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On the spurious eigenvalues for a concentric sphere in BIEM

The 32 nd National Conference on Theoretical and Applied Mechanics. On the spurious eigenvalues for a concentric sphere in BIEM. Reporter ： Shang-Kai Kao Ying-Te Lee , Jia-Wei Lee and Jeng-Tzong Chen Date ： 2008/11/28-29. National Taiwan Ocean University MSVLAB

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On the spurious eigenvalues for a concentric sphere in BIEM

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  1. The 32nd National Conference on Theoretical and Applied Mechanics On the spurious eigenvalues for a concentric sphere in BIEM Reporter：Shang-Kai Kao Ying-Te Lee , Jia-Wei Lee and Jeng-Tzong Chen Date：2008/11/28-29 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1

  2. Outline • Introduction • Problem statement • Mathematical analysis • Numerical example • Conclusions 2

  3. Outline • Introduction • Problem statement • Mathematical analysis • Numerical example • Conclusions 3

  4. Motivation • 1-D real-part BEM, CTAM31 • 2-D Journal of Sound and Vibration, 2002 doubly-connected membrane, CTAM31 • 3-D Computational Mechanics, 2002 4

  5. BIEM and Null-field Integral Equation Interior problem Exterior problem Degenerate (separate) form 5 5

  6. Outline • Introduction • Problem statement • Mathematical analysis • Numerical example • Conclusions 6

  7. Problem statement 7

  8. Outline • Introduction • Problem statement • Mathematical analysis • Numerical example • Conclusions 8

  9. 3D degenerate kernels 9

  10. 3D degenerate kernels 10

  11. Null-field Integral equation - UT 11

  12. Dirichlet B.C. (fixed-fixed) - UT 12

  13. It’s a special case that a=0.5b. Eigenvalue (k)-(fixed-fixed) 13

  14. Neumann B.C. (free-free) - UT 14

  15. Eigenvalue (k )-(free-free) 15

  16. The eigenvalues by using BIM and SVD U kernel L kernel T kernel M kernel 16

  17. Outline • Introduction • Problem statement • Mathematical analysis • Numerical example • Conclusions 17

  18. SVD updating terms U L Dirichlet B.C. (fixed-fixed)-True 18

  19. SVD updating terms T M Neumann B.C. (free-free)-True 19

  20. SVD updating document U T Singular formulation-Spurious 20

  21. SVD updating document L M Hypersingular formulation-Spurious 21

  22. Outline • Introduction • Problem statement • Mathematical analysis • Numerical example • Conclusions 22

  23. Conclusions • There are still spurious eigenvalues by using BIEM to deal with concentric sphere problems. • True eigenvalues are dependent on problems and spurious eigenvalues are dependent on methods. • Spurious eigenvalues are dependent on the inner boundary. 23

  24. ~Thanks for your kind attentions~ 24

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