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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.

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Presentation on theme: "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."— Presentation transcript:

1 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 2 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

3 3 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

4 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 5 BIEM and Null-field Integral Equation Interior problem Exterior problem Degenerate (separate) form 5

6 6 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

7 7 Problem statement

8 8 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

9 9 3D degenerate kernels

10 10 3D degenerate kernels

11 11 Null-field Integral equation - UT

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

13 13 Eigenvalue (k)-(fixed-fixed) n012345678 6.283198.9868211.526913.975916.365118.711621.025723.314125.5816 12.566415.450518.1920.834223.409825.933128.414830.862633.282 18.849621.808224.645927.39630.079332.709435.295937.84640.3649 25.132728.132431.029233.847236.602539.306341.966944.590747.1825 n012345678 6.283196.572017.111587.845048.71689.68210.707711.770812.8557 12.566412.721413.026113.471114.043714.729415.513316.380617.3173 18.849618.954419.162519.471119.845820.371820.953321.614122.3481 25.132725.211825.369225.603825.913726.296826.750227.27127.8561 It’s a special case that a =0.5 b.

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

15 15 Eigenvalue (k )-(free-free) n012345678 1.840273.151184.389965.574546.717537.831128.9249510.0056 6.572016.911527.553628.438879.5010110.677711.916513.177814.4344 12.721412.885213.208713.68414.301415.050415.920416.900517.9775 18.954419.062119.27619.593720.011520.525321.130521.822822.5981 n012345678 6.283198.9868211.526913.975916.365118.711621.025723.314125.5816 12.566415.450518.1920.834223.409825.933128.414830.862633.282 18.849621.808224.645927.39630.079332.709435.295937.84640.3649 25.132728.132431.029233.847236.602539.306341.966944.590747.1825

16 16 The eigenvalues by using BIM and SVD 12345678910 6.2806.5707.1107.8508.7208.9909.680 U kernel 12345678910 1.8403.1504.3905.5706.2806.5706.7206.9107.5507.830 T kernel 12345678910 4.1606.2806.5706.6807.1107.8408.7208.9909.0309.680 L kernel 12345678910 1.8403.1504.1604.3905.5706.5706.6906.7206.9107.550 M kernel

17 17 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

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

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

20 20 Singular formulation -Spurious SVD updating document U T

21 21 Hypersingular formulation -Spurious SVD updating document L M

22 22 Outline Introduction Problem statement Mathematical analysis Numerical example Conclusions

23 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 24 ~Thanks for your kind attentions~

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