Image method for the Green’s functions of annulus and half-plane Laplace problems
description
Transcript of Image method for the Green’s functions of annulus and half-plane Laplace problems
October, 22, 2008 p.1
Image method for the Green’s functions of annulus and half-plane Laplace
problems
Reporter: Shiang-Chih ShiehAuthors: Shiang-Chih Shieh, Ying-Te Lee and
Jeng-Tzong ChenDepartment of Harbor and River Engineering,
National Taiwan Ocean UniversityOct.22, 2008
October, 22, 2008 p.2
Outline
IntroductionProblem statementsAnalytical solution
Method of Fundmental Solution (MFS) Trefftz method
Equivalence of Trefftz method and MFSSemi-analytical solutionNumerical examplesConclusions
October, 22, 2008 p.3
),(
Trefftz method
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
1( )
M
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, 22, 2008 p.4
Method of Fundamental Solution (MFS)
1( ) ( , )
N
j jj
u x w U x s
( , ) ln , ,jU x s r r x s j N
Interior problem
exterior problem
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 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
( )2 0u xÑ = ( )2 0u xÑ =
D
u(x)
~x
s
Du(x)
~x
r
~s
is the number of complete functions MN is the number of source points in the MFS
1( )
M
j jj
u x c
j
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.6
Optimal source location
z
Ä
1z
Ä
2z
e
3z
e
4z
e
5z
e
6z
Ä
7z
ÄK
8z
ÄK
MFS (special case)Image method
Conventional MFS Alves CJS & Antunes PRS
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.7
Problem statements
Case 1 Annular
a
b
Governing equation :
),( R
xxxG ),(),(2
Boundary condition :
Fixed-Free boundary
u2=0t1=0
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.8
Problem statements
),( R
Governing equation :
xxxG ),(),(2
Dirichlet boundary condition :
( , ) 0,G x x Bz = Î
Case 2 half-plane problem
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
u1=0
u2=0
October, 22, 2008 p.9
Problem statements
Case 3 eccentric problem
b
a
e
R
Governing equation :
xxxG ),(),(2
Dirichlet boundary condition :
( , ) 0,G x x Bz = Î
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
u1=0
u2=0
October, 22, 2008 p.10
Present method- MFS (Image method)
z
Ä
1z
Ä
2z
e
3z
e
4z
e
5z
e
6z
Ä
7z
Ä
8z
Ä
2 2
4 4
6 6
8 8
8 6 8 6
8 4 8 4
8 2 8 2
8 8
inside
( , )
( , )
( , )
( , )
( , )
( , )
( , )
( , )
i i
i i
i i
i i
R
R
R
R
R
R
R
R
1 1
3 3
5 5
7 7
8 7 8 7
8 5 8 5
8 3 8 3
8 1 8 1
outside
( , )
( , )
( , )
( , )
( , )
( , )
( , )
( , )
i i
