1. National Taiwan Ocean University
MSVLAB
Department of Harbor and River Engineering
Null field integral approach for engineering problems with circular boundaries
J. T. Chen Ph.D.
終身特聘教授
Taiwan Ocean University
Keelung, Taiwan
Dec.26, 15:30-17:00, 2007
交通大學 機械系 邀請演講
交大機械2007.ppt

2. Overview of numerical methods
Domain
Boundary DE
PDE- variational IE
MFS,Trefftz method MLS, EFG 針 灸 把 脈 開刀 2

3. Number of Papers of FEM, BEM and FDM6 2 1
(Data form Prof. Cheng A. H. D.)

4. Growth of BEM/BIEM papers
(data from Prof. Cheng A.H.D.)

5. Top ten countries of BEM and dual BEM
USA (3569) , China (1372) , UK, Japan, Germany, France, Taiwan (580), Canada, South Korea, Italy (No.7)
Dual BEM (Made in Taiwan)
UK (124), USA (98), Taiwan (70), China (46), Germany, France, Japan, Australia, Brazil, Slovenia (No.3)
(ISI information Apr.20, 2007)

6. Top ten countries of FEM, FDM and Meshless methods
USA(9946), China(4107), Japan(3519), France, Germany, England, South Korea, Canada, Taiwan(1383), Italy
Meshless methods
USA(297), China(179), Singapore(105), Japan(49), Spain, Germany, Slovakia, England, Portugal, Taiwan(28)
FDM
USA(4041), Japan(1478), China(1411), England, France, Canada, Germany, Taiwan (559), South Korea, India
(ISI information Apr.20, 2007)

7. Active scholars on BEM and dual BEM
Aliabadi M H (UK, Queen Mary College)
Chen J T (Taiwan, Ocean Univ.)
113 SCI papers 479citing
Mukherjee S (USA, Cornell Univ.)
Leisnic D(UK, Univ. of Leeds)
Tanaka M (Japan, Shinshu Univ.)
Dual BEM (Made in Taiwan)
Aliabadi M H (UK, Queen Mary Univ. London)
Power H (UK, Univ. Nottingham)
(ISI information Apr.20, 2007)

8. Top 25 scholars on BEM/BIEM since 2001
NTOU/MSV
Taiwan
北京清華姚振漢教授提供(Nov., 2006)

9. Research topics of NTOU / MSV LAB using null-field BIEs
( ) ( )
Null-field BIEM
NUMPDE revision
Navier Equation
Laplace Equation
Helmholtz Equation
Biharmonic Equation
BiHelmholtz Equation
Elasticity
(holes and/or inclusions)
EABE
MRC,CMES
ASME JAM 2006
JSV
(Potential flow)
(Torsion)
(Anti-plane shear)
(Degenerate scale)
(Plate with circulr holes)
Screw dislocation
(Interior and exterior
Acoustics)
SH wave (exterior acoustics)
(Inclusions)
(Free vibration of plate)
Indirect BIEM
(Stokes flow)
JCA
CMAME 2007
ASME
EABE
JoM
(Free vibration of plate)
Direct BIEM
(Inclusion)
(Piezoleectricity)
(Beam bending)
Green function for
an annular plate
SDEE
ICOME 2006
SH wave
Impinging canyons
Degenerate kernel for ellipse
Torsion bar (Inclusion)
Imperfect interface
(Flexural wave of plate)
Green function of`circular inclusion (special case:static)
CMC
Image method
(Green function)
Added mass
SH wave
Impinging hill
Green function of half plane
(Hole and inclusion)
Previous work (done)
Interested topic
Present project
Effective conductivity
Water wave impinging circular cylinders
URL: 海洋大學工學院河工所力學聲響振動實驗室 Frame2007-c.ppt

10. Research collaborators
Dr. I. L. Chen (高海大) Dr. K. H. Chen (宜蘭大學)
Dr. S. Y. Leu Dr. W. M. Lee (中華技術學院)
Mr. S. K. Kao Mr. Y. Z. Lin Mr. S. Z. Hsieh
Mr. H. Z. Liao Mr. G. N. Kehr (2007)
Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao (2006)
Mr. A. C. Wu Mr. P. Y. Chen (2005)
Mr. Y. T. Lee Mr. S. K. Kao
Mr. S. Z. Shieh Mr. Y. Z. Lin

11. Top 25 scholars on BEM/BIEM since 2001
北京清華姚振漢教授提供(Nov., 2006)

12. 哲人日已遠 典型在宿昔 C B Ling (1909-1993) 林致平院士(數學力學家)
C B Ling (mathematician and expert in mechanics)
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 `

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

14. Motivation BEM / BIEM (mesh required) 除四舊
Numerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless method
BEM / BIEM (mesh required)
Treatment of singularity and hypersingularity
Boundary-layer effect
Convergence rate
Ill-posed model
除四舊
Mesh free for circular boundaries ?

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

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

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

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

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

20. 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.
To develop a systematic approach for solving Laplace problems with multiple holes is our goal.

