Image method for the Green’s functions of annulus and half-plane Laplace problems

October, 22, 2008 p.1

Image method for the Green’s functions of annulus and half-plane Laplace

problems

Reporter: Shiang-Chih ShiehAuthors: Shiang-Chih Shieh, Ying-Te Lee and

Jeng-Tzong ChenDepartment of Harbor and River Engineering,

National Taiwan Ocean UniversityOct.22, 2008

October, 22, 2008 p.2

Outline

IntroductionProblem statementsAnalytical solution

Method of Fundmental Solution (MFS) Trefftz method

Equivalence of Trefftz method and MFSSemi-analytical solutionNumerical examplesConclusions

October, 22, 2008 p.3

Trefftz method

1. Introduction2. Problem statements3. Analytical solution

4. Equivalence of Trefftz method and MFS

5. Semi-analytical solution6. Numerical examples7. Conclusions

j is the jth T-complete function

1, cos sinm mm and mr f r f

Interior problem:

ln , cos sinm mm and m

exterior problem:),(

October, 22, 2008 p.4

Method of Fundamental Solution (MFS)

1( ) ( , )

u x w U x s

( , ) ln , ,jU x s r r x s j N

Interior problem

exterior problem

October, 22, 2008 p.5

Trefftz method and MFS

Method Trefftz method MFS

Definition

Figure caption

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

Match B. C. Determine cj Determine wj

( , ) lnU x s r

1( ) ( , )

u x w U x s

( )2 0u xÑ = ( )2 0u xÑ =

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

October, 22, 2008 p.6

Optimal source location

MFS (special case)Image method

Conventional MFS Alves CJS & Antunes PRS

October, 22, 2008 p.7

Problem statements

Case 1 Annular

Governing equation :

xxxG ),(),(2

Boundary condition :

Fixed-Free boundary

u2=0t1=0

October, 22, 2008 p.8

Problem statements

xxxG ),(),(2

Dirichlet boundary condition :

( , ) 0,G x x Bz = Î

Case 2 half-plane problem

October, 22, 2008 p.9

Problem statements

Case 3 eccentric problem

xxxG ),(),(2

Dirichlet boundary condition :

( , ) 0,G x x Bz = Î

October, 22, 2008 p.10

Present method- MFS (Image method)

8 6 8 6

8 4 8 4

8 2 8 2

inside

8 7 8 7

8 5 8 5

8 3 8 3

8 1 8 1

outside

October, 22, 2008 p.11

MFS-Image group( )

( ) ( )

( )( )

RU ,xcos m

aU ,x ln R cos m

æ ö÷ç ÷= - -ç ÷ç ÷÷çè

1 11 1

( , )cos (

( , ) ln cos ( )

R b bR

aU x R

3 31 3

( , ) ln cos ( )

, )cos ( )

b R b bR

( , ) 1

1( , ) ln )

U x Rm

a R aR

RU x a

1( , ) ln cos ( )

U x Rm

R a a aR R

RU x a m

October, 22, 2008 p.12

MFS-Image group

( , )c

1( , ) ln cos ( )

os ( )

U x bm

b R b b bR

aU x R

1( , ) ln cos (

( , ) 1c

os ( )m

U x Rm

R a a a a

RU x a m

1( , )

, )cos ( )

ln cos )

aU x R m

U s x bm

b R b bR

R b R a a

( , ) 1cos ( )

1( , ) ln c

os ( )

a RR a aR

RU x b m

2 2 4 2 41

1 9 8 74 4

2 2 4 2 41

2 10 8 64 4

2 2 24 41

3 11 8 52 2 4 2 4

2 2 24 41

4 12 8 42 2 4 2 4

, ............ ( )

, ........... ( )

, ..... ( )

, .... ( )

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 Rb bR R R

a a a a a

a R a R a Ra aR R R

b b b b b

4 4 4 4 41

5 13 8 32 2 4 2 4

4 4 4 4 41

6 14 8 22 2 4 2 4

4 4 44 41

7 15 8 14 4 4 4 4

4 4 44 41

8 16 84 4 4 4 4

, ..... ( )

