1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations Reporter: Chou K. H. Advisor: Chen J. T. Date: 2008/07/11 Place: HR2 307

2 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem and boundary density  Adaptive observer system  Linear algebraic equation Numerical examples  Green’s function for the concentrated force problems  Green’s function for the screw dislocation problems Conclusions

3 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem for the kernel decomposition  Fourier expansion for the boundary density  Adaptive observer system  Linear algebraic equation Numerical examples  Green’s function for the concentrated force problems  Green’s function for the screw dislocation problems Conclusions

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 Present approach Fourier expansion Advantages of degenerate kernel 1.No principal value 2.Well-posed 3.Exponential convergence 4.Free of boundary-layer effect 5.Mesh-free generation Degenerate kernel

6 Literature review Laplace problem [Chen, Shen and Wu, 2005] Helmholtz problem [Chen, Chen, Chen and Chen, 2007] biharmonic problem [Chen, Hsiao and Leu, 2006] anti-plane piezoelectricity problem [Chen and Wu, 2006] Green’s function for Laplace [Chen, Ke and Liao, 2008], Helmholtz [Chen and Ke, 2008] and biharmonic problems [Chen and Liao, 2008] Green’s function for the screw dislocation problem (present work)

7 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem for the kernel decomposition  Fourier expansion for the boundary density  Adaptive observer system  Linear algebraic equation Numerical examples  Green’s function for the concentrated force problems  Green’s function for the screw dislocation problems Conclusions

8 Green third identity ???

9 Superposition technique Free field Typical BVP

10 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem for the kernel decomposition  Fourier expansion for the boundary density  Adaptive observer system  Linear algebraic equation Numerical examples  Green’s function for the concentrated force problems  Green’s function for the screw dislocation problems Conclusions

11 Addition theorem for the radial-based fundamental solution

12 Addition theorem for the angle-based fundamental solution

13 Boundary density discretization Fourier series Ex. constant element Fourier series expansions - boundary density

14 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem for the kernel decomposition  Fourier expansion for the boundary density  Adaptive observer system  Linear algebraic equation Numerical examples  Green’s function for the concentrated force problems  Green’s function for the screw dislocation problems Conclusions

15 Adaptive observer system Source point Collocation point

16 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem for the kernel decomposition  Fourier expansion for the boundary density  Adaptive observer system  Linear algebraic system Numerical examples  Green’s function for the concentrated force problems  Green’s function for the screw dislocation problems Conclusions

17 Linear algebraic system

18 Flowchart of the present approach Typical BVP (addition theorem) Null-field boundary integral equation Potential of domain point Fundamental solution Series form Close form Problem of the fundamental solution Superposition technique Original problem

19 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem for the kernel decomposition  Fourier expansion for the boundary density  Adaptive observer system  Linear algebraic system Numerical examples  Green’s function for the concentrated force problems  Green’s function for the screw dislocation problems Conclusions

20 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Neumann boundary condition

21 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Neumann boundary condition

22 The Green’s function of the annular ring

23 The Green’s function of the annular ring Null-field BIE approach (addition theorem and superposition technique) (M=50) Null-field BIE approach (Green’s third identity) [Chen and Ke, CMC, 2008]

24 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Neumann boundary condition

25 An eccentric ring

26 An eccentric ring Null-field BIE approach (addition theorem and superposition technique) (M=50) Melnikov’s method [Melnikov and Melnikov (2001)] Null-field BIE approach (Green’s third identity) [Chen and Ke, CMC, 2008]

27 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Neumann boundary condition

28 An infinite plane with an aperture subjected to the Neumann boundary condition

29 An infinite plane with an aperture subjected to the Neumann boundary condition Null-field BIE approach (addition theorem and superposition technique) (M=50) Image method

30 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane problem with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Neumann boundary condition

