1. 國立臺灣海洋大學河海工程研究所結構組 碩士班畢業論文口試1
A study of free terms and rigid body modes
in the dual BEM
對偶邊界元素法中自由項與剛體運動項之研究
指導教授：陳正宗
研 究 生：陳韋誌
中華民國九十年六月十五日AM10:40~12:00
河工二館307室

2. The structure of thesis2
The structure of thesis
This thesis
Laplace equation、 Navier equation
(2-D、3-D)
Free term
[Chap 2、3]
Jump term
[Chap 2]
Rigid body mode
+ complementary sol.
[Chap 4]
Closed-form
2. Series-form
1. Closed-form
1-D rod
1-D beam
2-D circular membrane
Bump- contour
approach
2. Limiting approach
1. Bump-contour approach

3. 3
PART I : Free terms
The dual boundary integral equations for 2-D problem with the interior point x
Singular integral equation
Free terms
Hypersingular integral equation

4. The methods of deriving the free terms using BEM4
The methods of deriving the free terms using BEM
Method (1) : The limiting process B '
The DBIE for 2-D problem with the domain x B '
The null-field integral equations for 2-D problem

5. 5 The methods of deriving the free terms using BEM
Method (2) : The bump-contour approach
The DBIE with the domain point x
for 2-D problem
The null-field integral equations
for 2-D problem

6. 6 The methods of deriving the free terms using BEM
Method (2) : The bump-contour approach
The DBIE with the domain point x
for 3-D problem

7. 7
The steps of using bump-contour approach for Laplace problem in the dual BEM
Four kernel functions
the unknown density functions：
Using Taylor’s series expansion to find
1. Deriving the free terms with a smooth boundary
2. Deriving the jump terms for a general boundary

8. The explicit form of kernels for Laplace problem8
The explicit form of kernels for Laplace problem
The governing equation：
The explicit form of kernels：
Kernel function
2-D case
3-D case
Remark

9. 9 B '
Method (1)
Method (2)

10. 10
Free terms of different kinds of potential across smooth boundary for the Laplace problem
Kernel function
Density function -t u
Free term
Method (1) (2-D)
no jump
Method (2) (2-D)
Method (1) (3-D)
Method (2) (3-D)

11. 11
The steps of using bump-contour approach for elasticity problem in the dual BEM
Four kernel functions
1. Taylor’s series expansion to find
2. Using the linear elastic property to find
1. Deriving the free terms with a smooth boundary
2. Deriving the jump terms for a general boundary

12. The relationship between displacement and traction field12
The relationship between displacement and traction field
G.E. :
The strain-displacement relations：
The Hook’s law：
Cauchy formula：
(traction and stress)
We can obtain the properties on a smooth boundary : OK OK OK

13. 13
Free terms of different kinds of potential across smooth boundary 2-D elasticity problem
Kernel function
Density function
Free term
(2-D)
(Navier)
i=k=1
no jump
i=2, k=1
i=1, k=2
i=k=2
(Laplace)

14. 14
Free terms of different kinds of potential across smooth boundary for 3-D elasticity problem
Kernel function
Density function
Free term
(3-D) (Navier)
i=k=1
 no jump
i=2, k=1  0
i=3, k=1
i=1, k=2
i=k=2

15. 15
Free terms of different kinds of potential across smooth boundary for 3-D elasticity problem
Kernel function
Density function
Free term
(3-D) (Navier)
i=3, k=2
no jump
i=1, k=3
i=2, k=3
i=k=3
(3-D)
(Laplace)

16. 16 PART II : Jump terms The limiting approach in conjunction with
the degenerate kernels
Domain

17. 17 The limiting approach in conjunction with the degenerate kernels
Hypersingular integral equation
Jump terms
singular integral equation

18. 18 ρ R θ

19. in the degenerate kernels [1]19
Symbol and series form
in the degenerate kernels [1]

20. in the degenerate kernels [2]20
Series form
in the degenerate kernels [2]

21. 21 The contour of degenerate kernels for 2-D Laplace equation
U kernel
T kernel
L kernel
M kernel
Closed-form
Series-form

22. 22 converge The free terms and jump terms by using the degenerate
kernel in conjunction with the limiting approach[1]
converge
By using the generalized functions :
where
are the high order terms with a finite value.

