94 學年度第 2 學期碩士論文口試 national taiwan ocean university msvlab department of...

194 學年度第 2 學期碩士論文口試

National Taiwan Ocean University

MSVLABDepartment of Harbor and River

Engineering

Null-field approach for Null-field approach for multiple circular inclusion multiple circular inclusion

problems in anti-plane problems in anti-plane piezoelectricitypiezoelectricity

Reporter: An-Chien WuAn-Chien Wu

Advisor: Jeng-Tzong ChenJeng-Tzong Chen

Date: 2006/06/292006/06/29

Place: HR2 307HR2 307

2MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering

OutlineOutline

• Motivation and literature review• Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations

◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique

• Numerical examples• Conclusions• Further studies


OutlineOutline

• Motivation and literature reviewMotivation and literature review• Unified formulation of null-field approach ◎ Boundary integral equations and null-field integral equations




MotivationMotivation

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


MotivationMotivation

BEM / BIEMBEM / BIEM

Improper integralImproper integral

Singularity & hypersingularitySingularity & hypersingularity RegularityRegularity

Bump contourBump contour Limit processLimit process Fictitious Fictitious boundaryboundary

Collocation Collocation pointpoint

Fictitious BEMFictitious BEM

Null-field approachNull-field approach

CPV and HPVCPV and HPVIll-posedIll-posed

Guiggiani (1995)Guiggiani (1995) Gray and Manne (199Gray and Manne (1993)3)

Waterman (1965)Waterman (1965)

Achenbach Achenbach et al.et al. (1988) (1988)


Present approachPresent approach

(s, x)iK

(s, x)eK

(s, x(x) (s) (s))B

dBKj y=ò

Fundamental solutionFundamental solution

(s, x), s x

(s, x), x s

i

i

K

K

ìï ³ïíï >ïîln x s-

No principal valueNo principal value

Advantages of degenerate kernel1. No principal value2. Well-posed3. Exponential convergence4. Free of boundary-layer effect

Degenerate kernelDegenerate kernel

CPV and HPVCPV and HPV


Engineering problem with holes, Engineering problem with holes, inclusions and cracksinclusions and cracks

Straight boundaryStraight boundary

Degenerate boundaryDegenerate boundary

Circular inclusionCircular inclusion

Elliptic holeElliptic hole

[Mathieu [Mathieu function]function]

[Legendre polynomia[Legendre polynomial]l]

[Chebyshev polynomial][Chebyshev polynomial]

[Fourier series][Fourier series]


Literature review – analytical solutions Literature review – analytical solutions for problems with circular boundariesfor problems with circular boundariesKey pointKey point Main applicationMain application AuthorAuthor

Conformal mappingConformal mapping Torsion problemTorsion problemIn-plane electrostaticsIn-plane electrostaticsAnti-plane elasticityAnti-plane elasticity

Chen & Weng (2001)Chen & Weng (2001)Emets & Onofrichuk (1996)Emets & Onofrichuk (1996)Budiansky & Carrier (1984)Budiansky & Carrier (1984)Steif (1989)Steif (1989)Wu & Funami (2002)Wu & Funami (2002)Wang & Zhong (2003)Wang & Zhong (2003)

Bi-polar coordinateBi-polar coordinate Electrostatic potentialElectrostatic potentialElasticityElasticity

Lebedev Lebedev et al.et al. (1965) (1965)Howland & Knight (1939)Howland & Knight (1939)

MMööbius transformatiobius transformationn

Anti-plane piezoelectricity & Anti-plane piezoelectricity & elasticityelasticity

Honein Honein et al.et al. (1992) (1992)

Complex potential Complex potential approachapproach

Anti-plane piezoelectricityAnti-plane piezoelectricity Wang & Shen (2001)Wang & Shen (2001)

Those Those analytical methodsanalytical methods are only limited to are only limited to doubly connected regionsdoubly connected regions even to even toconformal connected regionsconformal connected regions..


Literature review - Fourier series Literature review - Fourier series approximationapproximation

AuthorAuthor Main applicationMain application Key pointKey pointLingLing

(1943)(1943)

Torsion of a circular tubeTorsion of a circular tube

Caulk Caulk et al.et al.

(1983)(1983)

Steady heat conduction with Steady heat conduction with circular holescircular holes

Special BIEMSpecial BIEM

Bird and SteeleBird and Steele

(1992)(1992)

Harmonic and biharmonic problHarmonic and biharmonic problems with circular holesems with circular holes

Trefftz methodTrefftz method

Mogilevskaya Mogilevskaya et al.et al.

(2002)(2002)

Elasticity problems with circular Elasticity problems with circular holes holes oror inclusions inclusions

Galerkin methodGalerkin method

However, no one employed the However, no one employed the null-field approachnull-field approach and and degenerate degenerate kernelkernel to fully capture the circular boundary. to fully capture the circular boundary.


