1. 95 學年度第 2 學期碩士論文口試 1 Derivation of the Green’s function for Laplace and Helmholtz problems with circular boundaries by using the null-field integral equation approach Reporter : Ke J. N. Advisor : Chen J. T. Committee members : Chen I. L., Lee W. M., Leu S. Y. & Chen K, H.

2. National Taiwan Ocean University Department of Harbor and River Engineering 2 Outlines Motivation and literature review Derivation of the Green’s function Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Numerical examples Green ’ s function for Laplace problems Green ’ s function for Helmholtz problems Conclusions

3. National Taiwan Ocean University Department of Harbor and River Engineering 3 Outlines Motivation and literature review Motivation and literature review Derivation of the Green’s function Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Numerical examples Green ’ s function for Laplace problems Green ’ s function for Helmholtz problems Conclusions

4. National Taiwan Ocean University Department of Harbor and River Engineering 4 Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Ill-posed model Convergence rate

5. National Taiwan Ocean University Department of Harbor and River Engineering 5 Engineering problem with arbitrary geometries Degenerate boundary Circular boundary Straight boundary Elliptic boundary (Fourier series) (Legendre polynomial) (Chebyshev polynomial) (Mathieu function)

6. National Taiwan Ocean University Department of Harbor and River Engineering 6 Literature review Derivation of the Green’s function Successive iteration method Modified potential method Trefftz bases Melnikov, 2001, “Modified potential as a tool foor computing Green’s functions in continuum mechanics”, Computer Modeling in Engineering Science Boley, 1956, “A method for the construction of Green’s functions,”, Quarterly of Applied Mathematics Wang and Sudak, 2007, “Antiplane time-harmonic Green’s functions for a circular inhomogeneity with an imperfect interface”, Mechanics Research Communications

7. National Taiwan Ocean University Department of Harbor and River Engineering 7 Outlines Motivation and literature review Derivation of the Green’s function Derivation of the Green’s function Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Numerical examples Green ’ s function for Laplace problems Green ’ s function for Helmholtz problems Conclusions

8. National Taiwan Ocean University Department of Harbor and River Engineering 8 Null-field integral approach to construct the Green’s function Governing equation: Fundamental solution Boundary condition: Governing equation: Original Problem Auxiliary system Green’s third identity BIE for Green’s function Subjected to given B. C.

9. National Taiwan Ocean University Department of Harbor and River Engineering 9 Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form

10. National Taiwan Ocean University Department of Harbor and River Engineering 10 Expansions of fundamental solution (2D) Laplace problem-- Helmholtz problem-- O s x x Neumann factor

11. National Taiwan Ocean University Department of Harbor and River Engineering 11 Laplace problem-- Helmholtz problem-- O s x x U(s,x) T(s,x) L(s,x) M(s,x)

12. National Taiwan Ocean University Department of Harbor and River Engineering 12 Boundary density discretization Fourier series Ex. constant element Fourier series expansions - boundary density

13. National Taiwan Ocean University Department of Harbor and River Engineering 13 Adaptive observer system Source point Collocation point

14. National Taiwan Ocean University Department of Harbor and River Engineering 14 Vector decomposition technique for potential gradient Concentric case (special case) : Non-concentric case: True normal direction Source point Collocation point

15. National Taiwan Ocean University Department of Harbor and River Engineering 15 Linear algebraic equation Collocation point

16. National Taiwan Ocean University Department of Harbor and River Engineering 16 Take free body

17. National Taiwan Ocean University Department of Harbor and River Engineering 17 Image technique for solving half- plane problems Dirichlet boundary condition Neumann boundary condition Mirror

18. National Taiwan Ocean University Department of Harbor and River Engineering 18 Flowchart of present method Degenerate kernel Fourier series Null-field equation Algebraic system Fourier Coefficients Potential Analytical Numerical

