#### Transcript A linkage of Trefftz method and method of fundamental

```ICCES Special Symposium on Meshless & Other Novel
Computational Methods in Suzhou, China
A linkage of Trefftz method and method of
fundamental solutions for annular Green’s
Jeng-Tzong Chen
Department of Harbor and River Engineering,
National Taiwan Ocean University
08:00-08:20, Oct. 15, 2008
October, 13-17, 2008 p.1

Outline
Introduction
Problem statements
Present method
 MFS (image method)
 Trefftz method
Equivalence of Trefftz method and MFS
Numerical examples
Conclusions
October 13-17, 2008 p.2

Trefftz method
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
NT
u ( x)   c j
j 1

j
j
is the jth T-complete function
Interior problem:
exterior problem:
ln  , 
m
cos m  and 
m
sin m 
1, r
m
m
cos m f and r sin m f
October 13-17, 2008 p.3

MFS
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
NM
u ( x )   w jU ( x , s j )
j 1
exterior problem
Interior problem
U ( x , s )  ln r , r  x  s j , j  N
October 13-17, 2008 p.4

Trefftz method and MFS
Method
u ( x)   c j
j 1
j
D
~
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
MFS
u ( x )   w jU ( x , s j )
j 1
r
s
u(x)
x
4.
NM
NT
Figure
caption
Introduction
Problem statements
Present method
5.
6.
Trefftz method
Definition
1.
2.
3.
s
~
u(x)
D
x
~
Base

j
, (T-complete function)
U ( x , s )  ln r
, r=|x-s|
G. E.
Ñ u (x ) = 0
Ñ u (x ) = 0
Match B. C.
Determine cj
Determine wj
2
2
NT is the number of complete functions
N M is the number of source points in the MFS
October 13-17, 2008 p.5

Optimal source location
Conventional MFS
Image method
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
Ä e Ä e
Ä
e
Ä
e
s8 s6 s 4 s 2
s0
s1
s3
s5 s7
Ä
Alves CJS & Antunes PRS
MFS (special case)
October 13-17, 2008 p.6

Problem statements
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
Governing equation :
2
Ñ G ( s , x ) = d ( x - s ), x Î W
a
b
BCs:
1. fixed-fixed boundary
2. fixed-free boundary
3. free-fixed boundary
October 13-17, 2008 p.7

Present method- MFS (Image method)
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
inside
o u tsid e
s 2 ( R 2 , 2 )
s1 ( R 1 ,  1 )
 s 4 ( R 4 , 4 )
 s3 ( R3 , 3 )
s6 ( R 6 , 6 )
s5 ( R5 , 5 )
 s8 ( R8 , 8 )
s 4 i 2 ( R 4 i 2 , 4 i 2 )
 s 4 i ( R 4 i , 4 i )
 s7 ( R7 , 7 )
Ä e Ä e
Ä
e
Ä
e
Ä


s8 s6 s 4 s 2
s0
s1
s3
s5 s7
s 4 i3 ( R 4 i3 , 4 i3 )
 s 4 i 1 ( R 4 i 1 ,  4 i 1 )
October 13-17, 2008 p.8

MFS-Image group

Ä e
s4 s2
Ä
s0
 s0 ( R0 , 0 )

1 R0 m

ln
b

( ) c os m ( 0   ), b  R 0


b
m 1 m

2
2
2
2
U ( s0 , x )   b 2
b ba
b b i 1

1
m
R1 ln R, R 5  ( 2 )........
 R a c osRm4 i(3 0R )(, aa 2) R 0
0
R
0
m 1 m 0 R

0
 0
2
2
2
2
2
s 2 ( R 2 ,  2a)
a a
a a i 1
R2 
, R6 
.......
