November, 28-29, 2008 p.1 A linkage of Trefftz method and method of fundamental solutions for...

November, 28-29, 2008 p.1

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

functions using addition theorem

Shiang-Chih ShiehAuthors: Shiang-Chih Shieh, Ying-Te Lee, Shang-Ru Yu and Je

ng-Tzong Chen Department of Harbor and River Engineering,

National Taiwan Ocean UniversityNov.28, 2008

The 32nd Conference on Theoretical and Applied Mechanics

November, 28-29, 2008 p.2

Outline

Introduction

Problem statements

Present method MFS (image method) Trefftz method

Equivalence of Trefftz method and MFS

Numerical examples

Conclusions

November, 28-29, 2008 p.3

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:

November, 28-29, 2008 p.4

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

November, 28-29, 2008 p.5

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( ) ( , )

N

j jj

u x w U x s

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( )

N

j jj

u x c

j

November, 28-29, 2008 p.6

Optimal source location




MFS (special case)Image method

Conventional MFS Alves CJS & Antunes PRS

Not good Good

November, 28-29, 2008 p.7

Problem statements

a

b

Governing equation :

BCs:

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




November, 28-29, 2008 p.8

Present method- MFS (Image method)




……

November, 28-29, 2008 p.9

b

a

MFS-Image group




1

1

)(cos)(1

ln

)(cos)(1

ln

),(

m

m

m

m

mRm

R

mR

mxU

'

R

aR

R

a

R

R

a

2

''

'

R

bR

b

R

R

b 2

''

November, 28-29, 2008 p.10

MFS-Image group

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

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




November, 28-29, 2008 p.11

Analytical derivation




November, 28-29, 2008 p.12

Numerical solution




a

b

November, 28-29, 2008 p.13

Interpolation functions

a

b




November, 28-29, 2008 p.14

Trefftz Method

PART 1




November, 28-29, 2008 p.15

Boundary value problem

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

PART 2




November, 28-29, 2008 p.16

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




November, 28-29, 2008 p.17

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

November, 28-29, 2008 p.18

The same





November, 28-29, 2008 p.19





Trefftz method MFS

ln ,jx s j N- Î

Equivalence

addition theorem

November, 28-29, 2008 p.20

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

November, 28-29, 2008 p.21




fixed-free boundary




m=20 N=20

November, 28-29, 2008 p.22




free-fixed boundary




m=20 N=20

November, 28-29, 2008 p.23

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 )

November, 28-29, 2008 p.24

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

November, 28-29, 2008 p.25

Numerical examples- convergence rate

Image method Trefftz methodConventional MFS




Best Worst

November, 28-29, 2008 p.26

Optimal location of MFS

Depends on loading

Depends on geometry


4. Equivalence of Trefftz and MFS


November, 28-29, 2008 p.27

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




November, 28-29, 2008 p.28

Conclusions

Optimal image group points depend on loading

Frozen image point depends on geometry




November, 28-29, 2008 p.29

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

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