1. Boundary integral formulation and boundary element analysis for multiply-connected Helmholtz problems
地點: 河工二館會議室
時間:
口試生: 劉立偉 同學
指導教授: 陳正宗 教授 1

2. Outlines Motivation Methods of solution
Singular & hypersingular BIEs (BEMs)
Null-field BIEs (BEMs)
Fictitious BIEs (BEMs)
CHIEF method
SVD updating techniques
Numerical experiments
Conclusions & further research 2

3. Motivation
Kitahara (1985) Point out that spurious eigenvalues always appear in BEM
Chang (1999) Numerical verification
Lin (2001) 1. Discrete system
2. Burton & Miller method
Liu (2002) Continuous system
2. CHIEF method
3. SVD updating techniques
4. Truly multiply-connected problem 3

4. Singular and hypersingular BIEs (direct BIEs)
Dual boundary integral equations
Singular BIE
(UT formulation)
Hypersingular BIE
(LM formulation)
Degenerate kernel functions
R<ρ
ρ<R ρ R θ
Aleph
s =(R,)
x=(,) 4

5. Eigenproblem of an annular domain
s =(R,)
x=(,)
Eigenequation:
True eigenequation
Spurious eigenequation
by using singular BIE
Eigenequation:
by using hypersingular BIE 5

6. Singular and hypersingular BEMs (direct BEMs)1
r1=0.5 m
r2=2.0 m k 6

7. Eigenequations for annular domains with different boundary conditions
Spurious eigenvalues depend on formulations (UT or LM)
True eigenvalues depend on boundary conditions (Dirichlet, Neumann or Robin ) 7

8. Null-field BIEs (direct BIEs)
UT formulation
Spurious eigenequation
True eigenequation
LM formulation 
s =(R,)
x=(,)
In order to avoid the singularity, the null-field BIEs are used. The collocation points are located in the out of the domain instead of boundary.
The results of null-field BIEs are different in spurious eigenequations.
Spurious eigenequation
True eigenequation 8

9. Null-field BEMs (direct BEMs)1
1=0.45 m
2=2.2 m s s s k 9

10. Fictitious BIEs (indirect BIEs)
single-layer potential approach
double-layer potential approach
s =(R,)
x=(,) 10

11. Fictitious BEMs (indirect BEMs)
r1=0.5 m
r2=2.0 m   
1=0.45 m
2=2.2 m s s s
r’1=0.4 m
r’2=2.2 m 11

12. CHIEF method CHEEF point (Chen et al. 2001 JASA)  CHIEF point
(Chen et al JSV)
trivial (identity) 12

13. CHIEF or CHEEF valid ?
CHIEF valid
CHEEF invalid 1 s k 13

14. Two CHIEF points for spurious eigenvalues of multiplicity two1 s k 14

15. CHIEF flowchart Filtering technique for spurious eigenvalue ks Failure
Jn(k1)=0
sin n(1- 3)=0
Add a CHIEF point (r1 , 1)
No ( exterior point)
r1< r1
Yes (interior point)
Failure
Yes No
Overcome the spurious eigenvalue of multiplicity one
Add another interior point (r3 , 3)
Jn(k3)=0
Success (overcome the spurious eigenvalues of multiplicity two) 15

16. SVD updating techniques: Direct BEMs
SVD updating term for true eigenvalue
True eigenvalues 1 s s s k 16

17. SVD updating techniques: Direct BEMs
SVD updating documents for spurious eigenvalue
Spurious eigenvalues 1 s s k 17

18. SVD updating techniques: Indirect BEMs
SVD updating document for true eigenvalues
True eigenvalues 1 k 18

19. SVD updating techniques: Indirect BEMs
SVD updating term for spurious eigenvalues
Spurious eigenvalues s s 19

20. Numerical experiments
A circular domain with two unequal holes 1
[U]
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 - . 8 7 6 5 4 3 2 1 +
BEM
FEM
mode 1
mode 2
mode 3
mode 4
mode 5
k=4.82
k=4.790
k=4.801
k=6.619
k=6.72
k=7.82
k=6.634
k=7.797 k1 k3 k2 s k4 k5 k 2
[U] k
Each mode shape of BEM can be obtained by superimposing the other two mode shapes of FEM.
The spurious eigenvalues which depends on the right inner boundary will appears first. 20

21. Spurious boundary mode F1
ei1q
ei0q
ei0q
ei1q 21

22. Numerical experiments
A circular domain with two equal holes
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 - . 8 7 6 5 4 3 2 1 +
BEM
FEM
mode 1
mode 2
mode 3
mode 4
mode 5
k=4.50
k=6.37
k=7.16
k=4.453
k=4.512
k=6.267
k=6.269
k=6.930
Nagaya & Poltorak 1989 22

23. Spurious boundary modes F1 and F2
ei1q
ei0q
ei1q
ei0q 23

24. Conclusions & further research
Spurious eigenvalues depend on the inner boundaries, inner collocation points and inner fictitious boundaries by using the singular and hypersingular BIEs (BEMs), the null-field BIEs (BEMs) and fictitious BIEs (BEMs), respectively.
The criterion of CHIEF method were discussed.
SVD updating techniques were successful employed to deal with the spurious eigenvalues in the direct and indirect BEMs.
SVD updating terms (true eigenvalues)
SVD updating documents (spurious eigenvalues)
Direct BEMs:
SVD updating documents (true eigenvalues)
SVD updating terms (spurious eigenvalues)
Indirect BEMs: 24

25. Further research:
The ill-posed problem arises from the singularity free approaches. Hence, it is deserves further study to overcome the ill-posed problem in depth.
The extension to exterior acoustics for multiple radiators and scatters using the similar algorithm can be conducted to examine the occurrence of the fictitious frequency.
Whether the spurious modes in the UT and LM formulations are the same or not needs further investigation. 25