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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
2
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
3
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
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
5
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
Governing equation
Governing equation
0)()( 22 xuk
Fundamental solution
)()(),( 00 krYkriJxsU
6
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
8
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
24
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
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
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
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
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
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
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
31
3. Mathematical analysis
2. Problem statements
1. Introduction
4. Numerical examples
Outlines
5. Conclusions
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
The EndThanks for your
attention
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