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

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

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

Spurious eignesolutions in BIE (BEM and NBIE) RealImaginaryComplex Saving CPU timeYes No Spurious eigenvaluesAppear No Complex Spurious eigenvalues Appear Simply-connected problem Multiply-connected problem (Fundamental solution) 4

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

Governing equation Fundamental solution 6

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

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

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

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

For the multiply-connected problem 11

For the multiply-connected problem 12

For the Dirichlet B.C., 13

SVD technique 14

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

Effect of the eccentricity e on the possible eigenvalues e k Former five true eigenvalues 7.66 Former two spurious eigenvalues

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

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

19 The existence of the spurious eigenvalue by boundary mode

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

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

22 The existence of the spurious eigenvalue by boundary mode 22

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

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

Minimum singular value of the annular circular membrane for fixed-fixed case using UT formulate k k k=4.86 k=

Effect of the eccentricity e on the possible eigenvalues e k Former five true eigenvalues 7.66 Former two spurious eigenvalues

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

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

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

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

Conclusions The spurious eigenvalue occur for the doubly-connected membrane, even the complex fundamental solution are used. The spurious eigenvalue of the doubly-connected membrane are 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

The End Thanks for your attention