1

Study on the Green’s functions for Laplace problems with circular and

spherical boundaries by using the image

method Reporter: H.C. Shieh

Adviser: Dr. J.T. ChenDepartment of Harbor and River Engineering,

National Taiwan Ocean UniversityJuly 25, 2009

碩士學位論文口試報告

2

FrameMotivation and literature review

Two-dimensional Green’s functionGreen’s

function

Conclusions

MFS (Image method)

Trefftz method

BVP without sources

3

Numerical methods

Numerical methods

Boundary Element MethodFinite Element Method Meshless Method

4

Method of fundamental solutions

MN

jjj xsUcxu

1

),()(

is the fundamental solution),( xsU

Interior case Exterior case

This method was proposed by Kupradze in 1964.

5

Optimal source location

Conventional MFS Alves & Antunes

GoodNot Good

?

6

The simplest image method

Neumann boundary Neumann boundary conditioncondition DirichletDirichlet boundary conditio boundary conditionn

Mirror

7

Conventional method to determine the image location

R

R’

O

a rr’

aOR

ORa

ORa

ORa

PR

RP

'''

2

''

OR a aOR

a OR OR

P

AB

aa

O R’RO

PPLord Kelvin(1824~1907) (1949, 相似三角形 )

Greenberg (1971, 取巧法 )

8

Image location (Chen and Wu, 2006)

a

s 's2'

'ss

R

a

aR

R

a

R

1

1ln cos ( )

s

m

sm

ma

RR

m

1

1ln cos ( )

m

m

a mm

R

a

a s2

''s

s

aR

R

a

R

R

a

1

1ln cos ( )

ms

m

a mm

R

a

1

1ln cos ( )

m

m

a

RR m

m

Rigid body term

's

u=0

u=0

9

2-D Degenerate kernal

1

1

),(cos)(1

ln

,)(cos)(1

lnln

m

m

m

m

RmR

m

RmRm

Rr

s( , )R q

R

r

rx( , )r f

x( , )r f

o

iU

eU

References:

W. C. Chen, A study of free terms and rigid body modes in the dual BEM, NTOU Master Thesis, 2001.

C. S. Wu,Degenerate scale analysis for membrane and plate problems using the meshless method and boundary element method, NTOU Master Thesis, 2004

rsxU ln),(

10

Addition theorem & degenerate kernel

Addition theorem Subtraction theorem

sxsx eee

sxsxsx sinsincoscos)cos( sin( ) sin cos cos sinx s x s x s

/x s x se e e cos( ) cos cos sin sinx s x s x s sin( ) sin cos cos sinx s x s x s

Degenerate kernel for Laplace problem

1-D

2-D

RmR

m

RmRm

Rr

m

m

m

m

,)(cos)(1

ln

,)(cos)(1

lnln

1

1

sxifxs

sxifsxr

,

,

sx

11

3-D degenerate kernel

11 0

11 0

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

( )!1

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

( )!

nni m m

m n n nn m

nne m m

m n n nn m

n mU m P P R

R n m R

r n m RU m P P R

n m

1, 02 , 1,2,...,m

mm

s ( , , )( , , )

xs R

x

exterior

x

interior

12

Outline

Motivation and literature reviewDerivation of 2-D Green’s function

by using the image methodTrefftz method and MFS

Image method (special MFS)Trefftz method

Equivalence of solutions derived by Trefftz method and MFS

Boundary value problem without sourceConclusions

13

Eccentric annulus

b

a

d

s

sR

Governing equation:

2 ( , ) ( ),G x s x s x

Dirichlet boundary condition:

1 2( , ) 0,G x s x B B= Î U

Case 1

B1

B2

14

Eccentric problem

b

a

d

sGoverning equation:

2 ( , ) ( ),G x s x s x

Dirichlet boundary condition:

1 2( , ) 0,G x s x B B= Î U

Case 2

B1

B2

15

Half plane with circular hole problem

( , )sR

Governing equation:

2 ( , ) ( ),G x s x s x

Dirichlet boundary condition: s

1 2( , ) 0,G x s x B B= Î U

u1=0

u2=0

Case 3

B1

B2

16

Bipolar coordinates

221 lnln rr

1r 2r 1r

2r

221 lnln rr

x

121 lnln rr1r

2r1cs 2cs

17

Bipolar coordinates

Eccentric annulus A half plane with a hole An infinite plane with double holes

focus

18

Annular (EABE, 2009) to eccentric case

s s1s2s4s3

s6s5

…. ….

Image point+

-

Source point

sxsxGm ln2

1),(

4 3 4 2 4 1 41

lim ln ln ln lnN

i i i iN

i

x s x s x s x s remainder term

0),(1

BxsxG 0),(

2

BxsxG

19

Series of images

-4 -2 0 2

x

-1

0

1

y

Gra p h 1

Im ag e p o in t

The final images

sc1 sc2

4 3 4 2 4 1 41

1 1 2 2

1( , ) {ln lim ln ln ln ln

2( ) ln ( ) ln ( )}

N

i i i iN

i

c c

G x s x s x s x s x s x s

c N x s c N x s e N

20

Numerical approach to determine c1(N), c2(N) and e(N)

( ) 0-6-2.696 10 e N

1( )c N -0.8625

2 ( )c N -0.1375

0 2 4 6 8 10

N

-1

-0 .8

-0.6

-0.4

-0.2

0

V alu e

C o effic ien tc 1(N )

c 2(N )

e (N )

Coefficients sc1 sc2

21

Contour plot of eccentric annulus problem

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Image method Analytical solution

(bipolar coordinates )

Dirichlet boundary for the eccentric case 1

22

Analytical derivation of locationfor the two frozen points

d

a

b

2cR

2cs 1cs

1cRc2

1 1 1

2

11

2

11

1

2

1ln ( ) cosln ( ,)c c c

cc

c

cc

m

c

mRa m

m as x s a R

R a aR

a R R

ff¥

=- -å® - = >

ß

= Þ =

2 21

2

2 2 2

1ln ( ) cos ( )ln ,m

mc c

c

c c c

aR mx a R

ms

Rs ff

¥

=- -- <å® =

cRR cc 212

4 2 2 4 2 2 2 2 42 2 2

2

a a b b a d dc

b d

d

- + +=

-Þ

-

23

Eccentric case

4 3 4 2 4 1

1 1 2 2

41

1( , ) ln lim ln ln ln ln

( )

2

l ( )n l (n )

N

i i i i

c

i

c

NG x s x s x s x s x s x

c N s c N s e

s

x Nx

1 2 and c cs s focuses

Image sources

True source

24

Contour plot of eccentric annulus

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Image method

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Null-field BIE approach (addition theorem and superposition technique)

25

A half plane with a circular hole

4 3 4 2 4 1

1 1 2 2

41

1( , ) ln lim ln ln ln ln

( )

2

l ( )n l (n )

N

i i i i

c

i

c

NG x s x s x s x s x s x

c N s c N s e

s

x Nx

image source

true source

1 2 and c cs s focuses

26

Contour plot of half plane problem

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Image method Null-field BIE approach (addition theorem and superposition technique)

27

Linking of MFS and image method

s

Ä

1s

Ä

2s

e

3s

e

4s

e

5s

e

6s

Ä

7s

ÄK

8s

ÄK

MFS (special case) Conventional MFS

1

1( ) ln ( , )

2

N

j jj

u x x s w U x s

4 3 4 21

1( ) {ln

2

lim ln ln

( ) ( ) ln }

N

i iNi

u x x s

x s x s

c N d N

s

28

Image method versus MFS

N N

Ku p

K

N

large

3 3

Ku p

K

optimal

All the strength need to be determined.

