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On spurious eigenvalues of doubly-connected membrane

On spurious eigenvalues of doubly-connected membrane. Reporter: I. L. Chen Date: 07. 29. 2008 Department of Naval Architecture, National Kaohsiung Institute of Marine Technology. 1. Outlines. 1. Introduction. 2. Problem statements. 3. Mathematical analysis. 4. Numerical examples.

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On spurious eigenvalues of doubly-connected membrane

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  1. On spurious eigenvalues of doubly-connected membrane Reporter: I. L. Chen Date: 07. 29. 2008 Department of Naval Architecture, National Kaohsiung Institute of Marine Technology 1

  2. Outlines 1. Introduction 2. Problem statements 3. Mathematical analysis 4. Numerical examples 5. Conclusions 2

  3. Outlines 1. Introduction 2. Problem statements 3. Mathematical analysis 4. Numerical examples 5. Conclusions 3

  4. (Fundamental solution) Simply-connected problem Multiply-connected problem Spurious eignesolutions in BIE (BEM and NBIE) 4

  5. Outlines 1. Introduction 2. Problem statements 3. Mathematical analysis 4. Numerical examples 5. Conclusions 5

  6. Governing equation Governing equation Fundamental solution 6

  7. a e b Multiply-connected problem a = 2.0 m b = 0.5 m e=0.0～1.0 m Boundary condition: Outer circle: Inner circle 7

  8. Outlines 1. Introduction 2. Problem statements 3. Mathematical analysis 4. Numerical examples 5. Conclusions 8

  9. Boundary integral equation and null-field integral equation Interior problem Exterior problem Degenerate (separate) form 9

  10. cosnθ, sinnθ boundary distributions kth circular boundary Degenerate kernel and Fourier series x Expand fundamental solution by using degenerate kernel s O x Expand boundary densities by using Fourier series 10

  11. For the multiply-connected problem 11

  12. For the multiply-connected problem 12

  13. For the Dirichlet B.C., 13

  14. SVD technique 14

  15. k=4.86 k=7.74 k k Minimum singular valueof the annular circular membrane for fixed-fixed case using UT formulate 15

  16. 7.66 Former two spurious eigenvalues 4.86 k e Effect of the eccentricity e on the possible eigenvalues • Former five true eigenvalues 16

  17. For any point , we obtain the null-field response Eigenvalue of simply-connected problem By using the null-field BIE, the eigenequation is a True eigenmodeis : ,where . 17

  18. b a The existence of the spurious eigenvalue by boundary mode For the annular casewith fix-fix B.C. 18

  19. The existence of the spurious eigenvalue by boundary mode 19

  20. The eigenvalue of annular case with fix-fix B.C. Spurious eigenequation True eigenequation 20

  21. b a The eigenvalue of annular case with free-free B.C. 21

  22. The existence of the spurious eigenvalue by boundary mode 22 22

  23. The eigenvalue of annular case with free-free B.C. Spurious eigenequation True eigenequation 23

  24. Outlines 1. Introduction 2. Problem statements 3. Mathematical analysis 4. Numerical examples 5. Conclusions 24

  25. k=4.86 k=7.74 k k 25 Minimum singular valueof the annular circular membrane for fixed-fixed case using UT formulate

  26. Effect of the eccentricity e on the possible eigenvalues 7.66 Former two spurious eigenvalues 4.86 • Former five true eigenvalues k e 26

  27. b a t Innerboundary Outerboundary Fourier coefficients ID Boundary mode (true eigenvalue) Real part of Fourier coefficients for the first true boundary mode ( k =2.05, e = 0.0) 27

  28. k=4.81 Outer boundary (trivial) Innerboundary b a k=7.66 Outer boundary (trivial) Innerboundary Fourier coefficients ID Boundary mode (spurious eigenvalue) Dirichlet B.C. using UT formulate 28

  29. Boundary mode (spurious eigenvalue) Neumann B.C. using UT formulation T kernel k=4.81 ( ) real-par T kernel k=7.75 ( ) real-part

  30. Boundary mode (spurious eigenvalue) Neumann B.C. using LM formulate M kernel k=4.81 ( ) real-par M kernel k=7.75 ( ) real-part

  31. Outlines 1. Introduction 2. Problem statements 3. Mathematical analysis 4. Numerical examples 5. Conclusions 31

  32. Conclusions • The spurious eigenvalue occur for the doubly-connected membrane, even the complex fundamental solution are used. • The spurious eigenvalue of the doubly-connected membraneare true eigenvalue of simple-connected membrane. • The existence of spurious eigenvalue are proved in an analytical manner by using the degenerate kernels and the Fourier series. 32

  33. The End Thanks for your attention

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