1 applications of addition theorem and superposition technique to problems with circular boundaries...
Post on 18-Dec-2015
217 views
TRANSCRIPT
![Page 1: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/1.jpg)
1
Applications of addition theorem and superposition Applications of addition theorem and superposition technique to problems with circular boundaries technique to problems with circular boundaries subject to concentrated forces and screw dislocationssubject to concentrated forces and screw dislocations
Reporter: Chou K. H.Advisor: Chen J. T.Date: 2008/07/11Place: HR2 307
![Page 2: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/2.jpg)
2
Outline
Motivation and literature review Derivation of the Green’s function
Superposition technique Addition theorem and boundary density Adaptive observer system Linear algebraic equation
Numerical examples Green’s function for the concentrated force problems Green’s function for the screw dislocation problems
Conclusions
![Page 3: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/3.jpg)
3
Outline
Motivation and literature review Derivation of the Green’s function
Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic equation
Numerical examples Green’s function for the concentrated force problems Green’s function for the screw dislocation problems
Conclusions
![Page 4: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/4.jpg)
4
Motivation
Numerical methods for engineering problemsNumerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless methodFDM / FEM / BEM / BIEM / Meshless method
BEM / BIEMBEM / BIEM
Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity
Boundary-layer Boundary-layer effecteffect
Ill-posed modelIll-posed modelConvergence Convergence raterate
![Page 5: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/5.jpg)
5
Present approach
(s, x)iK
(s, x)eK
(s, x(x) (s) (s))B
dBKj y=ò
Fourier expansionFourier expansion
(s, x), s x
(s, x), x s
i
e
K
K
ìï ³ïíï >ïî0
1
cos sinm mm
a a m b mq q¥
=
+ +å
Advantages of degenerate kernel1. No principal value2. Well-posed3. Exponential convergence4. Free of boundary-layer effect5. Mesh-free generation
Degenerate kernelDegenerate kernel
![Page 6: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/6.jpg)
6
Literature review
0 ut
0u0
n
u
Laplace problem [Chen, Shen and Wu, 2005]
Helmholtz problem [Chen, Chen, Chen and Chen, 2007]
biharmonic problem [Chen, Hsiao and Leu, 2006]
anti-plane piezoelectricity problem [Chen and Wu, 2006]
Green’s function for Laplace [Chen, Ke and Liao, 2008], Helmholtz [Chen and Ke, 2008]
and biharmonic problems [Chen and Liao, 2008]
Green’s function for the screw dislocation problem (present work)
Ä
![Page 7: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/7.jpg)
7
Outline
Motivation and literature review Derivation of the Green’s function
Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic equation
Numerical examples Green’s function for the concentrated force problems Green’s function for the screw dislocation problems
Conclusions
![Page 8: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/8.jpg)
8
Green third identity
22
(s, )(s, ),
s
GG
n
xx
¶¶
(s, )2 (x, ) (s, x) (s, ) (s) (s, x) (s) ( , x), x
i i
ii i i
B Bs
GG T G dB U dB U D B
n
xp x x x
¶= - + Î È
¶ò ò
Äx
11
(s, )(s, ),
s
GG
n
xx
¶¶(s, )
(s, ), ii
s
GG
n
xx
¶
¶
???
![Page 9: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/9.jpg)
9
Superposition technique
11,
ww
n
¶¶
x
22 ,
ww
n
¶¶
, ii
ww
n
¶¶
11 11 ,
ww
n
¶¶
x
11 22 ,
ww
n
¶¶ 1
1, ii
ww
n
¶¶
22 11 ,
ww
n
¶¶
22 22 ,
ww
n
¶¶ 2
2 , ii
ww
n
¶
¶
Free field Typical BVP
![Page 10: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/10.jpg)
10
Outline
Motivation and literature review Derivation of the Green’s function
Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic equation
Numerical examples Green’s function for the concentrated force problems Green’s function for the screw dislocation problems
Conclusions
![Page 11: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/11.jpg)
11
Addition theorem for the radial-based fundamental solution
irzz sx ln)ln(
1
1
),(cos)(1
ln
,)(cos)(1
lnln
m
m
m
m
RmR
m
RmRm
Rr
s( , )R q
R
r
rx( , )r f
x( , )r f
o
iU
eU
y
Rr
fq
sz
xzr j
x
![Page 12: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/12.jpg)
12
Addition theorem for the angle-based fundamental solution
1
1
1( ) sin ( ),
( , ; , )1 R
( ) sin ( ),
m
m
m
m
m Rm R
R
m Rm
rq q f r
j r f q
f p q f rr
¥
=
¥
=
ìïï + - £ïïï=íïï - - - >ïïïî
å
å
Ij
o( , )x r f
( , )x r fEj
),( Rs
1
1
1
ln( ) ln( ) ln(1 )
1ln(1 ) ( )
1 R( )
1 R( ) [cos ( ) sin ( )]
sx s x
x
ms s
mx x
im
im
m
m
zz z z
z
z z
z m z
e
m e
m i mm
q
fr
q f q fr
¥
=
¥
=
¥
=
- = + -
- =-
=-
=- - + -
å
å
å
y
Rr
fq
sz
xzr j
x
![Page 13: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/13.jpg)
13
Boundary density discretization
Fourier Fourier seriesseries
Ex . constant Ex . constant elementelement
01
01
(s) ( cos sin ), s
(s) ( cos sin ), s
n nn
n nn
u a a n b n B
t p p n q n B
q q
q q
¥
=
¥
=
= + + Î
= + + Î
å
å
Fourier series expansions - boundary density
![Page 14: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/14.jpg)
14
Outline