i i
i i
i i
R
R
R
R
R
R
R
R
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.11
MFS-Image group( )
( ) ( )
( )( )
11
1
1
1
m
m
m
mm
RU ,xcos m
n b
aU ,x ln R cos m
m R
R ,
bz
r
z
z
z
z q
z
q
f
q
z
f
¥
=
¥
+=
æ ö÷ç ÷= - -ç ÷ç ÷÷çè
Ä
= -¶
ø
¶+
å
å
1 11 1
1
1
1
1
1
2
1
1
1
1
( , )cos (
( , ) ln cos ( )
(
)
, )m
m
m
mm
R
R b bR
b
U x
aU x R
bm
mm
R
R
R
R
1
3
1 3
3 31 3
3 3
2 2
23 2
3 2
1
( ,
(
)
( , ) ln cos ( )
, )cos ( )
m
m
mm
m
U x
a
R
b R b bR
U x
R
R m
R
bm
R
R
R
b
m
a
2 2
11
2 2
2
2
22
21
( , ) 1
( , )
cos
1( , ) ln )
( )
cos (
m
m
m
m
m
U x Rm
b
R
a R aR
R a R
m
b
RU x a
m a
4 4 4
2 2
44 2
1
4 4
11
44
1
1
( , )
1( , ) ln cos ( )
1cos
( , )
( )m
mm
m
m
U x Rm
b b
R
R a a aR R
a R R
RU x a m
m a
b
1zÄ
2ze
3ze
4ze
5ze
6zÄ
7zÄK
8zÄK
October, 22, 2008 p.12
MFS-Image group
1zÄ
2ze
3ze
4ze
5ze
6zÄ
7zÄK
8zÄK
5 5
2 2 2
45 2
5
1
4
1
5
5
5
5
5
1
( , )c
1( , ) ln cos ( )
(
os ( )
, )
m
m
m
m
m
U x bm
R
b R b b bR
R b R
R
aU x R
a
mR
R
m
66
1
6 6
2 2 2
6 6
11
66 2
3 3
1( , ) ln cos (
( , ) 1c
( , )
)
os ( )m
mm
m
m
U x Rm
R
R a a a a
RU x a m
b
Ra R
b
a
b
m
R R
7 7
22 2
67 2
7 71
2
7
7
6
1
1 7
7
(
1( , )
, )cos ( )
ln cos )
(
(
, )
m
mm
m
m
aU x R m
R
b R
U s x bm
R
m R
b R b bR
R b R a a
8
8 8
42
88 4
5
8
1
8
8
5
11
( , ) 1cos ( )
1( , ) ln c
, )
os ( )
(
m
m
m
m
m
R
a RR a aR
U
RU x b m
m
x R
a
a
b
m
b
R
a
R
2 2 4 2 41
1 9 8 74 4
2 2 4 2 41
2 10 8 64 4
2 2 24 41
3 11 8 52 2 4 2 4
2 2 24 41
4 12 8 42 2 4 2 4
, ............ ( )
, ........... ( )
, ..... ( )
, .... ( )
ii
ii
ii
ii
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 Rb bR R R
a a a a a
a R a R a Ra aR R R
b b b b b
R
4 4 4 4 41
5 13 8 32 2 4 2 4
4 4 4 4 41
6 14 8 22 2 4 2 4
4 4 44 41
7 15 8 14 4 4 4 4
4 4 44 41
8 16 84 4 4 4 4
, ..... ( )
, ...... ( )
, ...... ( )
, ...... ( )
ii
ii
ii
ii
b b b b bR R
a R a R a a R a
a a a a aR R R
b R b R b b R b
b R b R b Rb bR R R
a a a a a
a R a R a Ra aR R R
b b b b b
8 7 8 6 8 5 8 41
8 3 8 2 8 1 8
1( , ) {ln lim[ (ln ln ln ln
2ln ln ln ln )]}
N
m i i i iN i
i i i i
G x x x x x x
x x x x
z z z z z zp
z z z z- - - -®¥ =
- - -
= - + - - - - - - -å
- - + - + - + -
October, 22, 2008 p.13
Analytical derivation
z
Ä
1z
Ä
2z
e
3z
e
4z
eLL
ez
Ä
cz
e
8 7 8 6 8 5 8 41
8 3 8 2 8 1 8
1( , ) {ln lim[ (ln ln ln ln
2ln ln ln ln ) ( )ln ( )]}
N
i i i iN i
i i i i
G x x x x x x
x x x x c N e N
z z z z z zp
z z z z r- - - -®¥ =
- - -
= - + - - - - - - -å
- - + - + - + - + +
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
5z
e