OutlineOutline

• Motivation and literature review• Unified formulation of null-field approachUnified formulation of null-field approach ◎ Boundary integral equations and null-field integral equationsBoundary integral equations and null-field integral equations




Boundary integral equation and Boundary integral equation and null-field integral equationnull-field integral equation

Interior case Exterior case

cD

D D

x

xx

xcD

s

s

(s, x) ln x s ln

(s, x)(s, x)

n

(s)(s)

n

U r

UT

jy

= - =

¶=

¶

¶=

¶

0 (s, x) (s) (s) (s, x) (s) (s), x c

B BT dB U dB Dj y= - Îò ò

(x) . . . (s, x) (s) (s) . . . (s, x) (s) (s), xB B

C PV T dB R PV U dB Bpj j y= - Îò ò

2 (x) (s, x) (s) (s) (s, x) (s) (s), xB BT dB U dB Dpj j y= - Îò ò

x x

2 (x) (s, x) (s) (s) (s, x) (s) (s), xB BT dB U dB D Bpj j y= - Î Èò ò

0 (s, x) (s) (s) (s, x) (s) (s), x c

B BT dB U d D BBj y= - Î Èò ò

Degenerate (separate) formDegenerate (separate) form


Expansions of fundamental solution Expansions of fundamental solution and boundary densityand boundary density

(s, x) (s) (x), s x

(s, x)(s, x) (x) (s), x s

ij j

j

ej j

j

U A B

UU A B

ìï = ³ïïï=íï = >ïïïî

å

å

01

01

(s) ( cos sin ), s

(s) ( cos sin ), s

L

n nn

L

n nn

a a n b n B

p p n q n B

j q q

y q q

=

=

= + + Î

= + + Î

å

å

Degenerate kernel – fundamental solutionDegenerate kernel – fundamental solution

Fourier series expansion – boundary densityFourier series expansion – boundary density


Convergence rate between present Convergence rate between present method and conventional BEMmethod and conventional BEM

(s, x) interior(s, x)

(s, x) exterior

i

e

UU

U

ìïï=íïïî

Degenerate kernelDegenerate kernel

Fourier series expansionFourier series expansion

Fundamental Fundamental solutionsolution

Boundary Boundary densitydensity

Convergence Convergence raterate

Present methodPresent method Conventional BEMConventional BEM

Two-point functionTwo-point function

(s, x) ln ln x sU r= = -

Constant, linear, Constant, linear, quadratic elementsquadratic elements

Exponential convergenceExponential convergence Linear convergenceLinear convergence


Degenerate (separate) form of Degenerate (separate) form of fundamental solution (2-D)fundamental solution (2-D)

s( , )R q

R

r

rx( , )r f

x( , )r f

1

1

1( , ; , ) ln ( ) cos ( ),

(s, x) ln1

( , ; , ) ln ( ) cos ( ),

i m

m

e m

m

U R R m Rm R

U rR

U R m Rm

rq r f q f r

q r f r q f rr

¥

=

¥

=

ìïï = - - ³ïïïï= =íïï = - - >ïïïïî

å

å

o

iU

eU

s

x

2

s x

(s, x)(s, x)

n

(s, x)(s, x)

n

(s, x)(s, x)

n n

UT

UL

UM

¶º

¶

¶º

¶

¶º

¶ ¶


OutlineOutline


◎ Adaptive observer systemAdaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique



Adaptive observer systemAdaptive observer system

collocation pointcollocation point

0 , 01 , 1k , k2 , 2


OutlineOutline


◎ Adaptive observer system ◎ Linear algebraic equationLinear algebraic equation ◎ Vector decomposition technique



Linear algebraic equationLinear algebraic equation

{ }

0

1

2

N

ì üï ïï ïï ïï ïï ïï ïï ïï ï=í ýï ïï ïï ïï ïï ïï ïï ïï ïî þ

M

y

y

y y

y

[ ]

00 01 0

10 11 1

0 1

N

N

N N NN

é ùê úê úê ú= ê úê úê úê úë û

U U U

U U UU

U U U

L

L

M M O M

L

Column vector of Fourier coefficientsColumn vector of Fourier coefficients((NthNth routing circle) routing circle)

0B

1B

Index of collocation circleIndex of collocation circle

Index of routing circle Index of routing circle

2B

NB

[ ]{ } { }[ ]=U Ty j


Explicit form of each submatrix and Explicit form of each submatrix and vectorvector

0 1 11 1 1 1 1

0 1 12 2 2 2 2

0 1 13 3 3 3 3

0 1 12 2 2 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

c c s Lc Lsjk jk jk jk jkc c s Lc Ls

jk jk jk jk jkc c s Lc Ls

jk jk jk jk jkjk

c c s Lc Lsjk L jk L jk L jk L jk

U U U U U

U U U U U

U U U U U

U U U U U

ff ff f

ff ff f

ff ff f

ff ff

é ù=ê úë ûU

L

L

L

M M M O M M

L 20 1 1

2 1 2 1 2 1 2 1 2 1

( )

( ) ( ) ( ) ( ) ( )L

c c s Lc Lsjk L jk L jk L jk L jk LU U U U U

f

ff ff f+ + + + +

é ùê úê úê úê úê úê úê úê úê úê úê úê úë ûL

{ } { }0 1 1

Tk k k k kk L Lp p q p q= Ly

1f

2f

3f

2Lf

2 1Lf +

Fourier coefficientsFourier coefficients

Truncated terms of Truncated terms of Fourier seriesFourier series

Number of collocation pointsNumber of collocation points


Physical meaning of influence Physical meaning of influence coefficients andcoefficients and

mthmth collocation point on th collocation point on the e jthjth circular boundary circular boundary

jthjth circular boundary circular boundary xm

m

coscos nnsinsin nnboundary distributionboundary distribution

kthkth circular boundary circular boundary

( )ncjk mU f ( )ns

jk mU f


OutlineOutline


◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition techniqueVector decomposition technique



Vector decomposition technique for Vector decomposition technique for potential gradientpotential gradient

x

z

z x-

nt

t

n

True normal vectorTrue normal vector

(s, x) 1 (s, x)(s, x) cos( ) cos( )

2

U ULr

pz x z x

r r f¶ ¶

= - + - +¶ ¶

(s, x) 1 (s, x)(s, x) cos( ) cos( )

