bipolar coordinates, image method and method of fundamental solutions

April, 8-12, 2009 p.1海大陳正宗終身特聘教授

Bipolar coordinates, image method and method of fundamental solutions

Jeng-Tzong Chen

Department of Harbor and River Engineering,National Taiwan Ocean University

15:30-15:50, April 10, 2009

ICCES 09 in Phuket, Thailand


Prof. Wen Hwa Chen 60th birthday symposium

My Ph.D. Committeemember

April 8-12, 2009 p.3海大陳正宗終身特聘教授

Outline

Introduction

Problem statements

Present method MFS (image method) Trefftz method

Equivalence of Trefftz method and MFS

(2-D and 3-D annular cases)

Numerical examples

Conclusions


Trefftz method

1. Introduction2. Problem statements3. Present method

4. Equivalence of Trefftz method and MFS

5. Numerical examples6. Conclusions

1( )

TN

j jj

u x c

j is the jth T-complete function

ln , cos sinm mm and m

exterior problem:


MFS




1( ) ( , )

MN

j jj

u x w U x s

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

Interior problem

exterior problem


Trefftz method and MFS

Method Trefftz method MFS

Definition

Figure caption

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

G. E.

Match B. C. Determine cj Determine wj

( , ) lnU x s r

1( ) ( , )

MN

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 TN

MN is the number of source points in the MFS




1( )

TN

j jj

u x c

j


b

a

New point of view of image locationinstead of Kelvin concept (Chen and Wu, IJMEST 2006)

1

1

)(cos)(1

ln

)(cos)(1

ln

),(

m

m

m

m

mRm

R

mR

mxU

'2

''

R a aR

a R R

'

R

bR

b

R

R

b 2

''


Numerical examples - convergence rate

Image method Trefftz methodConventional MFS




Best Worst

Collocation point

Source point

True source point


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




Trefftz method MFS

1, cos , sin

ln , cos , sin

0,1,2,3, ,

m m

m m

m m

m m

m

r f r f

r r f r f- -

= ¥L

ln ,jx s j N- Î

Equivalence

addition theorem

2-D 3-DTrue source


Key point

Addition theorem (Degenerate kernel)

2-D

RmR

m

RmRm

R

rm

m

m

m

),(cos1

ln

),(cos1

ln

ln

1

1

3-D

11 0

11 0

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

( )!1

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

( )!

nnm m

m n n nn m

nnm m

m n n nn m

n mm P P R

R n m R

r n m Rm P P R

n m

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

mm

s( , )R q

R

r

rx( , )r f

x( , )r f

o

iU

eU


Annular cases (Green’s functions)

2-D

EABE 2009

xy

z

3-D

a

b

ab ),,( R

),( R

u1=0 u1=0u2=0

u2=0

True source


Numerical examples - case 1

(a) Trefftz method (b) Image method

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

fixed-fixed boundary (u=0)

m=20 N=20

10 terms

100 terms

Melnikov and Arman, 2001Computer unfriendly


Trefftz solution Image solution

Annular

circle

Annular sphere

Analytical solutions

1

4 3where i

i

b bw

R a

Addition

theorem

Addition

theorem


Analytical and numerical approaches to determine the strength

2-D 3-D

0 10 20 30 40 50

N

-12

-8

-4

0

c(N

) &

d(N

)

an a ly tic c (N )n u m erica l c (N )an a ly tic d (N )n u m erica l 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 )

4 3 4 21

4 1 4

1( , ) {ln lim ( ln ln

2ln ( )ln ) ln }( ) ln

N

i iN

m

i i c d

G x x

c N d N

x x

x x x x

x x x xp

x x x x

- -®¥=

-

= - + - + -

- - - - + - + -

å4 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

w w w wG x s

x s x s x s x s x s

c N d N

2

2

ln lnln ln( )

ln ln

R ac

RN bN

a a b

ln ln

n ln( )

l

b R

bd

aN

((

))

N R aa

b R b ac N

( )

(( )

)

N a b Rd

a

bN

b R a


Bipolar coordinates

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

focus


Animation – eccentric case

2

The final images

terminate at the

focus

46

7

1

5

3

(0,0.75)

4 3 4 2 4 1 41

1( , ) ln lim ln ln ln ln

2

( ) ln ( ) ln ( ) ,

N

i i i

c

iN

d

iG x x x x x x

c N x d N x e N

c

true source

image sources

and c d focus

d


Animation - a half plane with a circular hole

1

2

3

4x

The final imagesterminate at thefocus

4 3 4 2 4 1 41

1( , ) ln lim ln ln ln ln

2

( ) ln ( ) ln ( ) ,

N

i i i

c

iN

d

iG x x x x x x

c N x d N x e N

d

c

image source

true source

and c d focus


Animation- an infinite plane with double holes

1

234

The final images terminate at the focus

1d 2d

Multipole expansion

1

1

2

2

4 3 4 2 4 1 4

11

2

12

1

2

1

( ) ( )

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 M

j j

d

G x x f Nx x x f Nx x

x x N xN

x

xd

1f2f

true source

1 2 and f f focus

image source

2dand1dMultipoles


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 ),( R

a

b


A half plane with a circular hole

-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 (40+2 points)

u1=0

u2=0

),( R

a

h


An infinite plane with double holes

ab

h

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


Conclusions

The analytical solutions derived by the Trefftz method and MFS were proved to be mathematically equivalent for the annular Green’s functions (2D and 3D) after using addition theorem (degenerate kernel).

We can find final two frozen image points which are focuses in the bipolar coordinates.

The image idea provides the optimal location of MFS and only at most 4 by 4 matrix is required.


