R Ω P N T s N f dv = M T fs K ss = R Ω P ( N s) T D s N sdv K ff = R Ω P ( N f ) T D f N f dv K sf = R Ω P ( N s) T Q N f dv = K T sf f bs Neumann BC on frame f bf Neumann BC on fluid phase M fs M ff u f» fbs f bf Material parameters: In vacuo frame: Es, ν, ρ s Fluid: ρ 0, γ, Pr, η, P 0 Biot porous: α, φ, σ, Λ, Λ Detailed expressions: ρ ss = ρ s + ρ a iσφ 2 G(ω) ω ρ ff = φρ 0 + ρ a iσφ 2 G(ω) ω ρ sf = ρ a + iσφ 2 G(ω) ω with: ρ a = φρ 0 (α 1)» 1 G (ω) = 1 + 4iα2 ηρ 2 0ω σ 2 Λ 2 φ 2 Reduction methods for structural acoustic problems 8 / 28

11 (us u f ) porous element Us-Uf porous element implementation with: Implementation - Frequency domain:»» Kss K sf ω 2 Mss M sf us = K fs K ff M ss = ρ ss RΩ P N T s NsdV M ff = ρ ff RΩ P N T f N f dv M sf = ρ sf R Ω P N T s N f dv = M T fs K ss = R Ω P ( N s) T D s N sdv K ff = R Ω P ( N f ) T D f N f dv M fs M ff u f» fbs f bf Material parameters: In vacuo frame: Es, ν, ρ s Fluid: ρ 0, γ, Pr, η, P 0 Biot porous: α, φ, σ, Λ, Λ Detailed expressions: σ s = D sε s + Qε f K sf = R Ω P ( N s) T Q N f dv = K T sf f bs Neumann BC on frame f bf Neumann BC on fluid phase σ f = Q T ε s + D f ε f with: D s (Es, ν, φ, γ, P 0,Λ, Λ, σ, α, η, ρ 0, ω,pr) D f (φ, γ, P 0,Λ,Λ, σ, α, η, ρ 0, ω, Pr) Q (φ, γ, P 0, Λ,Λ, σ, α, η, ρ 0, ω, Pr) Reduction methods for structural acoustic problems 8 / 28

12 (us u f ) porous element Normal incidence impedance of porous covered wall Poroelastic layer on rigid wall: Material properties: Infinite lateral dimensions Unit plane wave excitation / Normal incidence 7.62cm thick porous layer Problem illustration: Wall acoustic impedance (Pa.s/m) Frame Fluid Porous Es = 800.8kPa c 0 = 343m/s φ = 0.9 ν s = 0.4 γ = 1.4 σ = 25kNs/m 4 ρ s = 30kg/m 3 Pr = 0.71 α = 7.8 ρ 0 = 1.21kg/m 3 Λ = 226µm η = Ns/m 2 Λ = 226µm Wall acoustic normal incidence impedance Re(Z) Computed - 10 elmt Im(Z) Computed - 10 elmt Re(Z) Computed - 1 elmt Im(Z) Computed - 1 elmt Re(Z) Exact Im(Z) Exact Frequency (Hz) [J.-F. Deü, W. Larbi, R. Ohayon, Vibration and transient response of structural-acoustic interior coupled systems with dissipative interface, CMAME, 2008] Reduction methods for structural acoustic problems 9 / 28

13 (us u f ) p F coupling element Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 10 / 28

14 (us u f ) p F coupling element Poro acoustic coupling: [(u s,u f ),p F ] Problem equations:»» Kss K sf ω 2 Mss K fs K ff `ˆKF M fs M sf M ff us u f ω 2 ˆM F ˆpF =» fbs f bf = ˆf bf Coupling on Γ PF through Neumann BC: f bs = f Fs = (1 φ)c sf p F f bf = f Ff = φc ff p F f bf = f PF = ω 2 ρ 0 c0 2 ``1 φ CT sf u s + φcff T u f with: C sf = R Γ (N PF s) T nn F ds C ff = R (N Γ f ) T nn F ds PF Implementation - Frequency domain: 02 K ss K sf ` φ C sf M ss M sf 0 u s f 4K fs K ff φc ff 5 ω 2 4 M fs M ff 0 5A 4u f 5 = 4f bf K F (1 φ)csf T φcff T K F p F f bf Reduction methods for structural acoustic problems 11 / 28