R

(
)
4i2
2
2
 R
R0
R0 b
10 b R 2 m
ln b   (2 ) 2 cos m ( 2  2 )
2
2
b0 b
m  1 mb R
b R 0
b R 0 b i 1
U ( s2R
, x )  , R 
... R 4 i 1 
( 2)
3
7
2
2
2
2
aln a   1 a( R 2 )am cos m ( a ) a
2
2
2
2
2
a0 a 2
m  1 ma R
a R 0
a R 0 a i 1
R4 
,
R

...
R

(
)
2
4i
2
2
2
2
a
R2 b 2
a8
b b
b
b

 R2 
R0
a
R0
 s 4 ( R 4 , 4 )
Ä
s3
e
s1
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of
Trefftz method and
MFS
Numerical examples
Conclusions

5.
6.
s1 ( R1 ,  1 )

 ln R1 

U ( s1 , x )  
 ln R 
1

R0
b

b
R1
 R1 


m 1


m 1
b
1
(
b
m R1
1
(
a
m R1
) cos m ( 1   )
m
) cos m ( 1   )
m
2
R0
 s3 ( R3 , 3 )

N

1 é
ù 1

ln
R

G m ( x , s ) =  êln x - 1s - Rå4 (m ln x - s 4 i - 3 + ln x - s 4 i - 2 - ln x - s 4 i - 1 - ln x - 3s 4 i 
ú
ln
b   ( i = )1 co s m ( 4   )
m 1 m
2
p

ë
û

b
m 1 m
U ( s3 , x )  

U ( s4 , x )  
1


1
R
ln
R

m
4

 ln a 
3
 ( ) cos m ( 4   )

m 1 m

a
m 1 m
2
2
2
R2
b
b
a
R4
a
a


R



 R4 
 R4  2 R0
3
b
R
R
R1
a
R1
b
3
2
)
(
b
R3
(
a
R3
) cos m ( 3   )
m
) cos m ( 3   )
R3 
m
b
2
a
2
R0
October 13-17, 2008 p.9

Analytical derivation
Ä
L
s4i
G ( x, s) =
N
1 é
êln x - s - å (ln x - s 4 i2 p êë
i= 1
3
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
Ä e
Ä
e
Ä
s4 s2
s0
s1
s3
+ ln x - s 4 i-
2
Ä
L
s 4 i- 1
ù
- ln x - s 4 i- 1 - ln x - s 4 i ) + c ( N ) + d ( N ) lnr ú
ú
û
October 13-17, 2008 p.10

Numerical solution
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
a
ìï r = a Þ G ( x a , s )
ï
í
ïï r = b Þ G ( x b , s )
î
b
N
1 é
êln x a - s - å (ln x a - s 4 i- 3 + ln x a - s 4 i- 2 - ln x a - s 4 i- 1 - ln x a - s 4 i )+ c ( N ) + d ( N ) ln a ]= 0
2 p êë
i= 1
N
1 é
êln x b - s - å (ln x b - s 4 i- 3 + ln x b - s 4 i- 2 - ln x b - s 4 i- 1 - ln x b - s 4 i )+ c ( N ) + d ( N ) ln b ]= 0
G ( xb , s ) =
2 p êë
i= 1
G ( xa , s ) =
N
ìï
ü
ï
ï ln x a - s - å (ln x a - s 4 i- 3 + ln x a - s 4 i- 2 - ln x a - s 4 i- 1 - ln x a - s 4 i )ï
ï
ï
ïï
i= 1
ïï +
í
ý
N
ï
ï
ï ln x - s - å (ln x - s
ï
ï
b
b
4 i - 3 + ln x b - s 4 i - 2 - ln x b - s 4 i - 1 - ln x b - s 4 i ) ï
ïïî
ïïþ
i= 1
é1
ê
ê1
ë
ln a ùìïï c ( N ) ïü
ï
úí
ý=
úï d ( N ) ï
ln b ûî
ï
ï
þ
ì
ïï
í
ïîï
0 ïü
ï
ý
0 ïþ
ï
October 13-17, 2008 p.11

Interpolation functions
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
G ( x a , s ) = G ( xb , s ) = 1
