1. A Case Study for a Coupled System of Acoustics and Structures Deng Li (Japan Research Institute, Tokyo Institute of Technology) Craig C. Douglas (UK, Yale University) Takashi, Kako (University of Electro-Communications) Ichiro, Hagiwara (Tokyo Institute of Technology) March 23, 2006 CS521 in UK This research was supported in part by National Science Foundation grants EIA-0219627, ACI-0324876, ACI-0305466, and OISE-0405349

2. OUTLINE Basic Idea Background Mathematical Analysis Discretization by FEM Perturbation Method Error Estimation Application on Nastran Software Numerical Results Future Work This research was supported in part by National Science Foundation grants

3. Basic Idea Using uncoupled eigen-pairs (eigenvalue and eigenvector) to calculate coupled eigen-pairs. coupled eigen-pairs: Acoustic and Structure coupled system Uncoupled eigen-pairs: Acoustic system Structure system This research was supported in part by National Science Foundation grants

4. Background We study a numerical method to calculate the eigen-frequencies of the coupled vibration between an acoustic field and a structure. A typical example of the structure in our present study is a plate which forms a part of the boundary of the acoustic region, its application of this research is a problem to reduce a noise inside a car which is caused by an engine or other sources of the sound. More in detail, the interior car noises such as a booming noise or a road noise are structural-acoustic coupling phenomena. Our present study was motivated by the work Our present study was motivated by the work of Hagiwara, where they developed intensively the sensitivity analysis based on the eigenvalue calculation and applied the results to the design of motor vehicles with a lower inside noise. This research was supported in part by National Science Foundation grants

5. Mathematical Analysis (1) 3D Coupled Problem where Ω0 : a three-dimensional acoustic region, S0 : a plate region, Γ0=∂Ω0 ＼ S0 : a part of the boundary of the acoustic field, of the acoustic field, ∂S0 : the boundary of the plate, P0 : the acoustic pressure in Ω0, U0 : the vertical plate displacement, c : the sound velocity, ρ0 : the air mass density, D : the flexural rigidity of plate, ρ1 : the plate mass density, n : the outward normal vector on ∂Ω from Ω0, σ : the outward normal vector on ∂S0 from S0. This research was supported in part by National Science Foundation grants

6. Mathematical Analysis (2) 2D Coupled Problem apply Fourier mode decomposition to P and U in the z direction: This research was supported in part by National Science Foundation grants

7. Mathematical Analysis (3) Introduce Parameter  : This research was supported in part by National Science Foundation grants

8. Discretization by FEM (1) Ka and Ma: the stiffness and mass matrices for the acoustic field, Ka and Ma: the stiffness and mass matrices for the acoustic field, Kp and Mp: the stiffness and mass matrices for the plate, L and LT :the coupling matrices. The precise definitions of Ka, Ma, Kp, Mp, L and LT are as follows: This research was supported in part by National Science Foundation grants

9. Discretization by FEM (2) This research was supported in part by National Science Foundation grants

10. Perturbation Method (1) Introduce Parameter  : This research was supported in part by National Science Foundation grants

11. Perturbation Method (2) There are two orthonormality conditions for the eigenvector: This research was supported in part by National Science Foundation grants

12. Perturbation Method (3) Perturbation Series This research was supported in part by National Science Foundation grants

13. Error Estimation We give the order of convergence for the error between exact and approximate eigenvalues using the standard result of Babuska and Osborn, where we assume a certain regularity condition for the corresponding inhomogeneous problem. For 2D coupled eigenvalue problem, we obtain the order estimate After a few calculation, we can get the similar order estimation: This research was supported in part by National Science Foundation grants

14. Application on Nastran Software(1) Ortho-normality Condition for Eigenvector This research was supported in part by National Science Foundation grants

15. Application on Nastran Software (2) How to Get the Coefficient This research was supported in part by National Science Foundation grants

16. Numerical Results(1) Exact Solution Coupled Eigenvalue Problem This research was supported in part by National Science Foundation grants

17. Numerical Results(2) Exact Solution Acoustic Eigenvalue Structure Eigenvalue This research was supported in part by National Science Foundation grants

18. Parameters in Example This research was supported in part by National Science Foundation grants

19. Numerical Results(3) Example 1 of Perturbation Analysis: The First Eigenvalue of Type 1 the first eigenvalue error= the first eigenvalue error= 0.0 6.250000000 0.001 6.25000044476.2500004453 -6.0E-10 0.01 6.2500444660 6.2500444652 8.0E-10 0.1 6.2544465978 6.2544466536 -5.58E-8 0.5 6.3611649457 6.3611542771 1.06686E-5 0.8 6.53458226096.5345070074 7.52535E-5 1.0 6.69447657566.6946597826 -1.832070E-4 This research was supported in part by National Science Foundation grants

20. Numerical Results(4) Example 2 of Perturbation Analysis: The First Eigenvalue of Type 2 the first eigenvalue error= the first eigenvalue error= 0.0 0.071111110.0 7111111 0.0 0.001 0.0711111090.071111109 0.0 0.01 0.071110909 0.071110907 2E-9 0.1 0.071090951 0.071090759 1.92E-7 0.5 0.07061816 0.070602315 1.5846E-5 0.8 0.06989274 0.069808592 8.4151E-5 1.0 0.069251598 0.069085726 1.65871E-4 This research was supported in part by National Science Foundation grants

21. Numerical Results (5) Relationship between Error and a Number of Used Eigen-pairs A number of used eigen-pairs jError 1 6.261615E-4 3 -9.130154E-5 5 -1.621518E-4 10 -1.83207E-4 100 -1.865037E-4 1000 -1.865071E-4 10000 -1.865071E-4 This research was supported in part by National Science Foundation grants

22. Numerical Results (6) A Special Case A Special Case An approaching phenomenon of eigenvalues which cannot be described by FEM but can be described by the perturbation method. This research was supported in part by National Science Foundation grants

23. Numerical Results(7) Exact Result This research was supported in part by National Science Foundation grants

24. Numerical Results(8) FEM Result This research was supported in part by National Science Foundation grants

25. Numerical Results(9) compare the results This research was supported in part by National Science Foundation grants

26. Future Work We expect to obtain a mathematically rigorous estimation of the magnitude of the convergence radius of the perturbation series. We need consider how to modify the perturbation series in the case of eigenvalue is not simple. This research was supported in part by National Science Foundation grants