october, 13-17, 2008 p.1 海大陳正宗終身特聘教授 a linkage of trefftz method and method of...

October, 13-17, 2008 p.1海大陳正宗終身特聘教授

A linkage of Trefftz method and method of fundamental solutions for annular Green’s

functions using addition theorem

Jeng-Tzong Chen

Department of Harbor and River Engineering,National Taiwan Ocean University

08:00-08:20, Oct. 15, 2008

ICCES Special Symposium on Meshless & Other Novel Computational Methods in Suzhou, China

October 13-17, 2008 p.2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

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

1, cos sinm mm and mr f r f

Interior problem:

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


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

0s

Ä

1s

e

2s

e

3s

Ä

4s

Ä

)( 4 3 4 2 4 1 41

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

2

N

i i i ii

G x s x s x s x s x s x s c N d N lnrp - - -

=

é ùê ú= - - - + - - - - - + +åê úë û

LL4 1is -

Ä

4is

Ä





Numerical solution

( ) ]

( ) ]

4 3 4 2 4 1 41

4 3 4 2 4 1 41

1( , ) ln ln ln ln ln ( ) ( ) ln 0

2

1( , ) ln ln ln ln ln ( ) ( ) ln 0

2

N

a a a i a i a i a ii

N

b b b i b i b i b ii

G x s x s x s x s x s x s c N d N a

G x s x s x s x s x s x s c N d N b

p

p

- - -=

- - -=

éê= - - - + - - - - - + + =åêëéê= - - - + - - - - - + + =åêë

( )

( )

4 3 4 2 4 1 41

4 3 4 2 4 1 41

ln ln ln ln ln1 ln ( ) 0

1 ln ( ) 0ln ln ln ln ln

N

a a i a i a i a ii

N

b b i b i b i b ii

x s x s x s x s x sa c N

b d Nx s x s x s x s x s

- - -=

- - -=

ì üï ïï ï- - - + - - - - -åï ï é ùì ü ì üï ï ï ïï ïï ï ï ï ï ïê ú+ =í ý í ý í ýê úï ï ï ï ï ïï ï ï ïë ûî þ î þï ï- - - + - - - - -åï ïï ïï ïî þ

( , )

( , )a

b

a G x s

b G x s

r

r

ì = Þïïíï = Þïî




a

b


Interpolation functions

( , ) ( , ) 1a bG x s G x s= =

ln ln ln ln( , ) ( , ) ( ) ( , ) ( ) ( , ),

ln ln ln lnm m b m aa b

G x s G x s G x s G x s a bb a b a

r - - r= - - £ r £

- -

a

b

)( 4 3 4 2 4 1 41

2 20 02 2

1

2 202 2

0


2

ln ln 1( ) ln ( ) ( ) cos ( )ln ln

ln ln 1( ) ln ( ) ( )ln ln

N

i i i ii

mN N

m

mN N

m

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

R Ra ab m

b a m ba b

Rb a aR

b a m Ra b

p

rq f

r

- - -=

¥

=

ì éïï ê= - - - + - - - - -åíï êëïîæ öé ù ÷ç- ÷ç ê ú ÷- - -åç ÷ê úç ÷- ç ÷ç ë ûè ø

é ù- ê ú- - ê ú- ë û1cos ( )m q f

¥

=

üæ öï÷ïç ÷ïç ÷-åç ý÷ç ï÷ç ÷ç ïè øïþ

( )

0 0 0

4 3 4 2 4 1 41

ln ln ln ln1( , ) ln 2 ln ( )ln ( )ln

2 ln ln ln ln

1ln ln ln ln

2

N

i i i ii

R R a b RG x s x s N b

a b a b a

x s x s x s x s

rp

p - - -=

é ù- -ê úÞ = - - - -ê ú- -ë û

- - + - - - - -å





Trefftz Method

( )( )

( )

1

1

1

1

m

m

m

m

ln R cos m ,Rm R

U s,xR

ln cos m ,Rm

rq f r

r q f rr

¥

=

¥

=

ìï æ öï ÷ç- - ³åï ÷ç ÷çï è øïï=íï æ öï ÷çï - - <÷å çï ÷ç ÷ï è øïî

( ) ( )

( )

( )

( )

1

1

2

1 1

2

1 1

2

m

m

m

m

u x U s,x

ln R cos m ,Rm R

u xR

ln cos m ,Rm

p

rq f r

p

r q f rp r

¥

=

¥

=

=

ì é ùï æ öï ê ú÷çï - - ³÷çï ê ú÷çè øï ê úï ë ûï=í é ùï æ öï ê ú÷çï - ÷ - <çï ê ú÷ç ÷çï è øê úï ë ûïî

