1. 報 告 者：蕭嘉俊 指導教授：陳正宗 博士 呂學育 博士 日 期：2005/6/16 地 點：河工二館307
A semi-analytical approach for Stokes flow and plate problems with circular boundaries
報 告 者：蕭嘉俊
指導教授：陳正宗 博士
呂學育 博士
日 期：2005/6/16
地 點：河工二館307

2. Outlines Introduction Direct boundary integral equation method
Indirect boundary integral equation method
Numerical examples
Conclusions

3. Outlines Introduction Direct boundary integral equation method
Indirect boundary integral equation method
Numerical examples
Conclusions

4. Engineering problems with arbitrary boundaries
Degenerate boundary
(Chebyshev polynomials)
Elliptic boundary
(Mathieu function)
Circular boundary
(Fourier series)
Straight boundary
(Legendre polynomials)
Degenerate boundary
Circular boundary

5. Fictitious boundary method
Motivation
BEM/BIEM
Singular
Improper integral
Desingular (Regular)
Direct
Indirect (Interior)
Null-field approach
Contour
Fictitious boundary method
Limiting process
: collocation point
Fictitious boundary

6. Fictitious boundary method
Motivation
BEM/BIEM
Singular
Improper integral
Desingular (Regular)
Direct
Indirect (Exterior)
Contour
Limiting process
Null-field
Fictitious boundary method
: collocation point
Fictitious boundary
Degenerate kernel
Field point
Present approach
ill-posed
CPV & HPV

7. Literature review Torsion bar with circular holes Laplace problems
Steady state heat conduction of tube
(Shen)
Electromagnetic wave
Engineering problems
Helmholtz problems
Membrane vibration
Water wave and Acoustic problems
(Chen)
Plane elasticity：Airy stress function
Biharmonic problems
Solid mechanics：plate problem
Fluid mechanics：Stokes flow
(Hsiao)

8. 2. Solid mechanics (Plate problem):
Literature review
Plane elasticity:
Jeffery (1921), Howland and Knight (1939),
Green (1940) and Ling (1948)
2. Solid mechanics (Plate problem):
Bird and Steele (1991)
3. Viscous flow (Stokes Flow):
Kamal (1966), DiPrima and Stuart (1972), Mills (1977) and Ingham and Kelmanson (1984)

9. Purpose
A semi-analytical approach in conjunction with Fourier series, degenerate kernels and adaptive observer system is extended to biharmonic problems.
Advantages：
1. Mesh free.
2. Accurate.
3. Free of CPV and HPV.

10. Outlines Introduction Direct boundary integral equation method
Indirect boundary integral equation method
Numerical examples
Conclusions
Further research

11. Problem statement Governing equation: Essential boundary condition:
:lateral displacement,
:slope
Natural boundary condition:
: moment,
: shear force

12. Direct boundary integral equations
BIEs are derived from the Rayleigh-Green identity：
BIE for the domain point
Null-field integral equation
Interior problem

13. Boundary integral equation for the domain point
Displacement
Slope
Displacement
Moment
Displacement
Shear force
: Poisson ratio

14. Null-field integral equation
Displacement
Slope
Displacement
Moment
Displacement
Shear force
: Poisson ratio

15. Relation among the kernels
Continuous (Separable form of degenerate kernel)
is the fundamental solution, which satisfies

16. Degenerate kernels x x r s O

17. Fourier series
The boundary densities are expanded in terms of Fourier series:
M: truncating terms of Fourier series

18. Adaptive observer system
: Collocation point
: Radius of the jth circle
: Origin of the jth circle
: Boundary of the jth circle

19. Vector decomposition for normal derivative
True normal direction
Tangential direction
Radial direction
: normal derivative
: tangential derivative

20. Linear algebraic system
Null-field integral equations for and formulations
H: number of circular boundaries
Collocation circle index
Routing circle index

21. Analytical Numerical Flowchart of the present method Potential
Degenerate kernels
Fourier series
Analytical
Collocation method
Matching B.C.
Adaptive observer system
Numerical
Linear algebraic system
Potential
Fourier coefficients
BIE for domain point