, ...... ( )

b b b b bR R

a R a R a a R a

a a a a aR R R

b R b R b b R b

b R b R b Rb bR R R

a a a a a

a R a R a Ra aR R R

b b b b b

8 7 8 6 8 5 8 41

8 3 8 2 8 1 8

1( , ) {ln lim[ (ln ln ln ln

2ln ln ln ln )]}

m i i i iN i

i i i i

G x x x x x x

x x x x

z z z z z zp

z z z z- - - -®¥ =

= - + - - - - - - -å

- - + - + - + -

October, 22, 2008 p.13

Analytical derivation

8 7 8 6 8 5 8 41

8 3 8 2 8 1 8

1( , ) {ln lim[ (ln ln ln ln

2ln ln ln ln ) ( )ln ( )]}

i i i iN i

i i i i

G x x x x x x

x x x x c N e N

z z z z z zp

z z z z r- - - -®¥ =

= - + - - - - - - -å

- - + - + - + - + +

October, 22, 2008 p.14

Numerical solution

8 7 8 6 8 5 8 41

8 3 8 2 8 1

8 7 8 6 8

1( , ) {ln lim [ (ln ln ln ln

ln ln ln ln 8 ) ( )ln ( )]} 0

( , ) 1{ln lim [ (ln ln ln

i i i ix x N i

i i i x x

i i iNx x x x

G x x x x x x

x x x x i c N x e N

G xx x x x

z z z z z zp

z z z z

zz z z z

- - - -= ®¥ =

- - - =

- - -®¥= =

Þ - + - - - - - - -å

- - + - + - + - + + =

¶ ¶Þ - + - - - - -

¶ ¶ 5 8 41

8 3 8 2 8 1

ln ln ln ln 8 ) ( )ln ( )]} 0b

i i i x x

x x x x i c N x e N

z z z z

- - - =

- - + - + - + - + + =

8 7 81

ln ln ln ln 1( ) 0

( ) 0ln (1)ln ln ln

i ix xi x x

i i x xi

x s x x xc N

e Nxx s x x x x

ì üé ùï ïï ï é ùê ú- - - + -åï ïï ï ê úê úë ûï ï ì ü ìï ï ïê úï ïï ï ï ï+ =í ý í ý íê ú¶ ¶ï ï ï ïì üé ù¶ï ï ê úï ïî þï ïï ïê ú- - - + -ï ïå ê úí ý ¶ ¶ï ïï ïê ú ë û¶ï ïë ûï ïî þï ïï ïî þ

üïï ïýï ïï ïî þ

( , ) 0

( , ) 0a

ì = Þ =ïïíï = Þ =ïî

October, 22, 2008 p.15

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

0 10 20 30 40 50

an a ly tic e (N )n u m e rica l e (N )an a ly tic c (N )n u m e rica l c (N )

e(N)=-0.1

c(N)=-0.159

October, 22, 2008 p.16

Trefftz Method

( )( )

ln R cos m ,Rm R

ln cos m ,Rm

rq f r

r q f rr

ìï æ öï ÷çï ÷- - ³çå ÷ï ç ÷ï ÷çè øï=íï æ öï ÷ï ç- - <÷åï ç ÷ï ç ÷è øïî

( ) ( )

u x U x,

ln R cos m , Rm R

ln cos m , Rm

rq f r

r q f rp r

ì é ùï æ öï ê ú÷çï ÷ç- - ³ï ê ú÷çï ÷÷çê úï è øïï ë û=íï é ùï æ öï ê ú÷çï ÷- - <çê ú÷ï ç ÷çï è øê úï ë ûïî