G ( x, s) = Gm ( x, s) - (
a
G ( x, s) =
b
ln r - ln a
ln b - ln a
)G m ( xb , s ) - (
N
1 ìïï é
í êln x - s - å (ln x - s 4 i2 p ïïî êë
i= 1
3
ln b - ln r
ln b - ln a
+ ln x - s 4 i-
2
) G m ( x a , s ), a £ r £ b
- ln x - s 4 i-
1
- ln x - s 4 i
æ
2
R0 N
ln r - ln a çç
- (
) çln b ( 2 ) ln b - ln a çç
a
è
m
ö
2
÷
1 éê a N R 0 ù
ú cos m ( q - f ) ÷
÷
( 2)
å
÷
ê
ú
÷
m
b
m= 1
÷
ëb
û
ø
æ
2
R0 N
ln b - ln r çç
- (
) çln R ( 2 ) ln b - ln a çç
a
è
m
öü
ï
2
÷
ïï
1 éê a N a ù
÷
ú
( 2)
cos m ( q - f ) ÷
å
ý
÷
êb
ú
÷
R
m= 1 m
÷ïï
0û
ë
ø
ïþ
Þ G ( x, s) =
-
)
¥
¥
ù
R
ln R 0 - ln a
ln b - ln R 0
1 é
êln x - s - 2 N ln 0 - (
) ln b - (
) ln r ú
ú
2 p êë
a
ln b - ln a
ln b - ln a
û
1
2p
N
å
(ln
x - s 4 i-
3
+ ln x - s 4 i-
2
- ln x - s 4 i-
1
- ln x - s 4 i
)
i= 1
October 13-17, 2008 p.12

Trefftz Method
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
PART 1
m
ìï
¥
ö
r ÷
ï ln R - å 1 æ
ç ÷ cos m (q - f ) , R ³ r
ï
ç ÷
ï
m= 1 m ç
èR ø
ïï
U (s , x ) = í
m
ï
ö
¥
1æ
R
ï
ç ÷
ï ln r - å
cos m (q - f ) , R < r
ç ÷
ï
÷
÷
m= 1 m ç
ïïî
èr ø
2 p u ( x ) = U (s , x )
ïì
ï
ï
ï
ï
ï
u ( x ) = ïí
ï
ï
ï
ï
ï
ïïî
1 éê
ln R 2 p êê
ë
é
1 ê
êln r 2p ê
ë
¥
å
m= 1
¥
å
m= 1
m
ù
1 ær ö
çç ÷
cos m (q - f )ú
÷
ú ,R ³ r
÷
m çè R ø
ú
û
m
ù
ö
1 æ
÷ cos m (q - f )ú , R < r
çç R ÷
ú
÷
m çè r ÷
ø
ú
û
October 13-17, 2008 p.13

Boundary value problem
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
u 1 = - u1
NT
u2 = - u2
GT ( x , s ) = å c j F j ,
j= 1
PART 2
ïìï
í
ïï
î
é- ln b (ln a - ln R ) ù
0 ú
ê
ê 2 p (ln a - ln b ) ú
p 0 ïü
ï = ê
ú
ý ê
ú
p 0 ïþ
ê- (ln b - ln R 0 )
ú
ï
ê
ú
êë2 p (ln b - ln a )
ú
û
interior
 1,  cos m  ,  sin m 
exterior
 ln  , 
m
¥
G T ( x , z ) = p 0 + p 0 ln r + å éê( p m r
ë
m= 1
m
+ pm r
m
m
- m
é
ù
é
ù
êcos m q êR m - a m ( a ) m ú
ú
ê
ú
ê 0
ú
R0 û
ë
ê
ú
ê
ú
2m
2m
2mp b - a
ìï p m ü
ê
ú
ï
ï
ï = ê
ú
í
ý
ê
ú
ïï p m ïï
é
ù
a m
î
þ ê m m
m
m R0 m ú
ê
ú
a b cos m q b (
) - a (
)
ê
ê
úú
R
b
0
ë
ûú
ê
ê
ú
2m
2m
2mp b - a
ê
ú
ë
û
(
)
(
)
cos m  , 
m
sin m 
) cos m f + ( q m r
ïìï
í
ïï
î
m
+ qm r
- m
) sin m f ù
= - u ( x)
ú
û
é
é
ùù
êa m b m sin m q êb m ( a ) m - a m ( R 0 ) m úú
ê
ê
úú
R0
b
ë
ûú
ê
ê
ú
2m
2m
2mp b - a
ê
ú
q m ïü
ï = ê
ú
ý
ê
ú
é
ù
q m ïþ
ï
êsin m q êb m ( R 0 ) m - a m ( a ) m ú
ú
ê
ú
ê
ú
b
R0
ë
û
ê
ú
ê
ú
2m
