1 eigenanalysis for a concentric sphere using bem and biem 報告學生：高聖凱...

1

Eigenanalysis for a concentric sphere using BEM and BIEM

報告學生：高聖凱指導教授：陳正宗終身特聘教授

時間： 2008/06/24

2

Problem statement

b

a

2 2 ( ) 0,x x k u

where is the wavenumberk

3

3D degenerate kernels

0 0

0

(2)

sin( ) cos( )( , )

4 4

(2 1) ( ) ( ) (cos ) (2 1) ( ) ( ) (cos )4

(2 1) ( ) (cos ) ( ) ( )4

(2 1) ( ) ( ) (cos4

ikr

n n n n n nn n

n n n nn

n n n

e ik kr i krU s x

r kr

ikn j kR j k p i n j kR y k p

ikn j kR p j k iy k

ikn j kR h k p

0

(2)

0 0

)

( )!(2 1) ( ) ( ) (cos( )) (cos( )) cos ( ) ,

4 ( )!

1, 0

2,

n

nm m

n n m n nn m

m

ik n mn j kR h k p p m

n m

mwhere R and

otherwise

(2)

0 0

( )!( , ) (2 1) ( ) ( ) (cos( )) (cos( )) cos ( ) ,

4 ( )!

n

m mn n m n n

n m

ik n mU s x n j k h kR p p m

n m

where R

4

3D degenerate kernels

2(2)

0 0

2(2)

0 0

( , ) ( , )

( )!(2 1) '( ) ( ) (cos( )) (cos( )) cos ( ) ,

4 ( )!

( )!(2 1) ( ) '( ) (cos( )) (cos( )) cos ( )

4 ( )!

s

nm m

n n m n nn m

nm m

n n m n nn m

T s x U s xn

ik n mn j kR h k p p m where R

n m

ik n mn j k h kR p p m

n m

, where R

2(2)

0 0

2(2)

0 0

( , ) ( , )

( )!(2 1) ( ) '( ) (cos( )) (cos( )) cos ( ) ,

4 ( )!

( )!(2 1) '( ) ( ) (cos( )) (cos( )) cos ( )

4 ( )!

x

nm m

n n m n nn m

nm m

n n m n nn m

L s x U s xn


n m


n m

, where R

3

(2)

0 0

3(2)

0 0

( , ) ( , )

( )!(2 1) '( ) '( ) (cos( )) (cos( )) cos ( ) ,

4 ( )!

( )!(2 1) '( ) '( ) (cos( )) (cos( )) cos (

4 ( )!

s x

nm m

n n m n nn m

nm m

n n m n nn m

M s x U s xn n


n m


n m

) , where R

5

Null-field Integral equation - UT

2 2 2 2

1 1 1 1 2 2 2 20 0 0 0 0 0 0 0

22(2) 1 2

1 10 0 0 00 0

0

( )!(2 1) '( ) ( ) (cos( )) (cos( )) cos ( ) cos( ) cos sin( )

4 ( )!

n vm m w

n n m n n vw vn m v w

Tu dS Ut dS Tu dS Ut dS

ik n mn j kR h k p p m A p w R d d

n m

ik

2

(2) 1 21 1

0 0 0 00 0

2(2)

20 0

( )!(2 1) ( ) ( ) (cos( )) (cos( )) cos ( ) cos( ) cos sin( )

4 ( )!

( )!(2 1) '( ) ( ) (cos( ))

4 ( )!

n vm m w


nm

n n m n nn m

n mn j kR h k p p m B p w R d d

n m

ik n mn j kR h k p p

n m

22 2

20 00 0

(2) 2 22 2

0 0 0 0

(cos( )) cos ( ) cos( ) cos sin( )

( )!(2 1) ( ) ( ) (cos( )) (cos( )) cos ( ) cos( ) cos sin( )

4 ( )!

vm w

vw vv w

n vm m w


m A p w R d d

ik n mn j kR h k p p m B p w R d

n m

2

0 0

d

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

ia k A j ka h ka p m ia kB j ka h ka p m

ib k A j kb h ka p m ib kB j kb h ka 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 m

nm n n n nm n n nn m

ia k A j kb h ka p m ia kB j kb h ka p m

ib k A j kb h kb p m ib kB j kb h kb p

0 0

( )) cos( )n

n mm

6

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

1 (2) 2 (2) (2) (2) 1

(2) (2) 21 (2) 2 (2)

0 ( ) ( ) ( ) ( ) 0( ) ( ) ( ) ( )

0( ) ( ) ( ) ( )0 ( ) ( ) ( ) ( )

nm n n nm n n n n n n nm

n n n n nmnm n n nm n n

B j ka h ka B j kb h ka j ka h ka j kb h ka B

j kb h ka j kb h kb BB j kb h ka B j kb h kb

1 (2) (2)

2 (2) (2)

(2) (2) (2) (2)

(2) (2) (2)

