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The 32 nd National Conference on Theoretical and Applied Mechanics. On the spurious eigenvalues for a concentric sphere in BIEM. Reporter ： Shang-Kai Kao Ying-Te Lee , Jia-Wei Lee and Jeng-Tzong Chen Date ： 2008/11/28-29. National Taiwan Ocean University MSVLAB - PowerPoint PPT Presentation
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1
On the spurious eigenvalues for a concentric sphere in
BIEMReporter： Shang-Kai Kao
Ying-Te Lee , Jia-Wei Lee and Jeng-Tzong Chen
Date： 2008/11/28-29
The 32nd National Conference on Theoretical and Applied Mechanics
National Taiwan Ocean University
MSVLABDepartment of Harbor and River Engineering
2
Outline• Introduction
• Problem statement
• Mathematical analysis
• Numerical example
• Conclusions
3
Outline• Introduction
• Problem statement
• Mathematical analysis
• Numerical example
• Conclusions
4
Motivation
• 1-D
real-part BEM, CTAM31
• 2-D
Journal of Sound and Vibration, 2002
doubly-connected membrane, CTAM31
• 3-D
Computational Mechanics, 2002
5
BIEM and Null-field Integral Equation
Interior problem Exterior problem
cD
D D
x
xx
xcD
x x
Degenerate (separate) formDegenerate (separate) form
4 ( ) ( , ) ( ) ( ) ( , ) ( ) ( ),B B
u x T s x u s dB s U s x t s dB s x D 2 ( ) . . . ( , ) ( ) ( ) . . . ( , ) ( ) ( ),
B Bu x C PV T s x u s dB s R PV U s x t s dB s x B
Bc
BBDxsdBstxsUsdBsuxsT ),()(),()()(),(0
B
( , )
( , )( , )
( )( )
ikr
s
s
eU s x
rU s x
T s xn
u st s
n
5
2 2( ) ( , ) 4 ( )k U s x x s
6
Outline• Introduction
• Problem statement
• Mathematical analysis
• Numerical example
• Conclusions
7
Problem statement
a
b
where is the wavenumberk
0.5a b
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Outline• Introduction
• Problem statement
• Mathematical analysis
• Numerical example
• Conclusions
9
3D degenerate kernels
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3D degenerate kernels
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Null-field Integral equation - UT
12
Dirichlet B.C. (fixed-fixed) - UT
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Eigenvalue (k)-(fixed-fixed)n 0 1 2 3 4 5 6 7 8
6.28319 8.98682 11.5269 13.9759 16.3651 18.7116 21.0257 23.3141 25.5816
12.5664 15.4505 18.19 20.8342 23.4098 25.9331 28.4148 30.8626 33.282
18.8496 21.8082 24.6459 27.396 30.0793 32.7094 35.2959 37.846 40.3649
25.1327 28.1324 31.0292 33.8472 36.6025 39.3063 41.9669 44.5907 47.1825
n 0 1 2 3 4 5 6 7 8
6.28319 6.57201 7.11158 7.84504 8.7168 9.682 10.7077 11.7708 12.8557
12.5664 12.7214 13.0261 13.4711 14.0437 14.7294 15.5133 16.3806 17.3173
18.8496 18.9544 19.1625 19.4711 19.8458 20.3718 20.9533 21.6141 22.3481
25.1327 25.2118 25.3692 25.6038 25.9137 26.2968 26.7502 27.271 27.8561
It’s a special case that a=0.5b.
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Neumann B.C. (free-free) - UT
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Eigenvalue (k )-(free-free)
65.96 10
n 0 1 2 3 4 5 6 7 8
1.84027 3.15118 4.38996 5.57454 6.71753 7.83112 8.92495 10.0056
6.57201 6.91152 7.55362 8.43887 9.50101 10.6777 11.9165 13.1778 14.4344
12.7214 12.8852 13.2087 13.684 14.3014 15.0504 15.9204 16.9005 17.9775
18.9544 19.0621 19.276 19.5937 20.0115 20.5253 21.1305 21.8228 22.5981
n 0 1 2 3 4 5 6 7 8
6.28319 8.98682 11.5269 13.9759 16.3651 18.7116 21.0257 23.3141 25.5816
12.5664 15.4505 18.19 20.8342 23.4098 25.9331 28.4148 30.8626 33.282
18.8496 21.8082 24.6459 27.396 30.0793 32.7094 35.2959 37.846 40.3649
25.1327 28.1324 31.0292 33.8472 36.6025 39.3063 41.9669 44.5907 47.1825
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The eigenvalues by using BIM and SVD
1 2 3 4 5 6 7 8 9 106.280 6.570 7.110 7.850 8.720 8.990 9.680
U kernel
1 2 3 4 5 6 7 8 9 101.840 3.150 4.390 5.570 6.280 6.570 6.720 6.910 7.550 7.830
T kernel
1 2 3 4 5 6 7 8 9 104.160 6.280 6.570 6.680 7.110 7.840 8.720 8.990 9.030 9.680
L kernel
1 2 3 4 5 6 7 8 9 101.840 3.150 4.160 4.390 5.570 6.570 6.690 6.720 6.910 7.550
M kernel
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Outline• Introduction
• Problem statement