2

T TM r

pz x z x

r r f¶ ¶

= - + - +¶ ¶

Special case (concentric case) :Special case (concentric case) : z x=

(s, x)(s, x)

ULr r

¶=

¶(s, x)

(s, x)T

M r r¶

=¶

Non-concentric case:Non-concentric case:

(x)2 (s, x) (s) (s) (s, x) (s) (s), x

n(x)

2 (s, x) (s) (s) (s, x) (s) (s), xt

B B

B B

M dB L dB D

M BdB L d D

B

B

r r

ff

jp j y

jp j y

¶= - Î

¶¶

= - Î¶

È

È

ò ò

ò ò


Flowchart of present methodFlowchart of present method

0 [ (s, x) (s) (s, x) (s)] (s)B

T U dBj y= -òDegenerate kernelDegenerate kernel Fourier seriesFourier series

Adaptive observer systemAdaptive observer system

Collocating point to Collocating point to construct compatible construct compatible

boundary data boundary data relationship relationship

Continuity of Continuity of displacement and displacement and

equilibrium of tractionequilibrium of traction

Linear algebraic systemLinear algebraic system Fourier coefficientsFourier coefficients

Potential of domain pointPotential of domain point

Vector decompositionVector decomposition

Potential gradientPotential gradient

AnalyticalAnalytical

NumericalNumerical


Comparisons of conventional BEM Comparisons of conventional BEM and present method and present method

Boundarydensity

discretization

Auxiliarysystem

FormulationObserversystem

Singularity ConvergenceBoundary

layereffect

ConventionalBEM

Constant,linear,

quadratic…elements

Fundamentalsolution

Boundaryintegralequation

Fixedobserversystem

CPV, RPVand HPV

Linear Appear

Presentmethod

Fourierseries

expansion

Degeneratekernel

Null-fieldintegralequation

Adaptiveobserversystem

Disappear Exponential Eliminate


OutlineOutline



• Numerical examplesNumerical examples• Conclusions• Further studies


Numerical examplesNumerical examples

• Anti-plane piezoelectricity problems

(EABE, 2006, accepted)(EABE, 2006, accepted)

• In-plane electrostatics problems

(??)(??)

• Anti-plane elasticity problems

(ASME-JAM, 2006, accepted)(ASME-JAM, 2006, accepted)



• Anti-plane piezoelectricity problemsAnti-plane piezoelectricity problems




Problem statementProblem statement

xE¥zxs¥

yE¥

zys¥

kB

1B

2B

1 1

1 1

,,

M M

M Mw tF Y

2 2

2 2

,,

M M

M Mw tF Y

,,

M Mk kM Mk k

w tF Y

,,

I Ik kI Ik k

w tF Y

1 1

1 1

,,

I I

I Iw tF Y

2 2

2 2

,,

I I

I Iw tF Y

,,

k k

k k

w t¥ ¥

¥ ¥F Y 1 1

1 1

,,

w t¥ ¥

¥ ¥F Y

2 2

2 2

,,

w t¥ ¥

¥ ¥F Y

,,

M Mk k k kM Mk k k k

w w t t¥ ¥

¥ ¥- -

F - F Y - Y

1 1 1 1

1 1 1 1

,,

M M

M Mw w t t¥ ¥

¥ ¥- -

F - F Y - Y

2 2 2 2

2 2 2 2

,,

M M

M Mw w t t¥ ¥

¥ ¥- -

F - F Y - Y

= +

+


Analogy between anti-plane deformation and in-Analogy between anti-plane deformation and in-plane electrostatics for anti-plane piezoelectricityplane electrostatics for anti-plane piezoelectricity

Anti-plane shear Anti-plane shear deformationdeformation

Constitutive equations for Constitutive equations for anti-plane piezoelectricityanti-plane piezoelectricity

In-plane In-plane electrostaticselectrostatics

z-displacement w Electric potential

Strain zi Electric field Ei

Stresszi

Electric displacement Di

Shear modulus Dielectric constant

Strain-disp.zi = w,i

ElectricityEi = – ,i

Constitutive lawzi = zi

Constitutive lawDi = Ei

Coupling effectCoupling effectzizi = c= c4444 zizi – e – e1515 E Eii

DDii = e = e1515 zizi + + 1111 E Eii

Shear modulus Shear modulus cc4444

Piezoelectric constant Piezoelectric constant ee1515

Dielectric constant Dielectric constant 1111


,,

I Ik kI Ik k

w tF Y

1 1

1 1

,,

I I

I Iw tF Y

2 2

2 2

,,

I I

I Iw tF Y

Linear algebraic systemLinear algebraic system

For the exterior problem of matrixFor the exterior problem of matrix

{ } { }M M M M¥ ¥é ù é ù- = -ê ú ê úë û ë ûU t t T w w

{ } { }M M M M¥ ¥é ù é ù- = -ê ú ê úë û ë ûU Ψ Ψ T Φ Φ

For the interior problem of each inclusionFor the interior problem of each inclusion

{ } { }I I I Ié ù é ù=ê ú ê úë û ë ûU t T w

{ } { }I I I Ié ù é ù=ê ú ê úë û ë ûU Ψ T Φ

The continuity of displacementThe continuity of displacement,M I

kw w on B=

,M Ikon BF =F

The equilibrium of tractionThe equilibrium of traction,M I

zr zr kon Bs s=

,M Ir r kD D on B=

44 44 15 15

15 15 11 11

M M M

I I M

M M I

I I I

M

M I M I M

I

M I M I I

é ùì üï ïï ïê úï ïê úï ïï ïê úï ïï ïê úï ïê úï ïï ïê úï ïïê úïïí ýê úïê úïïê úïïê úïê úïïê úïê úïïê úïê úïîë ûï

