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Analyzing Spurious Eigenvalues in Doubly-Connected Membranes

This research report explores spurious eigenvalues in doubly-connected membranes, detailing mathematical analysis, numerical examples, and conclusions. The study investigates boundaries, fundamental solutions, and the impact of eccentricity on eigenvalues. It verifies the existence of spurious eigenvalues with various boundary conditions and techniques, emphasizing the distinction from true eigenvalues. The analysis involves degenerate kernels, Fourier series, and boundary integral equations. The conclusions highlight the occurrence and nature of spurious eigenvalues in the context of multiply-connected problems and the analytical proof using specialized methods.

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Analyzing Spurious Eigenvalues in Doubly-Connected Membranes

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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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