1. M M S S V V 0 MSVLAB, HRE, NTOU 博士班資格考口試報告 報 告 人 : 李應德 先生 指導教授 : 陳正宗 教授 時 間 : 2007 年 07 月 27 日 地 點 : 河工二館 307 室 Null-field Integral Equation Approach Using Multipole Expansion and Their Applications 博士班資格考口試報告

2. M M S S V V 1 MSVLAB, HRE, NTOU 博士班資格考口試報告 Outlines 1. Introduction 2. Problem statements and formulation 3. Special treatment for some problems 4. Preliminary results 5. Multipole expansion 6. Further works

3. M M S S V V 2 MSVLAB, HRE, NTOU 博士班資格考口試報告 Outlines 1. Introduction 2. Problem statements and formulation 3. Special treatment for some problems 4. Preliminary results 5. Multipole expansion 6. Further works

4. M M S S V V 3 MSVLAB, HRE, NTOU 博士班資格考口試報告 Numerical methods Finite Difference Method Finite Element Method Boundary Element Method Meshless Method operation pulse acupuncture

5. M M S S V V 4 MSVLAB, HRE, NTOU 博士班資格考口試報告 Pitfalls of conventional BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Ill-posed system Convergence rate Bump contour Limit process Guiggiani (1995) Gray and Manne (1993) Fictitious BEM T-matrix Achenbach et al. (1988) Waterman (1965) Relative quantity σ BEM Kisu (1988) Sladek (1991) Linear order Quadratic order Cubic order

6. M M S S V V 5 MSVLAB, HRE, NTOU 博士班資格考口試報告 Literature review (circular boundaries) Key pointMain applicationAuthor Conformal mappingTorsion problem In-plane electrostatics Anti-plane elasticity Chen & Weng (2001) Emets & Onofrichuk (1996) Budiansky & Carrier (1984) Steif (1989) Wu & Funami (2002) Wang & Zhong (2003) Bi-polar coordinateElectrostatic potential Elasticity Lebedev et al. (1965) Howland & Knight (1939) Möbius transformationAnti-plane piezoelectricity & elasticity Honein et al. (1992) Complex potential approachAnti-plane piezoelectricityWang & Shen (2001) Those analytical methods are only limited to doubly connected regions. Analytical solutions for problems with circular boundaries

7. M M S S V V 6 MSVLAB, HRE, NTOU 博士班資格考口試報告 Literature review (Fourier series ) AuthorMain applicationKey point Ling (1943) Torsion of a circular tube Caulk et al. (1983) Steady heat conduction with circular holes Special BIEM Bird and Steele (1992) Harmonic and biharmonic problems with circular holes Trefftz method Mogilevskaya et al. (2002) Elasticity problems with circular holes or inclusions Galerkin method However, no one employed the null-field approach and degenerate kernel to fully capture the circular boundary. Fourier series approximation

8. M M S S V V 7 MSVLAB, HRE, NTOU 博士班資格考口試報告 Literature review (Fourier series ) AuthorMain application Sloan et al. (1975) Prove that it is equivalent to iterated Petrov- Galerkin approximation Kress (1985) Prove it combined with integral equation have convergence of exponential order Chen et al. (2005) Applied it to solve engineering problem with circular boundaries Chen et al. (2006) Link Trefftz method and method of fundamental solutions However, its application in practical problems seem to have taken a back seat to other methods. Degenerate kernel approximation

9. M M S S V V 8 MSVLAB, HRE, NTOU 博士班資格考口試報告 Literature review (FMM ) AuthorMain application Rokhlin (1983)Potential theory (First introducing) Amini and Profit (1999)Scattering theory Chen and Chen (2004)2-D exterior acoustics Liu et al. (2006)Combine with MFS to solve potential problem

10. M M S S V V 9 MSVLAB, HRE, NTOU 博士班資格考口試報告 Present approach Fundamental solution No principal value Advantages of present approach 1.No principal value 2.Well-posed system 3.Exponential convergence 4.Free of mesh Degenerate kernel CPV and HPV

11. M M S S V V 10 MSVLAB, HRE, NTOU 博士班資格考口試報告 Outlines 1. Introduction 2. Problem statements and formulation 3. Special treatment for some problems 4. Preliminary results 5. Multipole expansion 6. Further works

12. M M S S V V 11 MSVLAB, HRE, NTOU 博士班資格考口試報告 Problem statements B0B0 B1B1 B2B2 B3B3 BiBi B4B4 a0a0 a1a1 a2a2 a3a3 a4a4 aiai x y Governing equation: Helmholtz problem: Laplace problem: Biharmonic problem: BiHelmholtz problem: Elasticity problem:

13. M M S S V V 12 MSVLAB, HRE, NTOU 博士班資格考口試報告 Inclusion problems A circular bar with circular holesEach circular inclusion problem B0B0 B1B1 B2B2 B3B3 BiBi B4B4 B0B0 B1B1 B2B2 B3B3 BiBi B4B4 Satisfy

