image method for the green’s functions of annulus and half-plane laplace problems
DESCRIPTION
Image method for the Green’s functions of annulus and half-plane Laplace problems. Reporter: Shiang-Chih Shieh Authors: Shiang-Chih Shieh, Ying-Te Lee and Jeng-Tzong Chen Department of Harbor and River Engineering, National Taiwan Ocean University Oct.22, 2008. Outline. Introduction - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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:),(
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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
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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
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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
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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
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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
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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
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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
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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
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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
- - + - + - + -
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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
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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
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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
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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
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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
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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
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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
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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
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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
![Page 22: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/22.jpg)
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
![Page 23: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/23.jpg)
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
![Page 24: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/24.jpg)
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
![Page 25: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/25.jpg)
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
![Page 26: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/26.jpg)
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
![Page 27: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/27.jpg)
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-
= =
![Page 28: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/28.jpg)
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)
![Page 29: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/29.jpg)
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)
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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
![Page 31: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/31.jpg)
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
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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
![Page 33: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/33.jpg)
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
![Page 34: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/34.jpg)
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
![Page 35: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/35.jpg)
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
![Page 36: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/36.jpg)
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
![Page 37: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/37.jpg)
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
![Page 38: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/38.jpg)
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
![Page 39: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/39.jpg)
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
![Page 40: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/40.jpg)
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
![Page 41: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/41.jpg)
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
![Page 42: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/42.jpg)
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
![Page 43: Image method for the Green’s functions of annulus and half-plane Laplace problems](https://reader036.vdocument.in/reader036/viewer/2022081515/56815295550346895dc0bd70/html5/thumbnails/43.jpg)
October, 22, 2008 p.43
Thanks for your kind attentions
You can get more information from our website
http://msvlab.hre.ntou.edu.tw/
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