• Mathematical analysis
• Numerical example
• Conclusions
18
Dirichlet B.C. (fixed-fixed)-True
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
400
410
420
430
440
450
The
det
erm
ent o
f th
e in
flue
nce
mat
rice
for
U k
erne
l
T 6 .2 8 0(6 .2 8 3 )
T 6 .5 7 0(6 .5 7 2 )
T 7 .1 1 0(7 .1 1 1 )
T 7 .8 5 0(7 .8 4 5 )
T 8 .7 2 0(8 .7 1 7 )
T 9 .6 8 0(9 .6 8 2 )
U
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
670
680
690
700
710
720
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
for
L k
erne
l
T 6 .2 8 0(6 .2 8 3 )
T 6 .5 7 0(6 .5 7 2 )
T 7 .1 1 0(7 .1 1 1 )
T 7 .8 4 0(7 .8 4 5 )
T 8 .7 2 0(8 .7 1 7 )
T 9 .6 8 0(9 .6 8 2 )
L
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
760
770
780
790
800
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
T 6 .2 8 0(6 .2 8 3 )
T 6 .5 7 0(6 .5 7 2 )
T 7 .1 1 0(7 .1 1 1 )
T 7 .8 5 0(7 .8 4 5 )
T 8 .7 2 0(8 .7 1 7 )
T 9 .6 8 0(9 .6 8 2 )
SVD updating terms U
L
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Neumann B.C. (free-free)-True
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
1000
1010
1020
1030
1040
1050
1060
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
T 1 .8 4 0(1 .8 4 0 )
T 3 .1 5 0(3 .1 5 1 )
T 4 .3 9 0(4 .3 9 0 )
T 5 .5 7 0(5 .5 7 5 )
T 6 .5 7 0(6 .5 7 2 )
T 6 .7 2 0(6 .7 1 8 )
T 6 .9 1 0(6 .9 1 2 )
T 7 .5 5 0(7 .5 5 4 )
T 7 .8 3 0(7 .8 3 1 )
T 8 .9 2 0(8 .9 2 5 )
T 8 .4 4 0(8 .4 3 9 )
T 9 .5 0 0(9 .5 0 1 )
SVD updating terms T
M
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
650
660
670
680
690
700
710
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
for
T k
erne
l
T 1 .8 4 0(1 .8 4 0 )
T 3 .1 5 0(3 .1 5 1 )
T 4 .3 9 0(4 .3 9 0 )
T 5 .5 7 0(5 .5 7 5 )
T 6 .5 7 0(6 .5 7 2 )
T 6 .7 2 0(6 .7 1 8 )
T 6 .9 1 0(6 .9 1 2 )
T 7 .5 5 0(7 .5 5 4 )
T 7 .8 3 0(7 .8 3 1 )
T 8 .9 2 0(8 .9 2 5 )
T 8 .4 4 0(8 .4 3 9 ) T
9 .5 0 0(9 .5 0 1 )
T
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
920
940
960
980
1000
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
for
M k
erne
l
T 1 .8 4 0(1 .8 4 0 )
T 4 .3 9 0(4 .3 9 0 )
T 3 .1 5 0(3 .1 5 1 )
T 5 .5 7 0(5 .5 7 5 )
T 6 .5 7 0(6 .5 7 2 )
T 6 .7 2 0(6 .7 1 8 )
T 6 .9 1 0(6 .9 1 2 )
T 7 .5 5 0(7 .5 5 4 )
T 7 .8 3 0(7 .8 3 1 )
T 8 .4 4 0(8 .4 3 9 )
T 8 .9 2 0(8 .9 2 5 )
T 9 .5 0 0(9 .5 0 1 )
M
20
Singular formulation -Spurious
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
696
700
704
708
712
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
S 6 .2 8 0(6 .2 8 3 )
S 8 .9 9 0(8 .9 8 7 )
SVD updating document U T
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
400
410
420
430
440
450
The
det
erm
ent
of th
e in
flue
nce
mat
rice
for
U k
erne
l
S 6 .2 8 0(6 .2 8 3 )
S 8 .9 9 0(8 .9 8 7 )
U
0 2 4 6 8 10
T h e w av e n u m b e r ( k )
650
660
670
680
690
700
710
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
for
T k
erne
l
S 6 .2 8 0(6 .2 8 3 )
S 8 .9 9 0(8 .9 8 7 )
T
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Hypersingular formulation -Spurious
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
940
950
960
970
980
990
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
S 4 .1 6 0(4 .1 6 3 )
S 6 .6 8 0(6 .6 8 4 )
S 9 .0 3 0(9 .0 2 8 )
SVD updating document L M
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
670
680
690
700
710
720
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
for
L k
erne
l
S 4 .1 6 0(4 .1 6 3 )
S 6 .6 8 0(6 .6 8 4 )
S 9 .0 3 0(9 .0 2 8 )
L
0 2 4 6 8 10
T h e w a v e n u m b e r ( k )
920
940
960
980
1000
The
det
erm
ent
of t
he i
nflu
ence
mat
rice
for
M k
erne
l
S 4 .1 6 0(4 .1 6 3 )
S 6 .6 9 0(6 .6 8 4 )
S 9 .0 3 0(9 .0 2 8 )
M
22
Outline• Introduction
• Problem statement
• Mathematical analysis
• Numerical example
• Conclusions
23
Conclusions
• There are still spurious eigenvalues by using BIEM to deal with concentric sphere problems.
• True eigenvalues are dependent on problems and spurious eigenvalues are dependent on methods.
• Spurious eigenvalues are dependent on the inner boundary.
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~Thanks for your kind attentions~