2m
2mp b - a
ê
ú
ë
û
(
(
)
)
October 13-17, 2008 p.14

u1
u2  u2
u2
u1 = 0
u1   u1
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
u2 = 0
PART 1 + PART 2 :
G ( x, s) = u + u
ìï
ï
ï
ï
ï
ï
u ( x ) = ïí
ï
ï
ï
ï
ï
ï
ïî
u ( x) =
é
1 ê
ln R 2 p êê
ë
m
ù
ö
1 ær ÷
ç
cos
m
q
f
(
)úú , R ³ r
å
çç ÷
÷
m= 1 m è R ø
ú
û
m
é
ù
¥
ö
1 ê
1æ
R÷
ú
ç
cos m (q - f )ú , R < r
ç ÷
êln r - å
÷
ç
÷
2p ê
m= 1 m è r ø
ú
ë
û
¥
¥
1 ìï
é
í p 0 + p 0 ln r + å ê( p m r
ë
2 p ïïî
m= 1
G ( x, s) =
1
m
+ pm r
¥
- m
ln x - s + p 0 + p 0 ln r + å éë( p m r
m= 1
2p
{
) cos m f + ( q m r
m
+ pmr
- m
m
+ qm r
- m
) cos m f + ( q m r
m
ü
ï
) sin m f ù
úý
û
ïïþ
+ qmr
- m
) sin m f ù
û
}
October 13-17, 2008 p.15

1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
Equivalence of solutions derived by Trefftz
method and MFS
5.
6.
Trefftz method
G ( x, s) =
1
ln x {
2p
¥
s + p 0 + p 0 ln r + å éë( p m r
m= 1
é- ln b (ln a - ln R ) ù
0 ú
ê
ê
ìï p 0 üï
(ln a - ln b ) úú
ï
ï = ê
í
ý ê
ú
ïï p 0 ïï
ú
î
þ ê- (ln b - ln R 0 )
ê
ú
Image method
êë é(ln b - ln a )N
úû
1
G ( x, s) =
êln x - s - å (ln x - s 4 i 2 p êë
i= 1
m
+ pmr
- m
) cos m f + ( q m r
m
+ qmr
- m
) sin m f ù
û
}
é
R
ln a - ln R 0 ù
Equivalence
ê- (2 N ln 0 + ln b
)ú
ìï c ( N ) ü
ïï
ï
í
ý=
ïîï d ( N ) ïþ
ï
3
ê
ê
ê ln b êêë (ln b + ln x - s 4 i - 2 - ln
a
ln R 0
ln a )
x - s 4 i-
1
ln a - ln b ú
ú
ú
ú
ú
û
- ln x - s 4 i )
+ c ( N ) + d ( N ) ln r ]
October 13-17, 2008 p.16

Equivalence of Trefftz method and MFS
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
Trefftz method series expand
æ 2m a m a m
a m r m
R m a m
m m
m m
R m r mö
2m
çb ( ) ( )
a b ( ) ( )
a b ( 0) ( )
÷
b ( 0) ( ) ÷
ç
÷
ç
R0
r
R0
b
b
r
÷
b
b
ç
÷
+
cos( m f )
ç
÷
2m
2m
2m
2m
2m
2m
2m
2m
ç - a + b
÷
- a + b
- a + b
- a + b
ç
÷
÷
çç
÷
è
ø
The same
Image method series expand
R2
R6
a
2
a
2
a
2
a
2
a
(
2
a
2m
2
)
a b
m
2m
m
m
R0 r
R0r
=
( 2 ) ) + ... =
) + (
) + (
) + ... = (
) + (
e (
2m
2m
2
a 2m
(b ) - (a )
R0r b
R0r b
R0r
r
r
1- ( )
b
m
m
m
m
2
m
2
R8
m
a R0
m
a R0
m
a
m
2m
R0
m
)
m
2
r
b r
=
2m
2m
a 2m
(b ) - (a )
1- ( )
b
(
2
2
2
2
a R a 2 m
) + ...] = ( 2 ) + ( 2 ) + ( 2 0 ( 2 ) ) + ... =
) + (
Ä - [(
b r b
b r
b r
r
r
R4
a R0
m
2m
2
a r
a r
m
m
)
m
2
R0
b R0
=
=
...
+
)
)
(
(
+
)
) + ( 2
) + ...] = ( 2
) + (
Ä - [(
2m
