1. Study on the degenerate problems in BEM
邊界元素法中退化問題之探討
指導教授：陳正宗 教授
研究生：林書睿
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
時間: 5/21, 2002
地點: 河工二館307室

2. Mathematical essence—rank deficiency
Motivation
Four pitfalls in BEM
Why numerical instability occurs in BEM ?
(1) degenerate scale
(2) degenerate boundary
(3) fictitious frequency
Why spurious eigenvalues appear ?
(4) true and spurious eigenvalues
Mathematical essence—rank deficiency
(How to deal with ?)

3. Outlines Degenerate scale for torsion bar problems
Degenerate boundary problems
True and spurious eigensolution for interior eigenproblem
Fictitious frequency for exterior acoustics
Conclusions and further research

4. Four pitfalls in BEM 1.Degenerate scale for torsion bar problems
2.Degenerate boundary problems
3.True and spurious eigensolution for interior eigenproblem
4.Fictitious frequency for exterior acoustics

5. The degenerate scale for torsion bar using BEM
Error (%) of
torsional
rigidity
125 5 a
Previous approach : Try and error on a
Present approach : Only one trial

6. Determination of the degenerate scale by trial and error
Direct searching for the degenerate scale
Trial and error---detecting zero singular value by using SVD
[Lin (2000) and Lee (2001)]

7. Degenerate scale for torsion bar problems with arbitrary cross sections
An analytical way to determine the degenerate scale
The existence of degenerate scale for the two-dimensional Laplace problem xd x sd s Bd B

8. where is boundary density function
Degenerate scale for torsion bar problems with arbitrary cross sections
where is boundary density function
Mapping
properties
Expansion ratio , where

9. Adding a rigid body term c in the fundamental solution
For arbitrary cross section, expansion ratio is

10. Original degenerate scale
Degenerate scale for torsion bar problems with arbitrary cross sections
0.184
0.50
Unregularized
Regularized c=1.0
Shifting
Normal scale
Original degenerate scale
New degenerate scale

11. Original degenerate scale
Degenerate scale for torsion bar problems with arbitrary cross sections
Unregularized
Regularized
Shifting
0.85
0.31
( ): exact solution of the degenerate scale 2a
0.4
0.8
1.2
1.6 2 a
0.1
0.2
0.3
0.5 1 S q u r e c o s t i n
Conventional BEM (UT formulation)
Adding a rigid body term (c=1.0)
Normal scale
Original degenerate scale
New degenerate scale s

12. Determination of the degenerate scale for the two-dimensional Laplace problems
Cross
Section
Normal scale
Torsional rigidity
Reference equation
, where , x on B, .
1.4480
1.4509
(N.A.)
(N.A.)
(6.1538)
Expansion ratio
(0.5)
(0.5)
(N.A.)
(N.A.)
(0.85)
Degenerate scale
R= (1.0)
= (2.0)
a= (N.A.)
h= (N.A.)
a= (0.85) 2a R a b h
Note: Data in parentheses are exact solutions.
Data marked in the shadow area are derived by using the polar coordinate.

13. Three regularization techniques to deal with degenerate scale problems in BEM
Hypersingular formulation (LM equation)
Adding a rigid body term (U*(s,x)=U(s,x)+c)
CHEEF concept

14. Regularization techniques are not necessary.
Numerical results
Normal scale
( =3.0, =1.0)
Degenerate scale
( =1.5, =0.5)
Analytical
solution
8.4823
0.5301
2.249
1.174
U T
Conventional BEM
(3.30%)
(268.10%)
2.266 (0.76%)
(75.21%)
L M
formulation
(9.22%)
(2.31%)
Add a rigid
body term
c=1.0
(2.26%)
1.1721(0.19%)
c=2.0
(2.36%)
(0.17%)
CHEEF concept
(6.53%)
CHEEF POINT (2.0, 2.0)
(0.18%)
CHEEF POINT
(5.0, 5.0)
cross
section 2a
Torsion
rigidity
Square
Ellipse
method
(a=1.0)
(a=0.85)
Regularization techniques are not necessary.
Regularization techniques are not necessary.
Note: data in parentheses denote error.

15. Regularization techniques are not necessary.
Numerical results
Normal scale
h=3.0
Degenerate scale
h=2.07
Analytical
solution
3.1177
0.7067
0.9609
U T
Conventional BEM
(2.09%)
(57.08%)
(0.83%)
(94.73%)
L M
formulation
(3.25%)
(0.82%)
Add a rigid
body term
c=1.0
(0.45%)
(2.78%)
c=2.0
(0.61%)
(2.84%)
CHEEF concept
(5.46%)
CHEEF POINT (15.0, 15.0)
(3.51%)
CHEEF POINT (20.0, 20.0)
cross
section b a h
Torsion
rigidity
Keyway
Triangle
(a=2.0)
(a=1.05)
method
Regularization techniques are not necessary.
Regularization techniques are not necessary.
Note: data in parentheses denote error.

