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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
2
MMSS VV
MSVLAB, HRE, NTOU
Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions
3
MMSS VV
MSVLAB, HRE, NTOU
Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions
4
MMSS VV
MSVLAB, HRE, NTOU
Concentric sphere
a
b
2 2 ( ) 0,x x k u
where is the wavenumberk
0.5a b
5
MMSS VV
MSVLAB, HRE, NTOU
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
6
MMSS VV
MSVLAB, HRE, NTOU
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
7
MMSS VV
MSVLAB, HRE, NTOU
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
8
MMSS VV
MSVLAB, HRE, NTOU
Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions
9
MMSS VV
MSVLAB, HRE, NTOU
Eccentric sphere
2 2 ( ) 0,x x k u
where is the wavenumberk
0.8a b
a
b
10
MMSS VV
MSVLAB, HRE, NTOU
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
11
MMSS VV
MSVLAB, HRE, NTOU
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
12
MMSS VV
MSVLAB, HRE, NTOU
Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions
13
MMSS VV
MSVLAB, HRE, NTOU
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
14
MMSS VV
MSVLAB, HRE, NTOU
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
15
MMSS VV
MSVLAB, HRE, NTOU
Problem statement
3.85
2.1
0.41
2 ( ) ( )u x x
2 2 ( ) ( ), 0k u x x k
16
MMSS VV
MSVLAB, HRE, NTOU
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
17
MMSS VV
MSVLAB, HRE, NTOU
The endThe endThanks for your kind attention.Thanks for your kind attention.
Welcome to visit the web site of MSVLAB: Welcome to visit the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlabhttp://ind.ntou.edu.tw/~msvlab