1. 1 Some recent results of the null-field integral equation approach for engineering problems with circular J. T. Chen Ph.D. Taiwan Ocean University Keelung, Taiwan November, 14-18, 2006 icome2006heife.ppt National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering

2. 2 Research collaborators Dr. I. L. Chen Dr. K. H. Chen Dr. I. L. Chen Dr. K. H. Chen Dr. S. Y. Leu Dr. W. M. Lee Dr. S. Y. Leu Dr. W. M. Lee Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. A. C. Wu Mr.P. Y. Chen Mr. A. C. Wu Mr.P. Y. Chen Mr. Y. T. Lee Mr. Y. T. Lee

3. 3 URL: http://ind.ntou.edu.tw/~msvlab E-mail: jtchen@mail.ntou.edu.tw 海洋大學工學院河工所力學聲響振動實驗室 `` Elasticity & Crack Problem 李應德 Laplace Equation Research topics of NTOU / MSV LAB on null-field BIE (2003-2006) Navier Equation Null-field BIEM Biharmonic Equation Previous research and project Future work 蕭嘉俊 05’ (Plate with circulr holes) (Stokes ’ flow) BiHelmholtz EquationHelmholtz Equation 沈文成 05 ’ (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) 吳安傑 06 ’ (Inclusion) (Piezoleectricity) 陳柏源 06 ’ (Beam bending) 李應德 Torsion bar (Inclusion) 廖奐禎、李應德 Image method (Green function) 柯佳男 Green function of half plane (Hole and inclusion) 陳佳聰 05 ’ (Interior and exterior Acoustics) 陳柏源 05 ’ SH wave (exterior acoustics) (Inclusions) 李為民、李應德 (Plate vibration) ASMEMRC,CMESEABE ASME JoM EABE

4. 4 哲人日已遠 典型在宿昔 (1909-1993) 省立中興大學第一任校長 林致平校長 ( 民國五十年 ~ 民國五十二年 ) 林致平所長 ( 中研院數學所 ) 林致平院士 ( 中研院 ) PS: short visit (J T Chen) of Academia Sinica 2006 summer ` 林致平院士 ( 數學力學家 )

5. 5 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

6. 6 Motivation Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM (mesh required) Treatment of singularity and hypersingularity Boundary-layer effect Ill-posed model Convergence rate Mesh free for circular boundaries ?

7. 7 Motivation and literature review Fictitious BEM BEM/BIEM Null-field approach Bump contour Limit process Singular and hypersingular Regular Improper integral CPV and HPV Ill-posed Fictitious boundary Collocation point

8. 8 Present approach 1.No principal value 2. Well-posed 2. Well-posed 3. No boundary-layer effect 3. No boundary-layer effect 4. Exponetial convergence 4. Exponetial convergence Advantages of degenerate kernel Degenerate kernel Fundamental solution CPV and HPV No principal value

9. 9 Engineering problem with arbitrary geometries Degenerate boundary Circular boundary Straight boundary Elliptic boundary (Fourier series) (Legendre polynomial) (Chebyshev polynomial) (Mathieu function)

10. 10 Motivation and literature review Analytical methods for solving Laplace problems with circular holes Conformal mapping Bipolar coordinate Special solution Limited to doubly connected domain Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics Honein, Honein and Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics

11. 11 Fourier series approximation Ling (1943) - torsion of a circular tube Ling (1943) - torsion of a circular tube Caulk et al. (1983) - steady heat conduction with circular holes Caulk et al. (1983) - steady heat conduction with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Mogilevskaya et al. (2002) - elasticity problems with circular boundaries Mogilevskaya et al. (2002) - elasticity problems with circular boundaries

12. 12 Contribution and goal However, they didn’t employ the null-field integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. However, they didn’t employ the null-field integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. To develop a systematic approach for solving Laplace problems with multiple holes is our goal. To develop a systematic approach for solving Laplace problems with multiple holes is our goal.

