1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to screw dislocations Reporter: Chou K. H. Advisor: Chen J. T. Data: 2008/06/24 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  Screw dislocation problem with a circular hole subject to Dirichlet or Neumann boundary condition  Screw dislocation problem with a circular inclusion  Screw dislocation problems with two circular holes subject to Numann boundary condition Conclusions

3 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  Screw dislocation problem with a circular hole subject to Dirichlet or Neumann boundary condition  Screw dislocation problem with a circular inclusion  Screw dislocation problems with two circular holes subject to Numann boundary condition 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 Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Bump contour Limit process Fictitious boundary Collocation point Fictitious BEM Null-field approach CPV and HPV Ill-posed Guiggiani (1995) Gray and Manne (1993) Waterman (1965) Achenbach et al. (1988)

6 Present approach Fundamental solution No principal value 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 CPV and HPV

7 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  Screw dislocation problem with a circular hole subject to Dirichlet or Neumann boundary condition  Screw dislocation problem with a circular inclusion  Screw dislocation problems with two circular holes subject to Numann boundary condition Conclusions

8 Green’s third identity ???

9 Superposition technique

10 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  Screw dislocation problem with a circular hole subject to Dirichlet or Neumann boundary condition  Screw dislocation problem with a circular inclusion  Screw dislocation problems with two circular holes subject to Numann boundary condition Conclusions

11 Addition theorem

12 Addition theorem Similarly, modify

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 and boundary density  Adaptive observer system  Linear algebraic equation Numerical examples  Screw dislocation problem with a circular hole subject to Dirichlet or Neumann boundary condition  Screw dislocation problem with a circular inclusion  Screw dislocation problems with two circular holes subject to Numann boundary condition 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 and boundary density  Adaptive observer system  Linear algebraic system Numerical examples  Screw dislocation problem with a circular hole subject to Dirichlet or Neumann boundary condition  Screw dislocation problem with a circular inclusion  Screw dislocation problems with two circular holes subject to Numann boundary condition Conclusions

17 Linear algebraic system

18 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem and boundary density  Adaptive observer system  Linear algebraic system Numerical examples  Screw dislocation problem with a circular hole subject to Dirichlet or Neumann boundary condition  Screw dislocation problem with a circular inclusion  Screw dislocation problems with two circular holes subject to Numann boundary condition Conclusions

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

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

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

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

23 Screw dislocation problem with a circular inclusion

24 Superposition technique

25 Screw dislocation problem with a circular inclusion Smith data (1968)Present approach (M=50)

26 Parseval’s sum

27 Screw dislocation problems with two circular holes subject to Numann boundary condition

28 Screw dislocation problems with two circular holes subject to Numann boundary condition Present approach

29 Screw dislocation problems with two circular holes subject to Numann boundary condition Present approach

30 Outline Motivation and literature review Derivation of the Green’s function  Superposition technique  Addition theorem and boundary density  Adaptive observer system  Linear algebraic system Numerical examples  Screw dislocation problem with a circular hole subject to Dirichlet or Neumann boundary condition  Screw dislocation problem with a circular inclusion  Screw dislocation problems with two circular holes subject to Numann boundary condition Conclusions

31 Conclusions A systematic approach using addition theorem and superposition technique for screw dislocation problems has been successfully proposed. A systematic approach using addition theorem and superposition technique for screw dislocation problems has been successfully proposed. Five goals of singularity free, boundary-layer effect free, exponential convergence well-posed model and mesh-free generation are achieved. Five goals of singularity free, boundary-layer effect free, exponential convergence well-posed model and mesh-free generation are achieved. The results demonstrate the superiority of present approach over the conventional BEM. The results demonstrate the superiority of present approach over the conventional BEM.

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