PART 1

October, 22, 2008 p.17

Boundary value problem

1 1t t=-2 2u u=-

PART 2

1( , ) ,

T j jj

G x s c F=

sin,cos,lnexterior

sin,cos,1interior

( , ) ln ( cos ( sin ( )m m m mT m m m m

mG x p p p p ) m q q ) m u xz r r r f r r f

¥ - -

é ù= + + + + + =-å ê úë û

1 ln ln12

cos ( )

cos ( ) ( )

m m m m m m m

am a R

m b app Ra

a b m b aR b

sin ( )

sin ( ) ( )

m m m m m m m

am a R

m b aqq Ra

a b m b aR b

October, 22, 2008 p.18

2u1 1t t=-

22 uu 1 0t =

2 0u =

PART 1 + PART 2 :

1 1ln cos ,

1( ) ln ( cos ( sin

m m m mm m m m

G x u u

R m Rm R

u x p p p p ) m q q ) m

rq f r

r q f rp r

r r r f r r fp

ì é ùï æ öï ê ú÷çï ÷ç- - ³ï åê ú÷çï ÷÷ê úçï è øï ê úï ë û=íï é ùï æ öï ê ú÷çï ÷- - <å çê úï ÷ç ÷çï è øê úï ë ûïî

é ù= + + + + +ê úë û1

¥ì üï ïï ïåí ýï ïï ïî þ

{ }0 01

1( , ) ln ln ( cos ( sin

2m m m m

m m m mm

G x x p p p p ) m q q ) mz z r r r f r r fp

é ù= - + + + + + +å ë û

October, 22, 2008 p.19

Equivalence of solutions derived by using Trefftz method and MFS for annular problem

i i iN

i i i i i

G x ζ x ζ x ζ x ζ x ζπ

x ζ x ζ x ζ x ζ x ζ

a R ρ a ρ b

8 7 8 6 8 51

8 4 8 3 8 2 8 1 8

1( , ) l n l i m {l n l n l n

l n l n l n l n l n }

l n l n l n ,

p lna ln R

ì ü é ù-ï ïï ï ê ú=í ý ê úï ï -ë ûï ïî þ

MFS(Image method)

Trefftz method

The same

October, 22, 2008 p.20

Trefftz method series expand

m m m m

acos m a R

m b ap

p Raa b cos m b a

é ùé ùæ öê ú÷ê úç ÷-çê ú÷ê úç ÷çè øê úê úë ûê úê úì ü +ï ïï ï ê ú=í ý ê úï ï é ùæ öï ï æ öî þ ê ú÷ê úç ÷ç÷+çê ÷ úç÷ê ú÷ç ç÷ç è øê úè øê úë ûê úê ú+ë û

m m m m

asinm a R

m b aq

q Raa b sinm b a

é ùé ùæ öê ú÷ê úç ÷-çê ú÷ê úç ÷çè øê úê úë ûê úê úì ü +ï ïï ï ê ú=í ý ê úï ï é ùæ öï ï æ öî þ ê ú÷ê úç ÷ç÷+çê ÷ úç÷ê ú÷ç ç÷ç è øê úè øê úë ûê úê ú+ë û

2 2 2 2 2 2 2 2

( ) ( )

cos( )

m m m m

m m m m m

m m m m m m m m

R R R a Rm

b a b a b a b a

Without loss of generality

October, 22, 2008 p.21

8 7 2 2 441 9

2 2 4 4

8 3 4 4 445 13

8 7 8 34

( )( ) ( ) ( ) ( )

( ) ( )

m m m m

RR R a b

aR R b b bb

a Ra R a R a b

aR R b b bb

2 2 2 2 2 24

( )( ) ( )1 ( )

mm m m m m m

m m m m m mm

R b a Ra b a b a b ab

2 2 2 2 2 2 2 2

( ) ( )

cos( )

m m m m

m m m m m

m m m m m m m m

R R R a Rm

b a b a b a b a

Image method series expansion

Trefftz series expansion

z 1zÄ

October, 22, 2008 p.22

2 2 2 2 2 2 2 2

( ) ( )

cos( )

m m m m

m m m m m

m m m m m m m m

R R R a Rm

b a b a b a b a

2 108 6 4

24 4 4

6 148 2 2 2 4

8 6 8 24

( ) ( ) ( ) ( )1 ( )

