1 Applications of addition theorem and superposition technique to problems with circular boundaries...

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


Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic equation


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

Boundary-layer Boundary-layer effecteffect

Ill-posed modelIll-posed modelConvergence Convergence raterate

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

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




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




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




Conclusions

15

Adaptive observer system

Source pointSource point

Collocation pointCollocation point

16

Outline


Superposition technique Addition theorem for the kernel decomposition Fourier expansion for the boundary density Adaptive observer system Linear algebraic system


Conclusions

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

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

19

Outline




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





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





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



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





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



Image method

30

Numerical examples


n A half-plane problem with an aperture





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



Melnikov’s method [Melnikov and Melnikov (2001)]

-2 -1 0 1 2 3 40

1

2

3

4

5

6



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



Melnikov’s approach [Melnikov and Melnikov (2006)]

-1 0 1 2 3 40

1

2

3

4



35

Numerical examples







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¶= - +

¶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

39

Numerical examples





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







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






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


-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


-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




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.

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

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

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

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