19. National Taiwan Ocean University Department of Harbor and River Engineering 19 Outlines Motivation and literature review Derivation of the Green’s function Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Numerical examples Green ’ s function for Laplace problems Green ’ s function for Helmholtz problems Conclusions

20. National Taiwan Ocean University Department of Harbor and River Engineering 20 Numerical examples Laplace problems Laplace problems Eccentric ring (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane problem with a circular hole and a half- circular inclusion Helmholtz problems An infinite matrix containing a circular inclusion with a concentrated force in the matrix An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion Special cases and parameter study Special cases and parameter study An infinite matrix containing two circular inclusions with a concentrated force in the matrix An infinite matrix containing two circular inclusions with a concentrated force in the matrix

21. National Taiwan Ocean University Department of Harbor and River Engineering 21 Present study for Laplace equation a b 0u  0u  Annular case (Chen P. Y.) Analytical Green ’ s function Eccentric ring (Ke J. N.) Semi-Analytical Green ’ s function b a

22. National Taiwan Ocean University Department of Harbor and River Engineering 22 Numerical examples Laplace problems Laplace problems Eccentric ring Eccentric ring (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane problem with a circular hole and a half- circular inclusion Helmholtz problems An infinite matrix containing a circular inclusion with a concentrated force in the matrix An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion Special cases and parameter study Special cases and parameter study An infinite matrix containing two circular inclusions with a concentrated force in the matrix An infinite matrix containing two circular inclusions with a concentrated force in the matrix

23. National Taiwan Ocean University Department of Harbor and River Engineering 23 Eccentric ring

24. National Taiwan Ocean University Department of Harbor and River Engineering 24 Eccentric ring Potential contour using the Melnikov ’ s method Potential contour using the present method (M=50)

25. National Taiwan Ocean University Department of Harbor and River Engineering 25 Numerical examples Laplace problems Laplace problems Eccentric ring (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane problem with a circular hole and a half- circular inclusion Helmholtz problems An infinite matrix containing a circular inclusion with a concentrated force in the matrix An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion Special cases and parameter study Special cases and parameter study An infinite matrix containing two circular inclusions with a concentrated force in the matrix An infinite matrix containing two circular inclusions with a concentrated force in the matrix

26. National Taiwan Ocean University Department of Harbor and River Engineering 26 A half plane with an aperture subjected to Dirichlet boundary condition

27. National Taiwan Ocean University Department of Harbor and River Engineering 27 Result of a half-plane problem with an aperture subjected to Dirichlet boundary condition Potential contour using the Melnikov ’ s method Potential contour using the present method (M=50)

28. National Taiwan Ocean University Department of Harbor and River Engineering 28 A half plane with an aperture subjected to Robin boundary condition

29. National Taiwan Ocean University Department of Harbor and River Engineering 29 Result of a half-plane problem with an aperture subjected to Robin boundary condition Potential contour using the present method (M=50) Potential contour using the Melnikov ’ s method

30. National Taiwan Ocean University Department of Harbor and River Engineering 30 Numerical examples Laplace problems Laplace problems Eccentric ring (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane problem with a circular hole and a half- circular inclusion A half-plane problem with a circular hole and a half- circular inclusion Helmholtz problems An infinite matrix containing a circular inclusion with a concentrated force in the matrix An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion Special cases and parameter study Special cases and parameter study An infinite matrix containing two circular inclusions with a concentrated force in the matrix An infinite matrix containing two circular inclusions with a concentrated force in the matrix

31. National Taiwan Ocean University Department of Harbor and River Engineering 31 A half-plane problem with a circular hole and a half-circular inclusion Half -plane Inclusion Matrix A circular hole Source of Green’s function Material conductivity

32. National Taiwan Ocean University Department of Harbor and River Engineering 32 Result of a half-plane problem with a circular hole and a half-circular inclusion Contour plot by using the null-field integral equation approach Contour plot by using the Melikov's approach (2006)