Only three coefficients are required to be determined.

Conventional MFS Image method

29

Outline

Motivation and literature reviewDerivation of 2-D Green’s function

by using the image methodTrefftz method and MFS

Image method (special MFS)Trefftz method

Equivalence of solutions derived by Trefftz method and MFS

Boundary value problem without sourcesConclusions

30

Trefftz method

The method was proposed by Trefftz in 1926.

TN

jjjcxu

1

)(

mm mm sin,cos,1 mm mm sin,cos,ln

Interior case Exterior case

is the jth T-complete functionj

31

Trefftz method and MFS

Method Trefftz method MFS

Definition

Figure sketch

Base , (T-complete function) , r=|x-s|

G. E.

Match B. C. Determine cj Determine wj

( , ) lnU x s r

1( ) ( , )

N

j jj

u x w U x s

( )2 0u xÑ = ( )2 0u xÑ =

D

u(x)

~x

s

Du(x)

~x

r

~s

is the number of complete functions MN is the number of source points in the MFS

1( )

M

j jj

u x c

j

32

Derivation of 3-D Green’s function by using the image method

Interior problem

Exterior problem

33

The weighting of the image source in the

3-D problem

y

z

1a

x

y

1 a

x

z

),,(2

sR

as ),,( sRs

),,( sRs

),,(2

sR

as

Interior problem Exterior problem

1sR

a

1sR

a

34

The image group

22 15 91

1 5 9 4 32 3

2 32 1

2 6 10 4 22

2 32 13 7 11

3 7 11 4 12 3

2

4 8 2

, ( ), ( ) ( )

, ( ), ( ) ( )

, ( ), ( ) ( )

, (

nn

nn

nn

aR a RR b b b b b b bw w w w

b R b R a b R a R aa a a a a a a a a

w w w wR bR R b b R R b R baR a R a Rb b b b b b b

w w w wbR a b R a a b R a a a aa a a a

w wb b b

3

2 112 43

), ( ) ( )nn

a a a a aw w

b b b b b b

2 2 2 2 21

1 5 4 32 2

2 2 2 2 21

2 6 4 22 2

2 2 2 2 21

3 7 4 12 2 2 2 2

2 2 2 2 21

4 8 42 2 2 2 2

, ........ ( )

, ....... ( )

, ... ( )

, ... ( )

nn

nn

nn

nn

b b b b bR R R

R R a R a

a a a a aR R R

R R b R b

b R b R b b R bR R R

a a a a a

a R a R a a R aR R R

b b b b b

Obtain image weighted

Obtain image location

s

Ä

1s

Ä

2s

e

3s

e

4s

e

5s

e

6s

Ä

7s

ÄK

8s

ÄK

35

Interpolation functions

a

b

1),(),( sxGsxG ba

( ) ( )( , ) ( , ) ( , ) ( , ),

( ) ( )m m b m a

b a a bG x s G x s G x s G x s a b

b a b a

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

1 1( , ) lim

4

( ) 1

( ) ( )

Ni i i i

Ni i i i i

N Ns s

s s

w w w wG x s

x s x s x s x s x s

R a a b Ra a

b R b a b R b a

36

Analytical derivation

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

1 1 ( )( , ) lim ( )

4

Ni i i i

Ni i i i i

w w w w d NG x s c N

x s x s x s x s x s

37

Numerical solution

a

b

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

1 1 ( )( , ) ( ) 0

4

1 1 ( )( , ) ( ) 0

4

Ni i i i

aia a i a i a i a i

Ni i i i

bib b i b i b i b i

w w w w d NG x s c N

x s x s x s x s x s a

w w w w d NG x s c N

x s x s x s x s x s b

0

0

)(

)(1

1

11

1

1

14

4

14

14

24

24

34

34

14

4

14

14

24

24

34

34

Nd

Nc

b

a

sx

w

sx

w

sx

w

sx

w

sx

sx

w

sx

w

sx

w

sx

w

sx

iib

i

ib

i

ib

i

ib

i

b

iia

i

ia

i

ia

i

ia

i

a

38

Numerical and analytic ways to determine c(N) and d(N)

