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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MSVLAB, HRE, NTOU

Outline Eigenproblem for concentric sphere Eigenproblem for eccentric sphere Linton and Evans’ method Conclusions

4

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MSVLAB, HRE, NTOU

Concentric sphere

a

b

2 2 ( ) 0,x x k u

where is the wavenumberk

0.5a b

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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

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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

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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

9

MMSS VV

MSVLAB, HRE, NTOU

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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MSVLAB, HRE, NTOU

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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