15 (us u f ) p F coupling element Poro acoustic coupling: [(u s,u f ),p F ] Problem equations:»» Kss K sf ω 2 Mss K fs K ff `ˆKF M fs M sf M ff us u f ω 2 ˆM F ˆpF =» fbs f bf = ˆf bf Coupling on Γ PF through Neumann BC: f bs = f Fs = (1 φ)c sf p F f bf = f Ff = φc ff p F f bf = f PF = ω 2 ρ 0 c0 2 ``1 φ CT sf u s + φcff T u f with: C sf = R Γ (N PF s) T nn F ds C ff = R (N Γ f ) T nn F ds PF Implementation - Frequency domain: ` φ C sf u s 40 0 φc ff 5 ω A 4u f 5 = (1 φ)csf T φcff T 0 p F 0 Reduction methods for structural acoustic problems 11 / 28

16 (us u f ) p F coupling element Frequency response of partly treated rigid acoustic cavity Rigid cavity - partly covered: (0.4m 0.6m 0.75m) cavity ( ) acoustic elements (0.4m 0.6m 0.05m) porous layer on one wall (8 12 4) u s u f porous elts (8 12 4) quad coupling elts Corner harmonic excitation Frame Fluid Porous Es = 4.4MPa c 0 = 343m/s φ = 0.94 ν s = 0 γ = 1.4 σ = 40kNs/m 4 ρ s = 130kg/m 3 Pr = 0.71 α = 1.06 ρ 0 = 1.21kg/m 3 Λ = 56µm η = Ns/m 2 Λ = 110µm Mean quadratic pressure in cavity: Acoustic cavity (Davidsson) - Rigid partly covered with porous Poro-acoustic coupled cavity mesh: Mean quadratic pressure in cavity (db) Rigid cavity 130 Brennan eq. fluid UsUf porous formulation Frequency (Hz) [P. Davidsson, Structure-acoustic analysis: Finite element modelling and reduction methods, Ph.D Thesis, 2004] Reduction methods for structural acoustic problems 12 / 28

17 Complete concrete car coupled problem Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 13 / 28

18 Complete concrete car coupled problem Frequency response in the EC and PC of the concrete car Elasto-poro-acoustic problem: Porous layer 2 acoustic cavities: EC and PC A wireframe in between cavities A porous layer on a wall of the PC Corner harmonic excitation in EC DOFs Wireframe Complete concrete car - With and without porous 200 Mean quadratic pressure (db) Engine cavity Passenger cavity Passenger cavity - porous layer Acoustic cavities Frequency (Hz) Reduction methods for structural acoustic problems 14 / 28

19 Objectives of the reduction method Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 15 / 28

20 Objectives of the reduction method Objectives of the applied reduction method Complete model: DOFs First objective: reduction of the non dissipative part Reduction to interfaces Reduced model: DOFs (8700 for porous) Reduction methods for structural acoustic problems 16 / 28

21 Objectives of the reduction method Objectives of the applied reduction method Complete model: DOFs First objective: reduction of the non dissipative part Reduction to interfaces Reduced model: DOFs (8700 for porous) Reduction methods for structural acoustic problems 16 / 28

22 Objectives of the reduction method Objectives of the applied reduction method Complete model: DOFs First objective: reduction of the non dissipative part Reduction to interfaces Reduced model: DOFs (8700 for porous) Reduction methods for structural acoustic problems 16 / 28

23 CMS: Craig and Bampton reduction method Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 17 / 28

24 CMS: Craig and Bampton reduction method Craig and Bampton reduction method: Outline C & B subdomain partitioning: Kii KT 1i K2i T K 1i K ω 2 4 Mii MT 1i M2i T M 1i M A K 2i 0 K 22 M 2i 0 M u i 3 u 1 5 = u f i C & B transformation: u i I ii 0 0 u i 4u 1 5 = 4Ψ 1 Φ α 1 5 u 2 Ψ 2 0 Φ 2 α 2 with for s=1..2: Ψ s = K 1 ii K is: Static modes Φ s: First normal modes of fixed interface eigenvalue problem K ssφ s = ω 2 sm ssφ s α s: Modal coordinates (<<nb nodal coordinates) Fi 1 Interface: i 2 [R. Craig, M. Bampton, Coupling of substructures for dynamic analysis, AIAA, 1968] Reduction methods for structural acoustic problems 18 / 28

25 CMS: Craig and Bampton reduction method Craig and Bampton reduction method: Outline Resulting system for n substructures: K iiα M iiα M 0 ω1 2 1i T α... M T ni α u i f i M 1iα I α 1 ω C A 4. 5 = ωn 2 M niα 0... I n α n 0 With: K iiα = K ii + P n s=1 KT si Ψ s M iiα = M ii + P n s=1 ˆMT si Ψ s + Ψ T s M siα = ˆΦ T s `Msi + M ssψ s `Msi + M ssψ s Reduction methods for structural acoustic problems 19 / 28