T -U 0 0 0 0 0 0 w

0 0 T -U 0 0 0 0 t

0 0 0 0 T -U 0 0 w

0 0 0 0 0 0 T -U t

I 0 -I 0 0 0 0 0 Φ

0 c 0 c 0 e 0 e Ψ

0 0 0 0 I 0 -I 0 Φ

0 e 0 e 0 -ε 0 -ε Ψ

ì üï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ï ïï ï ïï ï ïí ýï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ïî þïþï ï ï

a

0

b

0=0

0

0

0

,,

M Mk k k kM Mk k k k

w w t t¥ ¥

¥ ¥- -

F - F Y - Y

1 1 1 1

1 1 1 1

,,

M M

M Mw w t t¥ ¥

¥ ¥- -

F - F Y - Y

2 2 2 2

2 2 2 2

,,

M M

M Mw w t t¥ ¥

¥ ¥- -

F - F Y - Y


Two circular inclusions embedded in a Two circular inclusions embedded in a piezoelectric matrix under such loadingspiezoelectric matrix under such loadings

xE¥

yE¥

zxs¥

zys¥

1r

2r

b

d

x

y


0 60 120 180 240 300 360

(degree)

- 1

- 0 . 5

0

0 . 5

1

M z

=5 10 7 N /m 2

E =10 6 V /me M15 /e I15=3.0

d/r1=10.0

d/r1=1.0

d/r1=0.1

d/r1=0.02

d/r1=0.01

Tangential stress distribution for different ratios Tangential stress distribution for different ratios d/rd/r11 with with rr22=2=2rr11, e, e1515

MM//ee1515II=3.0=3.0 and and =90=90°°

Chao & Chang’s data (199Chao & Chang’s data (1999)9)

Present method (L=20)Present method (L=20)


0 60 120 180 240 300 360

(degree)

- 3

- 2

- 1

0

1

2

3

E M E

=5 10 7 N /m 2

E =10 6 V /me M15 /e I15=3.0

d/r1=10.0

d/r1=1.0

d/r1=0.1

d/r1=0.02

d/r1=0.01

Tangential electric field distribution for different Tangential electric field distribution for different ratios ratios d/rd/r11 with with rr22=2=2rr11, e, e1515

MM//ee1515II=3.0=3.0 and and =90=90°°





MM//ee1515II=-5.0=-5.0 and and =90=90°°

0 60 120 180 240 300 360

(degree)

- 9

- 6

- 3

0

3

6

9

M z

=5 10 7 N /m 2

E =10 6 V /me M15 /e I15=-5.0

d/r1=10.0

d/r1=1.0

d/r1=0.1

d/r1=0.02

d/r1=0.01





MM//ee1515II=-5.0=-5.0 and and =90=90°°

0 60 120 180 240 300 360

(degree)

-12

-8

-4

0

4

8

12

E M E

=5 10 7 N /m 2

E =10 6 V /me M15 /e I15=-5 .0

d/r1=10.0

d/r1=1.0

d/r1=0.1

d/r1=0.02

d/r1=0.01




Parseval’s sum for Parseval’s sum for rr22=2=2rr11, , d/rd/r11=0.01=0.01, , =90=90° and e° and e

1515MM//ee1515

II=5.0=5.0

0 10 20 30

Term s of Fourier series (L)

2E-006

2.4E-006

2.8E-006

3.2E-006

3.6E-006

4E-006

Pa

rse

val's

sum

0 10 20 30

Term s of Fourier series (L)

5.4E-006

5.6E-006

5.8E-006

6E-006

6.2E-006

Pa

rse

val's

sum

0 10 20 30

Term s of Fourier sere is (L)

6.8E-007

7E-007

7.2E-007

7.4E-007

7.6E-007

Pa

rse

val's

sum

22

0[ ( )]f d

pq qò

2 2 20

1

2 ( )L

n nn

a a bp p=

+ +åB&

Parseval’s sumParseval’s sum1Mw

2Mw

1Mt

2Mt

0 10 20 30


6E-007

6.4E-007

6.8E-007

7.2E-007

7.6E-007

8E-007

8.4E-007

Pa

rse

val's

sum


0 10 20 30


7E+011

8E+011

9E+011

1E+012

1.1E+012

Par

seva

l's s

um

0 10 20 30


3.1E+013

3.2E+013

3.3E+013

3.4E+013

3.5E+013

Par

seva

l's s

um

0 10 20 30


1.2E+012

1.4E+012

1.6E+012

1.8E+012

2E+012

2.2E+012

Par

seva

l's s

um

0 10 20 30


7E+012

7.5E+012

8E+012

8.5E+012

9E+012

9.5E+012

Par

seva

l's s

umParseval’s sum for Parseval’s sum for rr22=2=2rr11, , d/rd/r11=0.01=0.01, , =90=90° and e° and e

1515MM//ee1515

II=5.0=5.0

22

0[ ( )]f d

pq qò

2 2 20

1

2 ( )L

n nn

a a bp p=

+ +åB&

Parseval’s sumParseval’s sum

1MF

2MF

1MY

2MY



MM//ee1515II=-5.0=-5.0 and and =0=0°°

0 60 120 180 240 300 360

(degree)

- 3

0

3

6

M z

=5 10 7 N /m 2

E =10 6 V /me M15 /e I15=-5 .0

d/r1=10.0

d/r1=1.0

d/r1=0.1

d/r1=0.05

d/r1=0.01




0 60 120 180 240 300 360

(degree)