6z
Ä
7z
Ä
8z
Ä
October, 22, 2008 p.14
Numerical solution
8 7 8 6 8 5 8 41
8 3 8 2 8 1
8 7 8 6 8
1( , ) {ln lim [ (ln ln ln ln
2
ln ln ln ln 8 ) ( )ln ( )]} 0
( , ) 1{ln lim [ (ln ln ln
2
a
a
b b
N
i i i ix x N i
i i i x x
i i iNx x x x
G x x x x x x
x x x x i c N x e N
G xx x x x
x x
z z z z z zp
z z z z
zz z z z
p
- - - -= ®¥ =
- - - =
- - -®¥= =
Þ - + - - - - - - -å
- - + - + - + - + + =
¶ ¶Þ - + - - - - -
¶ ¶ 5 8 41
8 3 8 2 8 1
ln
ln ln ln ln 8 ) ( )ln ( )]} 0b
N
ii
i i i x x
x
x x x x i c N x e N
z
z z z z
-=
- - - =
- -å
- - + - + - + - + + =
( )
( )
( )
( )
8 7 81
8 7 81
ln ln ln ln 1( ) 0
( ) 0ln (1)ln ln ln
aa
b
x xb
N
i ix xi x x
N
i i x xi
x s x x xc N
e Nxx s x x x x
x
z z
z z=
-== =
- ==
ì üé ùï ïï ï é ùê ú- - - + -åï ïï ï ê úê úë ûï ï ì ü ìï ï ïê úï ïï ï ï ï+ =í ý í ý íê ú¶ ¶ï ï ï ïì üé ù¶ï ï ê úï ïî þï ïï ïê ú- - - + -ï ïå ê úí ý ¶ ¶ï ïï ïê ú ë û¶ï ïë ûï ïî þï ïï ïî þ
L
L
üïï ïýï ïï ïî þ
( , ) 0
( , ) 0a
b
a G x
b G x
r z
r z
ì = Þ =ïïíï = Þ =ïî
a
b
u2=0
t1=0
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.15
Numerical and analytic ways to determine c(N) and e(N)
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
0 10 20 30 40 50
N
-0 .2
-0.16
-0.12
-0.08
c(N
) an
d e(
N)
an a ly tic e (N )n u m e rica l e (N )an a ly tic c (N )n u m e rica l c (N )
e(N)=-0.1
c(N)=-0.159
October, 22, 2008 p.16
Trefftz Method
( )( )
( )
1
1
1
1
m
m
m
m
ln R cos m ,Rm R
U x,R
ln cos m ,Rm
z z
z
zz
rq f r
z
r q f rr
¥
=
¥
=
ìï æ öï ÷çï ÷- - ³çå ÷ï ç ÷ï ÷çè øï=íï æ öï ÷ï ç- - <÷åï ç ÷ï ç ÷è øïî
( ) ( )
( )
( )
( )
1
1
2
1 1
2
1 1
2
m
m
m
m
u x U x,
ln R cos m , Rm R
u xR
ln cos m , Rm
z zz
zz
p z
rq f r
p
r q f rp r
¥
=
¥
=
=
ì é ùï æ öï ê ú÷çï ÷ç- - ³ï ê ú÷çï ÷÷çê úï è øïï ë û=íï é ùï æ öï ê ú÷çï ÷- - <çê ú÷ï ç ÷çï è øê úï ë ûïî
å
å
PART 1
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.17
Boundary value problem
1 1t t=-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
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
0
0
1 ln ln12
a Rpp
2 2
2 2
cos ( )
2
cos ( ) ( )
2
m m m
m m
m
m m m m m m m
m m
am a R
R
m b app Ra
a b m b aR b
m b a
2 2
2 2
sin ( )
2
sin ( ) ( )
2
m m m
m m
m
m m m m m m m
m m
am a R
R
m b aqq Ra
a b m b aR b
m b a
October, 22, 2008 p.18
1t
2u1 1t t=-
22 uu 1 0t =
2 0u =
PART 1 + PART 2 :
( )
( )
( )
1
1
0 0
( , )
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 u u
R m Rm R
u xR
m Rm
u x p p p p ) m q q ) m
z zz
zz
z
rq f r
p
r q f rp r
r r r f r r fp
¥
=
¥
=
- -
=
= +
