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1
On spurious eigenvalues of doubly-connected membrane
Reporter: I. L. Chen Date: 07. 29. 2008
Department of Naval Architecture, National Kaohsiung Institute of Marine Technology
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2
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
![Page 3: 1 On spurious eigenvalues of doubly-connected membrane Reporter: I. L. Chen Date: 07. 29. 2008 Department of Naval Architecture, National Kaohsiung Institute.](https://reader031.fdocuments.in/reader031/viewer/2022032800/56649d3e5503460f94a16fe5/html5/thumbnails/3.jpg)
3
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
![Page 4: 1 On spurious eigenvalues of doubly-connected membrane Reporter: I. L. Chen Date: 07. 29. 2008 Department of Naval Architecture, National Kaohsiung Institute.](https://reader031.fdocuments.in/reader031/viewer/2022032800/56649d3e5503460f94a16fe5/html5/thumbnails/4.jpg)
Spurious eignesolutions in BIE (BEM and NBIE)
Real Imaginary Complex
Saving CPU time Yes Yes No
Spurious eigenvalues Appear Appear No
Complex
Spurious eigenvalues Appear
Simply-connected problem
Multiply-connected problem
(Fundamental solution))()(),( 00 krYkriJxsU
4
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5
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
![Page 6: 1 On spurious eigenvalues of doubly-connected membrane Reporter: I. L. Chen Date: 07. 29. 2008 Department of Naval Architecture, National Kaohsiung Institute.](https://reader031.fdocuments.in/reader031/viewer/2022032800/56649d3e5503460f94a16fe5/html5/thumbnails/6.jpg)
Governing equation
Governing equation
0)()( 22 xuk
Fundamental solution
)()(),( 00 krYkriJxsU
6
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Multiply-connected problem
01 u
02 u
ID
01 u
a
be
a = 2.0 mb = 0.5 me=0.0~ 1.0 mBoundary condition:Outer circle:
Inner circle
02 u
2B
1B
01 u
7
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8
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
![Page 9: 1 On spurious eigenvalues of doubly-connected membrane Reporter: I. L. Chen Date: 07. 29. 2008 Department of Naval Architecture, National Kaohsiung Institute.](https://reader031.fdocuments.in/reader031/viewer/2022032800/56649d3e5503460f94a16fe5/html5/thumbnails/9.jpg)
Interior problem Exterior problem
cD
D D
x
xx
xcD
x x
Degenerate (separate) formDegenerate (separate) form
DxsdBstxsUsdBsuxsTxuBB
),()(),()()(),()(2
BxsdBstxsUVPRsdBsuxsTVPCxuBB
),()(),(...)()(),(...)(
Bc
BBDxsdBstxsUsdBsuxsT ),()(),()()(),(0
B
Boundary integral equation and null-field integral equation
s
s
n
sust
n
xsUxsT
krHixsU
)()(
),(),(
2
)(),(
)1(0
9
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Degenerate kernel and Fourier series
,,,2,1,,)sincos()(1
0 NkBsnbnaast kkn
kn
kn
kk
,,,2,1,,)sincos()(1
0 NkBsnqnppsu kkn
kn
kn
kk
s
Ox
R
kth circularboundary
cosnθ, sinnθboundary distributions
eU
x
iU
Expand fundamental solution by using degenerate kernel
Expand boundary densities by using Fourier series
,)),(cos()()()(2
),(
,)),(cos()()()(2
),(,
),(
RmkRJkYkiJxsU
RmkJkRYkRiJxsU
xsU
nnnn
E
nnnn
I
10
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For the multiply-connected problem