- 6

- 4

- 2

0

2

4

6

8

E M E

=5 10 7 N /m 2

E =10 6 V /me M15 /e I15=-5.0

d/r1=10.0

d/r1=1.0

d/r1=0.1

d/r1=0.05

d/r1=0.01


MM//ee1515II=-5.0=-5.0 and and =0=0°°




-10 -8 -6 -4 -2 0 2 4 6 8 10

e M15 /e I15

-40

-20

0

20

40

M z

=5 10 7 N /m 2

E =10 6 V /md/r1=10.0

d/r1=1.0

d/r1=0.1

d/r1=0.05

Stress concentrations as a function of the ratio Stress concentrations as a function of the ratio of piezoelectric constants with of piezoelectric constants with =0=0°°




-10 -8 -6 -4 -2 0 2 4 6 8 10

e M15 /e I15

- 5

0

5

1 0

1 5

2 0

2 5

E M E

=5 10 7 N /m 2

E =10 6 V /md/r1=10.0

d/r1=1.0

d/r1=0.1

d/r1=0.05

Electric field concentrations as a function of the Electric field concentrations as a function of the ratio of piezoelectric constants with ratio of piezoelectric constants with =0=0°°




Stress concentrations as a function of the ratio Stress concentrations as a function of the ratio of piezoelectric constants with of piezoelectric constants with =0=0°°

-10 -8 -6 -4 -2 0 2 4 6 8 10

e M15 /e I15

-80

-60

-40

-20

0

20

M z

=5 10 7 N /m 2

d/r1=0.01E =10 4 V /m

E =10 3 V /m

E =10 2 V /m

E =10 V /m




Electric field concentrations as a function of the Electric field concentrations as a function of the ratio of piezoelectric constants with ratio of piezoelectric constants with =0=0°°

-10 -8 -6 -4 -2 0 2 4 6 8 10

e M15 /e I15

-20

0

20

40

60

E M E

E =10 6 V /md/r1=0.05

=10 10 7 N /m 2

=5 10 7 N /m 2

=0 N /m 2

=-5 10 7 N /m 2

=-10 10 7 N /m 2




Contour of shear stress Contour of shear stress zxzx when when d/rd/r11=0.01=0.01

Wang & Shen’s data (2001)Wang & Shen’s data (2001)Present method (L=20)Present method (L=20)

-3 -2 -1 0 1 2 3 4 5 6

x

-3

-2

-1

0

1

2

3

y


Contour of shear stress Contour of shear stress zyzy when when d/rd/r11=0.01=0.01

Wang & Shen’s data (2001)Wang & Shen’s data (2001)Present method (L=20)Present method (L=20)

-3 -2 -1 0 1 2 3 4 5 6

x

-3

-2

-1

0

1

2

3

y


Contour of electric potentialContour of electric potentialwhen when d/rd/r11=0.01=0.01


-3 -2 -1 0 1 2 3 4 5 6

x

-3

-2

-1

0

1

2

3

y 0

0

-0.6

-1.2

-1.8

0.6

1.2

1.8

-0.6

0.6


Stress distributionStress distributionwith with rr22=2=2rr11 and and d/rd/r11=0.01=0.01 in in

two-directions loadingstwo-directions loadings

Pre

sent

met

hod

(L=

20)

Pre

sent

met

hod

(L=

20)

0 100 200 300 400 (degree)

- 4

- 2

0

2

4

zr

10

-7

Mzr

Izr

0 100 200 300 400 (degree)

-15

-10

-5

0

5

10

15

z

10

-7

Mz

IzW

ang

& S

hen’

s da

ta (

2001

)W

ang

& S

hen’

s da

ta (

2001

)


Electric displacement distributionElectric displacement distributionwith with rr22=2=2rr11

and and d/rd/r11=0.01=0.01 in two-directions loadings in two-directions loadings

Pre

sent

met

hod

(L=

20)

Pre

sent

met

hod

(L=

20)

Wan

g &

She

n’s

data

(20

01)

Wan

g &

She

n’s

data

(20

01)

0 100 200 300 400 (degree)

- 5

0

5

Dr

102

D MrD Ir

0 100 200 300 400 (degree)

- 6

- 4

- 2

0

2

4

6

D

102

D MD I


Stress distributionStress distributionwith with rr22=2=2rr11 and and d/rd/r11=0.01=0.01 in in

two-directions loadingstwo-directions loadings

Pre

sent

met

hod

(L=

20)

Pre

sent

met

hod

(L=

20)

Wan

g &

She

n’s

data

(20

01)

Wan

g &

She

n’s

data

(20

01)

0 100 200 300 400 (degree)

- 4

- 2

0

2

4

zr

10

-7

Mzr

Izr

0 100 200 300 400 (degree)

-15

-10

-5

0

5

10

15

z

10

-7

Mz

Iz


Electric displacement distributionElectric displacement distributionwith with rr22=2=2rr11

and and d/rd/r11=0.01=0.01 in two-directions loadings in two-directions loadings

Pre

sent

met

hod

(L=

20)

Pre

sent

met

hod

(L=

20)

Wan

g &

She

n’s

data

(20

01)

Wan

g &

She

n’s

data

(20

01)

0 100 200 300 400 (degree)

- 5

0

5

Dr

102

D MrD Ir

0 100 200 300 400 (degree)

- 6

- 4

- 2

0

2

4

6

D

102

D MD I




• In-plane electrostatics problemsIn-plane electrostatics problems



Analogy between anti-plane deformation and in-Analogy between anti-plane deformation and in-plane electrostatics for anti-plane piezoelectricityplane electrostatics for anti-plane piezoelectricity