ì é ùï æ öï ê ú÷çï ÷ç- - ³ï åê ú÷çï ÷÷ê úçï è øï ê úï ë û=íï é ùï æ öï ê ú÷çï ÷- - <å çê úï ÷ç ÷çï è øê úï ë ûïî
é ù= + + + + +ê úë û1
¥ì üï ïï ïåí ýï ïï ïî þ
{ }0 01
1( , ) ln ln ( cos ( sin
2m m m m
m m m mm
G x x p p p p ) m q q ) mz z r r r f r r fp
¥- -
=
é ù= - + + + + + +å ë û
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.19
Equivalence of solutions derived by using Trefftz method and MFS for annular problem
N
i i iN
i
i i i i i
ζ
G x ζ x ζ x ζ x ζ x ζπ
x ζ x ζ x ζ x ζ x ζ
a R ρ a ρ b
8 7 8 6 8 51
8 4 8 3 8 2 8 1 8
1( , ) l n l i m {l n l n l n
2
l n l n l n l n l n }
l n l n l n ,
0
0
1
12
p lna ln R
pz
p
ì ü é ù-ï ïï ï ê ú=í ý ê úï ï -ë ûï ïî þ
MFS(Image method)
Trefftz method
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
The same
October, 22, 2008 p.20
Trefftz method series expand
( )
( )
2 2
2 2
2
2
m
m m
m m
m
m mm
m m m m
m m
acos m a R
R
m b ap
p Raa b cos m b a
R b
m b a
z
z
z
z
q
p
q
p
é ùé ùæ öê ú÷ê úç ÷-çê ú÷ê úç ÷çè øê úê úë ûê úê úì ü +ï ïï ï ê ú=í ý ê úï ï é ùæ öï ï æ öî þ ê ú÷ê úç ÷ç÷+çê ÷ úç÷ê ú÷ç ç÷ç è øê úè øê úë ûê úê ú+ë û
( )
( )
2 2
2 2
2
2
m
m m
m m
m
m mm
m m m m
m m
asinm a R
R
m b aq
q Raa b sinm b a
R b
m b a
z
z
z
z
q
p
q
p
é ùé ùæ öê ú÷ê úç ÷-çê ú÷ê úç ÷çè øê úê úë ûê úê úì ü +ï ïï ï ê ú=í ý ê úï ï é ùæ öï ï æ öî þ ê ú÷ê úç ÷ç÷+çê ÷ úç÷ê ú÷ç ç÷ç è øê úè øê úë ûê úê ú+ë û
0
2 2 2
2
2 2 2 2 2 2 2 2
( ) ( )
cos( )
m m m m
m m m m m
m m m m m m m m
a b a
R R R a Rm
b a b a b a b a
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
Without loss of generality
October, 22, 2008 p.21
4 2
8 7 2 2 441 9
2
2 2 4 4
8 3 4 4 445 13
2
2 4
8 7 8 34
( )( ) ( ) ( ) ( )
1 ( )
( )( ) ( ) ( ) ( )
1 ( )
( ) ( )
1 ( )
m
m m m m
im
m
m m m m
im
m
i im
RR R a b
aR R b b bb
a Ra R a R a b
aR R b b bb
R a R
b bab
2 2
2 2 2 2 2 24
( )
( )( ) ( )1 ( )
mm m m m m m
m m m m m mm
R b a Ra b a b a b ab
2 2 2
2
2 2 2 2 2 2 2 2
( ) ( )
cos( )
m m m m
m m m m m
m m m m m m m m
a b a
R R R a Rm
b a b a b a b a
Image method series expansion
Trefftz series expansion
z 1zÄ
2ze
3ze
4ze
5ze
6zÄ
7zÄK
8zÄK
October, 22, 2008 p.22
2 2 2
2
2 2 2 2 2 2 2 2
( ) ( )
cos( )
m m m m
m m m m m
m m m m m m m m
a b a
R R R a Rm
b a b a b a b a
2
2 2 4
2 108 6 4
4
4
24 4 4
6 148 2 2 2 4
4
2 4
2
8 6 8 24
( )
( ) ( ) ( ) ( )1 ( )
( )
( ) ( ) ( ) ( )1 ( )
( ) ( )
1 ( )
m
m m m m
im
m
m m m m
im
m
i im
aRR R a a a
aR R bb
ab RR R a a a
ab R b R bb
a aR b R
ab
2 2 2 22 2
2 2 2 2 2 24
( ) ( ) ( )
( )( ) ( )1 ( )
m m m m m
