Reporter: H.C. Shieh Adviser: Dr. J.T. Chen Department of Harbor and River Engineering,

1. 碩士學位論文口試報告 Study on the Green’s functions for Laplace problems with circular and spherical boundaries by using the image method Reporter: H.C. Shieh Adviser: Dr. J.T. Chen Department of Harbor and River Engineering, National Taiwan Ocean University July 25, 2009

2. Frame Motivation and literature review Two-dimensional Green’s function MFS (Image method) Green’s Trefftz method function BVP without sources Conclusions

3. Numerical methods Numerical methods Finite Element Method Boundary Element Method Meshless Method

4. Method of fundamental solutions This method was proposed by Kupradze in 1964. is the fundamental solution Interior case Exterior case

5. Conventional MFS Alves & Antunes Optimal source location Not Good Good ?

6. The simplest image method Mirror Neumann boundary condition Dirichlet boundary condition

7. P P a a O R’ R O Conventional method to determine the image location P r’ a r R’ A B O Lord Kelvin(1824~1907) (1949,相似三角形) R Greenberg (1971,取巧法)

8. a a Image location (Chen and Wu, 2006) u=0 Rigid body term u=0

9. 2-D Degenerate kernal References: W. C. Chen, A study of free terms and rigid body modes in the dual BEM, NTOU Master Thesis, 2001. C. S. Wu,Degenerate scale analysis for membrane and plate problems using the meshless method and boundary element method, NTOU Master Thesis, 2004

10. Addition theorem & degenerate kernel x s

11. 3-D degenerate kernel s x interior x exterior

12. Outline • Motivation and literature review • Derivation of 2-D Green’s function by using the image method • Trefftz method and MFS • Image method (special MFS) • Trefftz method • Equivalence of solutions derived by Trefftz method and MFS • Boundary value problem without source • Conclusions

13. Eccentric annulus Case 1 Governing equation: B2 B1 a Dirichletboundary condition: b d

14. Eccentric problem Case 2 Governing equation: B2 B1 a Dirichletboundary condition: b d

15. Half plane with circular hole problem Case 3 Governing equation: u2=0 Dirichletboundary condition: B1 u1=0 B2

16. x Bipolar coordinates

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

18. + Image point - s2 s6 s4 s1 s5 s3 Annular (EABE, 2009) to eccentric case …. Source point …. s

19. The final images sc1 sc2 Series of images

20. sc1 sc2 Numerical approach to determine c1(N), c2(N) and e(N) Coefficients

21. Contour plot of eccentric annulus problem Dirichlet boundary for the eccentric case 1 Image method Analytical solution (bipolar coordinates )

22. Analytical derivation of locationfor the two frozen points

23. Eccentric case True source Image sources

24. Contour plot of eccentric annulus Image method Null-field BIE approach (addition theorem and superposition technique)

25. A half plane with a circular hole

26. Contour plot of half plane problem Image method Null-field BIE approach (addition theorem and superposition technique)

27. Linking of MFS and image method Conventional MFS MFS (special case) s

28. Image method versus MFS Image method Conventional MFS All the strength need to be determined. Only three coefficients are required to be determined. large

29. Outline • Motivation and literature review • Derivation of 2-D Green’s function by using the image method • Trefftz method and MFS • Image method (special MFS) • Trefftz method • Equivalence of solutions derived by Trefftz method and MFS • Boundary value problem without sources • Conclusions

30. Trefftz method The method was proposed by Trefftz in 1926. is the jth T-complete function Interior case Exterior case

31. r s u(x) u(x) D D Trefftz method and MFS is the number of complete functions is the number of source points in the MFS

32. Derivation of 3-D Green’s function by using the image method Interior problem Exterior problem

33. The weighting of the image source in the 3-D problem z z a 1 y y 1 a x x

34. The image group Obtain image weighted Obtain image location

35. a b Interpolation functions

36. Analytical derivation

37. a b Numerical solution

38. Numerical and analytic ways to determine c(N) and d(N) Coefficients

39. Derivation of 3-D Green’s function by using the Trefftz Method PART 2 PART 1 PART 1

40. Boundary value problem Interior: Exterior: PART 2

41. PART 1 + PART 2 :

42. Results Trefftz method (x-y plane) Image method (x-y plane)

43. Outline • Motivation and literature review • Derivation of 2-D Green’s function by using the image method • Trefftz method and MFS • Image method (special MFS) • Trefftz method • Equivalence of solutions derived by Trefftz method and MFS • Boundary value problem without sources • Conclusions

44. Trefftz solution Without loss of generality

45. s10 s10 s4 s4 s2 s2 s3 s3 s8 s8 s6 s6 s1 s1 s s s4 s4 s2 s2 s1 s1 s5 s5 s7 s7 s s s3 s3 s9 s9 Mathematical equivalencethe Trefftz method and MFS Trefftz method series expand Image method series expand

46. Equivalence of solutions derived by Trefftz method and image method (special MFS) 3-D True source Trefftz method MFS (image method) Equivalence addition theorem linkage

47. Outline • Motivation and literature review • Derivation of 2-D Green’s function by using the image method • Trefftz method and MFS • Image method (special MFS) • Trefftz method • Equivalence of solutions derived by Trefftz method and MFS • Boundary value problem without sources • Conclusions

48. An infinite plane with two circular holes (anti-symmetric BC) a=1.0 d=10 y u=V=V1=-1 u=V=V2=1 2c a a x B1 d B2

49. s4 s2 s1 s3 sc1 sc2 Animation - An infinite plane with two circular holes u=1 u=-1

50. Numerical approach to determine q(N), c1(N), c2(N) and e(N) q(N)=e(N)=0 Coefficients