1. April, 17-20, 2009 p.1 海大陳正宗終身特聘教授 Trefftz method, image method and method of fundamental solutions for Green’s functions with circular boundaries Jeng-Tzong Chen 陳正宗 終身特聘教授 Department of Harbor and River Engineering, National Taiwan Ocean University 14:30-15:00, April 18, 2009 (cmm2009.ppt) CMM 2009 東海大學 台中 台灣 National Taiwan Ocean University MSVLAB ( 海大河工系 ) Department of Harbor and River Engineering

2. April, 17-20, 2009 p.2 海大陳正宗終身特聘教授 Prof. Z C Li ’ s 70th birthday symposium 1989 MIT BEM 11 conference

3. April 8-12, 2009 p.3 海大陳正宗終身特聘教授 Outline Introduction Problem statements Present method  MFS (image method)  Trefftz method Equivalence of Trefftz method and MFS (2-D and 3-D annular cases) Numerical examples Conclusions

4. April 8-12, 2009 p.4 海大陳正宗終身特聘教授 Trefftz method 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions is the j th T-complete function exterior problem:

5. April 8-12, 2009 p.5 海大陳正宗終身特聘教授 MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Interior problem exterior problem

6. April 8-12, 2009 p.6 海大陳正宗終身特聘教授 Trefftz method and MFS MethodTrefftz methodMFS Definition Figure caption Base, (T-complete function), r=|x-s| G. E. Match B. C.Determine c j Determine w j D u(x) s D r is the number of complete functions is the number of source points in the MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

7. April 8-12, 2009 p.7 海大陳正宗終身特聘教授 b a New point of view of image location instead of Kelvin concept (Chen and Wu, IJMEST 2006)

8. April 8-12, 2009 p.8 海大陳正宗終身特聘教授 Optimal source location 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions MFS (special case) Image method Conventional MFS Alves CJS & Antunes PRS

9. April 8-12, 2009 p.9 海大陳正宗終身特聘教授 Numerical examples - convergence rate Image method Trefftz method Conventional MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Best Worst Collocation point Source point True source point

10. April 8-12, 2009 p.10 海大陳正宗終身特聘教授 Equivalence of solutions derived by Trefftz method and image method (special MFS) 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Trefftz methodMFS Equivalence addition theorem 2-D 3-D True source IMAGE

11. April 8-12, 2009 p.11 海大陳正宗終身特聘教授 Key point Addition theorem (Degenerate kernel) 2-D 3-D

12. April 8-12, 2009 p.12 海大陳正宗終身特聘教授 Annular cases (Green’s functions) 2-D EABE 2009 x y z 3-D a b a b u 1 =0 u 2 =0 True source

13. April 8-12, 2009 p.13 海大陳正宗終身特聘教授 Numerical examples - case 1 (a) Trefftz method (b) Image method Contour plot for the analytical solution (m=N). fixed-fixed boundary (u=0) m=20N=20 10 terms 100 terms Melnikov and Arman, 2001 Computer unfriendly

14. April 8-12, 2009 p.14 海大陳正宗終身特聘教授 Trefftz solutionImage solution Annular circle Annular sphere Analytical solutions Addition theorem Addition theorem u 1 =0 u 2 =0 u 1 =0

15. April 8-12, 2009 p.15 海大陳正宗終身特聘教授 Equivalence of solutions derived by Trefftz method and MFS 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Equivalence

16. April 8-12, 2009 p.16 海大陳正宗終身特聘教授 Image ID number 2, 6, 10, 14 … one group 4, 8, 12, 16 … one group 3, 7, 11, 15 … one group 1, 5, 9, 13 … one group

17. April 8-12, 2009 p.17 海大陳正宗終身特聘教授 The same 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Equivalence of solutions derived by Trefftz method and MFS

