reporter: h.c. shieh adviser: dr. j.t. chen department of harbor and river engineering,

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

FrameMotivation and literature review

Two-dimensional Green’s functionGreen’s

function

Conclusions

MFS (Image method)

Trefftz method

BVP without sources

Numerical methods

Boundary Element MethodFinite Element Method Meshless Method

Method of fundamental solutions

jjj xsUcxu

is the fundamental solution),( xsU

Interior case Exterior case

This method was proposed by Kupradze in 1964.

Optimal source location

Conventional MFS Alves & Antunes

GoodNot Good

The simplest image method

Neumann boundary Neumann boundary conditioncondition DirichletDirichlet boundary conditio boundary conditionn

Mirror

Conventional method to determine the image location

a rr’

OR a aOR

a OR OR

O R’RO

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

Greenberg (1971, 取巧法 )

Image location (Chen and Wu, 2006)

s 's2'

1ln cos ( )

Rigid body term

2-D Degenerate kernal

),(cos)(1

,)(cos)(1

s( , )R q

rx( , )r f

x( , )r f

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),(

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

,)(cos)(1

sxifxs

sxifsxr

3-D degenerate kernel

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

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

s ( , , )( , , )

exterior

interior

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

Eccentric annulus

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

Eccentric problem

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

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

Case 3

Bipolar coordinates

221 lnln rr

1r 2r 1r

221 lnln rr

121 lnln rr1r

2r1cs 2cs

Bipolar coordinates

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

Annular (EABE, 2009) to eccentric case

s s1s2s4s3

…. ….

Image point+

Source point

sxsxGm ln2

4 3 4 2 4 1 41

lim ln ln ln lnN

i i i iN

x s x s x s x s remainder term

BxsxG 0),(

Series of images

-4 -2 0 2

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 ( )}

i i i iN

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

c N x s c N x s e N

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

V alu e

C o effic ien tc 1(N )

c 2(N )

e (N )

Coefficients sc1 sc2

Contour plot of eccentric annulus problem

-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

(bipolar coordinates )

Dirichlet boundary for the eccentric case 1

Analytical derivation of locationfor the two frozen points

2cs 1cs

1ln ( ) cosln ( ,)c c c

m as x s a R

R a aR

=- -å® - = >

= Þ =

1ln ( ) cos ( )ln ,m

aR mx a R

=- -- <å® =

cRR cc 212

4 2 2 4 2 2 2 2 42 2 2

a a b b a d dc

- + +=

Eccentric case

4 3 4 2 4 1

1 1 2 2

1( , ) ln lim ln ln ln ln

l ( )n l (n )

i i i i

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

c N s c N s e

1 2 and c cs s focuses

Image sources

True source

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

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

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

A half plane with a circular hole

4 3 4 2 4 1

1 1 2 2

1( , ) ln lim ln ln ln ln

l ( )n l (n )

i i i i

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

c N s c N s e

image source

true source

1 2 and c cs s focuses

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

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

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

Linking of MFS and image method

MFS (special case) Conventional MFS

1( ) ln ( , )

u x x s w U x s

4 3 4 21

1( ) {ln

lim ln ln

( ) ( ) ln }

u x x s

x s x s

c N d N

Image method versus MFS

optimal

All the strength need to be determined.

Only three coefficients are required to be determined.

Conventional MFS Image method

Outline

Boundary value problem without sourcesConclusions

Trefftz method

The method was proposed by Trefftz in 1926.

jjjcxu

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

Interior case Exterior case

is the jth T-complete functionj

Trefftz method and MFS

Method Trefftz method MFS

Definition

Figure sketch

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

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

Interior problem

Exterior problem

The weighting of the image source in the

3-D problem

as ),,( sRs

),,( sRs

Interior problem Exterior problem

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

, ( ), ( ) ( )

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

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

, ........ ( )

, ....... ( )

, ... ( )

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

Interpolation functions

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

( ) ( )

Ni i i i

Ni i i i i

N Ns 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

Analytical derivation

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

1 1 ( )( , ) lim ( )

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

Numerical solution

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

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

x s x s x s x s x s a

x s x s x s x s x s b

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

0 2 4 6 8 10

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

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

PART 1 PART 2

PART 1

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

4 ( )!( , )

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

4 ( )!

m n n snn ms s

m msm n n sn

n mm P P R

R n m RG x s

Rn mm P P R

Boundary value problem

1( , )

T j jj

G x s c

Interior:

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

n PmPm

Exterior:

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

n PmPm

( 1)0000

( , ) [ (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

R b aAB a b R

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

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

PART 1 + PART 2 :

1G2G11 GG

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

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

4 ( )!( )

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

4 ( )!

m n n snn ms s

m msm n n sn

n mm P P R

R n m RG x

Rn mm P P R

( 1)0000

( , ) [ (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

C P m D P m

( 1)0000

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

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

nn m n m

nm n nm nn m

n m n m

nm n nm n

C P m D P m

Results

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

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

Outline

Trefftz solution

( 1)0000

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

nn m n m

nm n nm nn m

n m n m

nm n nm n

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 ( )! ( )

