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Image method for the Green’s functions of annulus and half-plane Laplace problems

Image method for the Green’s functions of annulus and half-plane Laplace problems. Reporter: Shiang-Chih Shieh Authors: Shiang-Chih Shieh, Ying-Te Lee and Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University Oct.22, 2008. Outline. Introduction

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Image method for the Green’s functions of annulus and half-plane Laplace problems

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  1. Image method for the Green’s functions of annulus and half-plane Laplace problems Reporter: Shiang-Chih Shieh Authors: Shiang-Chih Shieh,Ying-Te Lee and Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University Oct.22, 2008

  2. Outline • Introduction • Problem statements • Analytical solution • Method of Fundmental Solution (MFS) • Trefftz method • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions

  3. Interior problem: exterior problem: • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Trefftz method is the jth T-complete function

  4. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Method of Fundamental Solution (MFS) exterior problem Interior problem

  5. r s u(x) u(x) D D • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Trefftz method and MFS is the number of complete functions is the number of source points in the MFS

  6. Image method MFS (special case) • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Optimal source location Alves CJS & Antunes PRS Conventional MFS

  7. a b • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Problem statements Case 1 Annular Governing equation : t1=0 u2=0 Boundary condition : Fixed-Free boundary

  8. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Problem statements Case 2 half-plane problem Governing equation : u2=0 Dirichletboundary condition : u1=0

  9. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Problem statements Case 3 eccentric problem Governing equation : u1=0 u2=0 a Dirichletboundary condition : b e

  10. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Present method- MFS (Image method)

  11. MFS-Image group

  12. MFS-Image group

  13. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Analytical derivation

  14. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Numerical solution t1=0 a u2=0 b

  15. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Numerical and analytic ways to determine c(N) and e(N) e(N)=-0.1 c(N)=-0.159

  16. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Trefftz Method PART 1

  17. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Boundary value problem PART 2

  18. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions PART 1 + PART 2 :

  19. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Equivalence of solutions derived by using Trefftz method and MFS for annular problem MFS(Image method) The same Trefftz method

  20. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Trefftz method series expand Without loss of generality

  21. Image method series expansion Trefftz series expansion

  22. Image method series expansion Trefftz series expansion

  23. Image method series expansion Trefftz series expansion

  24. Image method series expansion Trefftz series expansion

  25. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Equivalence of solutions derived by Trefftz method and MFS Trefftz method MFS Equivalence addition theorem

  26. a • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Semi-analytical solution-case 2 b

  27. a • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions MFS-Image group

  28. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Frozen image location (0,-0.171) (frozen) (0,-5.828) (frozen) Successive images (20 points)

  29. y a x b • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Analytical derivation of location for the two frozen points (0.171 & 5.828)

  30. a b • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Series of images frozen The final two frozen images frozen

  31. a b • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Rigid body term

  32. Matching BCs to determine three coefficient

  33. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Numerical approach to determine c(N), d(N) and e(N)

  34. e • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Semi-analytical solution-case 3 y x a b u2=0 u1=0

  35. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions MFS-Image group

  36. e a b • Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Analytical derivation of location for the two frozen points

  37. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Numerical approach to determine c(N), d(N) and e(N) d(N)=-0.1375 c(N)=-0.8624

  38. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Numerical examples-case 1 Fixed-Free boundary for annular case m=20 (a) Trefftz method N=20 (b) Image method Contour plot for the analytical solution (m=N).

  39. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Numerical examples-case 2 Dirichlet boundary for half-plane case Present method-image Null-field BIE approach (addition theorem and superposition technique)(M=50) 40+2 points

  40. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Numerical examples-case 3 Dirichlet boundary for eccentric case image method analytical solution (bi-polar coordinate )

  41. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Conclusions • The analytical solutions derived by the Trefftz method and MFS were proved to be mathematically equivalent for the annular Green’s functions. • 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. • The image method can be seen as a special case for method of fundamental solution with optimal locations of sources.

  42. Introduction • Problem statements • Analytical solution • Equivalence of Trefftz method and MFS • Semi-analytical solution • Numerical examples • Conclusions Image method versus MFS large

  43. Thanks for your kind attentions You can get more information from our website http://msvlab.hre.ntou.edu.tw/ 垚淼2008研討會

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