Motivation and literature review Derivation of the Green’s function
Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic equation
Numerical examples Green’s function for the concentrated force problems Green’s function for the screw dislocation problems
Conclusions
![Page 15: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/15.jpg)
15
Adaptive observer system
Source pointSource point
Collocation pointCollocation point
![Page 16: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/16.jpg)
16
Outline
Motivation and literature review Derivation of the Green’s function
Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic system
Numerical examples Green’s function for the concentrated force problems Green’s function for the screw dislocation problems
Conclusions
![Page 17: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/17.jpg)
17
Linear algebraic system
s
(s, )0 (s, x) (s, ) (s) (s, x) (s)
B B
GT G dB U dB
n
xx
¶= -
¶ò ò
0B
1B
2B
NB
[ ] [ ]{ }GG
n
ì ü¶ï ïï ï =í ýï ï¶ï ïî þU T
[ ]
00 01 0
10 11 1
0 1
N
N
N N NN
U U U
U U UU
U U U
é ùê úê úê ú= ê úê úê úê úë û
L
L
M M O M
L
0
1
2
N
G
nG
nG
Gn
n
G
n
ì ü¶ï ïï ïï ïï ï¶ï ïï ïï ï¶ï ïï ïï ï¶ï ïì ü¶ ï ïï ïï ï ï ï¶=í ý í ýï ï ï ï¶ï ïî þ ï ï¶ï ïï ïï ïï ïï ïï ïï ï¶ï ïï ïï ï¶ï ïî þ
M
![Page 18: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/18.jpg)
18
Flowchart of the present approach
Typical BVP(addition theorem)
Null-field boundary integral equation
Potential of domain point
Fundamental solutionSeries formClose form
Problem of the fundamental solution
Superposition technique
Original problem
![Page 19: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/19.jpg)
19
Outline
Motivation and literature review Derivation of the Green’s function
Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic system
Numerical examples Green’s function for the concentrated force problems Green’s function for the screw dislocation problems
Conclusions
![Page 20: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/20.jpg)
20
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary con
dition A half-plane with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Neumann boundary condition(2) Neumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Neumann boundary con
dition
![Page 21: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/21.jpg)
21
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary con
dition A half-plane with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Neumann boundary condition(2) Neumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Neumann boundary con
dition
![Page 22: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/22.jpg)
22
The Green’s function of the annular ring
2 ( , ) ( )x d xÑ = -G x x
( , ) 0G x x =
( , ) 0G x x = 10b
4a )0,5.7( x
y
![Page 23: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/23.jpg)
23
The Green’s function of the annular ring
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Null-field BIE approach(addition theorem and
superposition technique)(M=50)
Null-field BIE approach(Green’s third identity)
[Chen and Ke, CMC, 2008]
![Page 24: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/24.jpg)
24
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary conditio
n A half-plane with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Neumann boundary condition(2) Neumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Neumann boundary con
dition
![Page 25: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/25.jpg)
25
An eccentric ring
2 ( , ) ( )x d xÑ = -G x x
4.0a
( 0.4,0)-
(0,0.75)x
x
y
1.0b =
( , ) 0G x x =
( , ) 0G x x =
![Page 26: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/26.jpg)
26
An eccentric ring
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Null-field BIE approach (addition theorem and
superposition technique) (M=50)
Melnikov’s method [Melnikov and Melnikov (2001)]
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
Null-field BIE approach(Green’s third identity)
[Chen and Ke, CMC, 2008]
![Page 27: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/27.jpg)
27
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary condi
tion A half-plane with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Neumann boundary condition(2) Neumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Neumann boundary con
dition
![Page 28: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/28.jpg)
28
An infinite plane with an aperture subjected to the Neumann boundary condition
2 ( , ) ( )x d xÑ = -G x x
x
y
( , )0
x
G x
n
x¶=
¶
1a (1.25,0)x
![Page 29: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/29.jpg)
29
An infinite plane with an aperture subjected to the Neumann boundary condition
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Null-field BIE approach(addition theorem and
superposition technique)(M=50)
Image method
![Page 30: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/30.jpg)
30
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary conditio
n A half-plane problem with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Neumann boundary condition(2) Neumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Neumann boundary con
dition
