1. 1 Eigenanalysis for a concentric sphere using BEM and BIEM 報告學生：高聖凱 指導教授：陳正宗終身特聘教授 時間： 2008/06/24

2. 2 Problem statement

3. 3 3D degenerate kernels

4. 4

5. 5 Null-field Integral equation - UT

6. 6 Dirichlet B.C. (fixed-fixed) - UT

7. 7 Eigenvalue n012345678 6.283198.986820.003160.030180.123580.270040.610641.099451.68051 12.566415.450511.526913.975916.365118.711621.025723.314125.5816 18.849621.808218.1920.834223.409825.933128.414830.862633.282 25.132728.132424.645927.39630.079332.709435.295937.84640.3649 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 25.132725.211725.369225.603825.913726.296826.750227.27127.8561

8. 8 Null-field Integral equation - LM

9. 9 Dirichlet B.C. (fixed-fixed) - LM

10. 10 Eigenvalue n012345678 0.003160.036820.150080.381020.731901.07051.70396 8.986824.163156.684199.028211.293413.512915.702217.869720.0207 15.450511.880714.579917.167519.68922.140424.558726.944129.3025 21.808218.411721.227723.945526.591129.181131.726434.23536.7126 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 25.132725.211725.369225.603825.913726.296826.750227.27127.8561

11. 11 Neumann B.C. (free-free) - UT

12. 12 Eigenvalue 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 6.283198.986820.003160.030180.123580.270040.610641.099451.68051 12.566415.450511.526913.975916.365118.711621.025723.314125.5816 18.849621.808218.1920.834223.409825.933128.414830.862633.282 25.132728.132424.645927.39630.079332.709435.295937.84640.3649

13. 13 Conclusions There are still spurious eigenvalues by using BEM and BIEM to deal with concentric sphere problems. True eigenvalues are dependent on problems and spurious eigenvalues are dependent on methods.

14. 14 ~Thank your kind attentions~