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

3. Boundary Element Method
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. Optimal source location
Not Good
Good
Conventional MFS
Alves & Antunes ?

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

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

8. Image location (Chen and Wu, 2006)
Rigid body term a
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
Subtraction theorem
Degenerate kernel for Laplace problem
1-D
2-D 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
Boundary value problem without source
Conclusions

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

14. Eccentric problem Case 2 Governing equation:
Dirichlet boundary condition: b d

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

16. Bipolar coordinates x

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

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

19. Series of images
The final images
sc1
sc2

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

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

22. Analytical derivation of location for 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
MFS (special case)
Conventional MFS s

28. Image method versus MFS
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
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. , (T-complete function)
Trefftz method and MFS
Method
Trefftz method
MFS
Definition
Figure sketch
Base
, (T-complete function)
, r=|x-s|
G. E.
Match B. C.
Determine cj
Determine wj s D
u(x) r D
u(x)
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
Interior problem
Exterior problem z z a 1 y y 1 a x x

34. The image group
Obtain image weighted
Obtain image location

35. Interpolation functionsb

36. Analytical derivation

37. Numerical solution a b

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 1
PART 2
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
Boundary value problem without sources
Conclusions

44. Trefftz solution
Without loss of generality

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

46. Equivalence of solutions derived by Trefftz method and image method (special MFS)
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
Boundary value problem without sources
Conclusions

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

49. Animation - An infinite plane with two circular holes
sc1
sc2 s4 s2

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

51. Contour plot of an infinite plane with two circular holes (antisymmetric case)
Image solution
bipolar coordinates
null-field BIEM

52. An infinite space with two cavities (anti-symmetric BC)z
u=V1=-1
u=V2=1
a=1.0
d=5.0 y B1 B2 x

53. Numerical approach to determine q(N), c1(N) and c2(N)
Coefficients

54. Contour plot of an infinite space with two spherical cavitiesy z x
x-y plane
Bispherical coordinates
Image method
Null-field BIE

56. 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
Boundary value problem without sources
Conclusions

57. Optimal location of MFS
Depends on loading (image location)
Depends on geometry (frozen image point)

58. Final images to bipolar (bispherical) focus
2-D
3-D
Bipolar coordinates
Bispherical coordinates

59. Equivalence of Trefftz method and MFS
MFS (image method)

60. Image solution for BVP without sourcesy x y x

61. Thanks for your kind attentions
The end
Thanks for your kind attentions
You can get more information from our website

62. A half plane with a circular hole
2007, Ke J. N.
2009, Image method a b

63. An infinite plane with two circular holes subject to Neumann boundary

64. Extra terms of complementary solutions
Source point
Frozen point
Two complementary solutions

65. The method provide of JW. Lee
Frozen point d1 d2
complementary
solutions

66. The total potential
Where the N=M

67. Null-field BIE approach (addition theorem and superposition technique)
Results
Image method)
Null-field BIE approach (addition theorem and superposition technique)

68. Conclusions
The analytical solutions derived by the Trefftz method and MFS were proved to be mathematically equivalent for Green’s functions of the concentric sphere.
In the concentric sphere case, we can find final two frozen image points (one at origin and one at infinity). Their singularity strength can be determined numerically and analytically in a consistent manner.
It is found that final image points terminate at the two focuses of the bipolar (bispherical) coordinates for all the cases

69. Numerical examples 1: Eccentric annulusb
Image method (50+2 points)

70. Numerical examples 3: An infinite plane with double holes
Image method ( point)

71. Animation- an infinite plane with double holes
Multipole expansion
t2=0
t1=0
and
Multipoles
The final images terminate at the focus

72. Equivalence of solutions derived by Trefftz method and MFS
Trefftz solution
The same
Image solution

73. The simplest MFS
1-D Rod l
where U(x,s) is the fundamental solution.

74. Present method- MFS (Image method)… …

75. An infinite space with two cavities (symmetric BC)
Obtain image location z
u=V1=1
u=V2=1 a a q q y d
Obtain image weighting B1 B2 x

76. The strength of two frozen points and q(N)
The strength of c1(N), c2(N) and q(N)

77. Contour plot of an infinite space with two spherical cavities (symmetric case)z x
x-y plane
3-D Bipolar coordinates
Bispherical coordinates
Image method
Null-field BIE

78. Illustrative examples – An eccentric annulus
u=V1=0
u=V2=1 B1 B2 c r1 r2

79. Numerical approach to determine q(N), c1(N), c2(N) and e(N)
e(N)=2 (exact)
c1(N)=1.44 (exact)
q(N)=0 (exact)
c2(N)=-1.44 (exact)

80. Contour plot of eccentric annulus
Image solution
bipolar coordinates
null-field BIEM

81. 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
Boundary value problem without source
Conclusions

82. Optimal source location
Not good
Good
Conventional MFS
Alves CJS & Antunes PRS

83. MFS-Image location and weighting-interior (Chen and Wu, 2006)z
u=0 y 1 b
The weighting of the image point x

84. MFS-Image location and weighting-exterior (Chen and Wu, 2006)1
The weighting of the image point

85. Chen and Wu-image method (2006)

86. Analytical derivation of location for the two frozen pointsb a x y
a=1, b=3
(0.171 & 5.828)

87. Numerical approach to determine c1(N), c2(N) and e(N)

88. Analytical derivation of location for the two frozen pointsb 2c