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A Case Study for a Coupled System of Acoustics and Structures

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 )

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A Case Study for a Coupled System of Acoustics and Structures

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  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 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, ∂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, 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= • 0.0 6.250000000 • 0.001 6.2500004447 6.2500004453 -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 6.5345822609 6.5345070074 7.52535E-5 • 1.0 6.6944765756 6.6946597826 -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= 0.0 0.07111111 0.0 7111111 0.0 0.001 0.071111109 0.071111109 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 j Error • 6.261615E-4 • -9.130154E-5 • -1.621518E-4 • -1.83207E-4 • -1.865037E-4 • -1.865071E-4 • -1.865071E-4 This research was supported in part by National Science Foundation grants

  22. Numerical Results (6) 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 ResultThis research was supported in part by National Science Foundation grants

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

  25. Numerical Results(9)compare the resultsThis 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

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