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Applications of Addition Theorem and Superposition Technique to Problems with Circular Boundaries Subject to Screw Dislo

This report discusses the application of the Addition Theorem and Superposition Technique in solving problems with circular boundaries that are subject to screw dislocations. It includes a literature review, derivation of the Green's function, numerical examples, and conclusions.

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Applications of Addition Theorem and Superposition Technique to Problems with Circular Boundaries Subject to Screw Dislo

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  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 2

  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 3

  4. Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Convergence rate Ill-posed model Motivation 4

  5. Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Fictitious BEM Bump contour Limit process Fictitious boundary Achenbach et al. (1988) Null-field approach Guiggiani (1995) Gray and Manne (1993) Collocation point CPV and HPV Ill-posed Waterman (1965) 5

  6. Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV • Advantages of degenerate kernel • No principal value • Well-posed • Exponential convergence • Free of boundary-layer effect • Mesh-free generation 6

  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 7

  8. Green’s third identity Green’s third identity Green’s third identity ??? 8

  9. Superposition technique 9

  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 10

  11. Addition theorem 11

  12. modify Addition theorem Similarly, 12

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

  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 14

  15. Adaptive observer system Source point Collocation point 15

  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 16

  17. Linear algebraic system 17

  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 18

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

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

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

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

  23. Screw dislocation problem with a circular inclusion 23

  24. Superposition technique 24

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

  26. Parseval’s sum 26

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

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

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

  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 30

  31. Conclusions • 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 convergencewell-posed model and mesh-free generation are achieved. • The results demonstrate the superiority of present approach over the conventional BEM. 31

  32. The end Thanks for your kind attention. Welcome to visit the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlab 32

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