23. 23 diverge The free terms and jump terms by using the degenerate
kernel in conjunction with the limiting approach[2]
By using the generalized functions :
By using the generalized functions :
diverge
where
are the high order terms with a finite value.

24. 24 The new definition of Hadamard principal value equivalence
The generalized definition of the Hadamard principal value :
equivalence
The finite part can be exacted.

25. 25
Jump terms of different kinds of potentials across smooth boundary for the Laplace problem
Kernel function
Density function -t u
Jump term
(2-D)
Method (1)
no jump
Method (2)
(3-D)

26. PART III : Rigid body mode26
PART III : Rigid body mode
UT formulation :
Neumann problem
Dirichlet problem
Non-unique solution
Physically realizable
Non-unique solution
Mathematically realizable
Rigid body mode
Spurious mode

27. The governing equation & fundamental solution27
The governing equation & fundamental solution
G.E.
Fundamental Solution
1-D rod
1-D beam
2-D circular membrane

28. The degenerate scale problem for a rod28
The degenerate scale problem for a rod
UT formulation :
Degenerate scale problem
A new degenerate scale :
Using L’Hopital’s rule

29. The degenerate scale problem for a beam29
The degenerate scale problem for a beam
UT formulation :
A new degenerate scale :
Degenerate scale problem

30. Using L’Hopital’s rule30
Using L’Hopital’s rule

31. 31 The degenerate scale problem for a circular membrane
The original eigenvalues are
Degenerate scale problem
R=1
Numerical analysis

32. Overcome the degenerate scale problem for a circular membrane32
Overcome the degenerate scale problem for a circular membrane
The eigenvalues are
A new degenerate scale
lnR+c=0

33. The stiffness using SVD technique33
The stiffness using SVD technique
Employing the SVD technique

34. 34 Fredholm’s alternative theorem For solving an algebraic system :
If H is a singular matrix, and u has a nontrival solution.
alternative theorem
Homogeneous sol.
where
the transpose conjugate matrix of H
if H is real

35. Fredholm’s alternative theorem and SVD updating technique35
Fredholm’s alternative theorem and SVD updating technique
UT formulation :
For Dirichlet problem
If U is a singular real matrix,
Fredholm’s alternative theorem
Spurious mode

36. For Dirichlet problem :36
A special case for a rod using SVD updating technique and Fredholm’s alternative theorem
For Dirichlet problem :
The spurious mode

37. 37
A special case for a rod using SVD updating technique and Fredholm’s alternative theorem
For Neumann problem
The rigid body mode
The spurious mode &
rigid body mode

38. 38
A special case for a beam using SVD updating technique and Fredholm’s alternative theorem
For Dirichlet problem
The spurious mode

39. 39
A special case for a beam using SVD updating technique and Fredholm’s alternative theorem
For Neumann problem
The rigid body mode

40. SVD updating technique :40
A circular membrane using SVD updating technique and Fredholm’s alternative theorem
SVD updating technique :
The spurious mode &
rigid body mode

41. 41 Linear algebraic system Using SVD technique, we can find by or
UT formulation :
Generalized coordinate
Based on Fredholm’s alternative theorem

42. 42 Linear algebraic system is singular matrix
By adding a matrix to [U]
Matrix level
We can deal with singular problem by UT formulation.
The unique solution is preserved.

43. 43
Conclusions
1. The free terms are contributed one half from L kernel and
the other half from M kernel by bump-contour approach
for 2-D potential problem.
2. The free terms are contributed one-third from L kernel
and two-thirds from M kernel by bump-contour
approach for 3-D potential problem.
3. The definition of Hadamard principal value can be
generalized,i.e.,

44. 44
Conclusions
4. The free terms for 2-D and 3-D elasticity problems are
derived by bump-contour approach .
5. The same stiffness can be derived no matter what the
rigid body mode and complementary solution are.
6. Based on the Fredholm’s alternative theorem and SVD
updating technique, the rigid body mode and spurious
mode can be obtained.

45. Further research 45 1.How to solve the corner problem and define the
normal vector on the boundary deserve research.
2. The jump terms by using the limiting approach in
conjunction with the degenerate kernels for 3-D
Laplace problem need to study.
3. How to eliminate the rigid body mode for the potential
and elasticity problems deserve research.
4. Based on Fredholm’s alternative theorem, the
degenerate boundary problem will only be solved by UT
formulation.