14. M M S S V V 13 MSVLAB, HRE, NTOU 博士班資格考口試報告 Interior case Exterior case Degenerate (separable) form BIE and null-field integral equation

15. M M S S V V 14 MSVLAB, HRE, NTOU 博士班資格考口試報告 Fundamental solution and Four kernels Helmholtz problem: Laplace problem:Biharmonic problem: BiHelmholtz problem: Elasticity problem: Relationship of four kernels: Fundamental solutions:

16. M M S S V V 15 MSVLAB, HRE, NTOU 博士班資格考口試報告 NumericalAnalytical Flowchart of present approach The problem with circular boundaries Null-field integral equation Degenerate kernel for fundamental solution Fourier series expansion for boundary density Adaptive observer system in the boundary integrations Collocating the point to boundary and matching boundary conditions Linear algebraic system Obtain the unknown Fourier coefficients Boundary integral equation for the domain point Displacement field

17. M M S S V V 16 MSVLAB, HRE, NTOU 博士班資格考口試報告 Degenerate kernel and Fourier series s O x kth circular boundary cosnθ, sinnθ boundary distributions x Expand fundamental solution by using degenerate kernel Expand boundary densities by using Fourier series In the real computation M number of terms

18. M M S S V V 17 MSVLAB, HRE, NTOU 博士班資格考口試報告 Keypoint of deriving degenerate kernels Laplace problem: Helmholtz problem: Elasticity problem: (Addition theorem) First derived

19. M M S S V V 18 MSVLAB, HRE, NTOU 博士班資格考口試報告 collocation point r 0, f 0 r 1, f 1 rk,fkrk,fk r2,f2r2,f2 Adaptive observer system

20. M M S S V V 19 MSVLAB, HRE, NTOU 博士班資格考口試報告 Collocation method and linear algebraic system B0B0 B1B1 B2B2 B3B3 BiBi B4B4 Index of collocation circle Index of routing circle

21. M M S S V V 20 MSVLAB, HRE, NTOU 博士班資格考口試報告 Outlines 1. Introduction 2. Problem statements and formulation 3. Special treatment for some problems 4. Preliminary results 5. Multipole expansion 6. Further works

22. M M S S V V 21 MSVLAB, HRE, NTOU 博士班資格考口試報告 Transformation of tensor components x 1 2 3 4 1’ 2’ 3’ 4’ 1, 2: transformed normal and tangential components 3, 4: original normal and tangential componentss

23. M M S S V V 22 MSVLAB, HRE, NTOU 博士班資格考口試報告 Domain superposition Incident SH-wave (a) Incident wave field (b) Radiation field + =

24. M M S S V V 23 MSVLAB, HRE, NTOU 博士班資格考口試報告 Image method for half-plane problem (a) Real problem Incident-SH wave t=0t=0 t=0t=0 t=0t=0 t=0t=0 t=0t=0 Image incident-SH wave (b) Extended problem Image

25. M M S S V V 24 MSVLAB, HRE, NTOU 博士班資格考口試報告 Outlines 1. Introduction 2. Problem statements and formulation 3. Special treatment for some problems 4. Preliminary results 5. Multipole expansion 6. Further works

26. M M S S V V 25 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 1: A circular bar with an eccentric inclusion R1R1 R0R0 exex Ratio: Torsional rigidity: G T : total torsion rigidity G M : torsion rigidity of matrix G I : torsion rigidity of inclusion

27. M M S S V V 26 MSVLAB, HRE, NTOU 博士班資格考口試報告 Results of Example 1 Torsional rigidity versus number of Fourier series terms Torsional rigidity versus shear modulus of inclusion

28. M M S S V V 27 MSVLAB, HRE, NTOU 博士班資格考口試報告 Results of Example 1 Torsional rigidity of a circular bar with an eccentric inclusion

29. M M S S V V 28 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 2: (limiting case) A circular bar with one circular hole R 1 =0.3 R 0 =1 e x =0.5

30. M M S S V V 29 MSVLAB, HRE, NTOU 博士班資格考口試報告 Torsional rigidity of a circular bar with an eccentric hole Results of Example 2

31. M M S S V V 30 MSVLAB, HRE, NTOU 博士班資格考口試報告 Stress calculation t tmtm External diameter of the tube D:D: tm:tm:The maxium wall thickness (eccentricity)

32. M M S S V V 31 MSVLAB, HRE, NTOU 博士班資格考口試報告 Stress calculation along outer and inner boundary at boundaries for λ=0.3 and p=0.4 (0.0%) (0.1%) (0.0%) (0.4%) (0.0%) (0.3%) (0.0%) (1.5%) (0.6%)