0 ( ) ( ) ( ) ( )det 0

0 ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0

( ) ( ) ( ) ( ) ( )

nm n n n n

nm n n n n

n n n n n n n n

n n n n n n

B j ka h ka j kb h ka

B j kb h ka j kb h kb

j ka h ka j kb h kb j kb h ka j kb h ka

j kb h ka j ka h kb j kb h

( ) 0ka

(2) (2)( ) ( ) ( ) ( ) 0

( ) ( ) ( ) ( ) ( ) ( ) 0

( ) ( ) ( ) ( ) 0

n n n n

n n n n n n

n n n n

j ka h kb j kb h ka

j ka j kb iy kb j kb j ka iy ka

j ka y kb j kb y ka

(2)( ) ( ) 0

( ) 0n n

n

j kb h ka

j kb

2 1 (2) 2 2 (2)

0 0 0 0

2 1 (2) 2 2 (2)

0 0 0

0 ( ) ( ) (cos( )) cos( ) ( ) ( ) (cos( )) cos( )

0 ( ) ( ) (cos( )) cos( ) ( ) ( ) (cos( )) cos( )

n nm m


n nm m

nm n n n nm n n nn m m

a kB j ka h ka p m a kB j kb h ka p m

b kB j kb h ka p m b kB j kb h kb p m

0n

7

Eigenvalue( ) 0nj kb

n 0 1 2 3 4 5 6 7 8

6.28319 8.98682 0.00316 0.03018 0.12358 0.27004 0.61064 1.09945 1.68051

12.5664 15.4505 11.5269 13.9759 16.3651 18.7116 21.0257 23.3141 25.5816

18.8496 21.8082 18.19 20.8342 23.4098 25.9331 28.4148 30.8626 33.282

25.1327 28.1324 24.6459 27.396 30.0793 32.7094 35.2959 37.846 40.3649

( ) ( ) ( ) ( ) 0 n n n nj ka y kb j kb y kan 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 25.2117 25.3692 25.6038 25.9137 26.2968 26.7502 27.271 27.8561

8

Null-field Integral equation - LM

2 2 2 2

1 1 1 1 2 2 2 20 0 0 0 0 0 0 0

32(2) 1 2

1 10 0 0 00 0

0

( )!(2 1) '( ) '( ) (cos( )) (cos( )) cos ( ) cos( ) cos sin( )

4 ( )!

n vm m w


Mu dS Lt dS Mu dS Lt dS

ik n mn j kR h k p p m A p w R d d

n m

i

22

(2) 1 21 1

0 0 0 00 0

3(2)

20 0

( )!(2 1) ( ) '( ) (cos( )) (cos( )) cos ( ) cos( ) cos sin( )

4 ( )!

( )!(2 1) '( ) '( ) (cos(

4 ( )!

n vm m w


nm

n n m nn m

k n mn j kR h k p p m B p w R d d

n m

ik n mn j kR h k p

n m

22 2

20 00 0

2(2) 2 2

2 20 0 0 0

)) (cos( )) cos ( ) cos( ) cos sin( )

( )!(2 1) ( ) '( ) (cos( )) (cos( )) cos ( ) cos( ) cos si

4 ( )!

vm wn vw v

v w

n vm m w


p m A p w R d d

ik n mn j kR h k p p m B p w R

n m

2

0 0

n( )d d

2 3 1 (2) 2 2 1 (2)

0 0 0 0

2 3 2 (2) 2 2 2 (2)

0 0 0

0 '( ) '( ) (cos( )) cos ( ) '( ) (cos( )) cos

'( ) '( ) (cos( )) cos ( ) '( )

n nm m


n nm

nm n n n nm n nn m n m

ia k A j ka h ka p m ia k B j ka h ka p m

ib k A j kb h ka p m ib k B j kb h ka

0

(cos( )) cosmnp m

2 3 1 (2) 2 2 1 (2)

0 0 0 0

2 3 2 (2) 2 2 2 (2)

0 0 0

0 '( ) '( ) (cos( )) cos '( ) ( ) (cos( )) cos

'( ) '( ) (cos( ))cos '( ) ( )

n nm m


n nm

nm n n n nm n nn m n m

ia k A j kb h ka p m ia k B j kb h ka p m

ib k A j kb h kb p m ib k B j kb h kb

0

(cos( )) cosmnp m

9

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

2 2 1 (2) 2 2 2 (2)

0 0 0 0

2 2 1 (2) 2 2 2 (2)

0 0 0 0

0 ( ) '( ) (cos( )) cos ( ) '( ) (cos( )) cos

0 '( ) ( ) (cos( )) cos '( ) ( ) (co

n nm m


n nm m


a k B j ka h ka p m b k B j kb h ka p m

a k B j kb h ka p m b k B j kb h kb p

s( )) cos m

2 2 1 (2) 2 2 2 (2) 2 2 (2) 2 2 (2)

2 2 (2) 2 2 (2)2 2 1 (2) 2 2 2 (2)

