1. 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion problems in anti-plane piezoelectricity An-Chien Wu Reporter: An-Chien Wu Jeng-Tzong Chen Advisor: Jeng-Tzong Chen 2006/06/29 Date: 2006/06/29 HR2 307 Place: HR2 307

2. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 2 Outline Motivation and literature review Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations ◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique Numerical examples Conclusions Further studies

3. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 3 Outline Motivation and literature reviewMotivation and literature review Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations ◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique Numerical examples Conclusions Further studies

4. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 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. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 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. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 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 Degenerate kernel CPV and HPV

7. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 7 Engineering problem with holes, inclusions and cracks Straight boundary Degenerate boundary Circular inclusion Elliptic hole [Mathieu function] [Legendre polynomial] [Chebyshev polynomial] [Fourier series]

8. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 8 Literature review – analytical solutions for problems with circular boundaries Key point Main application Author Conformal mapping Torsion 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 coordinate Electrostatic potential Elasticity Lebedev et al. (1965) Howland & Knight (1939) Möbius transformation Anti-plane piezoelectricity & elasticity Honein et al. (1992) Complex potential approach Anti-plane piezoelectricity Wang & Shen (2001) Those analytical methods are only limited to doubly connected regions even to conformal connected regions.

9. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 9 Literature review - Fourier series approximation Author Main application Key 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.

10. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 10 Outline Motivation and literature review Unified formulation of null-field approachUnified formulation of null-field approach Boundary integral equations and null-field integral equations ◎ Boundary integral equations and null-field integral equations ◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique Numerical examples Conclusions Further studies

11. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 11 Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form

12. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 12 Expansions of fundamental solution and boundary density Degenerate kernel – fundamental solution Fourier series expansion – boundary density

13. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 13 Convergence rate between present method and conventional BEM Degenerate kernel Fourier series expansion Fundamental solution Boundary density Convergence rate Present method Conventional BEM Two-point function Constant, linear, quadratic elements Exponential convergence Linear convergence

14. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 14 Degenerate (separate) form of fundamental solution (2-D)

15. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 15 Outline Motivation and literature review Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations Adaptive observer system ◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique Numerical examples Conclusions Further studies

16. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 16 Adaptive observer system collocation point r 0, f 0 r 1, f 1 rk,fkrk,fk r2,f2r2,f2

17. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 17 Outline Motivation and literature review Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations ◎ Adaptive observer system Linear algebraic equation ◎ Linear algebraic equation ◎ Vector decomposition technique Numerical examples Conclusions Further studies

18. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 18 Linear algebraic equation Column vector of Fourier coefficients ( Nth routing circle) Index of collocation circle Index of routing circle

19. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 19 Explicit form of each submatrix and vector Fourier coefficients Truncated terms of Fourier series Number of collocation points

20. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 20 Physical meaning of influence coefficients and mth collocation point on the jth circular boundary jth circular boundary xmxm fmfm cos n q, sin n q boundary distribution kth circular boundary

21. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 21 Outline Motivation and literature review Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations ◎ Adaptive observer system ◎ Linear algebraic equation Vector decomposition technique ◎ Vector decomposition technique Numerical examples Conclusions Further studies

22. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 22 Vector decomposition technique for potential gradient True normal vector Special case (concentric case) : Non-concentric case:

23. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 23 Flowchart of present method Degenerate kernel Fourier series Adaptive observer system Collocating point to construct compatible boundary data relationship Continuity of displacement and equilibrium of traction Linear algebraic system Fourier coefficients Potential of domain point Vector decomposition Potential gradient Analytical Numerical

24. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 24 Comparisons of conventional BEM and present method Boundary density discretization Auxiliary system Formulation Observer system SingularityConvergence Boundary layer effect Conventional BEM Constant, linear, quadratic… elements Fundamental solution Boundary integral equation Fixed observer system CPV, RPV and HPV LinearAppear Present method Fourier series expansion Degenerate kernel Null-field integral equation Adaptive observer system DisappearExponentialEliminate

25. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 25 Outline Motivation and literature review Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations ◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique Numerical examplesNumerical examples Conclusions Further studies

26. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 26 Numerical examples Anti-plane piezoelectricity problems (EABE, 2006, accepted) In-plane electrostatics problems (??) (??) Anti-plane elasticity problems (ASME-JAM, 2006, accepted) (ASME-JAM, 2006, accepted)

27. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 27 Numerical examples Anti-plane piezoelectricity problemsAnti-plane piezoelectricity problems In-plane electrostatics problems Anti-plane elasticity problems

28. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 28 Problem statement =+ +

29. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 29 Analogy between anti-plane deformation and in- plane electrostatics for anti-plane piezoelectricity Anti-plane shear deformation Constitutive equations for anti-plane piezoelectricity In-plane electrostatics z-displacement w Electric potential F Strain g zi Electric field E i Stress s zi Electric displacement D i Shear modulus mDielectric constant e Strain-disp. g zi = w,i Electricity E i = – F,i Constitutive law s zi = m g zi Constitutive law D i = e E i Coupling effect s zi = c 44 g zi – e 15 E i D i = e 15 g zi + e 11 E i Shear modulus c 44 Piezoelectric constant e 15 Dielectric constant e 11

30. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 30 Linear algebraic system For the exterior problem of matrix For the interior problem of each inclusion The continuity of displacement The equilibrium of traction

31. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 31 Two circular inclusions embedded in a piezoelectric matrix under such loadings

32. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 32 Tangential stress distribution for different ratios d/r 1 with r 2 =2r 1, e 15 M / e 15 I =3.0 and b =90 ° Chao & Chang’s data (1999) Present method (L=20)

33. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 33 Tangential electric field distribution for different ratios d/r 1 with r 2 =2r 1, e 15 M / e 15 I =3.0 and b =90 ° Chao & Chang’s data (1999) Present method (L=20)

34. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 34 Tangential stress distribution for different ratios d/r 1 with r 2 =2r 1, e 15 M / e 15 I =-5.0 and b =90 ° Chao & Chang’s data (1999) Present method (L=20)

35. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 35 Tangential electric field distribution for different ratios d/r 1 with r 2 =2r 1, e 15 M / e 15 I =-5.0 and b =90 ° Chao & Chang’s data (1999) Present method (L=20)

36. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 36 Parseval’s sum for r 2 =2r 1, d/r 1 =0.01, b =90 ° and e 15 M / e 15 I =5.0 Parseval’s sum

37. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 37 Parseval’s sum for r 2 =2r 1, d/r 1 =0.01, b =90 ° and e 15 M / e 15 I =5.0 Parseval’s sum

38. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 38 Tangential stress distribution for different ratios d/r 1 with r 2 =2r 1, e 15 M / e 15 I =-5.0 and b =0 ° Chao & Chang’s data (1999) Present method (L=20)

39. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 39 Tangential electric field distribution for different ratios d/r 1 with r 2 =2r 1, e 15 M / e 15 I =-5.0 and b =0 ° Chao & Chang’s data (1999) Present method (L=20)

40. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 40 Stress concentrations as a function of the ratio of piezoelectric constants with b =0 ° Chao & Chang’s data (1999) Present method (L=20)

41. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 41 Electric field concentrations as a function of the ratio of piezoelectric constants with b =0 ° Chao & Chang’s data (1999) Present method (L=20)

42. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 42 Stress concentrations as a function of the ratio of piezoelectric constants with b =0 ° Chao & Chang’s data (1999) Present method (L=20)

43. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 43 Electric field concentrations as a function of the ratio of piezoelectric constants with b =0 ° Chao & Chang’s data (1999) Present method (L=20)

44. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 44 Contour of shear stress s zx when d/r 1 =0.01 Wang & Shen’s data (2001) Present method (L=20)

45. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 45 Contour of shear stress s zy when d/r 1 =0.01 Wang & Shen’s data (2001) Present method (L=20)

46. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 46 Contour of electric potential F when d/r 1 =0.01 Present method (L=20)

47. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 47 Stress distribution with r 2 =2r 1 and d/r 1 =0.01 in two-directions loadings Present method (L=20) Wang & Shen’s data (2001)

48. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 48 Electric displacement distribution with r 2 =2r 1 and d/r 1 =0.01 in two-directions loadings Present method (L=20) Wang & Shen’s data (2001)

49. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 49 Stress distribution with r 2 =2r 1 and d/r 1 =0.01 in two-directions loadings Present method (L=20) Wang & Shen’s data (2001)

50. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 50 Electric displacement distribution with r 2 =2r 1 and d/r 1 =0.01 in two-directions loadings Present method (L=20) Wang & Shen’s data (2001)

51. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 51 Numerical examples Anti-plane piezoelectricity problems In-plane electrostatics problemsIn-plane electrostatics problems Anti-plane elasticity problems

52. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 52 Analogy between anti-plane deformation and in- plane electrostatics for anti-plane piezoelectricity Anti-plane shear deformation Constitutive equations for anti-plane piezoelectricity In-plane electrostatics z-displacement w Electric potential F Strain g zi Electric field E i Stress s zi Electric displacement D i Shear modulus mDielectric constant e Strain-disp. g zi = w,i Electricity E i = – F,i Constitutive law s zi = m g zi Constitutive law D i = e E i Coupling effect s zi = c 44 g zi – e 15 E i D i = e 15 g zi + e 11 E i Shear modulus c 44 Piezoelectric constant e 15 Dielectric constant e 11

53. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 53 The dielectric system of two inclusions in the applied electric field

54. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 54 Patterns of the electric field for e 0 =2, e 1 =9 and e 2 =5 Emets & Onofrichuk (1996) Present method (L=20)

55. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 55 Patterns of the electric field for e 0 =3, e 1 =9 and e 2 =1 Emets & Onofrichuk (1996) Present method (L=20)

56. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 56 Numerical examples Anti-plane piezoelectricity problems In-plane electrostatics problems Anti-plane elasticity problemsAnti-plane elasticity problems

57. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 57 Two equal-sized holes r 2 =r 1 with centers on the x axis

58. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 58 Stress concentration of the problem containing two equal-sized holes

59. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 59 Stress concentration factors and errors between present method and conventional BEM d/r 1 0.010.20.40.60.81.0 Analytical solution Steif (1989) 14.22473.53492.76672.47582.32742.2400 PresentMethod L=1010.5096(26.12%)3.5306(0.12%)2.7664(0.01%)2.4758(0.00%)2.3274(0.00%)2.2400(0.00%) L=2013.3275(6.31%)3.5349(0.00%)2.7667(0.00%)2.4758(0.00%)2.3274(0.00%)2.2400(0.00%) BEMBEPO2D No. node =217.2500(49.03%)3.4532(2.31%)2.738(1.04%)2.4639(0.48%)2.3168(0.46%)2.2366(0.15%) =4110.2008(28.29%)3.5188(0.46%)2.7619(0.17%)2.4747(0.04%)2.3312(0.16%)2.2398(0.01%) Stress concentration factor

60. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 60 Convergence test and boundary- layer effect analysis

61. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 61 Two circular inclusions with centers on the y axis

62. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 62 Two circular inclusions with centers on the y axis Honein et al.’sdata (1992) Present method (L=20) Equilibrium of traction

63. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 63 Convergence test for stress concentration factor

64. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 64 Boundary-layer effect analysis for radial and tangential stresses

65. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 65 One hole surrounded by two circular inclusions r 3 =r 2 =2r 1

66. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 66 Tangential stress distribution along the hole with b =0 ° Chao & Young’s data (1998) Present method (L=20)

67. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 67 Tangential stress distribution along the hole with b =90 ° Chao & Young’s data (1998) Present method (L=20)

68. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 68 Three identical inclusions forming an equilateral triangle

69. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 69 Tangential stress distribution around the inclusion located at the origin Present method (L=20), agrees well with Gong’s data (1995)

70. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 70 Outline Motivation and literature review Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations ◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique Numerical examples ConclusionsConclusions Further studies

71. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 71 Conclusions A systematic approach using degenerate kernels and Fourier series for null-field integral equation has been successfully proposed to solve BVPs with circular inclusions.A systematic approach using degenerate kernels and Fourier series for null-field integral equation has been successfully proposed to solve BVPs with circular inclusions. According to numerical results, only few terms of Fourier series can achieve accurate solutions.According to numerical results, only few terms of Fourier series can achieve accurate solutions. Four goals of singularity free, boundary-layer effect free, exponential convergence and well- posed model are achieved.Four goals of singularity free, boundary-layer effect free, exponential convergence and well- posed model are achieved.

72. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 72 Conclusions The results demonstrate the superiority of present method over the conventional BEM.The results demonstrate the superiority of present method over the conventional BEM. Our semi-analytical results may provide a datum for other researchers’ reference.Our semi-analytical results may provide a datum for other researchers’ reference. The stress and electric field concentrations are dependent on the distance between the two inclusions, the mismatch in the material constants and the magnitude of mechanical and electromechanical loadings.The stress and electric field concentrations are dependent on the distance between the two inclusions, the mismatch in the material constants and the magnitude of mechanical and electromechanical loadings.

73. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 73 Conclusions A general-purpose program for solving Laplace problems with multiple circular inclusions of various radii, arbitrary positions and different material constants was developed.A general-purpose program for solving Laplace problems with multiple circular inclusions of various radii, arbitrary positions and different material constants was developed. Its possible applications in engineering are very broad, not only limited in this thesis.Its possible applications in engineering are very broad, not only limited in this thesis.

74. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 74 Outline Motivation and literature review Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations ◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique Numerical examples Conclusions Further studiesFurther studies

75. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 75 Further studies Extension to general boundaries.Extension to general boundaries. 2-D problems to 3-D problems.2-D problems to 3-D problems. Various loading types, e.g. concentrated forces, screw dislocations, torques, in- plane shears and tensions.Various loading types, e.g. concentrated forces, screw dislocations, torques, in- plane shears and tensions.screw dislocationsscrew dislocations Various inhomogeneous types, e.g. coated fibers and inclusions with imperfect interfaces.Various inhomogeneous types, e.g. coated fibers and inclusions with imperfect interfaces.coated fibers coated fibers

76. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 76 The end Thanks for your kind attention. Your comments will be highly appreciated. Welcome to the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlab

77. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 77 Derivation of degenerate kernels

78. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 78

79. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 79 An infinite medium containing one hole under the screw dislocation screw dislocationscrew dislocation

80. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 80 Coated inclusion Coated inclusion under the anti- plane shear stress Coated inclusion

81. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 81 Separable form of fundamental solution (1D) Separable property continuous discontinuous

82. MSVLAB National Taiwan Ocean University Department of Harbor and River Engineering 82