0 2 4 6 8 10

N

0

0.02

0.04

0.06

0.08

0.1c (N ) a n d d (N )

A n a ly tica l c (N )N u m er ica l c (N )A n a ly tica l d (N )N u m er ica l d (N )

Coefficients

39

Derivation of 3-D Green’s function by using the Trefftz Method

1G

2G

11 GG

22 GG

PART 1 PART 2

PART 1

11 0

11 0

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

4 ( )!( , )

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

4 ( )!

nnm m

m n n snn ms s

Fnn

m msm n n sn

n m

n mm P P R

R n m RG x s

Rn mm P P R

n m

40

Boundary value problem

1( , )

TN

T j jj

G x s c

11 GG

22 GG

Interior:

)(cos)sin(),(cos)cos(,1 m

n

nm

n

n PmPm

Exterior:

)(cos)sin(),(cos)cos(,1 )1()1(

m

n

nm

n

n PmPm

( 1)0000

1 0

( 1)

( , ) [ (cos )cos( ) (cos )cos( )

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

nn m n m

T nm n nm nn m

n m n m

nm n nm n

BG x s A A P m B P m

C P m D P m

00

00

4( )

4

s

s

s

s

R a

R b aAB a b R

R b a

2 1 2 1

1 2 1 2 1

2 1 2 1 2 1

1 2 1 2 1

( )!(cos )cos( )

4 ( )!

( )!(cos )cos( )

4 ( )!

n nmm s

nn n nsnm

n n nnm s mm

nn n ns

R an mP m

n m R b aA

B a b Rn mP m

n m R b a

2 1 2 1

1 2 1 2 1

2 1 2 1 2 1

1 2 1 2 1

( )!(cos )sin( )

4 ( )!

( )!(cos )sin( )

4 ( )!

n nmm s

nn n n

nm

n n nnm s mm

nn n ns

R an mP m

n m R b aC

D a b Rn mP m

n m R b a

PART 2

41

PART 1 + PART 2 :

1G2G11 GG

22 GG

( , ) ( , ) ( , )F TG x s G x s G x s

11 0

11 0

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

4 ( )!( )

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

4 ( )!

nnm m

m n n snn ms s

Fnn

m msm n n sn

n m

n mm P P R

R n m RG x

Rn mm P P R

n m

( 1)0000

1 0

( 1)

( , ) [ (cos )cos( ) (cos )cos( )

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

nn m n m

T nm n nm nn m

n m n m

nm n nm n

BG x s A A P m B P m

C P m D P m

( 1)0000

1 0

( 1)

1( , ) [ (cos )cos( ) (cos )cos( )

4

(cos )sin( ) (cos )sin( )],

nn m n m

nm n nm nn m

n m n m

nm n nm n

BG x s A A P m B P m

x s

C P m D P m

42

Results

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Trefftz method (x-y plane) Image method (x-y plane)

43

Outline

Motivation and literature reviewDerivation of 2-D Green’s function

by using the image methodTrefftz method and MFS

Image method (special MFS)Trefftz method

Equivalence of solutions derived by Trefftz method and MFS

Boundary value problem without sourcesConclusions

44

Trefftz solution

( 1)0000

1 0

( 1)

1( , ) [ (cos )cos( ) (cos )cos( )

4

(cos )sin( ) (cos )sin( )],

nn m n m

nm n nm nn m

n m n m

nm n nm n

BG x s A A P m B P m

x s

C P m D P m

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1

1 1 2 1 2 11 0

( )1 1( , ) + +

4 ( ) ( )

( )! + cos[ ( )] (cos )