0 ( ) '( ) ( ) '( ) ( ) '( ) ( ) '( )

'( ) ( ) '( ) ( )0 '( ) ( ) '( ) ( )

nm n n nm n n n n n n

n n n nnm n n nm n n

a k B j ka h ka b k B j kb h ka a k j ka h ka b k j kb h ka

a k j kb h ka b k j kb h kba k B j kb h ka b k B j kb h kb

1

2

0

0nm

nm

B

B

1 2 2 (2) 2 2 (2)

2 2 2 (2) 2 2 (2)

(2) (2) (2) (2)

0 ( ) '( ) ( ) '( )det 0

0 '( ) ( ) '( ) ( )

( ) '( ) '( ) ( ) '( ) ( ) ( ) '( ) 0

'( )

nm n n n n

nm n n n n

n n n n n n n n

n n

B a k j ka h ka b k j kb h ka

B a k j kb h ka b k j kb h kb


j kb h

(2) (2) (2)'( ) ( ) ( ) ( ) ( ) 0n n n nka j ka h kb j kb h ka

'( ) 0nj kb

(2) (2)( ) ( ) ( ) ( ) 0

( ) ( ) ( ) ( ) ( ) ( ) 0

( ) ( ) ( ) ( ) 0

n n n n

n n n n n n

n n n n

j ka h kb j kb h ka


j ka y kb j kb y ka

10

Eigenvalue'( ) 0nj kb

( ) ( ) ( ) ( ) 0 n n n nj ka y kb j kb y ka

n 0 1 2 3 4 5 6 7 8

0.00316 0.03682 0.15008 0.38102 0.73190 1.0705 1.70396

8.98682 4.16315 6.68419 9.0282 11.2934 13.5129 15.7022 17.8697 20.0207

15.4505 11.8807 14.5799 17.1675 19.689 22.1404 24.5587 26.9441 29.3025

21.8082 18.4117 21.2277 23.9455 26.5911 29.1811 31.7264 34.235 36.7126

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 25.2117 25.3692 25.6038 25.9137 26.2968 26.7502 27.271 27.8561

69.24 10 51.26 10

11

Neumann B.C. (free-free) - UT2 1 (2) 2 2 (2)

0 0 0 0

2 1 (2) 2 2 (2)

0 0 0 0

0 '( ) ( ) '( ) ( )

0 ( ) '( ) ( ) '( )

n n

nm n n nm n nn m n m

n n

nm n n nm n nn m n m

a A j ka h ka b A j kb h ka

a A j kb h ka b A j kb h kb

2 1 (2) 2 2 (2) 2 (2) 2 (2) 1

2 (2) 2 (2) 22 1 (2) 2 2 (2)

0 '( ) ( ) '( ) ( ) '( ) ( ) '( ) ( )

( ) '( ) ( ) '( )0 ( ) '( ) ( ) '( )

nm n n nm n n n n n n nm

n n n n nmnm n n nm n n

a A j ka h ka b A j kb h ka a j ka h ka b j kb h ka A

a j kb h ka b j kb h kb Aa A j kb h ka b A j kb h kb

0

0

1 2 (2) 2 (2)

2 2 (2) 2 (2)

(2) (2) (2) (2)

(2)

0 '( ) ( ) '( ) ( )det 0

0 ( ) '( ) ( ) '( )

'( ) ( ) ( ) '( ) ( ) '( ) '( ) ( ) 0

( ) ( )

nm n n n n

nm n n n n

n n n n n n n n

n n

A a j ka h ka b j kb h ka

A a j kb h ka b j kb h kb


j kb h ka j

(2) (2)'( ) '( ) '( ) '( ) 0n n n nka h kb j kb h ka

(2)( ) ( ) 0

( ) 0

n n

n

j kb h ka

j kb

(2) (2)'( ) '( ) '( ) '( ) 0

'( ) '( ) '( ) '( ) '( ) '( ) 0

'( ) '( ) '( ) '( ) 0

n n n n

n n n n n n

n n n n

j ka h kb j kb h ka


j ka y kb j kb y ka

12

Eigenvalue( ) 0nj kb

'( ) '( ) '( ) '( ) 0 n n n nj ka y kb j kb y ka

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 0.00316 0.03018 0.12358 0.27004 0.61064 1.09945 1.68051

12.5664 15.4505 11.5269 13.9759 16.3651 18.7116 21.0257 23.3141 25.5816

18.8496 21.8082 18.19 20.8342 23.4098 25.9331 28.4148 30.8626 33.282

25.1327 28.1324 24.6459 27.396 30.0793 32.7094 35.2959 37.846 40.3649

13

Conclusions

• There are still spurious eigenvalues by using BEM and BIEM to deal with concentric sphere problems.

• True eigenvalues are dependent on problems and spurious eigenvalues are dependent on methods.

14

~Thank your kind attentions~

1 eigenanalysis for a concentric sphere using bem and biem 報告學生：高聖凱...

Documents