18. April 8-12, 2009 p.18 海大陳正宗終身特聘教授 Analytical and numerical approaches to determine the strength 2-D 3-D

19. April 8-12, 2009 p.19 海大陳正宗終身特聘教授 Bipolar coordinates Eccentric annulusA half plane with a holeAn infinite plane with double holes focus

20. April 8-12, 2009 p.20 海大陳正宗終身特聘教授 Animation – eccentric case The final images terminate at the focus

21. April 8-12, 2009 p.21 海大陳正宗終身特聘教授 Eccentric annulus Image method (50+2 points) u 1 =0 u 2 =0 a b

22. April 8-12, 2009 p.22 海大陳正宗終身特聘教授 Animation - a half plane with a circular hole The final images terminate at the focus

23. April 8-12, 2009 p.23 海大陳正宗終身特聘教授 A half plane with a circular hole Image method (40+2 points) u 1 =0 u 2 =0 a h

24. April 8-12, 2009 p.24 海大陳正宗終身特聘教授 Animation- an infinite plane with double holes The final images terminate at the focus Multipole expansion and Multipoles

25. April 8-12, 2009 p.25 海大陳正宗終身特聘教授 An infinite plane with double holes a b h Image method (20+4+10 point) t 1 =0 t 2 =0

26. April 8-12, 2009 p.26 海大陳正宗終身特聘教授 Conclusions The analytical solutions derived by the Trefftz method and MFS were proved to be mathematically equivalent for the annular Green’s functions (2D and 3D) after using addition theorem (degenerate kernel). We can find final two frozen image points which are focuses in the bipolar coordinates. The image idea provides the optimal location of MFS and only at most 4 by 4 matrix is required. 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz and MFS 5.Numerical examples 6.Conclusions

27. April, 17-20, 2009 p.27 海大陳正宗終身特聘教授 Thanks for your kind attentions You can get more information from our website http://msvlab.hre.ntou.edu.tw/ (more than 120000 entries since 1999)

28. April 8-12, 2009 p.28 海大陳正宗終身特聘教授 Problem statements a b Governing equation : BCs: 1.fixed-fixed boundary 2.fixed-free boundary 3.free-fixed boundary 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

29. April 8-12, 2009 p.29 海大陳正宗終身特聘教授 Present method- MFS (Image method) 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

30. April 8-12, 2009 p.30 海大陳正宗終身特聘教授 MFS-Image group 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

31. April 8-12, 2009 p.31 海大陳正宗終身特聘教授 Analytical derivation 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

32. April 8-12, 2009 p.32 海大陳正宗終身特聘教授 Numerical solution 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions a b

33. April 8-12, 2009 p.33 海大陳正宗終身特聘教授 Interpolation functions a b 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

34. April 8-12, 2009 p.34 海大陳正宗終身特聘教授 Trefftz Method PART 1 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

35. April 8-12, 2009 p.35 海大陳正宗終身特聘教授 Boundary value problem PART 2 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

36. April 8-12, 2009 p.36 海大陳正宗終身特聘教授 PART 1 + PART 2 : 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

37. April 8-12, 2009 p.37 海大陳正宗終身特聘教授 Numerical examples-case 2 (a) Trefftz method (b) Image method Contour plot for the analytical solution (m=N). fixed-free boundary 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions m=20 N=20

38. April 8-12, 2009 p.38 海大陳正宗終身特聘教授 Numerical examples-case 3 (a) Trefftz method (b) Image method Contour plot for the analytical solution (m=N). free-fixed boundary 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions m=20 N=20

39. April 8-12, 2009 p.39 海大陳正宗終身特聘教授 Numerical and analytic ways to determine c(N) and d(N) Values of c(N) and d(N) for the fixed-fixed case. 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions

40. April 8-12, 2009 p.40 海大陳正宗終身特聘教授 Numerical examples- convergence 1.Introduction 2.Problem statements 3.Present method 4.Equivalence of Trefftz method and MFS 5.Numerical examples 6.Conclusions Pointwise convergence test for the potential by using various approaches.