1 Applications of addition theorem and Applications of addition theorem and superposition technique to problems with superposition technique to problems with circular boundaries subject to concentrated circular boundaries subject to concentrated forces and screw dislocations forces and screw dislocations Reporter: Chou K. H. Advisor: Chen J. T. Date: 2008/07/11 Place: HR2 307
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1 Applications of addition theorem and superposition technique to problems with circular boundaries subject to concentrated forces and screw dislocations.
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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
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
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
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
Advantages of degenerate kernel1. No principal value2. Well-posed3. Exponential convergence4. Free of boundary-layer effect5. Mesh-free generation
Degenerate kernelDegenerate kernel
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
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
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
¶
¶
???
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
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
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
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
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
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
15
Adaptive observer system
Source pointSource point
Collocation pointCollocation point
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
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
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
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
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
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]
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
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 =
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]
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
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
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
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
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
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]
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
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]
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
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 =
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)
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¶= - +
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
40
Screw dislocation problem with the circular hole subject to the Dirichlet boundary condition
1.5a =
0w =
x
y 0),(2 yxw
75.1x
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)
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
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)
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
45
Screw dislocation problem with a circular inclusion
x
y 0),(2 yxw
75.1x1.5a =
1.0Mm =
2.0Im=
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
¶ ¶- -
¶ ¶
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
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)
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
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
¶=
¶
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)
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)
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
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.
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.
56
The endThe end
Thanks for your kind attention.Thanks for your kind attention.
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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
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