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

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

s s1s2s4 s3 s5 s9s7

)(1121211

s s1 s3s2s4s6s8s10

s s1s2s4 s3 s5 s9s7

)(1)()(

s s1 s3s2s4s6s8s10

)(1)()(

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, , ,

j Nx s

Equivalence

addition theorem

linkage

True source

Outline

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

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

a=1.0 d=10

2 ( ) 0,u x x D

Animation - An infinite plane with two

circular holes

u=-1 u=1

s1 s2s3s4sc1 sc2

)lnln()(lim)( 21 ooNrrNqxu

)(ln)(ln)(

)lnlnln(ln

14142434

NesxNcsxNc

sxsxsxsx

)(Nq )(Nq

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

0 4 8N

c 1(N )

c 2(N )

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

436.0sinh

Coefficients

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

Image solution bipolar coordinates

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

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

null-field BIEM

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

An infinite space with two cavities (anti-

symmetric BC)

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

2 ( ) 0,u x x D

()(lim)(

rrNqxu

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

0 4 8N

V a lu e

c 1(N )

c 2(N )

q (N )

1)(lim

0)(lim,0)(lim 21

NcNcNN

Coefficients

Contour plot of an infinite space with two spherical cavities

Bispherical coordinates Image method Null-field BIE

( ) ( ) 0

0 (10 )

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

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

x-y planey

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

Outline

Optimal location of MFS

• Depends on loading (image location)

• Depends on geometry (frozen image point)

Final images to bipolar (bispherical) focus

2-D 3-D

Bipolar coordinates Bispherical coordinates

Equivalence of Trefftz method and MFS

Trefftz method MFS (image method)

Image solution for BVP without sources

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

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

The end

A half plane with a circular hole

2007, Ke J. N. 2009, Image method

An infinite plane with two circular holes subject to Neumann boundary

01 t02 t

Extra terms of complementary solutions

01 tt 2 0t t

( , )S R

4 3 4 2 4 1 4

1 1 2 2 1 2

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

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

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

Source point

Frozen point

The method provide of JW. Lee

Frozen point

jjMO xxdxxd

111 lnlimlnlnlimln

complementary solutions

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

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

4 3 4 2 4 1 4

1( ) {ln [ln ln ln ln ]

2( )ln ( )ln

( ) ln lim ln

i i i ii

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

f N s x f N s x

d N x x

Where the N=M

Results

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

Image method)

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

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

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

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

Image method (50+2 points)

u1=0u2=0 ( , )ss R

Numerical examples 3: An infinite plane with double holes

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

Image method (20+4+10 point)

t1=0t2=0

Animation- an infinite plane with double holes

The final images terminate at the focus

1ds 2ds

Multipole expansion

4 3 4 2 4 1 4

11 1 2

( ) ( )

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

ln lim ln ln

( )ln (

i i i ii

j jM Mj j

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

1fs2fs

true source

1 2 and c cs s focus

image source

2dsand1ds

Multipoles

t1=0 t2=0

Trefftz solution

( 1)0000

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

nn m n m

nm n nm nn m

n m n m

nm n nm n

C P m D P m

R b aAB a b R

Image solution

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

1 1 ( )( , ) lim ( )

Ni i i i

Ni i i i i

x s x s x s x s x s

The same

The simplest MFS

1-D Rod

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

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

Present method- MFS (Image method)

……

4 3 4 2 4 1 4

1 4 3 4 2 4 1 4

1 1( , ) lim

Ni i i i

i i i i i

w w w wG x s remainder term

x s x s x s x s x s

An infinite space with two cavities (symmetric BC)

u=V1=1 u=V2=1

q q Obtain image weighting4 4

4 34 4

4 54 2

4 24 1

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

4 24 5

4 14 2

1qs2qs

33sqw44sqw

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

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

x-y planey

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

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

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

Illustrative examples – An eccentric annulus

1a2 5 .b

x1c 2c

u=V1=0 u=V2=1

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

0 4 8N

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)

Contour plot of eccentric annulus

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

Image solution bipolar coordinates null-field BIEM

Outline

Boundary value problem without sourceConclusions

Optimal source location

Conventional MFS Alves CJS & Antunes PRS

Not good Good

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

),,( x

),,( Rs ),,('

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

m n n nn m

n m Rm P P

x s b n m b

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

' ( )! ( ')

m n n nn m

R n m bm P P

b x s b n m R

The weighting of the image point

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

),,( x

),,( Rs

),,('2

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

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

)'()'(

The weighting of the image point

Chen and Wu-image method (2006)

1ln ( ) cos ( ),

( , ) 1ln ( ) cos ( ),

mU x sR m R

1ln cos ( )

Analytical derivation of location for the two frozen points

2 12c cR b R 1 1 1

1ln ( ) cosln ( ,)c c c

m as x s a R

R a aR

=- -å® - = >

= Þ =

1ln ( ) cos ( )ln ,m

aR mx a R

=- -- <å® =

(0.171 & 5.828)

a=1, b=3

1 2 3 4 5 6 7 8 9 10N

)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

Analytical derivation of locationfor the two frozen points

1 22 2

,c cc c

a bR R d

a b d RR a

4 2 2 4 2 2 2 2 42 2 2

a a b b a d b d dc

- + - - +=

cRR cc 212

reporter: h.c. shieh adviser: dr. j.t. chen department of harbor and river engineering,

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