33. National Taiwan Ocean University Department of Harbor and River Engineering 33 Numerical examples Laplace problems Eccentric ring (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane problem with a circular hole and a half- ircular inclusion Helmholtz problems Helmholtz problems An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion Special cases and parameter study An infinite matrix containing two circular inclusions with a concentrated force in the matrix

34. National Taiwan Ocean University Department of Harbor and River Engineering 34 Present study for Helmholtz equation SH-wave problem (Chen P. Y.) Perfect interface boundary Inclusion Matrix SH-Wave Imperfect interface boundary Green ’ s function problem (Ke J. N.) SH-Wave Inclusion Matrix

35. National Taiwan Ocean University Department of Harbor and River Engineering 35 Numerical examples Laplace problems Eccentric ring (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane problem with a circular hole and a half- circular inclusion Helmholtz problems Helmholtz problems An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion Specialcases and parameter study Special cases and parameter study An infinite matrix containing two circular inclusions with a concentrated force in the matrix An infinite matrix containing two circular inclusions with a concentrated force in the matrix

36. National Taiwan Ocean University Department of Harbor and River Engineering 36 An infinite matrix containing a circular inclusion with a concentrated force at in the matrix is the shear modulus is the wave speed Interface condition is the imperfect interface parameter

37. National Taiwan Ocean University Department of Harbor and River Engineering 37 Take free body

38. National Taiwan Ocean University Department of Harbor and River Engineering 38 Distribution of for the quasi-static solution along the circular boundary Wang and Sudak ’ s solution The present solution 050100150200250300350 0 0.5 1 1.5 2 2.5 (Deg) (quasi-static) (static)

39. National Taiwan Ocean University Department of Harbor and River Engineering 39 Parameter study of for the stress response Wang and Sudak ’ s solution The present solution Bonding behavior Ideal bonding

40. National Taiwan Ocean University Department of Harbor and River Engineering 40 The distribution of displacement along the circular boundary for the case Wang and Sudak ’ s solution The present solution Dynamic effect

41. National Taiwan Ocean University Department of Harbor and River Engineering 41 Test of convergence for the Fourier series with a concentrated force in the inclusion imaginary part Terms of Fourier series (M) real part Parseval ’ s sum of real solution for Parseval ’ s sum of imaginary part for Terms of Fourier series (M) Parseval’s sum

42. National Taiwan Ocean University Department of Harbor and River Engineering 42 An infinite matrix containing a circular inclusion with a concentrated force at in the inclusion is the shear modulus is the wave speed is the imperfect interface parameter

43. National Taiwan Ocean University Department of Harbor and River Engineering 43 Distribution of for the quasi-static solution along the circular boundary (Deg) (quasi-static)

44. National Taiwan Ocean University Department of Harbor and River Engineering 44 Parameter study of for the stress response Bonding behavior

45. National Taiwan Ocean University Department of Harbor and River Engineering 45 The distribution of displacement along the circular boundary for the case of 050100150200250300350 0 1 2 3 4 5 6 k M a=1 k M a=2 k M a=3 k M a=4 k M a=5 Dynamic effect

46. National Taiwan Ocean University Department of Harbor and River Engineering 46 Numerical examples Laplace problems Eccentric ring (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane problem with a circular hole and a half- ircular inclusion Helmholtz problems Helmholtz problems An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion Special cases and parameter study Special cases and parameter study An infinite matrix containing two circular inclusions with a concentrated force in the matrix

47. National Taiwan Ocean University Department of Harbor and River Engineering 47 Special case of an ideally bonded case is the shear modulus is the wave speed is the imperfect interface parameter Imperfect bonding Ideal bonding

48. National Taiwan Ocean University Department of Harbor and River Engineering 48 The absolute amplitude of displacement by the present method

49. National Taiwan Ocean University Department of Harbor and River Engineering 49 Special case of cavity is shear modulus is wave speed is the imperfect interface parameter Imperfect bonding Cavity