( ) ( )

m m m m

aRR R a a a

aR R bb

ab RR R a a a

ab R b R bb

a aR b R

2 2 2 22 2

2 2 2 2 2 24

( ) ( ) ( )

( )( ) ( )1 ( )

m m m m m

m m m m m mm

a b a bb a

R Ra b a b a b ab

z 1zÄ

October, 22, 2008 p.23

22 2 4

8 5 2 2 443 11

44 4 4

8 1 4 4 447 15

8 5 8 14

( ) ( ) ( ) ( )1 ( )

m m m m

ab Ra a a

aR R b R b R bb

ab Ra a a

aR R b R b R bb

4 2 22 2

2 2 2 2 2 24

) ( ) ( ) ( )

( )( ) ( )1 ( )

m m m m m

m m m m m mm

a a ab a

b R R Ra b a b a b ab

2 2 2 2 2 2 2 2

( ) ( )

cos( )

m m m m

m m m m m

m m m m m m m m

R R R a Rm

b a b a b a b a

z 1zÄ

October, 22, 2008 p.24

2 2 2 2 2 2 2 2

( ) ( )

cos( )

m m m m

m m m m m

m m m m m m m m

R R R a Rm

b a b a b a b a

z 1zÄ

2 2 4 24 12

8 4 2 2 44

4 4 4 48 16

8 4 4 44

8 4 84

( )( ) ( ) ( ) ( )

m m m m

a Ra R a RR R a b

ab b bb

a Ra R a RR R a b

ab b bb

a R a R

2 2 2 2 2 24

) ( ) ( ) ( )

( )( ) ( )1 ( )

m m m m m

m m m m m mm

a R a Rb a

ba b a b a b ab

October, 22, 2008 p.25

Equivalence of solutions derived by Trefftz method and MFS

Trefftz method MFS

1, cos , sin

ln , cos , sin

0,1,2,3, ,

r f r f

r r f r f- -

ln ,jx s j N- Î

Equivalence

addition theorem

October, 22, 2008 p.26

Semi-analytical solution-case 2

October, 22, 2008 p.27

MFS-Image group

1ln ( ) cos ( ),

aR m a R

r q f rr

ìïï - - <åïïïÄ ® íïï - - <ï åïïî

1ln ( ) cos ( ),

Ra m a R

R a aR

a R Rz z

r q f rr

ìïï - - >åïïï® íïï - - >åïïïî

= Þ =

1ln ( ) cos ( ),

Ra m a R

R a aR

r q f rr

ìïï - - >åïïïÄ ® íïï - - >åïïïî

= Þ =

2 2 21

1ln ( ) cos ( ),

2 sin costan ,

cos cos

aR m a R

R m Rm R

b R RR

ìïï - - <åïïï® íïï - - <åïïïî-

4 3 41

1 1 1 1 13 4

1ln ( ) cos ( ),

2 sin costan ,

cos cos

aR m a R

R m Rm R

b R RR

ìïï - - <åïïïÄ ® íïï - - <åïïïî-

October, 22, 2008 p.28

Frozen image location1. Introduction2. Problem statements3. Analytical solution

0 2 4 6

Successive images (20 points)

(0,-5.828)(frozen)

(0,-0.171)(frozen)

October, 22, 2008 p.29

Analytical derivation of location for the two frozen points

2d cR b R 2

1ln ( ) cos ( )ln ,m

Ra mx a R

z yz f¥

=- -® - = >

= Þ =

1ln ( ) cos ,(n )ld d d

aR mx a R

m Ryz fz

=- - å® <-=

(0.171 & 5.828)