4 ( )! ( )

s s

s s

n n n n n n n nnmm s s

nn n n nn m s

R a a b RG x s

x s R b a R b a

R a a b a Rn mm P

n m R b a

Without loss of generality

45

Mathematical equivalence the Trefftz method

and MFS Trefftz method series expand

2 1 2 1 2 1 2 1

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

( ) ( ) ( )

n n n n n n n nm

nn n n n n n n n n n n n

R a a b a RP

b a R b a R b a b a

Image method series expand

1212

12

12

12

24

222

121

9

91

5

51

1

1

1

nn

nn

n

n

n

nn

n

nnn

n

nn

n

n

n

n

n

n

ab

R

ba

bR

b

Ra

aR

b

b

R

R

b

Rw

Rw

Rw

s s1s2s4 s3 s5 s9s7

)(1121211

1212

12

12

11

12

12

42

1

2

110

1016

612

2

nnnn

nn

n

n

nn

n

nnn

n

nn

n

n

n

n

n

n

n

abR

ba

baR

a

Rb

a

bR

a

R

a

R

aRw

Rw

Rw

s s1 s3s2s4s6s8s10

s s1s2s4 s3 s5 s9s7

)(1)()(

12121

12

12

12

112

12

144

44

2

2

122

22

1

7

71

3

3

nnn

nn

n

n

nn

nn

nn

nn

nn

nn

n

n

n

n

abR

a

baRb

a

Rb

a

a

b

Rb

a

a

b

Rw

Rw

s s1 s3s2s4s6s8s10

)(1)()(

12121

12

12

12

112

12

14

4

2

2

12

2

1

881

44

nnn

nn

n

n

nn

nn

nn

nn

nn

nn

n

n

n

n

ab

Ra

ba

bRa

b

Ra

b

a

b

Ra

b

aRw

Rw

46

Equivalence of solutions derived by Trefftz method and image method (special MFS)

Trefftz method MFS (image method)

1, cos( ) (cos ),

sin( ) (cos )

, , 0,1,2,3, , ,

1,2, , ,

n m

n

n m

n

m P

m P

m

n

r f q

r f q

= ¥

= ¥

K L

L

1,

j

j Nx s

-Î

-

Equivalence

addition theorem

linkage

3-D

True source

47

Outline

Motivation and literature reviewDerivation of 2-D Green’s function

by using the image methodTrefftz method and MFS

Image method (special MFS)Trefftz method

Equivalence of solutions derived by Trefftz method and MFS

Boundary value problem without sourcesConclusions

48

An infinite plane with two circular holes (anti-symmetric BC)

B2

y

x

B1

u=V=V1=-1 u=V=V2=1

a=1.0 d=10

2 ( ) 0,u x x D

d

a a2c

49

Animation - An infinite plane with two

circular holes

u=-1 u=1

s1 s2s3s4sc1 sc2

)lnln()(lim)( 21 ooNrrNqxu

)(ln)(ln)(

)lnlnln(ln

2211

14142434

NesxNcsxNc

sxsxsxsx

cc

N

iiiii

)(Nq )(Nq

50

Numerical approach to determine q(N), c1(N), c2(N) and e(N)

0 4 8N

-0 .6

-0 .4

-0 .2

0

0.2

0.4

0.6

c 1(N )

c 2(N )

e(N )q (N )q(N)=e(N)=0

436.0sinh

)(1

1

ac

VNc

436.0sinh

)(1

2

ac

VNc

Coefficients

51

Contour plot of an infinite plane with two circular holes (antisymmetric case)

Image solution bipolar coordinates

-10-8

-6-4

-20

24

68

-14

-12

-10 -8 -6 -4 -2 0 2 4 -10-8

-6-4

-20

24

68

-14

-12

-10 -8 -6 -4 -2 0 2 4

null-field BIEM

-10-8

-6-4

-20

24

68

-14

-12

-10 -8 -6 -4 -2 0 2 4

52

An infinite space with two cavities (anti-

symmetric BC)

