On the spurious eigenvalues for a concentric sphere in BIEM

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

8

Outline• Introduction

• Problem statement

• Mathematical analysis

• Numerical example

• Conclusions

9

3D degenerate kernels

10

3D degenerate kernels

11

Null-field Integral equation - UT

12

Dirichlet B.C. (fixed-fixed) - UT

13

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.

14

Neumann B.C. (free-free) - UT

15

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

16

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

17

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

19

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

21

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~