SVD application in non-uniqueness problem of BEM/BIEM
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SVD application in non-uniq ueness problem of BEM/BIEM Reporter : Kao S. K. Advisor : Chen J. T. Date: 2009/03/17 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering
description
SVD application in non-uniqueness problem of BEM/BIEM. Reporter : Kao S. K. Advisor : Chen J. T. Date: 2009/03/17. National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering. Outline. Eigenproblem for concentric sphere Eigenproblem for eccentric sphere - PowerPoint PPT Presentation
Transcript of SVD application in non-uniqueness problem of BEM/BIEM
SVD application in non-uniqueness problem of BEM/BIEM
Reporter : Kao S. K.Advisor : Chen J. T.
Date: 2009/03/17National Taiwan Ocean University
MSVLABDepartment of Harbor and River Engineering
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MSVLAB, HRE, NTOU
Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions
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Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions
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Concentric sphere
a
b
2 2 ( ) 0,x x k u
where is the wavenumberk
0.5a b
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Null-field Integral equation - UT2 2 2 2
1 1 1 1 2 2 2 20 0 0 0 0 0 0 0
0 Tu dS Ut dS Tu dS Ut dS
2 2 1 (2) 2 1 (2)
0 0 0 0
2 2 2 (2) 2 2 (2)
0 0
0 ( ) '( ) (cos( )) cos( ) ( ) ( ) (cos( )) cos( )
( ) '( ) (cos( )) cos( ) ( ) ( ) (cos
n nm m
nm n n n nm n n nn m n m
nm m
nm n n n nm n n nn m
ib k A j kb h kb p m ib kB j kb h kb p m
ia k A j ka h kb p m ia kB j ka h kb p
0 0
( )) cos( )n
n mm
2 2 1 (2) 2 1 (2)
0 0 0 0
2 2 2 (2) 2 2 (2)
0 0
0 ( ) '( ) (cos( )) cos( ) ( ) ( ) (cos( ))cos( )
( ) '( ) (cos( )) cos( ) ( ) ( ) (cos
n nm m
nm n n n nm n n nn m n m
nm m
nm n n n nm n n nn m
ib k A j ka h kb p m ib kB j ka h kb p m
ia k A j ka h ka p m ia kB j ka h ka p
0 0
( )) cos( )n
n mm
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Dirichlet B.C. (fixed-fixed)-True
0 2 4 6 8 10
T h e w av e n u m b er ( k )
400
410
420
430
440
450
The
det
erm
ent
of 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 av e n u m b er ( 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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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 av e n u m b er ( 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 av e n u m b er ( 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
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Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions
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Eccentric sphere
2 2 ( ) 0,x x k u
where is the wavenumberk
0.8a b
a
b
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Singular formulation-Spurious
0 2 4 6 8 10
300
310
320
330
340
0 2 4 6 8 10
340
360
380
400
420
440
Concentric sphere eccentric sphere
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Hypersingular formulation-Spurious
0 2 4 6 8 10
500
520
540
560
580
600
2.8 3.2 3.6 4 4.4 4.8 5.2
531
532
533
534
4 4.4 4.8 5.2 5.6 6
531
532
533
534
535
536
537
6 6.4 6.8 7.2 7.6
536
540
544
548
8 8.2 8.4 8.6 8.8 9
552
556
560
564
568
572
9 9.2 9.4 9.6 9.8 10
568
572
576
580
584
588
eccentric sphere
0 2 4 6 8 10
420
440
460
480
500
concentric sphere
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Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions
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Linton and Evans’ method (Multi-pole method)
Interior problem Exterior problem
1r
1o
2o
o
2o
o
r2r
1o
radiation
scattering
concentric sphere
eccentric sphere
rr
2r
2r
1r
r
2r
1r
1r
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Comparison between the present method and LE method []
( ) ( ) ( )i i
i ii B B
u x TudB s UtdB s
(1)( ) ( ) ( )i m
n n nn
u x c h x Y x
(2)
0 0
( ) ( ) cos( ) cosn
i mnm n i n n
n m
B j kR h k P m
(1)
0 0
( ) cos( ) cosn
i mnm n n
n m
C h k P m
Method Present method Linton and Evans’ method
Coordinate Adaptive observer system Multi-pole
Formulation
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Problem statement
3.85
2.1
0.41
2 ( ) ( )u x x
2 2 ( ) ( ), 0k u x x k
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2 1r r b
2
12r
b
1r 22 2
1 1
( )
( ) ( ),
imm m
mn nm
r H kr e
S b r r b
( )( ) ( ) i m nmn m nS b H kb e
11 1( ) in
n nr J kr e
1( )(1)2 1 1,i m n n
m m n nn
r H kb J kr e r b
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The endThe endThanks for your kind attention.Thanks for your kind attention.
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