4. Equivalence of Trefftz and MFS



Thanks for your kind attentions

You can get more information from our website

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


Optimal source location

0s

Ä

1s

e

2s

e

3s

Ä

4s

Ä

5s

e

6s

e

7s

Ä

8s

Ä




MFS (special case)Image method

Conventional MFS Alves CJS & Antunes PRS


Problem statements

a

b

Governing equation :

( )2 , ( ),G s x x s xd WÑ = - Î

BCs:

1. fixed-fixed boundary2. fixed-free boundary3. free-fixed boundary





Present method- MFS (Image method)

0s

Ä

1s

e

2s

e

3s

Ä

4s

Ä

5s

e

6s

e

7s

Ä

8s

Ä

2 2 2

4 4 4

6 6 6

8 8 8

4 2 4 2 4 2

4 4 4

inside

( , )

( , )

( , )

( , )

( , )

( , )i i i

i i i

s R

s R

s R

s R

s R

s R

1 1 1

3 3 3

5 5 5

7 7 7

4 3 4 3 4 3

4 1 4 1 4 1

outside

( , )

( , )

( , )

( , )

( , )

( , )i i i

i i i

s R

s R

s R

s R

s R

s R





MFS-Image group0s

ee1s

Ä

3s4s ÄÄ

00

0

0 0 0

1 0

0

01

0 0

1ln ( ) c

1ln ( ) c

( , )

(os ( )

,

os (

),

),m

m

m

m

aR m

s R

U s

Rb m b

a

Rb

m R

m

Rx

1 11 1

2

0

1 11

1 1 1

1

1

1 0

1

1ln ( ) cos (

1ln ( ) cos (

(

( ,

)

)

)

, )

m

m

m

m

aR

b

m

s

R mm R

R b bR

b R R

m R

R

U s x

22

1

2 2 2

2

22

1

2

22

0 0

1ln ( ) cos (

1ln ( ) cos

( ,

(

)

( , ))

)

m

m

m

m

Ra

R

m

s

b mm

m a

a R aR

R a R

U x b

R

s

44

1

2 2

44 4 02

1 1

4 4 4

44

1

4

( , )

1ln (

1ln ( ) cos

) co

(

s )

)

(( , )

m

m

m

m

Ra m

m a

a R a aR R R

R a R b

s R

Rb m

m bU s x

3 31 3

2 2

23 3 02

3 3 3

3

3 31 3

3 2

( , )

( , )1

ln ( ) cos (

1ln ( ) ( )

)

cos

m

m

m

m

bR m

m R

R b b bR R R

s R

U s xa

R mm R

b R R a

2s

2 2 2 2 21

1 5 4 32 2

0 0 0

2 2 2 2 21

2 6 4 22 2

0 0 0

2 2 2 2 210 0 0

3 7 4 12 2 2 2 2

2 2 2 2 210 0 0

4 8 42 2 2 2 2

, ........ ( )

, ....... ( )

, ... ( )

, ... ( )

i

i

i

i

i

i

i

i

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 aa R a R a a R a

R R Rb b b b b

( )4 3 4 2 4 1 41

1( , ) ln ln ln ln ln

2

N

m i i i ii

G x s x s x s x s x s x s- - -=

é ù= - - - + - - - - -åê ú

pë û





Analytical derivation





Numerical solution




a

b


Interpolation functions

a

b





Trefftz Method

PART 1





Boundary value problem

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

PART 2





1u

2u11 uu

22 uu 1 0u =

2 0u =

PART 1 + PART 2 :

( )

( )

( )

1

1

0 01

( , )

1 1ln cos ,

2

1 1ln cos ,

2

1( ) ln ( cos ( sin

2

m

m

m

m

m m m mm m m m

m

G x s u u

R m Rm R

u xR

m Rm

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

rq f r

p

r q f rp r

r r r f r r fp

¥

=

¥

=

¥ - -

=

= +

ì é ùï æ öï ê ú÷çï - - ³å ÷çï ê ú÷çè øï ê úï ë ûï=í é ùï æ öï ê ú÷çï - - <÷å çï ê ú÷ç ÷ï è øê úï ë ûïîì üï ïï é ù= + + + + +åí ýê úë ûïïî

ïïïþ





Equivalence of solutions derived by Trefftz method and MFS




Equivalence ( )

( )( )( )

0

0

0 0

ln ln ln

ln ln

ln ln

ln ln

b a R

a bp

p b R

b a

é ù- -ê úê úì ü -ï ïï ï ê ú=í ý ê úï ï - -ï ï ê úî þê ú-ê úë û

0 0

0

ln ln(2 ln ln )

( ) ln lnln ln( )(ln ln )

R a RN b

c N a a bb Rd Nb a

é ù-ê ú- +ì ü ê úï ï -ï ï =ê úí ý -ï ï ê úï ïî þ -ê ú

-ê úë û


The same




Equivalence of solutions derived by Trefftz method and MFS


Numerical examples-case 2



fixed-free boundary




m=20 N=20


Numerical examples-case 3



free-fixed boundary




m=20 N=20


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

Values of c(N) and d(N) for the fixed-fixed case.




0 10 20 30 40 50

N

-12

-8

-4

0

c(N

) &

d(N

)

an a ly tic c (N )n u m erica l c (N )an a ly tic d (N )n u m erica l d (N )

ba

Rab

a

RNNc

lnln

lnlnlnln)(

2

2

ab

RbNd

lnln

lnln)(


Numerical examples- convergence




Pointwise convergence test for the potential by using various approaches.

0 2 4 6 8 10

m

-0 .02

-0.01

0

0.01

0.02

u (6 ,/3 )

Im a g e m e th o dT re fftz m e th o dC o n v en tio n a l M F S

bipolar coordinates, image method and method of fundamental solutions

Documents

double holesimage method

circular holeimage method

image idea

ahan infinite plane

aba half plane

frozen image points

number of complete functions

annular greens functions