33. M M S S V V 32 MSVLAB, HRE, NTOU 博士班資格考口試報告 Stress calculation for point in the center line alnog lines and for λ=0.3 and p=0.4 (0.0%) (0.1%) (0.3%) (0.0%) (0.2%) (0.5%) (0.0%) (0.6%)

34. M M S S V V 33 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 3: Four cylinders a2b x y Incident wave

35. M M S S V V 34 MSVLAB, HRE, NTOU 博士班資格考口試報告 Results of the hydordynamic force Perrey-Debain et al.Present method This work was done some years ago, but my recollection was that we simply picked one example for comparison for scattering by multiple objects. We did find, however, that although the series solution in the paper is correct, some of the images in the Linton & Evans paper were incorrect. If you are using the images for comparison with your own work, it might be a good idea to check against the series solutions instead.

36. M M S S V V 35 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 4: Stress concentrated factor SS SS = +

37. M M S S V V 36 MSVLAB, HRE, NTOU 博士班資格考口試報告 Deformation = + undeformed deformed

38. M M S S V V 37 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 5: Lamé problem b a PePe PiPi B1B1 B2B2

39. M M S S V V 38 MSVLAB, HRE, NTOU 博士班資格考口試報告 Deformation

40. M M S S V V 39 MSVLAB, HRE, NTOU 博士班資格考口試報告 Outlines 1. Introduction 2. Problem statements and formulation 3. Special treatment for some problems 4. Preliminary results 5. Multipole expansion 6. Further works

41. M M S S V V 40 MSVLAB, HRE, NTOU 博士班資格考口試報告 Our goals x 1 2 3 4 1’ 2’ 3’ 4’ Tensor transformation x1x1 y1y1 xjxj yjyj xixi yiyi x2x2 y2y2 B1B1 B2B2 BiBi BjBj osos osos osos osos Adaptive observer system

42. M M S S V V 41 MSVLAB, HRE, NTOU 博士班資格考口試報告 Multipole expansion Expanding the kernel function where

43. M M S S V V 42 MSVLAB, HRE, NTOU 博士班資格考口試報告 Fast algorithm Generalized Minimum RESidual Method (GMRES) Solver: Generalized Conjugate RESidual Method

44. M M S S V V 43 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 1: Two semi-circular canyons Incident SH wave Reflected SH wave 3a3a t=0t=0t=0t=0t=0t=0 t=0t=0t=0t=0 a x y

45. M M S S V V 44 MSVLAB, HRE, NTOU 博士班資格考口試報告 Tsaur et al.Present method

46. M M S S V V 45 MSVLAB, HRE, NTOU 博士班資格考口試報告 Tsaur et al.Present method

47. M M S S V V 46 MSVLAB, HRE, NTOU 博士班資格考口試報告 Tsaur et al.Present method

48. M M S S V V 47 MSVLAB, HRE, NTOU 博士班資格考口試報告 Tsaur et al.Present method

49. M M S S V V 48 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 2: A circular hole x y h=1.5a a t=0t=0 t=0t=0

50. M M S S V V 49 MSVLAB, HRE, NTOU 博士班資格考口試報告 Lee and ManoogianPresent method

51. M M S S V V 50 MSVLAB, HRE, NTOU 博士班資格考口試報告 Lee and ManoogianPresent method

52. M M S S V V 51 MSVLAB, HRE, NTOU 博士班資格考口試報告 Lee and ManoogianPresent method

53. M M S S V V 52 MSVLAB, HRE, NTOU 博士班資格考口試報告 Lee and ManoogianPresent method

54. M M S S V V 53 MSVLAB, HRE, NTOU 博士班資格考口試報告 Outlines 1. Introduction 2. Problem statements and formulation 3. Special treatment for some problems 4. Preliminary results 5. Multipole expansion 6. Further works

55. M M S S V V 54 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 1: Perfect interface to imperfect interface R1R1 R0R0 exex

56. M M S S V V 55 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 2: Two circular holes under a unified tension R 1 (0,0)(-d,0) T T s

57. M M S S V V 56 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 3: An array of circular inclusions in an infinite plane x y

58. M M S S V V 57 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 4: A half-plane problem with a semi-circular hill Incident SH wave a y x

59. M M S S V V 58 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 5: Two circular holes under the screw dislocation ⊕ a2a2 x y a1a1 d

60. M M S S V V 59 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 6: A circular inclusion under the screw dislocation ⊕ a x y

61. M M S S V V 60 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 7: A circular hole under the edge dislocation a x y d

62. M M S S V V 61 MSVLAB, HRE, NTOU 博士班資格考口試報告 Example 8: A half-plane problem with two alluvial valleys Incident SH wave a 3a a y x Alluvial Matrix

63. M M S S V V 62 MSVLAB, HRE, NTOU 博士班資格考口試報告 Thanks for your kind attention The End