1

Equivalence between the Trefftz method and the method of fundamental

solutions for the Green’s function of concentric spheres using the addition

theorem and image concept J.T. Chen

Life-time Distinguished ProfessorDepartment of Harbor and River Engineering,

National Taiwan Ocean UniversitySep. 2-4, 2009

New Forest, UK

BEM/MRM 31

2

Outline

Numerical methods

Trefftz method and MFS Image method (special MFS)

Trefftz method

Equivalence of solutions derived by

Trefftz method and MFS

Conclusions

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 CJS & Antunes PRS

Not good Good

6

Optimal source location

Conventional MFS Alves & Antunes

GoodNot Good

?

7

The simplest image method

Neumann boundary Neumann boundary conditioncondition DirichletDirichlet boundary conditio boundary conditionn

Mirror

8

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)

Greenberg (1971)

9

Image location using degenerate kernel (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

10

Degenerate kernel-2D (addition theorem)

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

rsxU ln),(

11

Addition theorem & degenerate kernel

Addition theorem Subtraction theorem

( )ik x s ikx ikse e e sxsxsx sinsincoscos)cos(

sin( ) sin cos cos sinx s x s x s

( ) /ik x s ikx ikse 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

3-D next page

12

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

13

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

14

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

Interior problem

Exterior problem

15

Weightings 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

True source

Image source

16

Weighting and locations of succesive images

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

Weighting of successive images

Location of successive images

s

Ä

1s

Ä

2s

e

3s

e

4s

e

5s

e

6s

Ä

7s

ÄK

8s

ÄK

17

Derivation of analytical solution using interpolation functions

a

b

( , ) ( , ) 0a bG x s G x s ( ) ( )

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

18

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

Derivation of analytical solution using complementary solutions

19

Numerical approach by collocation BCs

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

20

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

c(N)

d(N)

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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

22

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

23

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

24

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)

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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

26

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

27

Mathematical equivalence the Trefftz method

and MFS Trefftz method series expansion

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 expansion

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

28

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

29

Equivalence of Trefftz method and MFS

3-D

Trefftz method MFS (image method)

30

Conclusions

The analytical solutions derived by using 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.

31

References

J. T. Chen, Y. T. Lee, S. R. Yu and S. C. Shieh, 2009, Equivalence between Trefftz method and method of fundamental solution for the annular Green’s function using the addition theorem and image concept, Engineering Analysis with Boundary

Elements, Vol.33, pp.678-688.

J. T. Chen and C. S. Wu, 2006, Alternative derivations for the Poisson integral formula, Int. J. Math. Edu. Sci. Tech, Vol.37, No.2, pp.165-185.

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