B2

y

z

x

B1

u=V1=-1 u=V2=1 a=1.0 d=5.0

2 ( ) 0,u x x D

2

2

1

1

14

4

14

14

24

24

34

34

21

)()(

)()11

()(lim)(

c

a

c

a

N

ii

i

i

i

i

i

i

i

oo

a

N

sx

Nc

sx

Nc

sx

w

sx

w

sx

w

sx

w

rrNqxu

53

Numerical approach to determine q(N), c1(N) and c2(N)

0 4 8N

-0 .4

0

0.4

0.8

1.2

V a lu e

c 1(N )

c 2(N )

q (N )

1)(lim

NqN

0)(lim,0)(lim 21

NcNcNN

Coefficients

54

Contour plot of an infinite space with two spherical cavities

Bispherical coordinates Image method Null-field BIE

2 2

6

( ) ( ) 0

0 (10 )

k u x

k

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8

-8-6

-4-2

02

46

8-8 -6 -4 -2 0 2 4 6 8

x-y planey

z

x

-8-6

-4-2

02

46

8-8 -6 -4 -2 0 2 4 6 8

55

56

Outline

Motivation and literature reviewDerivation of 2-D Green’s function

by using the image methodTrefftz method and MFS

Image method (special MFS)Trefftz method

Equivalence of solutions derived by Trefftz method and MFS

Boundary value problem without sourcesConclusions

57

Optimal location of MFS

• Depends on loading (image location)

• Depends on geometry (frozen image point)

58

Final images to bipolar (bispherical) focus

2-D 3-D

Bipolar coordinates Bispherical coordinates

59

Equivalence of Trefftz method and MFS

3-D

Trefftz method MFS (image method)

60

Image solution for BVP without sources

x

x

y

y

61

Thanks for your kind attentionsYou can get more information from our website

http://msvlab.hre.ntou.edu.tw/

The end

62

A half plane with a circular hole

2007, Ke J. N. 2009, Image method

a

b

63

An infinite plane with two circular holes subject to Neumann boundary

ab

d

s

01 t02 t

64

Extra terms of complementary solutions

x

y

01 tt 2 0t t

( , )S R

O1 O2

1 2

4 3 4 2 4 1 4

1 1 2 2 1 2

1( , ) {ln [ln ln ln ln ]

2( )ln ( )ln ( )ln ( )ln

N

i i i ii

c c O O

G x s s x s x s x s x s x

c N s x c N s x d N s x d N s x

Two complementary solutions

f1 f2

Source point

Frozen point

65

The method provide of JW. Lee

d1 d2

Frozen point

M

jjMO

M

jjMO xxdxxd

122

111 lnlimlnlnlimln

complementary solutions

66

The total potential

1 1 1 2

4 3 4 2 4 1 4

1 2 1 2

1( ) {ln [ln ln ln ln ]

2( )ln ( )ln ( )ln ( )ln

N

i i i ii

f f O O

u x s x s x s x s x s x

f N s x f N s x d N s x d N s x

1 1

1

2

4 3 4 2 4 1 4

1 2

11

21

1( ) {ln [ln ln ln ln ]

2( )ln ( )ln

( ) ln lim ln

( ) ln lim ln

N

i i i ii

f fM

O jM

jM

O jM

j

u x s x s x s x s x s x

f N s x f N s x

d N x x

d N x x

Where the N=M

67

Results

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Image method)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Null-field BIE approach (addition theorem and superposition technique)

68

Conclusions

The analytical solutions derived by the Trefftz method and MFS were proved to be mathematically equivalent for Green’s functions of the concentric sphere.

In the concentric sphere case, we can find final two frozen image points (one at origin and one at infinity). Their singularity strength can be determined numerically and analytically in a consistent manner.