Anti-plane shear Anti-plane shear deformationdeformation

Constitutive equations for Constitutive equations for anti-plane piezoelectricityanti-plane piezoelectricity

In-plane In-plane electrostaticselectrostatics

z-displacement w Electric potential

Strain zi Electric field Ei

Stresszi

Electric displacement Di

Shear modulus Dielectric constant

Strain-disp.zi = w,i

ElectricityEi = – ,i

Constitutive lawzi = zi

Constitutive lawDi = Ei

Coupling effectCoupling effectzizi = c= c4444 zizi – e – e1515 E Eii

DDii = e = e1515 zizi + + 1111 E Eii

Shear modulus Shear modulus cc4444

Piezoelectric constant Piezoelectric constant ee1515

Dielectric constant Dielectric constant 1111


The dielectric system of two inclusions The dielectric system of two inclusions in the applied electric fieldin the applied electric field

xE¥

yE¥

d

1r 2r

0e

1e 2ex

y


Patterns of the electric field for Patterns of the electric field for 00=2=2, ,

11=9=9 and and 22=5=5

-1 .5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

xE E¥¥= cos 45 , sin 45x yE E E E¥ ¥

¥ ¥= =o oyE E¥

¥=

Em

ets

& O

nofr

ichu

kE

met

s &

Ono

fric

huk

(199

6)(1

996)

Pre

sen

t m

eth

od (

L=20

)P

rese

nt

met

hod

(L=

20)


Patterns of the electric field for Patterns of the electric field for 00=3=3, ,

11=9=9 and and 22=1=1

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

xE E¥¥= cos 45 , sin 45x yE E E E¥ ¥

¥ ¥= =o oyE E¥

¥=

Em

ets

& O

nofr

ichu

kE

met

s &

Ono

fric

huk

(199

6)(1

996)

Pre

sen

t m

eth

od (

L=20

)P

rese

nt

met

hod

(L=

20)





• Anti-plane elasticity problemsAnti-plane elasticity problems


Two equal-sized holes Two equal-sized holes rr22==rr11 with with

centers on the centers on the xx axis axis

t ¥

2r

1r

e

2d

x

y


Stress concentration of the problem Stress concentration of the problem containing two equal-sized holescontaining two equal-sized holes

0 0.2 0.4 0.6 0.8 1

d/r1

0

5

10

15

z

/

Paul S . S te if (1989)

C . K . C hao and C . W . Young (1998)

P resent so lu tion

2d


Stress concentration factors and errors between Stress concentration factors and errors between present method and conventional BEMpresent method and conventional BEM

d/r1 0.01 0.2 0.4 0.6 0.8 1.0

Analytical solutionAnalytical solutionSteif (1989)Steif (1989) 14.224714.2247 3.53493.5349 2.76672.7667 2.47582.4758 2.32742.3274 2.24002.2400

PresentPresentMethodMethod

L=10L=1010.509610.5096(26.12%)(26.12%)

3.53063.5306(0.12%)(0.12%)

2.76642.7664(0.01%)(0.01%)

2.47582.4758(0.00%)(0.00%)

2.32742.3274(0.00%)(0.00%)

2.24002.2400(0.00%)(0.00%)

L=20L=2013.327513.3275(6.31%)(6.31%)

3.53493.5349(0.00%)(0.00%)

2.76672.7667(0.00%)(0.00%)

2.47582.4758(0.00%)(0.00%)

2.32742.3274(0.00%)(0.00%)

2.24002.2400(0.00%)(0.00%)

BEMBEMBEPO2DBEPO2D

No. nodeNo. node=21=21

7.25007.2500(49.03%)(49.03%)

3.45323.4532(2.31%)(2.31%)

2.7382.738(1.04%)(1.04%)

2.46392.4639(0.48%)(0.48%)

2.31682.3168(0.46%)(0.46%)

2.23662.2366(0.15%)(0.15%)

No. nodeNo. node=41=41

10.200810.2008(28.29%)(28.29%)

3.51883.5188(0.46%)(0.46%)

2.76192.7619(0.17%)(0.17%)

2.47472.4747(0.04%)(0.04%)

2.33122.3312(0.16%)(0.16%)

2.23982.2398(0.01%)(0.01%)S

tres

s co

ncen

trat

ion

fact

orS

tres

s co

ncen

trat

ion

fact

or


Convergence test and boundary-Convergence test and boundary-layer effect analysislayer effect analysis

2.04

2.08

2.12

2.16

2.2

2.24

2.28

Str

ess

co

nce

ntra

tion

fact

or

P . S . S te if (1989)Present m ethod

BEM -BEPO 2D

0

11 21 31 41 51 61 71

N um ber of degrees of freedom (nodes)0

0 5 10 15 20 25 30 35

N um ber of degrees of freedom (term s of Fourier series, L)0.01 0.1 1

/r1

1

10

z

/

Paul S . S te if (1989)P resent m ethod (L=10)P resent m ethod (L=20)

BEM -BEP O 2D (node=41)

2d

e


Two circular inclusions with centers Two circular inclusions with centers on the on the yy axis axis

t ¥

d

x

y

2r

1re

q

2m

1m

2 1

1

1 0

2 0

2

0.1

2 / 3

13 / 7

r r

d r

m m

m m

=

=

=

=

0m


Two circular inclusions with centers Two circular inclusions with centers on the on the yy axis axis

0 1 2 3 4 5 6 ( in radians)

- 2

0

2

4

Str

esse

s ar

ound

incl

usio

n of

rad

ius

r 1

Mzr /

Izr /

Mz /

Iz /

Hon

ein

Hon

ein

et a

l.et

al. ’

sdat

a (1

992)

’sda

ta (

1992

)