13. 13 Outlines (Direct problem) Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

14. 14 Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form

15. 15 R.P.V. (Riemann principal value) C.P.V.(Cauchy principal value) H.P.V.(Hadamard principal value) Definitions of R.P.V., C.P.V. and H.P.V. using bump approach Ó NTUCE

16. 16 Principal value in who’s sense Riemann sense (Common sense) Riemann sense (Common sense) Lebesgue sense Lebesgue sense Cauchy sense Cauchy sense Hadamard sense (elasticity) Hadamard sense (elasticity) Mangler sense (aerodynamics) Mangler sense (aerodynamics) Liggett and Liu’s sense Liggett and Liu’s sense The singularity that occur when the The singularity that occur when the base point and field point coincide are not integrable. (1983) base point and field point coincide are not integrable. (1983)

17. 17 Two approaches to understand HPV (Limit and integral operator can not be commuted) (Leibnitz rule should be considered) Differential first and then trace operator Trace first and then differential operator

18. 18 Bump contribution (2-D) Bump contribution (2-D) x s T x s U x s L x s M 0 0

19. 19 Bump contribution (3-D) Bump contribution (3-D) x x x x s s s s 0`0` 0

20. 20 Outlines (Direct problem) Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Degenerate scale Degenerate scale Conclusions Conclusions

21. 21 Gain of introducing the degenerate kernel Degenerate kernel Fundamental solution CPV and HPV No principal value?

22. 22 Expansions of fundamental solution and boundary density Degenerate kernel - fundamental solution Degenerate kernel - fundamental solution Fourier series expansions - boundary density Fourier series expansions - boundary density

23. 23 Separable form of fundamental solution (1D) Separable property continuous discontinuous

24. 24 Separable form of fundamental solution (2D)

25. 25 Boundary density discretization Fourier series Ex. constant element Present method Conventional BEM

26. 26 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

27. 27 Adaptive observer system collocation point

28. 28 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

29. 29 Vector decomposition technique for potential gradient Special case (concentric case) : Non-concentric case: True normal direction

30. 30 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

31. 31 Linear algebraic equation where Column vector of Fourier coefficients (Nth routing circle) Index of collocation circle Index of routing circle

32. 32 Physical meaning of influence coefficient kth circular boundary xmxm mth collocation point on the jth circular boundary jth circular boundary Physical meaning of the influence coefficient cosnθ, sinnθ boundary distributions

33. 33 Flowchart of present method Potential of domain point Analytical Numerical Adaptive observer system Degenerate kernel Fourier series Linear algebraic equation Collocation point and matching B.C. Fourier coefficients Vector decomposition Potential gradient

34. 34 Comparisons of conventional BEM and present method BoundarydensitydiscretizationAuxiliarysystemFormulationObserversystemSingularityConvergenceBoundarylayereffect ConventionalBEMConstant,linear,quadratic…elementsFundamentalsolutionBoundaryintegralequationFixedobserversystem CPV, RPV and HPV LinearAppear PresentmethodFourierseriesexpansionDegeneratekernelNull-fieldintegralequationAdaptiveobserversystemDisappearExponentialEliminate

35. 35 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

36. 36 Numerical examples Laplace equation (EABE 2005, EABE 2007) Laplace equation (EABE 2005, EABE 2007) (CMES 2005, ASME 2007) (CMES 2005, ASME 2007) (JoM 2007,MRC, 2007) (JoM 2007,MRC, 2007) Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation (JAM, ASME 2006 ) Biharmonic equation (JAM, ASME 2006 ) Plate vibration (JSV revision, 2006) Plate vibration (JSV revision, 2006)

37. 37 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

38. 38 Steady state heat conduction problems Case 1 Case 2

39. 39 Case 1: Isothermal line Exact solution (Carrier and Pearson) BEM-BEPO2D(N=21) FEM-ABAQUS (1854 elements) Present method (M=10)

40. 40 Relative error of flux on the small circle

41. 41 Convergence test - Parseval’s sum for Fourier coefficients Parseval’s sum

42. 42 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

43. 43 Electrostatic potential of wires Hexagonal electrostatic potential Two parallel cylinders held positive and negative potentials

44. 44 Contour plot of potential Exact solution (Lebedev et al.) Present method (M=10)

45. 45 Contour plot of potential Onishi’s data (1991) Present method (M=10)

46. 46 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

47. 47 Flow of an ideal fluid pass two parallel cylinders is the velocity of flow far from the cylinders is the velocity of flow far from the cylinders is the incident angle is the incident angle

48. 48 Velocity field in different incident angle Present method (M=10)

49. 49 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

50. 50 Torsion bar with circular holes removed The warping function Boundary condition where on Torque

51. 51 Axial displacement with two circular holes Present method (M=10) Caulk’s data (1983) ASME Journal of Applied Mechanics Dashed line: exact solution Solid line: first-order solution

52. 52 Torsional rigidity ?