It is found that final image points terminate at the two focuses of the bipolar (bispherical) coordinates for all the cases

69

Numerical examples 1: Eccentric annulus

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Image method (50+2 points)

u1=0u2=0 ( , )ss R

a

b

70

Numerical examples 3: An infinite plane with double holes

ab

d

x

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Image method (20+4+10 point)

t1=0t2=0

71

Animation- an infinite plane with double holes

s

1s2s

3s4s

The final images terminate at the focus

1ds 2ds

Multipole expansion

4 3 4 2 4 1 4

11 1 2

1 1 2

2

12

1

2

( ) ( )

1( , ) {ln [ln ln ln ln ]

2

ln lim ln ln

( )ln (

m

n

l

l

li n

)N

i i i ii

M

f f

d d

M

j jM Mj j

d

G x s x s x s x s x s x s

x x s x x s

f N x s f N

s N

x s

N d s

1fs2fs

true source

1 2 and c cs s focus

image source

2dsand1ds

Multipoles

t1=0 t2=0

72

Equivalence of solutions derived by Trefftz method and MFS

Trefftz solution

( 1)0000

1 0

( 1)

1( , ) [ (cos )cos( ) (cos )cos( )

4

(cos )sin( ) (cos )sin( )],

nn m n m

nm n nm nn m

n m n m

nm n nm n

BG x s A A P m B P m

x s

C P m D P m

00

00

1( )4

s

s

s

s

R a

R b aAB a b R

R b a

Image solution

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

1 1 ( )( , ) lim ( )

4

Ni i i i

Ni i i i i

w w w w d NG x s c N

x s x s x s x s x s

)(

)(

)(

)(

)(

abR

Rba

b

a

abR

aR

b

a

Nd

Nc

s

s

N

s

s

N

The same

73

The simplest MFS

1-D Rod

2211 ),(),()( PsxUPsxUxu 1P

2P

1s 2s

1s2s

1P2P

1s2s

2P1P

where U(x,s) is the fundamental solution.

0 l

74

Present method- MFS (Image method)

……

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

1 1( , ) lim

4

Ni i i i

mN

i i i i i

w w w wG x s remainder term

x s x s x s x s x s

75

An infinite space with two cavities (symmetric BC)

B2

y

z

x

B1

u=V1=1 u=V2=1

q q Obtain image weighting4 4

4 34 4

4 54 2

4 5

4 24 1

4 2

4 34

4 3

ii

i

ii

i

ii

i

ii

i

aww

d Raw

wRaw

wd R

aww

R

--

-

--

-

--

-

-

-

=-

=

=-

=

1 2

4 3 4 2 4 1 4 1 2

11 2 4 3 4 2 4 1 4

( ) ( )1 1( ) lim ( )

s sNs i i i i

Nio o i i i i c c

w w w w c N c Nu x q N

r r x s x s x s x s x s x s

Obtain image location2

4 34 4

2

4 24 5

2

4 14 2

2

44 3

ii

ii

ii

ii

aR d

d Ra

RR

aR d

d Ra

RR

--

--

--

-

= --

=

= --

=

1qs2qs

33sqw44sqw

d

a a

76

The strength of two frozen points and q(N)

The strength of c1(N), c2(N) and q(N)

0.1)( Nq

1( ) -1.691750044745917E-014c N

2 ( ) -1.720415903023510E-014c N

77

Contour plot of an infinite space with two spherical cavities (symmetric case)

3-D Bipolar coordinates

Bispherical coordinates

Image method Null-field BIE

2 2( ) ( ) 0

0

k u x

k

x-y planey

z

x

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8-8

-6-4

-20

24

68

-8 -6 -4 -2 0 2 4 6 8

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8

78

Illustrative examples – An eccentric annulus

1a2 5 .b

1d

y

x1c 2c

u=V1=0 u=V2=1

B1

B2

c c

r1

r2

79

Numerical approach to determine q(N), c1(N), c2(N) and e(N)