16. Four pitfalls in BEM 1.Degenerate scale for torsion bar problems
2.Degenerate boundary problems
3.True and spurious eigensolution for interior eigenproblem
4.Fictitious frequency for exterior acoustics

17. Degenerate boundary problems
Multi-domain BEM
u=0
r=1
interface
Subdomain 1
Subdomain 2
Dual BEM

18. Singular Value Decomposition
Conventional BEM in conjunction with SVD
Singular Value Decomposition
Rank deficiency originates from two sources:
(1). Degenerate boundary
(2). Nontrivial eigensolution
Nd=5
Nd=4
Nd=5

19. UT BEM + SVD (Present method) Multi-domain BEM Dual BEM versus k
Determinant versus k
Dual BEM
Determinant versus k

20. Two sources of rank deficiency (k=3.09)
Nd=5
None trivial sol.
Degenerate boundary
Eigensolution

21. k=3.14
k=3.82
k=4.48
UT BEM+SVD
k=3.09
k=3.84
k=4.50
FEM (ABAQUS)

22. Four pitfalls in BEM 1.Degenerate scale for torsion bar problems
2.Degenerate boundary problems
3.True and spurious eigensolution for interior eigenproblem
4.Fictitious frequency for exterior acoustics

23. True and spurious eigensolution for Interior eigenproblem
Complex-valued BEM true eigenvalues
Spurious eigenvalues?
MRM
Real-part BEM
Imaginary-part BEM

24. SVD structure for the influence matrices (true)
k=kt
Dirichlet problem
Neumann problem

25. SVD structure for the influence matrices (spurious)
k=ks
Singular formulation
Hypersingular formulation

26. True eigensolution for interior eigenproblems
Real-part BEM
SVD updating technique
Dirichlet problem
Neumann problem
20 constant elements

27. True eigensolution for interior eigenproblems
Imaginary-part BEM
SVD updating technique
Dirichlet problem
Neumann problem
8 constant elements

28. Spurious eigensolution for interior eigenproblems
Real-part BEM
The Fredholm alternative theorem and SVD updating technique
Singular formulation
Hypersingular formulation
20 constant elements

29. Spurious eigensolution for interior eigenproblem
Imaginary-part BEM
The Fredholm alternative theorem and SVD updating technique
Singular formulation
Hypersingular formulation
8 constant elements

30. Four pitfalls in BEM 1.Degenerate scale for torsion bar problems
2.Degenerate boundary problems
3.True and spurious eigensolution for interior eigenproblem
4.Fictitious frequency for exterior acoustics
t(a,0) ka

31. Fictitious frequency for exterior acoustics
Mathematical structure for updating matrix
Source of numerical instability---zero division by zero
A criterion to check the validity of CHIEF points
Numerical example

32. Mathematical structure for updating matrix
Proof
The subscript i denotes the use of interior degenerate kernel
for exterior problem

33. Mathematical structure for updating matrix
For the Dirichlet eigenproblem
or symmetry
or transpose symmetry

34. Mathematical structure for updating matrix
Multiplicity:
True boundary mode for
the Dirichlet eigenproblem
Fictitious boundary mode

35. Source of numerical instability---zero division by zero
For the Dirichlet problem
By pre-multiplying the regular modes
Solvable

36. Source of numerical instability---zero division by zero
By pre-multiplying the fictitious modes
Unsolvable

37. A criterion to check the validity of CHIEF points
Adding CHIEF points
New constraints

38. A criterion to check the validity of CHIEF points
The selected P CHIEF points are valid
(No change)
For the Neumann problem

39. Numerical example
: valid
Real-part
Imaginary-part
: invalid

40. Conclusions and further research
A more efficient technique was proposed to directly determine the degenerate scale since only one normal scale needs to be computed.
The conventional BEM in conjunction with SVD was applied to deal with rank-deficiency (degenerate boundary and nontrivial eigensolution) for the degenerate boundary eigenproblem.
By using the Fredholm alternative theorem and SVD techniques in conjunction with the dual formulations, the true and spurious eigenvalues in the complex-valued formulation,
the real-part, the imaginary-part BEMs and MRM were sorted out successfully.
In order to overcome the rank-deficiency problem due to fictitious frequency, the CHIEF method was reformulated in a unified manner by using the Fredholm alternative theorem and SVD technique.

41. Further research
Although the degenerate scale occurs in the Dirichlet problem of simply two-dimensional Laplace problems by using the BEM, there is no proof of the occurrence of degenerate scale for the problem with the mixed-type boundary condition.
The main drawback of the imaginary-part BEM seems to produce ill-conditioned matrices. While this is sometimes the case, it is hoped that further research can alleviate the drawback.
Whether the spurious (fictitious) modes in the UT and LM formulations are the same or not needs further investigation.