53. 53 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

54. 54 Infinite medium under antiplane shear The displacement Boundary condition Total displacement on

55. 55 Shear stress around the hole of radius a 1 (x axis) Shear stress σ zq around the hole of radius a 1 (x axis) Present method (M=20) Honein’s data (1992) Quarterly of Applied Mathematics

56. 56 Shear stress around the hole of radius a 1 Shear stress σ zq around the hole of radius a 1 Present method (M=20) Steele’s data (1992) Stress approach Displacement approach Honein’s data (1992) 5.348 5.349 4.647 5.345 13.13% 0.02% Analytical 0.06%

57. 57 Hole to inclusion Hole to inclusion Anti-plane piezoelectricity problems Anti-plane piezoelectricity problems In-plane electrostatics problems In-plane electrostatics problems Anti-plane elasticity problems Anti-plane elasticity problems

58. 58 Two circular inclusions with centers on the y axis Honein et al.’sdata (1992) Present method (L=20) Equilibrium of traction

59. 59 Convergence test and boundary-layer effect analysis

60. 60 Patterns of the electric field for e 0 =2, e 1 =9 and e 2 =5 Emets & Onofrichuk (1996) Present method (L=20)

61. 61 Three identical inclusions forming an equilateral triangle

62. 62 Tangential stress distribution around the inclusion located at the origin Present method (L=20), agrees well with Gong’s data (1995)

63. 63 Laplace equation Steady state heat conduction problems Steady state heat conduction problems Electrostatic potential of wires Electrostatic potential of wires Flow of an ideal fluid pass cylinders Flow of an ideal fluid pass cylinders A circular bar under torque A circular bar under torque An infinite medium under antiplane shear An infinite medium under antiplane shear Half-plane problems Half-plane problems

64. 64 Half-plane problems Dirichlet boundary condition (Lebedev et al.) Mixed-type boundary condition (Lebedev et al.)

65. 65 Dirichlet problem Exact solution (Lebedev et al.) Present method (M=10) Isothermal line

66. 66 Mixed-type problem Exact solution (Lebedev et al.) Present method (M=10) Isothermal line

67. 67 Numerical examples Laplace equation Laplace equation Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation Biharmonic equation

68. 68 Problem statement Doubly-connected domainMultiply-connected domain Simply-connected domain

69. 69 Example 1

70. 70 k1k1k1k1 k2k2k2k2 k3k3k3k3 k4k4k4k4 k5k5k5k5 FEM(ABAQUS)2.032.202.623.153.71 BEM (Burton & Miller) 2.062.232.673.223.81 BEM(CHIEF)2.052.232.673.223.81 BEM(null-field)2.042.202.653.213.80 BEM(fictitious)2.042.212.663.213.80 Present method 2.052.222.663.213.80 Analytical solution[19] 2.052.232.663.213.80 The former five true eigenvalues by using different approaches

71. 71 The former five eigenmodes by using present method, FEM and BEM

72. 72 Numerical examples Laplace equation Laplace equation Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation Biharmonic equation

73. 73 u=0..... x y Sketch of the scattering problem (Dirichlet condition) for five cylinders

74. 74 (a) Present method (M=20) (b) Multiple DtN method (N=50) The contour plot of the real-part solutions of total field for

75. 75 The contour plot of the real-part solutions of total field for (a) Present method (M=20) (b) Multiple DtN method (N=50)

76. 76 Fictitious frequencies

77. 77 Present method Soft-basin effect 14 18 3

78. 78 Numerical examples Laplace equation Laplace equation Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation Biharmonic equation

79. 79 Plate problems Geometric data: and on Essential boundary conditions: (Bird & Steele, 1991)