22. Stokes flow problems (Eccentric case)
Governing equation:
Essential boundary condition: on on
(Stationary)
: stream function
: normal derivative of stream function

23. Linear algebraic system
Unknown
Given
Unknown constant

24. Constraint equation
Vorticity:
Constraint:

25. Trapezoid integral Inner circle Outer circle Vector decomposition
Numerical
Analytical
Series sum
Trapezoid integral

26. Linear algebraic augmented system
Unknown

27. Outlines Introduction Direct boundary integral equation method
Indirect boundary integral equation method
Numerical examples
Conclusions
Further research

28. Indirect boundary integral equation
Indirect boundary integral equation is originated from the physical concept of superposition
: Vorticity
: single layer fictitious densities
: double layer fictitious densities

29. Stokes flow problems (Eccentric case)
Governing equation:
Essential boundary condition: on on
(Stationary)
: stream function
: normal derivative of stream function

30. Linear algebraic system
Unknown
Given
Unknown constant
:Collocation point

31. Constraint equation
Vorticity:
Constraint:

32. Trapezoid integral Inner circle Outer circle Vector decomposition
Numerical
Analytical
Series sum
Trapezoid integral

33. Linear algebraic augmented system
Unknown

34. Outlines Introduction Direct boundary integral equation method
Indirect boundary integral equation method
Numerical examples
Conclusions

35. Numerical examples Direct BIEM Indirect BIEM Plate problems
1. Annular plate
2. Circular plate with three circular holes
Stokes flow problems
1. Doubly-connected case
1. Simply-connected case
2. Doubly-connected case

36. Numerical examples Direct BIEM Indirect BIEM Plate problems
1. Annular plate
2. Circular plate with three circular holes
Stokes flow problems
1. Doubly-connected case
1. Simply-connected case
2. Doubly-connected case

37. Plate problems (Case 1) Exact solution： Geometric data: and
Essential boundary conditions:
and on
and on
Exact solution：

38. Contour plot of displacement
(No. of nodes=1,920, No. of elements=3,600)
Exact solution
FEM mesh
Present method (M=10)
FEM (ABAQUS)

39. Boundary densities for outer circle
Exact solution:
Exact solution:

40. Boundary densities for inner circle
Exact solution:
Exact solution:

41. Plate problems (Case 2) Geometric data: Essential boundary conditions:
and on
and on
and on
and on
(Bird & Steele, 1991)

42. Contour plot of displacement
Present method (N=21)
Present method (N=41)
Present method (N=61)
Present method (N=81)

43. 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)

44. Parseval sum for convergence

45. Parseval sum for convergence

46. Stokes flow problems (Case 1)
Boundary conditions:
Exact solution:
(Mills, 1977)
where

47. Contour plot of Streamline
Exact solution (Mills, 1977)
Analytical solution, P=100 (Wu, 2004)
Present method (N=161)
(Null-field + collocation)
Exact solution:
(Mills, 1977)
(Closed-form solution)
Analytical solution:
(Wu, 2004)
(Null-field equation solution)
(Trefftz solution)

48. Stokes flow problems (Case 2)
Governing equation:
Angular velocity:
Boundary conditions:
and on
and on
(Stationary)
Eccentricity:

49. Comparison of stream function
Kelmanson & Ingham (BIE)
Analytical solution
Present method
(Direct BIEM)
(Indirect BIEM)
n=80
n=160
n=320
Limit
n→∞
0.0
0.1066
0.1062
0.1061
0.1060
(N=5)
0.1
0.1052
0.1048
0.1047
0.1046
(N=7)
0.2
0.1011
0.1006
0.1005
(N=7)
0.3
0.0944
0.0939
0.0938
(N=7)
0.4
0.0854
0.0850
0.0848
0.0846
(N=9)
0.5
0.0748
0.0740
0.0739
0.0738
(N=11)
0.6
0.0622
0.0615
0.0613
0.0612
0.0611
(N=17)
0.7
0.0484
0.0477
0.0474
0.0472
(N=17)
0.8
0.0347
0.0332
0.0326
0.0322
(N=21)
0.9
0.0191
0.0175
0.0168
0.0163
0.0164
(N=31)
n: number of boundary nodes
N: number of collocation points

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

51. 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)