26 CMS: Craig and Bampton reduction method C & B implementation in FEAP Resulting system for subdomain n with its boundary interface: 02 «3 2» Kiin + M 6 Kni T 0 4 Ψn 7 5 M iin + ni Ψn+» «T 31 ω2 6 Ψ T Φ n `Mni + M nnψ n T Mni+ n M nnψ n 7A ω 2 n ˆΦT n `Mni + M nnψ n I n " pin α n # = " fin 0 # (1) (2) Computation steps: (2) α n = ω 2 `ωn 2 ω2 I n 1 ˆΦT n `Mni + M nnψ n pin " `Kiin + K T Ψn ni ω 2 ˆM iin + M T ni (1) Ψn + # ΨT n `Mni + M nnψ n ω 2 hˆφt n `Mni + M nnψ n T ω 2 `ω 2 n p in = f in ω2 I n 1 ˆΦT n `Mni + M nnψ n i n Interface: i Solving steps: Solve (1) p in Compute p n = hφ nω 2 `ω 2 n ω2 I n 1 ˆΦT n `Mni + M nnψ n + Ψn i p in Reduction methods for structural acoustic problems 20 / 28

27 CMS: Craig and Bampton reduction method C & B implementation in FEAP Problem equations: " `Kiin + K TΨn ni ω 2 ˆM iin + Mni T Solve Ψn + # ΨT n `Mni + M nnψ n ω hˆφt 2 n `Mni + M nnψ n T ω 2 `ωn 2 p in = f in ω2 I n 1 ˆΦT n `Mni + M nnψ n i Compute p n = hφ nω 2 `ω i n 2 ω 2 I n 1 ˆΦT n `Mni + M nnψ n + Ψn p in Steps of algorithm: 1. Loop on subdomains n: - Compute K nn & M nn - Eigenvalue problem: ω 2 n, Φ n - Loop on interface dofs K ni Ψ n = Knn 1 K ni M ni A n = M ni + M nnψ n - Compute B n = Φ T n An - Update: K iiα = K iiα + K T ni Ψn M iiα = M iiα +M T ni Ψn +ΨT n An 2. Interface sized problem: - Compute K ii & M ii - Update: K iiα = K iiα + K ii M iiα = M iiα + M ii 3. Loop on ω: - K FRF (ω) = K iiα ω 2 M iiα ω 4 P h i n Bn T `ω2 n ω 2 I n 1 Bn - Solve K FRF (ω) p i = f i p i - p n = hφ nω 2 `ω i n 2 ω2 I n 1 Bn + Ψ n p in Reduction methods for structural acoustic problems 21 / 28

28 CMS: Academic applications Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 22 / 28

29 CMS: Academic applications C & B reduction applied to enclosed cavity Rigid cavity: (0.4m 0.6m 0.75m) cavity ( ) acoustic elements Decomposed in 2 subdomains Comparison 20/50 normal modes Corner harmonic excitation Cavity mesh: Mean quadratic pressure (db) Mean quadratic pressure in cavity: Mean quadratic pressure in rigid cavity - 2 subdomains 120 Complete problem CB Reduction - 20 modes CB Reduction - 50 modes Frequency (Hz) Reduction methods for structural acoustic problems 23 / 28

30 CMS: Academic applications Reduction of acoustic part of porous treated cavity Porous treated cavity: (0.4m 0.6m 0.75m) cavity ( ) acoustic elements (0.4m 0.6m 0.05m) porous layer on one wall (8 12 4) hexahedric elements for porous Acoustic part reduced only Comparison 10/20/50 normal modes Corner harmonic excitation Mean quadratic pressure (db) Mean quadratic pressure in cavity: Reduced acoustic part of porous treated cavity Complete problem CB Reduction - 10 modes CB Reduction - 20 modes CB Reduction - 50 modes Frequency (Hz) Reduction methods for structural acoustic problems 24 / 28

31 CMS: Academic applications Reduction of acoustic part of porous treated cavity Porous treated cavity: Same cavity covered with porous layer 2 acoustic subdomains for reduction of acoustic part Comparison 50 normal modes Corner harmonic excitation Cavity and porous layer mesh: Mean quadratic pressure (db) Mean quadratic pressure in cavity: Reduced acoustic part (2 subdomains) of porous treated cavity Complete problem CB Reduction - 50 modes CB Reduction - 2 subdomains - 50 modes Frequency (Hz) Reduction methods for structural acoustic problems 25 / 28

32 CMS: Application to the concrete car Elements implementation (U s p F ) coupling element (u s u f ) porous element (u s u f ) p F coupling element Complete concrete car coupled problem Implementation of reduction method Objectives of the reduction method CMS: Craig and Bampton reduction method Applications CMS: Academic applications CMS: Application to the concrete car Conclusion - Pespectives Reduction methods for structural acoustic problems 26 / 28