m m m m m mm
a b a bb a
R Ra b a b a b ab
Image method series expansion
Trefftz series expansion
z 1zÄ
2ze
3ze
4ze
5ze
6zÄ
7zÄK
8zÄK
October, 22, 2008 p.23
2
22 2 4
8 5 2 2 443 11
4
44 4 4
8 1 4 4 447 15
2
2
8 5 8 14
( )
( ) ( ) ( ) ( )1 ( )
( )
( ) ( ) ( ) ( )1 ( )
( ) (
1 ( )
m
m m m m
im
m
m m m m
im
m
i im
ab Ra a a
aR R b R b R bb
ab Ra a a
aR R b R b R bb
ab R
ab
4 2 22 2
4
2 2 2 2 2 24
) ( ) ( ) ( )
( )( ) ( )1 ( )
m m m m m
m m m m m mm
a a ab a
b R R Ra b a b a b ab
Image method series expansion
2 2 2
2
2 2 2 2 2 2 2 2
( ) ( )
cos( )
m m m m
m m m m m
m m m m m m m m
a b a
R R R a Rm
b a b a b a b a
Trefftz series expansion
z 1zÄ
2ze
3ze
4ze
5ze
6zÄ
7zÄK
8zÄK
October, 22, 2008 p.24
Image method series expansion
2 2 2
2
2 2 2 2 2 2 2 2
( ) ( )
cos( )
m m m m
m m m m m
m m m m m m m m
a b a
R R R a Rm
b a b a b a b a
Trefftz series expansion
z 1zÄ
2ze
3ze
4ze
5ze
6zÄ
7zÄK
8zÄK
2
2 2 4 24 12
8 4 2 2 44
4
4 4 4 48 16
8 4 4 44
2 4
2
8 4 84
( )( ) ( ) ( ) ( )
1 ( )
( )( ) ( ) ( ) ( )
1 ( )
( ) (
1 ( )
m
m m m m
im
m
m m m m
im
m
i im
a Ra R a RR R a b
ab b bb
a Ra R a RR R a b
ab b bb
a R a R
bab
2 2
2 2
4
2 2 2 2 2 24
) ( ) ( ) ( )
( )( ) ( )1 ( )
m m m m m
m m m m m mm
a R a Rb a
ba b a b a b ab
October, 22, 2008 p.25
Equivalence of solutions derived by Trefftz method and MFS
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. 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, 22, 2008 p.26
Semi-analytical solution-case 2
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
a
1
2
3
4
5
6
7
8
b
October, 22, 2008 p.27
MFS-Image group
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
a
1
1
1ln ( ) cos ( ),
1ln ( ) cos ( ),
m
m
m
m
aR m a R
m R
Rm R
m
z z
z
z
z
q f
z
r q f rr
¥
=
¥
=
ìïï - - <åïïïÄ ® íïï - - <ï åïïî
11 1
1
11
1 11
2
11
1ln ( ) cos ( ),
1ln ( ) cos ( ),
m
m
m
m
Ra m a R
m aR
m Rm
R a aR
a R Rz z
q fz
r q f rr
¥
=
¥
=
ìïï - - >åïïï® íïï - - >åïïïî
= Þ =
e
32 3
1
33
2 31
2
33
2 2
1ln ( ) cos ( ),
1ln ( ) cos ( ),
m
m
m
m
Ra m a R
m aR
m Rm
R a aR
a R R
q fz
r q f rr
¥
=
¥
=
ìïï - - >åïïïÄ ® íïï - - >åïïïî
= Þ =
M
2 2 21
2
2
2 2 21
2
1 11
2 2
1 2
1ln ( ) cos ( ),
1ln ( ) cos ( ),
2 sin costan ,
cos cos
m
m
m
m
aR m a R
m R
R m Rm R
b R RR
Rz z
z
q f
zr
q f r
q qq
q q
¥
=
¥
=
-
ìïï - - <åïïï® íïï - - <åïïïî-
= =
e
4 3 41
4
4
4 3 41
4
1 1 1 1 13 4
1 1 3
1ln ( ) cos ( ),
1ln ( ) cos ( ),
2 sin costan ,
cos cos
m
m
m
m
aR m a R
m R
R m Rm R
b R RR
R
q f
zr
q f r
q qq
q q
¥
=
¥
=
-