å

å

PART 1





Boundary value problem

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

PART 2

1( , ) ,

TN

T j jj

G x s c F=

= å

mm

mmmm

mm

sin,cos,lnexterior

sin,cos,1interior

0 01

( , ) 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

( )( )

( )( )

0

0

0 0

ln ln ln

2 ln ln

ln ln

2 ln ln

b a R

a bp

p b R

b a

p

p

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

( )

( )

00

2 2

0

02 2

cos ( )

2

cos ( ) ( )

2

m m m

m m

m

m m m m m m m

m m

am R a

R

m b ap

p Raa b m b a

R b

m b a

q

p

q

p

é ùé ùê úê ú-ê úê úë ûê úê ú-ì ü ê úï ïï ï ê ú=í ý ê úï ï é ùï ïî þ ê úê ú-ê úê úë ûê úê ú-ê úë û

( )

( )

0

02 2

0

02 2

2

2

m m m m m m

m m

m

m m m m m

m m

Raa b sinm b ( ) a ( )

R b

m b aq

q R asinm b ( ) a ( )

b R

m b a

q

p

q

p

é ùé ùê úê ú-ê úê úë ûê úê ú-ì ü ê úï ïï ï ê ú=í ý ê úï ï é ùï ïî þ ê úê ú-ê úê úë ûê úê ú-ê úë û





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




{ }0 01

1( , ) ln ln ( cos ( sin

2m m m m

m m m mm

G x s x s p p p p ) m q q ) mr r r f r r fp

¥- -

=

é ù= - + + + + + +å ë û


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

{ }0 01

1( , ) ln ln ( cos ( sin

2m m m m

m m m mm

G x s x s p p p p ) m q q ) mr r r f r r fp

¥- -

=

é ù= - + + + + + +å ë û

Trefftz method

( )

]

4 3 4 2 4 1 41


2

( ) ( )

N

i i i ii

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

c N d N ln

p

r

- - -=

é= - - - + - - - - -ê å

êë+ +

Image method


Equivalence of Trefftz method and MFS

2 2 2

2 2 2 2 222 6 0 0

2 2 2 220 0 0

2 2

0 02 2 2 2 2

24 8 0 0 0

2 2 2 2 22

1

1

m mm

m mm m m m m

m mm

m mm

mm m m m m

mm

a a b( )

R R a a a a a R R( ) ( ) ... ( ) ( ) ( ( ) ) ...

aR R b R b ( b ) ( a )( )b

a R a R( )

R R a R a R a R a b[( ) ( ) ...] ( ) ( ) ( ( ) ) ...

ab b b b ( b ) (( )b

r rr r r r r

r rr r r r r

+ + = + + + = =--

Ä - + + = + + + = =--

e

2

2 2

2 2 2 2 2 22 0 0

2 2 2 2 2 2 223 7 0 0 0

02 2 2

20 0 0 0

2 2 2 2 221 5

1

1

m

m mm

mm m m m m

m mm

mm m

m m m m m

m

a )

a a( )

a a a a a b R R[( ) ( ) ...] ( ) ( ) ( ( ) ) ...

aR R b R b R b b R b ( b ) ( a )( )b

R( )R R a R a Rb( ) ( ) ... ( ) ( ) ( ( ) ) ...

aR R b b b b b ( b )( )b

r rr r r r r

rr r r r r r

Ä - + + = + + + = =---

-+ + = + + + = =

-e

2 2m m( a )-

The same




2 0 2 0

0 0

2 2 2 2 2 2 2 2

( ) ( ) ( ) ( )( ) ( ) ( ) ( )cos( )

m m m m m m mm m m m m m m

m m m m m m m m

a a aR a Rb a ba b bR R bb b b ma b a b a b a b

r rr r f

æ ö÷ç ÷ç ÷ç ÷ç ÷- - +ç ÷ç ÷- + - + - + - +ç ÷÷ç ÷çè ø

Trefftz method series expand

Image method series expand


Equivalence of solutions derived by Trefftz method and 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


Numerical examples-case 1

(a) Trefftz method (b) Image method

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

fixed-fixed boundary




m=20 N=20





fixed-free boundary




m=20 N=20





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 )


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


Numerical examples- convergence rate

Image method Trefftz methodConventional MFS




Best Worst


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. Convergence rate of Image method(best), Trefftz method and MFS(worst) due to optimal source locations in the image method


4. Equivalence of Trefftz and MFS


October, 13-17, 2008 p.26海大陳正宗終身特聘教授

Thanks for your kind attentions

You can get more information from our website

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

The 32nd Conference on Theoretical and Applied Mechanics

october, 13-17, 2008 p.1 海大陳正宗終身特聘教授 a linkage of trefftz method and method of...

Documents