![Page 31: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/31.jpg)
31
A half-plane problem with an aperture subjected to the Dirichlet boundary condition
1a
(2,1)
( , ) 0G x
( , ) 0G x
3
2 ( , ) ( )x d xÑ = -G x x
![Page 32: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/32.jpg)
32
A half-plane problem with an aperture subjected to the Dirichlet boundary condition
-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
Null-field BIE approach(addition theorem and
superposition technique)(M=50)
Melnikov’s method [Melnikov and Melnikov (2001)]
-2 -1 0 1 2 3 40
1
2
3
4
5
6
Null-field BIE approach(Green’s third identity)
[Chen and Ke, CMC, 2008]
![Page 33: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/33.jpg)
33
A half-plane problem with an aperture subjected to the Robin boundary condition
(0,3.5)
1a
( , )2 ( , )
x
G xG x
n
( , ) 0G x
2
2
2 ( , ) ( )x d xÑ = -G x x
x
y
![Page 34: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/34.jpg)
34
A half-plane problem with an aperture subjected to the Robin boundary condition
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
Null-field BIE approach(addition theorem and
superposition technique)(M=50)
Melnikov’s approach [Melnikov and Melnikov (2006)]
-1 0 1 2 3 40
1
2
3
4
Null-field BIE approach(Green’s third identity)
[Chen and Ke, CMC, 2008]
![Page 35: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/35.jpg)
35
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary conditio
n A half-plane with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Neumann boundary condition(2) Neumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Neumann boundary con
dition
![Page 36: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/36.jpg)
36
An infinite plane with a circular inclusion
x
y2 ( , ) ( )
M
pG x xx d x
mÑ =- -
1.1x=1.0a =
4.0Im=
1.0Mm =
![Page 37: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/37.jpg)
37
Stress distribution along the interface
0 100 200 30050 150 250 350
0
1
2
0.5
1.5
2.5
z r
W ang and Sudak, (2007)
present approach (M =50)
![Page 38: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/38.jpg)
38
Equivalence between the solution of Green’s third identity and that of superposition technique
+= 2 ( , )G x
( , )G x
1( , )G x
( , )2 ( , ) ( , ) ( , ) ( ) ( , ) ( ) ( , )B B s
G sG x T s x G s dB s U s x dB s U xnxp x x x¶= - +
¶ò ò
22 2 ( , )
2 ( , ) ( , ) ( , ) ( ) ( , ) ( ) (2)B B
s
G sG x T s x G s dB s U s x dB s
n
xp x x
¶= -
¶ò ò L L
Green’s third identity
Superposition technique1
1 1 ( , )2 ( , ) ( , ) ( , ) ( ) ( , ) ( ) ( , ) (1)
B Bs
G sG x T s x G s dB s U s x dB s U x
n
xp x x x
¶= - +
¶ò ò L
),(),(),( 21 xGxGxG
![Page 39: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/39.jpg)
39
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary conditio
n A half-plane with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) (1) Dirichlet boundary conditionDirichlet boundary condition (2) (2) Neumann boundary conditionNeumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Numann boundary cond
ition
![Page 40: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/40.jpg)
40
Screw dislocation problem with the circular hole subject to the Dirichlet boundary condition
1.5a =
0w =
x
y 0),(2 yxw
75.1x
![Page 41: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/41.jpg)
41
Screw dislocation problem with the circular hole subject to the Dirichlet boundary condition
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Smith data (1968)
(close form)
Present approach
(series form) (M=50)
![Page 42: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/42.jpg)
42
Screw dislocation problem with the circular hole subject to the Neumann boundary condition
1.5a =
0w
n
¶=
¶
x
y 0),(2 yxw
75.1x
![Page 43: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/43.jpg)
43
Screw dislocation problem with the circular hole subject to the Neumann boundary condition
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Smith data (1968)
(close form)
Present approach
(series form) (M=50)
![Page 44: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/44.jpg)
44
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary conditio
n A half-plane with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Neumann boundary condition(2) Neumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Neumann boundary con
dition
![Page 45: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/45.jpg)
45
Screw dislocation problem with a circular inclusion
x
y 0),(2 yxw
75.1x1.5a =
1.0Mm =
2.0Im=
![Page 46: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/46.jpg)
46
Take free body and Superposition technique
x
y 0),(2 yxw
75.1x1.5a =
1.0Mm =
2.0Im=
x
y 0),(2 yxw
75.1x1.5a =
1.0Mm =
x
y
1.5a =
,I
I ww
n
¶¶
,M
M ww
n
¶¶
x
y 0),(2 yxw
75.1x1.5a = x
y 0),(2 yxw
1.5a =
,sd
sd ww
n
¶¶
,M sd
M sd w ww w
n n
¶ ¶- -
¶ ¶
![Page 47: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/47.jpg)
47
Test convergence (Parseval’s sum)
0 10 20 30 40 50
T erm s o f F o u rie r se rie s (M )
0
1
2
3
4
Par
seva
l's s
um o
f
wM
n
0 10 20 30 40 50
T erm s o f F o u rie r se rie s (M )
2
4
6
8
Par
seva
l's s
um o
f w
M
![Page 48: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/48.jpg)