ìïï - - <åïïïÄ ® íïï - - <åïïïî-
= =
October, 22, 2008 p.28
Frozen image location1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
0 2 4 6
-6
-4
-2
0
Successive images (20 points)
(0,-5.828)(frozen)
(0,-0.171)(frozen)
October, 22, 2008 p.29
Analytical derivation of location for the two frozen points
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
b
ax
y
c
d
cR
2d cR b R 2
1
1ln ( ) cos ( )ln ,m
c c c
cc
d
m
d
cd
Ra mx a R
m
a
a R
a
R aR
R
z yz f¥
=- -® - = >
ß
= Þ =
å
1
1ln ( ) cos ,(n )ld d d
m
d dm
d
aR mx a R
m Ryz fz
¥
=- - å® <-=
2
2
2
6 1 0
3 2 2
c
c
c c
c
aR
b R
R R
R
(0.171 & 5.828)
October, 22, 2008 p.30
Series of images
The final two frozen images
a
b
1
42
3
6
54 1i
4i
lim
4 3
lim
4 2
, ln
, ln
N
N
i c
i d
i N x
i N x
(0, 5.828)d
frozen
frozen
(0, 0.171)c
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.31
Rigid body term
a
b
1
42
3
6
54 1i
4i
11
11 1 1
1
2
1 1
1ln ( ) cos ( ),
1ln ( ) cos ( ),
ln ln ln ln
m
m
m
m
aR m B
m R
Ra m B
m a
aR x s x s R a
R
z
z
z z
z
z y f r
z y f r
¥
=
¥
=
ìïï Ä Þ - - Îåïïïíïïï Þ - - Îåïïî
= \ - - - = -
e
Q
2 2 2 11
2
33 2 1
1
2
3 2 3 2
2
1ln ( ) cos ( ),
1ln ( ) cos ( ),
ln ln ln ln
m
m
m
m
aR m B
m R
Ra m B
m a
aR x s x s a R
R
z y f r
z y f r
¥
=
¥
=
ìïï Þ - - Îåïïïíïï Ä Þ - - Îï åïïî
= \ - - + - = -
e
Q
M
2ln ln ln ln .... ( )R a a R e N
4 3 4 2 4 1 41
1( , ) ln lim ln ln ln ln ( )ln ( )ln ( )
2
N
i i i i c dN iG x x x x x x c N x d N x e N
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.32
1 1 4 3 1 4 2 1 4 1 1 4 1 11
2 2 4 3 2 4 2 2 4 1 2 4 2 21
3
ln lim ln ln ln ln ( )ln ( )ln ( ) 0
1ln lim ln ln ln ln ( )ln ( )ln ( ) 0
2
ln
N
i i i i c dN i
N
i i i i c dN i
x x x x x c N x d N x e N
x x x x x c N x d N x e N
x
3 4 3 3 4 2 3 4 1 3 4 3 31
lim ln ln ln ln ( )ln ( )ln ( ) 0N
i i i i c dN ix x x x c N x d N x e N
1 1 4 3 1 4 2 1 4 1 1 41
1 1
2 2 4 3 2 4 2 2 4 1 2 4 2 21
3 3 4 3 3 4 2 3 4 1 3 41
ln ln ln ln lnln ln 1
ln ln ln ln ln ln ln
ln ln ln ln ln
N
i i i ii
c dN
i i i i ci
N
i i i ii
x x x x xx x
x x x x x x x
x x x x x
3 3
( ) 0
1 ( ) 0
ln ln 1 ( ) 0d
c d
c N
d N
x x e N
1x
2x 3x
Matching BCs to determine three coefficient
October, 22, 2008 p.33
1 2 3 4 5 6 7 8 9 10N
-0 .8
-0 .6
-0 .4
-0 .2
0
0.2
c(N
), d
(N)
and
e(N
)c (N )d (N )e (N )
Numerical approach to determine c(N), d(N) and e(N)
( ) 0-157.9713 10 e N
( )c N -0.26448
( )d N -0.73551
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.34
Semi-analytical solution-case 3
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