1 1 1, 1,1
0
2 1 2, 2,2
0
1 1 1, 1,1
0
2 1 2, 2,0
0 ( , ) cos( ) sin( ) ( )
( , ) cos( ) sin( ) ( )
( , ) cos( ) sin( ) ( )
( , ) cos( ) sin( )
n nB
n
n nB
n
n nB
n
n nn
U s x a n b n dB s
U s x a n b n dB s
T s x p n q n dB s
T s x p n q n
2
1 1
( )
,
BdB s
x B
1B
2B
1x
11
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For the multiply-connected problem
1 2 1, 1,1
0
2 2 2, 2,2
0
1 2 1, 1,1
0
2 2 2, 2,0
0 ( , ) cos( ) sin( ) ( )
( , ) cos( ) sin( ) ( )
( , ) cos( ) sin( ) ( )
( , ) cos( ) sin( )
n nB
n
n nB
n
n nB
n
n nn
U s x a n b n dB s
U s x a n b n dB s
T s x p n q n dB s
T s x p n q n
2
2 2
( )
,
BdB s
x B
1B
2B
2x
12
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For the Dirichlet B.C., 021 uu
1 1 1, 1,1
0
2 1 2, 2,2
0
1 1
0 ( , ) cos( ) sin( ) ( )
( , ) cos( ) sin( ) ( )
,
n nB
n
n nB
n
U s x a n b n dB s
U s x a n b n dB s
x B
1 2 1, 1,1
0
2 2 2, 2,2
0
2 2
0 ( , ) cos( ) sin( ) ( )
( , ) cos( ) sin( ) ( )
,
n nB
n
n nB
n
U s x a n b n dB s
U s x a n b n dB s
x B
13
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SVD technique
,0][
,2
,2
,1
,1
n
n
n
n
b
a
b
a
A
H
n
HA
00
000
00
00
][ 2
1
14
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0 2 4 6 8
0
0.2
0.4
0.6
0.8
0 2 4 6 8
0
0.2
0.4
0.6
k
1
k
1
0)(0 kJk=4.86
k=7.74
0)(1 kJ
Minimum singular value of the annular circular membrane for fixed-fixed case using UT formulate
15
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Effect of the eccentricity e on the possible eigenvalues
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
e
kFormer five true eigenvalues
7.66
Former two spurious eigenvalues
4.86
16
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Eigenvalue of simply-connected problem
a
By using the null-field BIE,
the eigenequation is
True eigenmode is :
n
n
b
a
,where . 022 nn ba
cx
cx
For any point , we obtain the null-field response cx
,3,2,1,0,
0)sincos)(()]()([
n
nbnakaJkaYkaiJ nnnnn
17
0)( kaJ n
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1B
2B
2x
18
The existence of the spurious eigenvalue by boundary mode
.0
)sincos)](()()[(
)sincos)](()()[(
)(),()()(),(
,2,22
,1,12
1 2
nnnnnn
nnnnnn
B B
ii
nbnakbiJkbYkabJ
nbnakaiJkaYkaaJ
stxsUsdBstxsU
For the annular case with fix-fix B.C.
nnn
nnn
nn
akbHkabJ
kaHkaaJa
ba
,1)1(
)1(
,2
2,1
2,1
)()(
)()(
0
a b
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1B
2B
1x
19
The existence of the spurious eigenvalue by boundary mode
.0
)sincos)](()()[(
)sincos)](()()[(
)(),()()(),(
,2,22
,1,12
1 2
nnnnnn
nnnnnn
B B
ee
nbnakbiJkbYkbbJ
nbnakbiJkbYkaaJ
stxsUsdBstxsU
nnn
nnn
nn
akbHkbbJ
kbHkaaJa
ba
,1)1(
)1(
,2
2,1
2,1
)()(
)()(
0
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The eigenvalue of annular case with fix-fix B.C.
1,1)1(
)1(
,2
2,1
2,1
,)()(
)()(
0
BxakbHkabJ
kaHkaaJa
ba
nnn
nnn
nn
2,1)1(
)1(
,2 ,)()(
)()(Bxa
kbHkbbJ
kbHkaaJa n
nn
nnn
.0)]()()()([
,0)(
.0)]()()()()[(
)()(
)()(
)()(
)()()1(
)1(
)1(
)1(
kbYkaJkaYkbJ
kaJ
kbYkaJkaYkbJkaJ
kbHkbJ
kbHkaJ
kbHkaJ
kaHkaJ
nnnn
n
nnnnn
nn
nn
nn
nn
Spurious eigenequation
True eigenequation
20
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The eigenvalue of annular case with free-free B.C.