80. 80 Contour plot of displacement Present method (N=101)Bird and Steele (1991) FEM (ABAQUS) FEM mesh (No. of nodes=3,462, No. of elements=6,606)

81. 81 Stokes flow problem Governing equation: Boundary conditions: and on and on Eccentricity: Angular velocity: (Stationary)

82. 82 Comparison for DOF of BIE (Kelmanson) DOF of present method BIE (Kelmanson) Present method Analytical solution (160) (320) (640) u1u1 (28) (36) (44) (∞) Algebraic convergence Exponential convergence

83. 83 Contour plot of Streamline for Present method (N=81) Kelmanson (Q=0.0740, n=160) Kamal (Q=0.0738) e Q/2 Q Q/5 Q/20 -Q/90 -Q/30 0 Q/2 Q Q/5 Q/20 -Q/90 -Q/30 0

84. 84 Numerical examples Laplace equation Laplace equation Eigen problem Eigen problem Exterior acoustics Exterior acoustics Biharmonic equation Biharmonic equation Plate vibration Plate vibration

85. 85 Free vibration of plate

86. 86 Comparisons with FEM

87. 87 Outlines Motivation and literature review Motivation and literature review Mathematical formulation Mathematical formulation Expansions of fundamental solution Expansions of fundamental solution and boundary density and boundary density Adaptive observer system Adaptive observer system Vector decomposition technique Vector decomposition technique Linear algebraic equation Linear algebraic equation Numerical examples Numerical examples Conclusions Conclusions

88. 88 Some findings `LaplaceHelmholtz Ling 1947 Analytical solution Bird & Steele 1991 房營光 1995 Analytical solution Lee & Manoogian 1 992 Caulk 1983 Naghdi 1991 Analytical solution Tsaur et al. 2004 Analytical solution Present method Present method (semi-analytical) Tsaur et al. ?

89. 89 Stress concentration at point B Present method Naghdi’s results Steele & Bird The two approaches disagree by as much 11%. The grounds for this discrepancy have not yet been identified. --ASME Applied Mechanics Review

90. 90 A half-plane problem with two alluvial valleys subject to the incident SH-wave Canyon Matrix 3a SH-Wave 房 [93] 將正弦和餘弦函數的正交特性使用錯誤，以 至於推導出錯誤的聯立方程，求得錯誤的結果。 -- 亞太學報

91. 91 Limiting case of two canyons Present method Tsaur et al.’s results [103]

92. 92 Inclusion Matrix h SH- Wave a x y A half-plane problem with a circular inclusion subject to the incident SH-wave

93. 93 Surface displacements of a inclusion problem under the ground surface Present method Tsaur et al.’s results [102] Manoogian and Lee’s results [62] When I solved this problem I could find no published results for comparison. I also verified my results using the limiting cases. I did not have the benefit of published results for comparing the intermediate cases. I would note that due to precision limits in the Fortran compiler that I was using at the time. --Private communication

94. 94 Conclusions A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve Laplace Helmholtz and Biharminic problems with circular boundaries. A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve Laplace Helmholtz and Biharminic problems with circular boundaries. Numerical results agree well with available exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for only few terms of Fourier series. Numerical results agree well with available exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for only few terms of Fourier series.

95. 95 Conclusions Four previous results were examined. Four previous results were examined.. `Laplace Helmholtz` Ling 1947 Analytical solution Bird & Steele 1991 房營光 1995 Analytical solution Lee & Manoogian1992 ?

96. 96 Conclusions Engineering problems with circular boundaries which satisfy the Laplace Helmholtz and Biharminic problems can be solved by using the proposed approach in a more efficient and accurate manner. Engineering problems with circular boundaries which satisfy the Laplace Helmholtz and Biharminic problems can be solved by using the proposed approach in a more efficient and accurate manner. Free of boundary-layer effect Free of boundary-layer effect Free of singular integrals Free of singular integrals Well posed Well posed Exponetial convergence Exponetial convergence

97. 97 The End Thanks for your kind attentions. Your comments will be highly appreciated. URL: http://msvlab.hre.ntou.edu.tw/ http://msvlab.hre.ntou.edu.tw/

98. 98