e
ba
u1=0
u2=0
1ze
2ze
3zÄ
4zÄ
5ze
6ze
7zÄ
8zÄ x
y
October, 22, 2008 p.35
MFS-Image group1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
11
21
1( , ) ln cos ( )
1( , ) ln cos )
( , )
(
m
m
m
m
R
aU x R m
m R
R e
U x bm b
e
m
1 1 11 1
1 1 11 1
1 1 1
1( , ) ln
1( ,
( ,
) ln c
cos ( )
)
os ( )
m
m
m
m
aU x R m
m R
bU x R m
m R
R
2
22 2 2
1
2
1
2
2 2
2
2
1( , ) ln
( ,
1( , ) ln cos ( )
cos ( )
)m
m
m
m
R
RU x b
RU x a
mb
m a
m
m
4
2
1
2
2
2
3
22
1
bR e
R ea
RR
bR e
e Ra
RR
2
5
42
6
32
7
62
8
5
bR e
e Ra
RR
bR e
R ea
RR
2
9
82
10
72
11
102
12
9
bR e
R ea
RR
bR e
R ea
RR
2
13
122
14
112
15
142
16
13
bR e
R ea
RR
bR e
R ea
RR
October, 22, 2008 p.36
Analytical derivation of location for the two frozen points
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
e
ab
c
d
cR
dR
eeR
bR
Ra
Rc
d
d
c
22
,
0)( 2
2222 a
eReba
R dd
October, 22, 2008 p.37
Numerical approach to determine c(N), d(N) and e(N)
0 10 20 30 40 50N
-1 .2
-0.8
-0.4
0
0.4
c(N
), d
(N)
and
e(N
)
c (N )d (N )e (N )
83.445 10 ( ) 0e N-- ´ Þ @
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
c(N)=-0.8624
d(N)=-0.1375
October, 22, 2008 p.38
Numerical examples-case 1Fixed-Free boundary for annular case
(a) Trefftz method (b) Image method
Contour plot for the analytical solution (m=N).
m=20 N=20
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.39
Numerical examples-case 2
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Present method-image40+2 points
Null-field BIE approach (addition theorem and superposition technique) (M=50)
Dirichlet boundary for half-plane case
October, 22, 2008 p.40
Numerical examples-case 3
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1 1.5 2 2.5 3-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
image method analytical solution
(bi-polar coordinate )
Dirichlet boundary for eccentric case
October, 22, 2008 p.41
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. The image method can be seen as a special case for method of fundamental solution with optimal locations of sources.
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
October, 22, 2008 p.42
Image method versus MFS
1. Introduction2. Problem statements3. Analytical solution
4. Equivalence of Trefftz method and MFS
5. Semi-analytical solution6. Numerical examples7. Conclusions
N N
Ku p
K
N
large 3 3
Ku p
K
optimal
October, 22, 2008 p.43
Thanks for your kind attentions
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