October, 22, 2008 p.30

Series of images

The final two frozen images

(0, 5.828)d

frozen

(0, 0.171)c

October, 22, 2008 p.31

Rigid body term

11 1 1

1ln ( ) cos ( ),

ln ln ln ln

aR m B

Ra m B

aR x s x s R a

z y f r

ìïï Ä Þ - - Îåïïïíïïï Þ - - Îåïïî

= \ - - - = -

2 2 2 11

33 2 1

3 2 3 2

1ln ( ) cos ( ),

ln ln ln ln

aR m B

Ra m B

aR x s x s a R

z y f r

ìïï Þ - - Îåïïïíïï Ä Þ - - Îï åïïî

= \ - - + - = -

2ln ln ln ln .... ( )R a a R e N

4 3 4 2 4 1 41

1( , ) ln lim ln ln ln ln ( )ln ( )ln ( )

i i i i c dN iG x x x x x x c N x d N x e N

October, 22, 2008 p.32

1 1 4 3 1 4 2 1 4 1 1 4 1 11

2 2 4 3 2 4 2 2 4 1 2 4 2 21

ln lim ln ln ln ln ( )ln ( )ln ( ) 0

1ln lim ln ln ln ln ( )ln ( )ln ( ) 0

i i i i c dN i

x x x x x c N x d N x e N

3 4 3 3 4 2 3 4 1 3 4 3 31

lim ln ln ln ln ( )ln ( )ln ( ) 0N

i i i i c dN ix x x x c N x d N x e N

1 1 4 3 1 4 2 1 4 1 1 41

2 2 4 3 2 4 2 2 4 1 2 4 2 21

3 3 4 3 3 4 2 3 4 1 3 41

ln ln ln ln lnln ln 1

ln ln ln ln ln ln ln

ln ln ln ln ln

i i i ii

i i i i ci

i i i ii

x x x x xx x

x x x x x x x

x x x x x

1 ( ) 0

ln ln 1 ( ) 0d

x x e N

Matching BCs to determine three coefficient

October, 22, 2008 p.33

1 2 3 4 5 6 7 8 9 10N

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

Numerical approach to determine c(N), d(N) and e(N)

( ) 0-157.9713 10 e N

( )c N -0.26448

( )d N -0.73551

October, 22, 2008 p.34

Semi-analytical solution-case 3

8zÄ x

October, 22, 2008 p.35

MFS-Image group1. Introduction2. Problem statements3. Analytical solution

1( , ) ln cos ( )

1( , ) ln cos )

aU x R m

U x bm b

1 1 11 1

1( , ) ln

) ln c

cos ( )

os ( )

aU x R m

bU x R m

22 2 2

1( , ) ln

1( , ) ln cos ( )

cos ( )

RU x b

RU x a

October, 22, 2008 p.36

Analytical derivation of location for the two frozen points

2222 a

October, 22, 2008 p.37

Numerical approach to determine c(N), d(N) and e(N)

0 10 20 30 40 50N

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

83.445 10 ( ) 0e N-- ´ Þ @

c(N)=-0.8624

d(N)=-0.1375

October, 22, 2008 p.38

Numerical examples-case 1Fixed-Free boundary for annular case

(a) Trefftz method (b) Image method

Contour plot for the analytical solution (m=N).

m=20 N=20

October, 22, 2008 p.39

Numerical examples-case 2

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

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

Present method-image40+2 points

Null-field BIE approach (addition theorem and superposition technique) (M=50)

Dirichlet boundary for half-plane case

October, 22, 2008 p.40

Numerical examples-case 3

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

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

image method analytical solution

(bi-polar coordinate )

Dirichlet boundary for eccentric case

October, 22, 2008 p.41

Conclusions

The analytical solutions derived by the Trefftz method and MFS were proved to be mathematically equivalent for the annular Green’s functions. 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. The image method can be seen as a special case for method of fundamental solution with optimal locations of sources.

October, 22, 2008 p.42

Image method versus MFS

large 3 3

optimal

October, 22, 2008 p.43

Thanks for your kind attentions

You can get more information from our website

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

垚淼 2008研討會

Image method for the Green’s functions of annulus and half-plane Laplace problems

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