50. National Taiwan Ocean University Department of Harbor and River Engineering 50 The absolute amplitude of displacement by the present method Cavity

51. National Taiwan Ocean University Department of Harbor and River Engineering 51 Parameter study for ideal bonding Fundamental solution is the shear modulus is the imperfect interface parameter

52. National Taiwan Ocean University Department of Harbor and River Engineering 52 Stress contours of and for the static solutions (a concentrated force in the matrix)

53. National Taiwan Ocean University Department of Harbor and River Engineering 53 Stress contours of and for the dynamic solutions (a concentrated force in the matrix)

54. National Taiwan Ocean University Department of Harbor and River Engineering 54 Stress contours of and for the static solutions (a concentrated force in the inclusion)

55. National Taiwan Ocean University Department of Harbor and River Engineering 55 Stress contours of and for the dynamic solutions (a concentrated force in the inclusion)

56. National Taiwan Ocean University Department of Harbor and River Engineering 56 Series-form & closed-form solutions for the static case (ideally bonded interface) Concentrated force in the matrixConcentrated force in the inclusion Stress distribution along the interface Closed-form solution (Wang and Sudak, 2007) Seires-form solution (Deg) Degenerate kernel (not easy) Fourier series (Poisson integral formula) (easy) 050100150200250300350 0 0.5 1 1.5 2 2.5 (Deg) (static)

57. National Taiwan Ocean University Department of Harbor and River Engineering 57 Numerical examples Laplace problems Eccentric ring (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition A half-plane problem with a circular hole and a half- ircular inclusion Helmholtz problems Helmholtz problems An infinite matrix containing a circular inclusion with a concentrated force in the matrix or inclusion Special cases and parameter study Special cases and parameter study An infinite matrix containing two circular inclusions with a concentrated force in the matrix An infinite matrix containing two circular inclusions with a concentrated force in the matrix

58. National Taiwan Ocean University Department of Harbor and River Engineering 58 An infinite matrix containing two circular inclusions with a concentrated force at in the matrix is the shear modulus is the wave speed is the imperfect interface parameter

59. National Taiwan Ocean University Department of Harbor and River Engineering 59 Distribution of of the matrix at the position of various

60. National Taiwan Ocean University Department of Harbor and River Engineering 60 The contour of the displacement for an infinite matrix containing two inclusions with a concentrated force at in the matrix for ideal bonding Potential contour using the present method (M=30)

61. National Taiwan Ocean University Department of Harbor and River Engineering 61 Outlines Motivation and literature review Derivation of the Green’s function Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Take free body Image technique for solving half-plane problems Numerical examples Green ’ s function for Laplace problems Green ’ s function for Helmholtz problems Conclusions Conclusions

62. National Taiwan Ocean University Department of Harbor and River Engineering 62 Conclusions After introducing the degenerate kernel, the BIE is nothing more than the linear algebra. We derived the analytic Green’s function for one inclusion problem by using the null-field integral equation. Also, the present approach can be utilized to construct semi-analytic Green’s functions for several circular inclusions.

63. National Taiwan Ocean University Department of Harbor and River Engineering 63 Conclusions Several examples, Laplace and Helmholtz problems were demonstrated to check the validity of the present formulation and the results match well with available solutions in the literature. A general-purpose program for deriving the Green’s function of Laplace or Helmholtz problems with arbitrary number of circular apertures and/or inclusions of arbitrary radii and various positions involving Dirichlet or Neumann or mixed boundary condition was developed.

64. National Taiwan Ocean University Department of Harbor and River Engineering 64 Further studies The imperfect circular interface is homogeneous nonhomogeneous. According to our successful experiences for half-plane problems, it is straightforward to quarter-plane problems.

65. National Taiwan Ocean University Department of Harbor and River Engineering 65 The end Thanks for your attentions. You can get more information on our website. http://msvlab.hre.ntou.edu.tw