Bipolar coordinates, image method and method of fundamental solutions

1. ICCES 09 in Phuket, Thailand Bipolar coordinates, image method and method of fundamental solutions Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University 15:30-15:50, April 10, 2009

2. Prof. Wen Hwa Chen 60th birthday symposium My Ph.D. Committee member

3. 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. exterior problem: • Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Trefftz method is the jth T-complete function

5. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions MFS exterior problem Interior problem

6. r s u(x) u(x) D D • Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Trefftz method and MFS is the number of complete functions is the number of source points in the MFS

7. a b New point of view of image locationinstead of Kelvin concept (Chen and Wu, IJMEST 2006)

8. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Numerical examples - convergence rate Collocation point Source point True source point Image method Trefftz method Conventional MFS Best Worst

9. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Equivalence of solutions derived by Trefftz method and image method (special MFS) 3-D 2-D True source Trefftz method MFS Equivalence addition theorem

10. Key point Addition theorem (Degenerate kernel) 2-D 3-D

11. z y x Annular cases (Green’s functions) 2-D 3-D u2=0 u2=0 b u1=0 u1=0 a b a EABE 2009 True source

12. Numerical examples - case 1 Melnikov and Arman, 2001 Computer unfriendly fixed-fixed boundary (u=0) 10 terms m=20 (a) Trefftz method N=20 (b) Image method Contour plot for the analytical solution (m=N). 100 terms

13. Analytical solutions Addition theorem Addition theorem

14. Analytical and numerical approaches to determine the strength 2-D 3-D

15. Bipolar coordinates focus Eccentric annulus A half plane with a hole An infinite plane with double holes

16. The final images terminate at the focus Animation – eccentric case

17. Animation - a half plane with a circular hole The final images terminate at the focus

18. Animation- an infinite plane with double holes Multipole expansion and Multipoles The final images terminate at the focus

19. Eccentric annulus u2=0 u1=0 a b Image method (50+2 points)

20. A half plane with a circular hole u2=0 h u1=0 a Image method (40+2 points)

21. b a h An infinite plane with double holes t2=0 t1=0 Image method (20+4+10 point)

22. Introduction • Problem statements • Present method • Equivalence of Trefftz and MFS • Numerical examples • Conclusions 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.

23. Thanks for your kind attentions You can get more information from our website http://msvlab.hre.ntou.edu.tw/

24. Image method MFS (special case) • Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Optimal source location Alves CJS & Antunes PRS Conventional MFS

25. a b • Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Problem statements Governing equation : • BCs: • fixed-fixed boundary • fixed-free boundary • free-fixed boundary

26. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Present method- MFS (Image method)

27. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions MFS-Image group

28. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Analytical derivation

29. a b • Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Numerical solution

30. a b • Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Interpolation functions

31. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Trefftz Method PART 1

32. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Boundary value problem PART 2

33. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions PART 1 + PART 2 :

34. Equivalence • Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Equivalence of solutions derived by Trefftz method and MFS

35. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Equivalence of solutions derived by Trefftz method and MFS The same

36. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Numerical examples-case 2 fixed-free boundary m=20 (a) Trefftz method N=20 (b) Image method Contour plot for the analytical solution (m=N).

37. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Numerical examples-case 3 free-fixed boundary m=20 (a) Trefftz method N=20 (b) Image method Contour plot for the analytical solution (m=N).

38. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Numerical and analytic ways to determine c(N) and d(N) Values of c(N) and d(N) for the fixed-fixed case.

39. Introduction • Problem statements • Present method • Equivalence of Trefftz method and MFS • Numerical examples • Conclusions Numerical examples- convergence Pointwise convergence test for the potential by using various approaches.