1. 1 SIMULATION OF VIBROACOUSTIC PROBLEM USING COUPLED FE / FE FORMULATION AND MODAL ANALYSIS Ahlem ALIA presented by Nicolas AQUELET Laboratoire de Mécanique de Lille Université des Sciences et Technologies de Lille Avenue Paul Langevin, Cité Scientifique 59655 Villeneuve d’Ascq, France

2. 2 Introduction StructureFluid The vibrations generate sound The sound engenders vibrations Main industrial concern in Vibroacoustics: Reduction of NOISE Actually, noise constitutes an important indicator of quality in many industrial products such as vehicles, machinery…

3. 3 Introduction  Analytical technique  Simple geometry  Under very restrictive hypothesis  Numerical methods  FEM / FEM  FEM / BEM

4. 4 Introduction  Classical FEM / FEM  Six Nodes / Wavelength  Application domain: Low Frequency Range DOFf

5. 5  FEM / FEM with Modal analysis  Modal analysis solves the vibroacoustic problem with some modes  Reduction of the problem size  The modal analysis is applied with a Lumped mass representation  Lumped Mass matrix consists of Zero-off diagonal terms  Advantage of this approach:  Reduction of the computational cost Introduction (100 modes in our problem versus 432 physical unknowns)

6. 6 Introduction  Model the vibroacoustic behavior of an acoustic cavity with one flexible wall boundary by using FEM/FEM with:  Modal analysis  Lumped mass representation Simply supported elastic plate Mechanical load Rigid wall

7. 7 Governing equations  Structure  Fluid Fluid (  f ) (f)(f) (  sf ) Structure (  s ) Vibroacoustic problem  Pressure Continuity  Normal Displacement Continuity BC at Coupling Interface P: pressure k=  /c wave number w: displacement  : stress n: interface normal (1) (2) (3) (4)

8. 8 The application of the FEM to the variational formulation of structure cavity system yields to the following linear system : Coupling system K s, M s : structural stiffness and mass matrices K f, M f : fluid matrices B: coupling matrix F s, F f : mechanical load, acoustical sources c: sound velocity,  f : fluid density M K N s, N f : structural and fluid shape function (5) (6)

9. 9  The problem (6) can be seen as an eigenvalue problem: (K -  2 M ) = F Coupling system  For a great number of DOF, solving the system directly is always hard in term of CPU time. (6)

10. 10 Purpose of the approach We search two matrices L and R verifying: - L and R contain the LEFT and RIGHT eigenvectors, respectively -  is a diagonal matrix containing the eigenvalues of: We obtain the physical unknowns of (K -  2 M ) = F by this relation: (7) (10) (8) (9) (11) (6)

11. 11  Efficient eigenvalue algorithms can’t be used  A symmetric form of eigenvalue problem is required (K -  2 M ) = F      p w Since is non-symmetric, Sandberg’s method enables us to make it symmetric by using Modal analysis Purpose of the approach (6)

12. 12 Classical modal analysis s, f : the structural and the fluid eigenvalues. X s, X f : the structural and the fluid eigenvectors. Cavity with stiff boundaries Structure in vacuum Solved Independently (12)(13)

13. 13 Classical modal analysis D s, D f are diagonal matrices containing the structural and the fluid eigenvalues.  s,  f represent the modal structural displacement and the modal fluid pressure.  s,  f are matrices containing some eigenvectors of structure and fluid  s,  f verify the following properties: (14) (15) (50 modes)

14. 14 Classical modal analysis Hence, the coupling system (5) can be rewritten as the reduced system (17):  s,  f w, p The system is reduced (the problem size is divided by 4) but it remains non symmetric (5) (14) (17) 432 physical unknowns 100 modal unknowns

15. 15 Modal analysis ( Sandberg Method) Symmetric system Non Symmetric system Transition matrix (17) (18) (19)

16. 16 Symmetric generalized eigenvalue problem Modal analysis ( Sandberg Method)  V  : right eigenvector matrix of the symmetric system (19) (20) (21)

17. 17  s,  f w, p  s,  f Modal analysis ( Sandberg Method) R: right eigenvector matrix of the original system The left eigenvector matrix “ L” is obtained in the same manner (18) (14) (22) (9)

18. 18 Modal analysis ( Sandberg Method) (K -  2 M ) = F &   is a diagonal matrix containing the eigenvalues of: (6) (11) (10) (9)

19. 19 Numerical results Coupling Interface Fluid Rigid Structure Elastic Structure

20. 20 Numerical results Cavity (c) Plate (b)  Discrete Kirchhoff Quadrilateral (DKQ) plate element thin plate  Kirchhoff theory  8-node brick isoparametric acoustic element

21. 21 Structure ( Frequency response)  Simply supported plate (0.5m  0.5m)  Unit punctual force (0.125m  0.125m) Variation of the displacement with the frequency at the load point Results given by Migeot et al (1) Numerical results (1) 2nd Worldwide Automotive Conference Papers,1-7

22. 22 Structure ( Natural frequencies) Structure: Simply supported plate (0.2m  0.2m) made of brass Natural frequencies of the plate Consistent and lumped mass matrices are in good agreement with analytical ones as long as low frequencies are considered (<50th mode).

23. 23 Cavity ( Natural frequencies) Rigid cavity (0.2m  0.2m  0.2m) FEM leads to good results below the 50th mode Natural frequencies of the rigid cavity

24. 24 Coupling problem Results Given by Lee et al (2) CPU Time Simply supported elastic plate Field point Pressure at the point (0.1,0.1,0.2) Numerical results (2) Engineering Analysis with Boundary Elements, 16 (1995) 305-315

25. 25 Structure ( Frequency response) Plate quadratic displacement of the structure In vaccum Plate-cavity (air) Coupling effect 854Hz

26. 26 Mean square pressure Frequency (Hz) Mean square velocity Frequency (Hz) Coupling problem (air) Comparison between the direct and the modal results Mean square pressure: cavityMean square velocity:structure

27. 27 Coupling problem (water) Comparison between the direct and the modal results Mean square pressure: cavityMean square velocity: structure Mean square velocity Frequency (Hz) Mean square pressure Frequency (Hz) Mean square velocity

28. 28 FEM / FEM -- FEM / BEM FEM-FEM FEM-BEM comparison Frequency (Hz) Pressure (dB) Simply supported elastic plate Field point

29. 29 Conclusion  FEM / FEM with modal analysis and lumped mass representation has been used to model a simple vibroacoustic problem.  A good representation of the mass is very essential to achieve accurate results.  Modal FEM / FEM with only small number of modes is less efficient for strong coupling.  More modes must be taken into account ( disadvantage)  Solution: Improve the numerical results by using Modal correction for diagonal system