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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
October 13-17, 2008 p.3海大陳正宗終身特聘教授
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:
October 13-17, 2008 p.4海大陳正宗終身特聘教授
MFS
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
1( ) ( , )
MN
j jj
u x w U x s
( , ) ln , ,jU x s r r x s j N
Interior problem
exterior problem
October 13-17, 2008 p.5海大陳正宗終身特聘教授
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. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
1( )
TN
j jj
u x c
j
October 13-17, 2008 p.6海大陳正宗終身特聘教授
Optimal source location
0s
Ä
1s
e
2s
e
3s
Ä
4s
Ä
5s
e
6s
e
7s
Ä
8s
Ä
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
MFS (special case)Image method
Conventional MFS Alves CJS & Antunes PRS
October 13-17, 2008 p.7海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
October 13-17, 2008 p.8海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
October 13-17, 2008 p.9海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
October 13-17, 2008 p.10海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
October 13-17, 2008 p.11海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
a
b
October 13-17, 2008 p.12海大陳正宗終身特聘教授
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
1( , ) ln ln ln ln ln
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
October 13-17, 2008 p.13海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
October 13-17, 2008 p.14海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
October 13-17, 2008 p.15海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
{ }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
October 13-17, 2008 p.16海大陳正宗終身特聘教授
Equivalence of solutions derived by Trefftz method and MFS
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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
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 ln
p
r
- - -=
é= - - - + - - - - -ê å
êë+ +
Image method
October 13-17, 2008 p.17海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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
October 13-17, 2008 p.18海大陳正宗終身特聘教授
Equivalence of solutions derived by Trefftz method and MFS
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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
October 13-17, 2008 p.19海大陳正宗終身特聘教授
Numerical examples-case 1
(a) Trefftz method (b) Image method
Contour plot for the analytical solution (m=N).
fixed-fixed boundary
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
m=20 N=20
October 13-17, 2008 p.20海大陳正宗終身特聘教授
Numerical examples-case 2
(a) Trefftz method (b) Image method
Contour plot for the analytical solution (m=N).
fixed-free boundary
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
m=20 N=20
October 13-17, 2008 p.21海大陳正宗終身特聘教授
Numerical examples-case 3
(a) Trefftz method (b) Image method
Contour plot for the analytical solution (m=N).
free-fixed boundary
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
m=20 N=20
October 13-17, 2008 p.22海大陳正宗終身特聘教授
Numerical and analytic ways to determine c(N) and d(N)
Values of c(N) and d(N) for the fixed-fixed case.
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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 )
October 13-17, 2008 p.23海大陳正宗終身特聘教授
Numerical examples- convergence
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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
October 13-17, 2008 p.24海大陳正宗終身特聘教授
Numerical examples- convergence rate
Image method Trefftz methodConventional MFS
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
Best Worst
October 13-17, 2008 p.25海大陳正宗終身特聘教授
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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz and MFS
5. Numerical examples6. Conclusions
October, 13-17, 2008 p.26海大陳正宗終身特聘教授
Thanks for your kind attentions
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The 32nd Conference on Theoretical and Applied Mechanics