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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
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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
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3MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
OutlineOutline
• Motivation and literature reviewMotivation 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
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4MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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5MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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)
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6MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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7MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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]
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8MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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..
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9MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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.
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10MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
◎ Adaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique
• Numerical examples• Conclusions• Further studies
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11MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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12MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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13MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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14MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
¶ ¶
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15MSVLABMSVLAB 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 systemAdaptive observer system ◎ Linear algebraic equation ◎ Vector decomposition technique
• Numerical examples• Conclusions• Further studies
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16MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
Adaptive observer systemAdaptive observer system
collocation pointcollocation point
0 , 01 , 1k , k2 , 2
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17MSVLABMSVLAB 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 equationLinear algebraic equation ◎ Vector decomposition technique
• Numerical examples• Conclusions• Further studies
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18MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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19MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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20MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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21MSVLABMSVLAB 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 techniqueVector decomposition technique
• Numerical examples• Conclusions• Further studies
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22MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
¶¶
= - ζ
È
È
ò ò
ò ò
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23MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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24MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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25MSVLABMSVLAB 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 examplesNumerical examples• Conclusions• Further studies
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26MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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)
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27MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
Numerical examplesNumerical examples
• Anti-plane piezoelectricity problemsAnti-plane piezoelectricity problems
• In-plane electrostatics problems
• Anti-plane elasticity problems
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28MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
= +
+
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29MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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30MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
,,
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
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31MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
![Page 32: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/32.jpg)
32MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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)
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33MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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°°
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 34: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/34.jpg)
34MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
Tangential stress distribution for different ratios Tangential stress distribution for different ratios d/rd/r11 with with rr22=2=2rr11, e, e1515
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
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 35: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/35.jpg)
35MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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=-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
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 36: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/36.jpg)
36MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
Term s of Fourier sere is (L)
6E-007
6.4E-007
6.8E-007
7.2E-007
7.6E-007
8E-007
8.4E-007
Pa
rse
val's
sum
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37MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
0 10 20 30
Term s of Fourier sere is (L)
7E+011
8E+011
9E+011
1E+012
1.1E+012
Par
seva
l's s
um
0 10 20 30
Term s of Fourier sere is (L)
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
Term s of Fourier sere is (L)
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
Term s of Fourier sere is (L)
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
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38MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
Tangential stress distribution for different ratios Tangential stress distribution for different ratios d/rd/r11 with with rr22=2=2rr11, e, e1515
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
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 39: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/39.jpg)
39MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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=-5.0=-5.0 and and =0=0°°
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 40: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/40.jpg)
40MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
-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°°
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 41: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/41.jpg)
41MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
-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°°
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 42: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/42.jpg)
42MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 43: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/43.jpg)
43MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
Chao & Chang’s data (199Chao & Chang’s data (1999)9)
Present method (L=20)Present method (L=20)
![Page 44: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/44.jpg)
44MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
![Page 45: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/45.jpg)
45MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
![Page 46: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/46.jpg)
46MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
Contour of electric potentialContour of electric potentialwhen when d/rd/r11=0.01=0.01
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 0
0
-0.6
-1.2
-1.8
0.6
1.2
1.8
-0.6
0.6
![Page 47: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/47.jpg)
47MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
)
![Page 48: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/48.jpg)
48MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
![Page 49: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/49.jpg)
49MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
![Page 50: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/50.jpg)
50MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
![Page 51: 94 學年度第 2 學期碩士論文口試 National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 1 Null-field approach for multiple circular inclusion](https://reader031.vdocuments.pub/reader031/viewer/2022032201/56649d0c5503460f949dfffc/html5/thumbnails/51.jpg)
51MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
Numerical examplesNumerical examples
• Anti-plane piezoelectricity problems
• In-plane electrostatics problemsIn-plane electrostatics problems
• Anti-plane elasticity problems
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52MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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53MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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54MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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)
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55MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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)
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56MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
Numerical examplesNumerical examples
• Anti-plane piezoelectricity problems
• In-plane electrostatics problems
• Anti-plane elasticity problemsAnti-plane elasticity problems
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57MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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58MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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59MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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60MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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61MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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62MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
)
Present method (L=20)Present method (L=20)
Equilibrium of tractionEquilibrium of traction
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63MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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64MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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65MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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66MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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)
Present method (L=20)Present method (L=20)
d d
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67MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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)
Present method (L=20)Present method (L=20)
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
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68MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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69MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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=
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70MSVLABMSVLAB 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• ConclusionsConclusions• Further studies
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71MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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.
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72MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
ConclusionsConclusions
• 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..
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73MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
ConclusionsConclusions
• 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.
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74MSVLABMSVLAB 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 studiesFurther studies
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75MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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.
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76MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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77MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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78MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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79MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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80MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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=
- - + + +
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81MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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
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(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
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82MSVLABMSVLAB National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
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