2m
2
2
2
a 2m
(b ) - (a )
b R0 b
b R0 b
b R0
R7
R3
1- ( )
b
r
e (
r
R1
m
r
m
) + (
r
R5
m
2
2
a r
m
) + ... = (
R0r
b
2
m
) + (
a r a
m
R0r a
b
2
b
2
m
2
) + (
2
2
R0r
b
2
a
a r
m
(
a
2
b
2
(
2
2
m
(
2
m
) ) + ... =
R0r
b
1- (
2
a
)
m
m
=
)
2m
- R0 r
(b )
2m
m
- (a)
2m
b
October 13-17, 2008 p.17

1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
Equivalence of solutions derived by Trefftz
method and MFS
5.
6.
Trefftz method
m
m
1, r co s m f , r sin m f
ln r , r
- m
co s m f , r
m = 0,1, 2, 3, L , ¥
- m
MFS
Equivalence
ln x - s j , j Î N
sin m f
October 13-17, 2008 p.18

Numerical examples-case 1
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
fixed-fixed boundary
m=20
(a) Trefftz method
N=20
(b) Image method
Contour plot for the analytical solution (m=N).
October 13-17, 2008 p.19

Numerical examples-case 2
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
fixed-free boundary
m=20
(a) Trefftz method
N=20
(b) Image method
Contour plot for the analytical solution (m=N).
October 13-17, 2008 p.20

Numerical examples-case 3
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
free-fixed boundary
m=20
(a) Trefftz method
N=20
(b) Image method
Contour plot for the analytical solution (m=N).
October 13-17, 2008 p.21

Numerical and analytic ways to
determine c(N) and d(N)
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
c(N) & d(N)
0
-4
-8
analytic c(N)
numerical c(N)
analytic d(N)
numerical d(N)
-12
0
10
20
30
40
50
N
Values of c(N) and d(N) for the fixed-fixed case.
October 13-17, 2008 p.22

Numerical examples- convergence
0.02
Image method
Trefftz method
Conventional MFS
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
0.01
u(6,/3)
0
-0.01
-0.02
0
2
4
6
8
10
m
Pointwise convergence test for the potential
by using various approaches.
October 13-17, 2008 p.23

Numerical examples- convergence rate
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
method and MFS
Numerical examples
Conclusions
5.
6.
Best
Image method
Worst
Trefftz method
Conventional MFS
October 13-17, 2008 p.24

Conclusions
1.
2.
3.
Introduction
Problem statements
Present method
4.
Equivalence of Trefftz
and MFS
Numerical examples
Conclusions
5.
6.
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
October 13-17, 2008 p.25