1 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 The 32 nd National Conference on Theoretical and Applied Mechanics National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering

2 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

3 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

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

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

6 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

7 Problem statement

8 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

9 3D degenerate kernels

10 3D degenerate kernels

11 Null-field Integral equation - UT

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

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

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

15 Eigenvalue (k )-(free-free) n n

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

17 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

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

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

20 Singular formulation -Spurious SVD updating document U T

21 Hypersingular formulation -Spurious SVD updating document L M

22 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

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.

24 ~Thanks for your kind attentions~