Equilibrium of tractionEquilibrium of traction


Convergence test for stress Convergence test for stress concentration factorconcentration factor

0 10 20 30Term s of Fourier series (L)

1.3

1.304

1.308

1.312

1.316

Str

ess

conc

entr

atio

n fa

ctor

in t

he m

atrix


Boundary-layer effect analysis for Boundary-layer effect analysis for radial and tangential stressesradial and tangential stresses

1E-006 1E-005 0.0001 0.001 0.01 0.1/r1

-0 .056

-0 .054

-0 .052

-0 .05

-0 .048

M zr/

Present m ethod (L=10)Present m ethod (L=20)

1E-006 1E-005 0.0001 0.001 0.01 0.1/r1

1.26

1.27

1.28

1.29

1.3

1.31

M z/

Present m ethod (L=10)Present m ethod (L=20)

e e


One hole surrounded by two One hole surrounded by two circular inclusions circular inclusions rr33==rr22=2=2rr11

x

yt ¥

d

2r

1r2m

3m

d

3r0m

b


0 60 120 180 240 300 360

(degree)

- 3

- 2

- 1

0

1

2

3

M z/

1/ 0= 2/ 0=0.0

1/ 0= 2/ 0=0.1

1/ 0= 2/ 0=1.0

1/ 0= 2/ 0=10.0

1/ 0= 2/ 0=

Tangential stress distribution along Tangential stress distribution along the hole with the hole with =0=0°°

Chao & Young’s data (199Chao & Young’s data (1998)8)


d d


Tangential stress distribution along Tangential stress distribution along the hole with the hole with =90=90°°

Chao & Young’s data (199Chao & Young’s data (1998)8)


0 60 120 180 240 300 360

(degree)

- 4

- 2

0

2

4

M z

/

1/ 0= 2/ 0=0.0

1/ 0= 2/ 0=0.1

1/ 0= 2/ 0=1.0

1/ 0= 2/ 0=10.0

1/ 0= 2/ 0=

d

d


Three identical inclusions forming Three identical inclusions forming an equilateral trianglean equilateral triangle

12d r=x

y

t ¥

2r

1r

2m

3m3r

0m

30o1m

30o


Tangential stress distribution around Tangential stress distribution around the inclusion located at the originthe inclusion located at the origin

Present method (L=20),Present method (L=20),agrees well with Gong’s data (1995)agrees well with Gong’s data (1995)

0 0.2 0.4 0.6 0.8 1

/

- 2

- 1

0

1

2

M z

/

1/ 0= 2/ 0= 3/ 0=0.0

1/ 0= 2/ 0= 3/ 0=0.5

1/ 0= 2/ 0= 3/ 0=2.0

1/ 0= 2/ 0= 3/ 0=5.0

1/ 0= 2/ 0= 3/ 0=


OutlineOutline



• Numerical examples• ConclusionsConclusions• Further studies


ConclusionsConclusions

• A A systematic approachsystematic approach using using degenerate kernelsdegenerate kernels and and Fourier seriesFourier series for for null-field integral equationnull-field integral equation has been successfully proposed to solve BVPs has been successfully proposed to solve BVPs with circular inclusions.with circular inclusions.

• According to numerical results, According to numerical results, only few terms of only few terms of Fourier seriesFourier series can achieve accurate solutions. can achieve accurate solutions.

• Four goals of Four goals of singularity freesingularity free, , boundary-layer effboundary-layer effect freeect free, , exponential convergenceexponential convergence and and well-posewell-posed modeld model are achieved. are achieved.



• The results demonstrate the The results demonstrate the superioritysuperiority of of present methodpresent method over the conventional BEM. over the conventional BEM.

• Our Our semi-analytical resultssemi-analytical results may provide a may provide a datumdatum for other researchers’ reference.for other researchers’ reference.

• The The stress and electric field concentrationsstress and electric field concentrations are are dependent on the dependent on the distancedistance between the two between the two inclusions, the mismatch in the inclusions, the mismatch in the material material constantsconstants and the magnitude of and the magnitude of mechanical and mechanical and electromechanical loadingselectromechanical loadings..



• A A general-purpose programgeneral-purpose program for solving Laplace for solving Laplace problems with multiple circular inclusions of varioproblems with multiple circular inclusions of various radii, arbitrary positions and different material us radii, arbitrary positions and different material constants was developed.constants was developed.

• Its possible Its possible applicationsapplications in engineering are in engineering are very very broadbroad, not only limited in this thesis., not only limited in this thesis.


OutlineOutline



• Numerical examples• Conclusions• Further studiesFurther studies


Further studiesFurther studies

• Extension to Extension to general boundariesgeneral boundaries..

• 2-D problems to2-D problems to 3-D 3-D problems.problems.

• Various loading typesVarious loading types, e.g. concentrated , e.g. concentrated forces, forces, screw dislocations, torques, in-, torques, in-plane shears and tensions.plane shears and tensions.

• Various inhomogeneous typesVarious inhomogeneous types, e.g. , e.g. coated fibers and inclusions with imperfect interfaces.


The endThe end

Thanks for your kind attention.Thanks for your kind attention.

Your comments will be highly appreciated.Your comments will be highly appreciated.

