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國立台灣海洋大學河工二館 307 教室 中華民國九十二年七月十五日

Spurious Eigensolutions and Fictitious Frequencies for Acoustic Problems with the Mixed-type Boundary Conditions by using BEM 邊界元素法於混合型邊界條件問題之 假根及虛擬頻率探討. 研 究 生： 林宗衛 指導老師： 陳正宗教授. 國立台灣海洋大學河工二館 307 教室 中華民國九十二年七月十五日. Outlines. Motivation. Boundary integral equations.

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國立台灣海洋大學河工二館 307 教室 中華民國九十二年七月十五日

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  1. Spurious Eigensolutions and Fictitious Frequencies for Acoustic Problems with the Mixed-type Boundary Conditions by using BEM邊界元素法於混合型邊界條件問題之假根及虛擬頻率探討 研 究 生： 林宗衛 指導老師： 陳正宗教授 國立台灣海洋大學河工二館307教室 中華民國九十二年七月十五日

  2. Outlines Motivation Boundary integral equations Techniques and treatments Examples Semi-analytical approach Conclusions Further research 02

  3. Outlines Motivation Boundary integral equations Techniques and treatments Examples Semi-analytical approach Conclusions Further research 03

  4. Motivation 04

  5. Outlines Motivation Boundary integral equations Techniques and treatments Examples Semi-analytical approach Conclusions Further research 05 05

  6. Governing equation where D : the domain of interest k : the wave number x : the domain point u(x): the acoustic potential 06

  7. Types of boundary conditions u=u t= =u Dirichlet Neumann Mixed-type 07

  8. The null-field integral formulations D Dc Dc D 8

  9. Dc: the complementary domain of D, x,s: the field and source point, u(s): the potential on the boundary t(s): the normal derivative of potential on the boundary U (s,x): the kernel function, T(s,x) M(s,x) L (s,x) 9

  10. U (s,x)kernel in different methods Complex-valued BEM: Real-part BEM: Imaginary-part BEM: MRM: 10

  11. Discretization (singular formulation) 11

  12. Rearrangement of the influences matrices 12

  13. Discretization(hypersingular formulation) 13

  14. Outlines Motivation Boundary integral equations Techniques and treatments Examples Semi-analytical approach Conclusions Further research 14

  15. Extraction of the true eigensolutions by using SVD updating terms kt: the true eigenvalues 15

  16. Detection of the spurious eigensolutions by using SVD updating documents by using the Fredhohm’s alternative theorem T: transpose ks: the spurious eigenvalues 16

  17. Burton & Miller method +) [ikU(s, x)+L(s, x)]u(s) = [ikT(s, x)+M(s, x)]t(s) 17

  18. CHIEF method 18

  19. Outlines Motivation Boundary integral equations Techniques and treatments Examples Semi-analytical approach Conclusions Further research 19

  20. Interior problems 20

  21. A finite rod x x s0=0 s1=1 0 1 Type I : u(0)=0; t(1)=0 Type II: t(0)=0; u(1)=0 B.C.: 21

  22. Comparison of the eigenfuctions 22

  23. S T T T T T T S S T T T S S S T T T Real-part BEM Imaginary-part BEM Complex-valued BEM 23

  24. A circular cavity Elements: N=30 u2=0 R=1 m t1=0 24

  25. Detection the eigenvalues using the complex-valued BEM 25

  26. True Updating term Updating document Spurious Detection of true and spurious eigenvalues using the real-part BEM 26

  27. Detection of true and spurious eigenvalues using the real-part BEM True Updating term Updating document Spurious 27

  28. The comparison of the spurious eigenvalues with different B.C. Neumann or Dirichlet Mixed-typed 28

  29. The spurious eigenvalues using different methods where 29

  30. The comparison for the former five eigenmodes Real –part BEM FEM 30

  31. The comparison for the former five eigenvalues 31

  32. Exterior problems 32

  33. A semi-infinite rod x s0=1 s1= 0 1 B.C.: mt(1)+u(1)=n 33

  34. Comparison of the fictitious frequencies 34

  35. The fictitious frequency of 1-D problem T(s,x)=0 U(s,x)=0 35

  36. A circular radiator Elements: N=30 R=1 m 36

  37. The positions of irregular for u(1,0) and their treatments x 37

  38. The positions of irregular for t(1,p) and their treatments x 38

  39. Comparison of the BEM and exact solutions for radiation problem BEM solution Analytical solution 39

  40. Outlines Motivation Boundary integral equations Techniques and treatments Examples Semi-analytical approach Conclusions Further research 40

  41. The null-field integral formulations 41

  42. Semi-analytical approach 42

  43. Interior problem Real-part BEM Imaginary-part BEM Complex-valued BEM 43

  44. Exterior problem Singular Formulation Hypersingular Formulation 44

  45. Outlines Motivation Boundary integral equations Techniques and treatments Examples Semi-analytical approach Conclusions Further research 45

  46. Conclusions The results in the numerical experiment match well with those in our semi-analytical results. The eigensolutions and fictitious frequencies were solved successfully by using semi-analytical approach and BEMs 46

  47. Conclusions The spurious eigenvalues and fictitious frequencies depend on the representation no matter what the given types of B.C. for the problem are specified. 47

  48. Outlines Motivation Boundary integral equations Techniques and treatments Examples Semi-analytical approach Conclusions Further research 48

  49. Further research It will be interesting to know that if spurious eigenvalues and fictitious frequencies locate at the zeros of the Mathieu functions in the shape of ellipse. The quantity of the fictitious frequencies depend on the modal participation factor. 49

  50. Further research To derive the fictitious frequencies analytically by using the complex-valued BEM for mixed-type boundary condition problem. The extension to multiple radiators & scatters and half-plane problems using the similar algorithm can be conducted to examine the occurrence of the fictitious frequencies. 50

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