1B
2B
2x
21
.0
)sincos)](()()[(
)sincos)](()()[(
)(),()()(),(
,2,22
,1,12
1 2
nnnnnn
nnnnnn
B B
ii
nqnpkbJikbYkabJ
nqnpkaJikaYkaaJ
stxsTsdBstxsT
nnn
nnn
nn
pkbHkabJ
kaHkaaJp
qp
,1)1(
)1(
,2
2,1
2,1
)()(
)()(
0
a b
![Page 22: 1 On spurious eigenvalues of doubly-connected membrane Reporter: I. L. Chen Date: 07. 29. 2008 Department of Naval Architecture, National Kaohsiung Institute.](https://reader031.fdocuments.in/reader031/viewer/2022032800/56649d3e5503460f94a16fe5/html5/thumbnails/22.jpg)
1B
2B
1x
22
The existence of the spurious eigenvalue by boundary mode
.0
)sincos)](()()[(
)sincos)](()()[(
)(),()()(),(
,2,22
,1,12
1 2
nnnnnn
nnnnnn
B B
ee
nqnpkbiJkbYkbJb
nqnpkbiJkbYkaJa
stxsTsdBstxsT
nnn
nnn
nn
pkbHkbJb
kbHkaJap
qp
,1)1(
)1(
,2
2,1
2,1
)()(
)()(
0
22
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The eigenvalue of annular case with free-free B.C.
1,1)1(
)1(
,2
2,1
2,1
,)()(
)()(
0
BxpkbHkabJ
kaHkaaJp
qp
nnn
nnn
nn
2,1)1(
)1(
,2 ,)()(
)()(Bxp
kbHkbJb
kbHkaJap n
nn
nnn
.0)]()()()([
,0)(
.0)]()()()()[(
)()(
)()(
)()(
)()()1(
)1(
)1(
)1(
kbYkaJkaYkbJ
kaJ
kbYkaJkaYkbJkaJ
kbHkbJ
kbHkaJ
kbHkaJ
kaHkaJ
nnnn
n
nnnnn
nn
nn
nn
nn
Spurious eigenequation
True eigenequation
23
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24
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
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Minimum singular value of the annular circular membrane for fixed-fixed case using UT formulate
0 2 4 6 8
0
0.2
0.4
0.6
0.8
0 2 4 6 8
0
0.2
0.4
0.6
k
1
k
1
0)(0 kJk=4.86
k=7.74
0)(1 kJ
25
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Effect of the eccentricity e on the possible eigenvalues
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
e
kFormer five true eigenvalues
7.66
Former two spurious eigenvalues
4.86
26
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a b
Real part of Fourier coefficients for the first true boundary mode ( k =2.05, e = 0.0)
Boundary mode (true eigenvalue)
1 11 21 31 41
-1
-0.8
-0 .6
-0 .4
-0 .2
0
0.2
Fourier coefficients ID
t Outer boundary Inner boundary
27
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Boundary mode (spurious eigenvalue)
Dirichlet B.C. using UT formulate
a b
1 11 21 31 41
-0.4
0
0.4
0.8
1.2
Outer boundary
(trivial)
Inner boundary
Outer boundary
(trivial)
Inner boundary
Fourier coefficients ID
k=4.81
k=7.66
1 11 21 31 41
-0.2
0
0.2
0.4
0.6
28
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Boundary mode (spurious eigenvalue)
Neumann B.C. using UT formulation
0.00 10.00 20.00 30.00 40.00
-1.00
0.00
1.00
0.00 10.00 20.00 30.00 40.00
-1.00
0.00
1.00T kernel k=4.81 ( ) real-par
T kernel k=7.75 ( ) real-part )803.3(1J
)405.2(0J
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Boundary mode (spurious eigenvalue)
Neumann B.C. using LM formulate
0.00 10.00 20.00 30.00 40.00
-1.00
0.00
1.00
0.00 10.00 20.00 30.00 40.00
-0.40
0.00
0.40 M kernel k=4.81 ( ) real-par
M kernel k=7.75 ( ) real-part )803.3(1J
)405.2(0J
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31
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
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Conclusions
The spurious eigenvalue occur for the doubly-connected membrane , even the complex fundamental solution are used.
The spurious eigenvalue of the doubly-connected membrane are true eigenvalue of simple-connected membrane.The existence of spurious eigenvalue are proved in an analytical manner by using the degenerate kernels and the Fourier series.
32
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The EndThanks for your
attention