31 A half-plane problem with an aperture subjected to the Dirichlet boundary condition

32 A half-plane problem with an aperture subjected to the Dirichlet boundary condition Null-field BIE approach (addition theorem and superposition technique) (M=50) Melnikov’s method [Melnikov and Melnikov (2001)] Null-field BIE approach (Green’s third identity) [Chen and Ke, CMC, 2008]

33 A half-plane problem with an aperture subjected to the Robin boundary condition

34 A half-plane problem with an aperture subjected to the Robin boundary condition Null-field BIE approach (addition theorem and superposition technique) (M=50) Melnikov’s approach [Melnikov and Melnikov (2006)] Null-field BIE approach (Green’s third identity) [Chen and Ke, CMC, 2008]

35 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Neumann boundary condition

36 An infinite plane with a circular inclusion

37 Stress distribution along the interface

38 Equivalence between the solution of Green’s third identity and that of superposition technique += Green’s third identity Superposition technique

39 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Numann boundary condition

40 Screw dislocation problem with the circular hole subject to the Dirichlet boundary condition

41 Screw dislocation problem with the circular hole subject to the Dirichlet boundary condition Smith data (1968) (close form) Present approach (series form) (M=50)

42 Screw dislocation problem with the circular hole subject to the Neumann boundary condition

43 Screw dislocation problem with the circular hole subject to the Neumann boundary condition Smith data (1968) (close form) Present approach (series form) (M=50)

44 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Neumann boundary condition

45 Screw dislocation problem with a circular inclusion

46 Take free body and Superposition technique

47 Test convergence (Parseval’s sum )

48 Screw dislocation problem with a circular inclusion Present approach (series form) (M=50) Smith data (1968) (close form)

49 Numerical examples Concentrated force problems  An annular case  An eccentric ring  An infinite plane with an aperture subjected to the Neumann boundary condition  A half-plane with an aperture (1) Dirichlet boundary condition (2) Robin boundary condition  An infinite plane with a circular inclusion Screw dislocation problems  An infinite plane with an aperture (1) Dirichlet boundary condition (2) Neumann boundary condition  An infinite plane with a circular inclusion  An infinite plane with two circular holes subject to the Numann boundary condition

50 Screw dislocation problems with two circular holes subject to the Neumann boundary condition

51 Screw dislocation problems with two circular holes subject to the Neumann boundary condition Present approach (series form) Present approach (series form)

52 Screw dislocation problems with two circular holes subject to the Neumann boundary condition Present approach (series form)

53 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem for the kernel decomposition  Fourier expansion for the boundary density  Adaptive observer system  Linear algebraic system Numerical examples  Green’s function for the concentrated force problems  Green’s function for the screw dislocation problems Conclusions

54 Conclusions A systematic approach with five advantage singularity free, boundary-layer effect free, exponential convergence, well-posed model and mesh-free generation was developed in this thesis. A systematic approach with five advantage singularity free, boundary-layer effect free, exponential convergence, well-posed model and mesh-free generation was developed in this thesis. The angle-based fundamental solution was successfully expanded into the separable form. The angle-based fundamental solution was successfully expanded into the separable form. Mathematical equivalence between the Green’s third identity and superposition technique for solving the Green’s function problem was successfully presented. Mathematical equivalence between the Green’s third identity and superposition technique for solving the Green’s function problem was successfully presented.

55 Further studies Extension to the imperfect interface. Derivation the Green’s third identity for the screw dislocation problems. Extension to the general boundaries. 2-D problems to 3-D problems.

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57 Literature review Solve the concentrated force problems 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 Null-field integral equation Chen and Ke, 2008, “Derivation of anti- plane Dynamic Green’s function for several circular inclusions with imperfect interfaces”, Computer modeling in Engineering Science

58 Literature review Solve the screw dislocation problems Image technique Inverse point method Circle theorem Sendeckyj, 1970, “Screw dislocation near circular inclusions”, Physica status solidi Dundurs, 1969, “Elastic interaction of dislocations with inhomogeneities”, Mathematical Theory of Dislocations Smith, 1968, “The interaction between dislocations and inhomogeneities-I”, International Journal of Engineering Sciences