Welcome to the web site of MSVLAB: Welcome to the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlabhttp://ind.ntou.edu.tw/~msvlab


Derivation of degenerate kernelsDerivation of degenerate kernels

s x( , ) R , ( , )i iz R e z eq fq r f r= =

x s x sln Re{ln }, Im{ln }sdr z z z zq= - = -

x sz z Rr> ® >

s s sx s x x x

1x x x

1ln( ) ln[ (1 )] ln( ) ln(1 ) ln( ) ( )m

m

z z zz z z z z

z z m z

¥

=

- = - = + - = - å

s

1 1 1x

1 1 1( ) ( ) ( ) [cos ( ) sin ( )]

im m m

im m m

z em i m

m z m Re m R

f

f

r rq f q f

¥ ¥ ¥

= = =

= = - + -å å å

1x s

1

1ln ( ) cos ( ),

ln Re{ln }1

ln ( ) cos ( ),

m

m

m

m

R m Rm R

r z zR

m Rm

rq f r

r q f rr

¥

=

¥

=

ìïï - - ³ïïïï= - =íïï - - >ïïïïî

å

å


1x s

1

1( ) sin ( ),

Im{ln }1

( ) sin ( ),

m

msd

m

m

m Rm R

z zR

m Rm

rp q q f r

q

f q f rr

¥

=

¥

=

ìïï + + - >ïïïï= - =íïï - - >ïïïïî

å

å

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

12

3

45

6


An infinite medium containing one An infinite medium containing one hole under the hole under the screw dislocation

12

3

45

6

3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3


Coated inclusion under the anti-plane shear stress

t

0m

1m2m

2r

1r

( )2 2 4 2 2 2 2 22 0 1 1 2 1 0 1 1 2 1 0 1 2 1 2 2

2 2 2 22 0 1 1 2 1 0 1 1 2

( )( ) ( )( ) ( )[( ) ( ) ]sin

( )( ) ( )( )Mrz

a a a a a

a a

t r m m m m m m m m m m r m r ms f

r m m m m r m m m m

- - - - + + + - + +=

- - + + +

( )2 2 4 2 2 2 2 22 0 1 1 2 1 0 1 1 2 1 0 1 2 1 2 2

2 2 2 22 0 1 1 2 1 0 1 1 2

( )( ) ( )( ) ( )[( ) ( ) ]cos

( )( ) ( )( )Mz

a a a a a

a aq

t r m m m m m m m m m m r m r ms f

r m m m m r m m m m

- - + - + + + + + -=

- - + + +

2 2 2 2 21 1 2 1 2 2

2 2 2 22 0 1 1 2 1 0 1 1 2

2 [( ) ( ) ]sin

( )( ) ( )( )Crz

a a a

a a

t m r m r ms f

r m m m m r m m m m- + +

=- - + + +

2 2 2 2 21 1 2 1 2 2

2 2 2 22 0 1 1 2 1 0 1 1 2

2 [( ) ( ) ]cos

( )( ) ( )( )Cz

a a a

a aq

t m r m r ms f

r m m m m r m m m m+ + -

=- - + + +

21 1 2

2 22 0 1 1 2 1 0 1 1 2

4sin

( )( ) ( )( )Frz

a

a a

t mms f

m m m m m m m m=

- - + + +2

1 1 22 2

2 0 1 1 2 1 0 1 1 2

4cos

( )( ) ( )( )Fz

a

a aq

t mms f

m m m m m m m m=

- - + + +


Separable form of fundamental solution (1D)

-10 10 20

2

4

6

8

10

Us,x

2

1

2

1

(x) (s), s x

(s, x)

(s) (x), x s

i ii

i ii

a b

U

a b

=

=

ìïï ³ïïïï=íïï >ïïïïî

å

å

1(s x), s x

1 2(s, x)12

(x s), x s2

U r

ìïï - ³ïïï= =íïï - >ïïïî

-10 10 20

-0.4

-0.2

0.2

0.4

Ts,x

s

Separable Separable propertyproperty

continuocontinuousus

discontidiscontinuousnuous

1, s x

2(s, x)1

, x s2

T

ìïï >ïïï=íï -ï >ïïïî


11 12 13

12 11 13

13

44

4

13 33

1 1

4

1 2

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

1 20 0 0 0 0 (2

2

2

)

xx xx

y

zy zy

zx

y yy

zz z

y

x

x

z

z

x y

c

c c c

c c c

c

c

c c

c c

s g

s

s g

s g

s g

s g

g

é ùì ü ìï ï ïê úï ï ïê úï ï ïï ï ïê úï ï ïï ï ïê úï ï ïê úï ï ïï ï ïï ï ïê ú=í ý íê úï ï ïï ï ïê úï ï ïï ï ê úï ï ê úï ïï ï ê úï ï -ê úï ïï ïî þ îê úë û

31

31

33

1

31 31 3

5

15

1

3

15

5

0 0

0 0

0 0

0 0

0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 02

02

x

y

x

yz

z

xx

yy

z

y

z

zzx

E

Ee

e

D e

e

e

e

E

D e e

e

e

D

g

g

g

g

g

ü é ùïï ê úïï ê úï ê úì üï ï ïï ê úï ïï ï ïï ï ïï ê ú-ý í ýê úï ï ïï ï ïê úï ï ïï ïê úî þï ïï ï ê úï ïï ï ê úï ï ê úë ûï ïï ïþ

ì üé ùï ïï ï ê úï ïï ï ê úí ýê úï ïï ï ê úï ï ê úï ïî þë û

1

33

1

11

0 0

0 0

0 0

2

z

xy

x

y

E

E

E

e

e

e

g

ì üï ïï ïï ïï ïï ï é ùì üï ï ï ïï ï ï ïê úï ï ï ïï ï ï ïï ï ê ú+í ý í ýê úï ï ï ïï ï ï ïê úï ï ï ïê úï ïë ûî þï ïï ïï ïï ïï ïï ïï ïî þ

94 學年度第 2 學期碩士論文口試 national taiwan ocean university msvlab department of...

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