0 4 8N

-1

0

1

2

V alu e

c 1(N )

c 2(N )

e(N )q (N )q(N)=0 (exact)

e(N)=2 (exact)

c1(N)=1.44 (exact)

c2(N)=-1.44 (exact)

80

Contour plot of eccentric annulus

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Image solution bipolar coordinates null-field BIEM

81

Outline

Motivation and literature reviewDerivation of 2-D Green’s function

by using the image methodTrefftz method and MFS

Image method (special MFS)Trefftz method

Equivalence of solutions derived by Trefftz method and MFS

Boundary value problem without sourceConclusions

82

Optimal source location

Conventional MFS Alves CJS & Antunes PRS

Not good Good

83

MFS-Image location and weighting-interior (Chen and Wu, 2006)

b

),,( x

1

1R

b

),,( Rs ),,('

2

R

bs

y

x

z

11 0

1 1 ( )!cos ( ) (cos ) (cos )

( )!

nnm m

m n n nn m

n m Rm P P

x s b n m b

1

1 0

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

' ( )! ( ')

nnm m

m n n nn m

R n m bm P P

b x s b n m R

R

bR

R

b

b

Rn

n

n

n 21

1'

)'(

The weighting of the image point

R

b

bR

b

b

R

1' 2

u=0

84

MFS-Image location and weighting-exterior (Chen and Wu, 2006)

),,( x

1

),,( Rs

),,('2

R

as

1R

a

Ra

RPPm

mn

mn

RsxR

a

R

aPPm

mn

mn

Rsx

n

nm

nn

n

m

m

n

n

nm

nn

n

m

m

n

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

)!(

)!(1

'

1

)(cos)(cos)(cos)!(

)!(11

1 0

11 0

aR

aR

a

R

R

a

Ra

R

R

an

n

n

n

n

n

n

n 2

1'

)'()'(

The weighting of the image point

1R

a

u=0

85

a

a

Chen and Wu-image method (2006)

1

1

1ln ( ) cos ( ),

( , ) 1ln ( ) cos ( ),

mss

m

ms s

m s

Rm R

mU x sR m R

m R

s

2

''s

s

aR

R

a

R

R

a

s's

2''

ss

R

a

aR

R

a

R

1

1ln cos ( )

ms

m

a mm

R

a

1

1ln cos ( )

m

m

a

RR m

m

1

1ln cos ( )

s

m

sm

ma

RR

m

1

1ln cos ( )

m

m

a mm

R

a

's

86

Analytical derivation of location for the two frozen points

b

ax

y

1cs

2cs

1cR

2 12c cR b R 1 1 1

2

11

2

11

1

2

1ln ( ) cosln ( ,)c c c

cc

c

cc

m

c

mRa m

m as x s a R

R a aR

a R R

ff¥

=- -å® - = >

ß

= Þ =

2 21

2

2 2 2

1ln ( ) cos ( )ln ,m

mc c

c

c c c

aR mx a R

ms

Rs ff

¥

=- -- <å® =

2

2

2

6 1 0

3 2 2

c

c

c c

c

aR

b R

R R

R

(0.171 & 5.828)

a=1, b=3

87

1 2 3 4 5 6 7 8 9 10N

-0 .8

-0 .6

-0 .4

-0 .2

0

0.2

c(N

), d

(N)

and

e(N

)c (N )d (N )e (N )

Numerical approach to determine c1(N), c2(N) and e(N)

( ) 0-157.9713 10 e N

1( )c N -0.26448

2 ( )c N -0.73551

)(1 Nc)(2 Nc

)(Ne

88

Analytical derivation of locationfor the two frozen points

d

ab

1cs

2cs

1cR

2cR

2 2

1 22 2

,c cc c

a bR R d

R R d

2 2 2

2 222

( )0c

c

a b d RR a

d

2c

4 2 2 4 2 2 2 2 42 2 2

2

a a b b a d b d dc

d

- + - - +=

cRR cc 212