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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
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Outline
Introduction
Problem statements
Present method MFS (image method) Trefftz method
Equivalence of Trefftz method and MFS
Numerical examples
Conclusions
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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:
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November, 28-29, 2008 p.4
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
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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. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
1( )
N
j jj
u x c
j
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Optimal source location
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
Not good Good
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November, 28-29, 2008 p.7
Problem statements
a
b
Governing equation :
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
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November, 28-29, 2008 p.8
Present method- MFS (Image method)
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
……
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November, 28-29, 2008 p.9
b
a
MFS-Image group
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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
''
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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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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November, 28-29, 2008 p.11
Analytical derivation
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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November, 28-29, 2008 p.12
Numerical solution
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
a
b
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November, 28-29, 2008 p.13
Interpolation functions
a
b
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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November, 28-29, 2008 p.14
Trefftz Method
PART 1
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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November, 28-29, 2008 p.15
Boundary value problem
1 1u u=-2 2u u=-
PART 2
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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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
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
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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
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November, 28-29, 2008 p.18
The same
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz method and MFS
5. Numerical examples6. Conclusions
Equivalence of solutions derived by Trefftz method and MFS
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November, 28-29, 2008 p.19
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
ln ,jx s j N- Î
Equivalence
addition theorem
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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
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November, 28-29, 2008 p.21
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
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November, 28-29, 2008 p.22
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
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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.
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 )
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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
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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
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Optimal location of MFS
Depends on loading
Depends on geometry
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz and MFS
5. Numerical examples6. Conclusions
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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
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Conclusions
Optimal image group points depend on loading
Frozen image point depends on geometry
1. Introduction2. Problem statements3. Present method
4. Equivalence of Trefftz and MFS
5. Numerical examples6. Conclusions
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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