52. Contour plot of Streamline for
Present method (N=21)
Present method (N=41)
Present method (N=81)
Kelmanson (Q=0.0740, n=160)

53. Contour plot of vorticity for
Present method (N=21)
Present method (N=41)
Kelmanson (n=160)

54. Contour plot of vorticity for
Present method (N=21)
Present method (N=41)
Kelmanson (n=160)

55. Numerical examples Direct BIEM Indirect BIEM Plate problems
1. Annular plate
2. Circular plate with three circular holes
Stokes flow problems
1. Doubly-connected case
1. Simply-connected case
2. Doubly-connected case

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

57. Comparison of n: number of boundary nodes
Kelmanson & Ingham
Analytical solution
Present method
(Direct BIEM)
(Indirect BIEM)
n=80
n=160
n=320
Limit
n→∞
0.0
0.1066
0.1062
0.1061
0.1060
(N=5)
0.1
0.1052
0.1048
0.1047
0.1046
(N=7)
0.2
0.1011
0.1006
0.1005
(N=7)
0.3
0.0944
0.0939
0.0938
(N=7)
0.4
0.0854
0.0850
0.0848
0.0846
(N=9)
0.5
0.0748
0.0740
0.0739
0.0738
(N=11)
0.6
0.0622
0.0615
0.0613
0.0612
0.0611
(N=17)
0.7
0.0484
0.0477
0.0474
0.0472
(N=17)
0.8
0.0347
0.0332
0.0326
0.0322
(N=21)
0.9
0.0191
0.0175
0.0168
0.0163
0.0164
(N=31)
n: number of boundary nodes
N: number of collocation points

58. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,222
No. of elements=2,244)
FEM result

59. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,230
No. of elements=2,260)
FEM result

60. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,245
No. of elements=2,290)
FEM result

61. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,257
No. of elements=2,314)
FEM result

62. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,397
No. of elements=3,228)
FEM result

63. Contour plot of streamline for
Separation point
Reattachment point
Present method (N=161)
FEM mesh (No. of nodes=1,734
No. of elements=3,218)
FEM result

64. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,735
No. of elements=3,220)
FEM result

65. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,635
No. of elements=3,010)
FEM result

66. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,196
No. of elements=2,162)
FEM result

67. Contour plot of streamline for
Present method (N=161)
FEM mesh (No. of nodes=1,254
No. of elements=2,268)
FEM result

68. Degree of separation and reattachment point versus eccentricity
Separation point
Reattachment point

69. Outlines Introduction Direct boundary integral equation method
Indirect boundary integral equation method
Numerical examples
Conclusions

70. Conclusions
Successful extension to biharmonic problems with circular boundaries.
Both direct and indirect BIEMs were presented to solve biharmonic problems.
Good agreement was obtained after compared with previous results, exact solution and ABAQUS data.
Stream function and vorticity were found to be independent of Poisson ratio as we predicted.

71. Conclusions
Boundary layer effect in the present method was also examined.
A general-purpose program for solving the biharmonic problems with circular boundaries was developed successfully.
Once engineering problems satisfy the biharmonic equation with circular boundaries, our present method can be used.

72. Conclusions Direct BIEM Indirect BIEM
Null-field integral equation available?
Direct BIEM
Indirect BIEM
Null-field !
No !

73. Thank you for your kind attention!
The end
Thank you for your kind attention!

74. Further research Viewpoint Finished Further research Direct BIEM
& formulation
Indirect BIEM
Single & double layer potentials
Triple & Quadruple layer potentials
Post processing
Lateral displacement
Stress or moment diagram
Boundary condition
Essential boundary condition
Natural boundary condition...etc.

75. Further research Viewpoint Finished Further research Degenerate scale
Simply-connected problem had finished by Wu (2004)
Doubly-connected & multiply-connected problems
Shape of domain
Circular domain
Arbitrary domain

76. : deflection of the circular plate General form : flexure rigidity
uniform pressure a B
w=constant
Governing equation:
Boundary condition:
Splitting method
Governing equation:
Boundary condition:
: deflection of the circular plate
General form
: flexure rigidity
Governing equation:
: uniform distributed load
Boundary condition:
: domain of interest