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

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

23. Definitions of R.P.V., C.P.V. and H.P.V. using bump approach
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

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

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

26. Bump contribution (2-D)s T s x x L s M s x x

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

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

29. Gain of introducing the degenerate kernel
Fundamental solution
CPV and HPV
interior
exterior
No principal value?

30. How to separate the region

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

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

33. Separable form of fundamental solution (2D)

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

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

36. Adaptive observer system
collocation point

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

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

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

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

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

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

43. Comparisons of conventional BEM and present method
Boundary
density
discretization
Auxiliary
system
Formulation
Observer
Singularity
Convergence
layer
effect
Conventional
BEM
Constant,
linear,
quadratic…
elements
Fundamental
solution
integral
equation
Fixed
observer
CPV, RPV
and HPV
Linear
Appear
Present
method
Fourier
series
expansion
Degenerate
kernel
Null-field
Adaptive
Disappear
Exponential
Eliminate

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

45. Numerical examples Laplace equation (EABE 2006, EABE2006,EABE 2007)
(CMES 2006, ASME 2007, JoM2007, CMC)
(MRC 2007, NUMPDE 2007,ASME 2007)
Eigen problem (JCA)
Exterior acoustics (CMAME 2007, SDEE 2008)
Biharmonic equation (JAM, ASME 2006, IJNME)
Plate vibration (JSV 2007)

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

47. Steady state heat conduction problems
Case 1
Case 2

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

49. Relative error of flux on the small circle

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

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

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

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

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

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

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

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

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

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

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

61. Torsional rigidity ?

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

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

64. Shear stress σzq around the hole of radius a1 (x axis)
Honein’s data (1992)
Quarterly of Applied Mathematics
Present method (M=20)

65. Shear stress σzq around the hole of radius a1
Stress approach
Analytical
Displacement approach
Steele’s data (1992)
Present method (M=20)
Honein’s data (1992)
Present method (M=20)
4.647
13.13%
5.349
0.02%
5.348
5.345
0.06%

66. Extension to inclusion
Anti-plane piezoelectricity problems
In-plane electrostatics problems
Anti-plane elasticity problems

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

68. Convergence test and boundary-layer effect analysis

69. Patterns of the electric field for e0=2, e1=9 and e2=5
Present method (L=20)
Emets & Onofrichuk
(1996)

70. Three identical inclusions forming an equilateral triangle

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

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

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

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

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

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

77. Problem statement Simply-connected domain Doubly-connected domain
Multiply-connected domain

78. Example 1

79. The former five true eigenvalues by using different approachesk1 k2 k3 k4 k5
FEM
(ABAQUS)
2.03
2.20
2.62
3.15
3.71
BEM
(Burton & Miller)
2.06
2.23
2.67
3.22
3.81
(CHIEF)
2.05
(null-field)
2.04
2.65
3.21
3.80
(fictitious)
2.21
2.66
Present method
2.22
Analytical solution[19]

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

81. Numerical examples
Laplace equation
Eigen problem
Exterior acoustics
Water wave
Biharmonic equation

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

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

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

85. Fictitious frequencies

86. Soft-basin effect
Present method 18 14 3

87. Water wave impinging four cylinders
1. Introduction
2. Problem statement
3. Method of solution
4. Numerical examples
5. Concluding remarks
Water wave impinging four cylinders

88. Maximum free-surface elevation
Perrey-Debain et al. (JSV 2003 )
Present method (2007)

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

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

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

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

93. Comparison for Algebraic convergence Exponential convergence (160)
BIE (Kelmanson)
Present method
Analytical solution
(28)
Algebraic convergence u1
(320)
(640)
(36)
Exponential convergence
(∞)
(44)
DOF of BIE (Kelmanson)
DOF of present method

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

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

96. Free vibration of plate

97. Comparisons with FEM

98. Disclaimer (commercial code)
The concepts, methods, and examples using our software are for illustrative and educational purposes only. Our cooperation assumes no liability or responsibility to any person or company for direct or indirect damages resulting from the use of any information contained here.
inherent weakness ?
misinterpretation ? User 當自強

99. BEM trap ? Why engineers should learn mathematics ?
Well-posed ?
Existence ?
Unique ?
Mathematics versus Computation
Some examples

100. Numerical phenomena (Degenerate scale)
Commercial ode output ?
Error (%) of
torsional
rigidity a 5
125

101. Numerical and physical resonance
Numerical resonance
radiation
incident
wave

102. Numerical phenomena (Fictitious frequency)
t(a,0)
A story of NTU Ph.D. students

103. Numerical phenomena (Spurious eigensolution)

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

105. Conclusions
A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve BVPs with arbitrary circular holes and/or inclusions.
Numerical results agree well with available exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for only few terms of Fourier series.

106. Conclusions Free of boundary-layer effect Free of singular integrals
Well posed
Exponetial convergence
Mesh-free approach

107. Thanks for your kind attentions.
The End
Thanks for your kind attentions.
Your comments will be highly appreciated.
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