33 CMS: Application to the concrete car Reduction method applied to the concrete car Reduced concrete car problem: 4 reduced acoustic components: EC(1) PC(3) Wireframe A 5cm porous layer on one wall FRF range [0,600] Hz Cavity mesh: Reduction methods for structural acoustic problems 27 / 28

34 CMS: Application to the concrete car Reduction method applied to the concrete car Reduced concrete car problem: 4 reduced acoustic components: EC(1) PC(3) Wireframe A 5cm porous layer on one wall FRF range [0,600] Hz EC: 50 modes 740 Hz PC: 75 modes 640 Hz Cavity mesh: Mean quadratic pressure (db) Mean quadratic pressure in EC and PC: Mean quadratic pressure in EC / PC+poro - 4 acoustic components Passenger cavity Engine cavity Passenger cavity - reduction Engine cavity - reduction Frequency (Hz) Reduction methods for structural acoustic problems 27 / 28

35 CMS: Application to the concrete car Reduction method applied to the concrete car Reduced concrete car problem: 4 reduced acoustic components: EC(1) PC(3) Wireframe A 5cm porous layer on one wall FRF range [0,600] Hz EC: 50 modes 740 Hz PC: 75 modes 640 Hz Cavity mesh: Relative error (%) Error on mean quadratic pressure: Relative error of 75 modes reduction Passenger cavity - smoothed Passenger cavity Engine cavity - smoothed Engine cavity Frequency (Hz.) Reduction methods for structural acoustic problems 27 / 28

36 CMS: Application to the concrete car Reduction method applied to the concrete car Reduced concrete car problem: 4 reduced acoustic components: EC(1) PC(3) Wireframe A 5cm porous layer on one wall FRF range [0,600] Hz EC: 50 modes 740 Hz PC: 75 modes 640 Hz Cavity mesh: CPU time (min) CPU computation time: Time with/without reduction on complete CC - 4ss - porous 500 CPU time - no reduction CPU time - 75 modes/ss Number frequency increments Reduction methods for structural acoustic problems 27 / 28

37 CMS: Application to the concrete car Reduction method applied to the concrete car Reduced concrete car problem: 4 reduced acoustic components: EC(1) PC(3) Wireframe A 5cm porous layer on one wall FRF range [0,600] Hz EC: 150 modes 1200 Hz PC: 250 modes 1000 Hz Cavity mesh: Mean quadratic pressure (db) Mean quadratic pressure in EC and PC: Mean quadratic pressure in EC / PC+poro - 4 acoustic components Passenger cavity Engine cavity Passenger cavity - reduction Engine cavity - reduction Frequency (Hz) Reduction methods for structural acoustic problems 27 / 28

38 CMS: Application to the concrete car Reduction method applied to the concrete car Reduced concrete car problem: 4 reduced acoustic components: EC(1) PC(3) Wireframe A 5cm porous layer on one wall FRF range [0,600] Hz EC: 150 modes 1200 Hz PC: 250 modes 1000 Hz Cavity mesh: Relative error (%) Error on mean quadratic pressure: Relative error of 250 modes reduction Passenger cavity - smoothed Passenger cavity Engine cavity - smoothed Engine cavity Frequency (Hz.) Reduction methods for structural acoustic problems 27 / 28

39 CMS: Application to the concrete car Reduction method applied to the concrete car Reduced concrete car problem: 4 reduced acoustic components: EC(1) PC(3) Wireframe A 5cm porous layer on one wall FRF range [0,600] Hz EC: 150 modes 1200 Hz PC: 250 modes 1000 Hz Cavity mesh: Relative error (%) Improvement on error: Relative error comparison between modes Passenger cavity Hz truncation Passenger cavity Hz truncation Engine cavity Hz truncation Engine cavity Hz truncation Frequency (Hz.) Reduction methods for structural acoustic problems 27 / 28

40 CMS: Application to the concrete car Reduction method applied to the concrete car Reduced concrete car problem: 4 reduced acoustic components: EC(1) PC(3) Wireframe A 5cm porous layer on one wall FRF range [0,600] Hz EC: 150 modes 1200 Hz PC: 250 modes 1000 Hz Cavity mesh: CPU time (min) CPU computation time: Time with/without reduction on complete CC - 4ss - porous 500 CPU time - no reduction CPU time - 75 modes/ss CPU time modes/ss Number of frequency increments Reduction methods for structural acoustic problems 27 / 28

41 Conclusion and perspectives Basic tools for coupled problems implemented (acoustic, porous and coupling elements) Efficient implementation of a reduction method for non-dissipative part of the problem Reduction method applied to the Smart-Structure Concrete car model Porous media and interfaces responsible of remaining dofs Further reduce interfaces with dofs condensation Work with Matlab on methods for reduction of the porous media Reduction methods for structural acoustic problems 28 / 28