48
Screw dislocation problem with a circular inclusion
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Present approach
(series form) (M=50)Smith data (1968)
(close form)
![Page 49: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/49.jpg)
49
Numerical examples
Concentrated force problems An annular case An eccentric ring An infinite plane with an aperture subjected to the Neumann boundary conditio
n A half-plane with an aperture
(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Robin boundary condition(2) Robin boundary condition
An infinite plane with a circular inclusion Screw dislocation problems
An infinite plane with an aperture(1) Dirichlet boundary condition (1) Dirichlet boundary condition (2) Neumann boundary condition(2) Neumann boundary condition
An infinite plane with a circular inclusion An infinite plane with two circular holes subject to the Numann boundary co
ndition
![Page 50: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/50.jpg)
50
Screw dislocation problems with two circular holes subject to the Neumann boundary condition
11.0a = 2 12.0a a=
d
y
x2 1
( , 0.01 )a d a+1 1( , 0.01 )a a-
0wn
¶=
¶
0wn
¶=
¶
![Page 51: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/51.jpg)
51
Screw dislocation problems with two circular holes subject to the Neumann boundary condition
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
50,0.2 1 Mad10.1 , 50d a M= =
Present approach
(series form)
Present approach
(series form)
![Page 52: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/52.jpg)
52
Screw dislocation problems with two circular holes subject to the Neumann boundary condition
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
10.01 , 50d a M= =
Present approach
(series form)
![Page 53: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/53.jpg)
53
Outline
Motivation and literature review Derivation of the Green’s function
Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic system
Numerical examples Green’s function for the concentrated force problems Green’s function for the screw dislocation problems
Conclusions
![Page 54: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/54.jpg)
54
Conclusions
A A systematic approachsystematic approach with five advantage with five advantage singularity free, boundary-layer effect free, singularity free, boundary-layer effect free, exponential convergence, well-posed model and exponential convergence, well-posed model and mesh-free generationmesh-free generation was developed in this thesis. was developed in this thesis.
The The angle-based fundamentalangle-based fundamental solution was solution was successfully expanded into the successfully expanded into the separable formseparable form..
Mathematical Mathematical equivalence equivalence between the between the Green’s third Green’s third identityidentity and and superposition techniquesuperposition technique for solving the for solving the Green’s function problem was successfully presented.Green’s function problem was successfully presented.
![Page 55: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/55.jpg)
55
Further studies
Extension to the imperfect interface.Derivation the Green’s third identity for the
screw dislocation problems.Extension to the general boundaries.2-D problems to 3-D problems.
![Page 56: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/56.jpg)
56
The endThe end
Thanks for your kind attention.Thanks for your kind attention.
Welcome to visit the web site of MSVLAB: Welcome to visit the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlabhttp://ind.ntou.edu.tw/~msvlab
![Page 57: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/57.jpg)
57
Literature review
Solve the concentrated force problems
Successive Successive iteration iteration methodmethod
Modified Modified potentialpotentialmethodmethod
Trefftz basTrefftz baseses
Melnikov, 2001, “Modified potential as a tool foor computing Green’s functions in continuum mechanics”, Computer Modeling in Engineering Science
Boley, 1956, “A method for the construction of Green’s functions,”, Quarterly of Applied Mathematics
Wang and Sudak, 2007, “Antiplane time-harmonic Green’s functions for a circular inhomogeneity with an imperfect interface”, Mechanics Research Communications
Null-field Null-field integral integral equationequationChen and Ke, 200
8, “Derivation of anti-plane Dynamic Green’s function for several circular inclusions with imperfect interfaces”, Computer modeling in Engineering Science
![Page 58: 1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations](https://reader035.vdocument.in/reader035/viewer/2022062421/56649d265503460f949fcc17/html5/thumbnails/58.jpg)
58
Literature review
Solve the screw dislocation problems
Image Image techniquetechnique
Inverse Inverse point point
methodmethod
Circle Circle theoremtheorem
Sendeckyj, 1970, “Screw dislocation near circular inclusions”, Physica status solidi
Dundurs, 1969, “Elastic interaction of dislocations with inhomogeneities”, Mathematical Theory of Dislocations
Smith, 1968, “The interaction between dislocations and inhomogeneities-I”, International Journal of Engineering Sciences