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Null-field approach for Laplace proNull-field approach for Laplace problems with circular boundaries usinblems with circular boundaries using degenerate kernelsg degenerate kernels
研究生:沈文成指導教授:陳正宗 教授時間: 10:30 ~ 12:00地點:河工二館 307 室
碩士論文口試
2
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
3
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
4
Motivation and literature reviewMotivation and literature review
Fictitious Fictitious BEMBEM
BEM/BEM/BIEMBIEM
Null-field Null-field approachapproach
Bump Bump contourcontour
Limit Limit processprocess
Singular and hypersiSingular and hypersingularngular
RegulRegularar
Improper Improper integralintegral
CPV and CPV and HPVHPV
Ill-Ill-posedposed
FictitiFictitious ous
bounboundarydary
CollocatCollocation ion
pointpoint
5
Present approachPresent approach
1. No principal 1. No principal valuevalue2. Well-2. Well-posedposed
(s, x)eK
(s, x)iK
Advantages of Advantages of degenerate kerneldegenerate kernel
(x) (s, x) (s) (s)BK dBj f=ò
DegeneratDegenerate kernele kernel
Fundamental Fundamental solutionsolution
CPV and CPV and HPVHPV
No principal No principal valuevalue
(x) (s)(x) (s) (s)B
db Baj f=ò 2
1 1( ), ( )x s x s
O O- -
(x) (s)a b
6
Engineering problem with arbitrary Engineering problem with arbitrary geometriesgeometries
Degenerate Degenerate boundaryboundary
Circular Circular boundaryboundary
Straight Straight boundaryboundary
Elliptic Elliptic boundaryboundary
a(Fourier (Fourier series)series)
(Legendre poly(Legendre polynomial)nomial)
(Chebyshev poly(Chebyshev polynomial)nomial)
(Mathieu (Mathieu function)function)
7
Motivation and literature reviewMotivation and literature review
Analytical methods for solving Laplace problems with circ
ular holesConformal Conformal mappingmapping
Bipolar Bipolar coordinatecoordinate
Special Special solutionsolution
Limited to doubly Limited to doubly connected domainconnected domain
Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications
Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics
Honein, Honein and Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics
8
Fourier series approximationFourier series approximation
Ling (1943) - Ling (1943) - torsiontorsion of a circular tube of a circular tube Caulk et al. (1983) - Caulk et al. (1983) - steady heat conducsteady heat conduc
tiontion with circular holes with circular holes Bird and Steele (1992) - Bird and Steele (1992) - harmonic and harmonic and
biharmonicbiharmonic problems with circular hol problems with circular holeses
Mogilevskaya et al. (2002) - Mogilevskaya et al. (2002) - elasticityelasticity pr problems with circular boundariesoblems with circular boundaries
9
Contribution and goalContribution and goal
However, they didn’t employ the However, they didn’t employ the nnull-field integral equationull-field integral equation and and degedegenerate kernelsnerate kernels to fully capture the ci to fully capture the circular boundary, although they all ercular boundary, although they all employed mployed Fourier series expansionFourier series expansion..
To develop a To develop a systematic approachsystematic approach f for solving Laplace problems with or solving Laplace problems with mmultiple holesultiple holes is our goal. is our goal.
10
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
11
Boundary integral equation and null-Boundary integral equation and null-field integral equationfield integral equation
2 (x) (s, x) (s) (s) (s, x) (s) (s), xB B
u T u dB U t dB Dp = - Îò ò0 (s, x) (s) (s) (s, x) (s) (s), x c
B BT u dB U t dB D= - Îò ò
s
s
(s, x) ln x s ln
(s, x)(s, x)
(s)(s)
U r
UT
ut
= - =
¶=
¶
¶=
¶
n
n
x
D
xcD
x
D xcD
Interior Interior casecase
Exterior Exterior casecase
Null-field integral Null-field integral equationequation
12
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
13
Expansions of fundamental solution Expansions of fundamental solution and boundary densityand boundary density
Degenerate kernel - fundamental Degenerate kernel - fundamental solutionsolution
Fourier series expansions - boundary Fourier series expansions - boundary densitydensity
1
1
1( , ; , ) ln ( ) cos ( ),
(s, x)1
( , ; , ) ln ( ) cos ( ),
i m
m
e m
m
U R R m Rm R
UR
U R m Rm
rq r f q f r
q r f r q f rr
¥
=
¥
=
ìïï = - - ³ïïïï=íïï = - - >ïïïïî
å
å
01
01
(s) ( cos sin ), s
(s) ( cos sin ), s
M
n nn
M
n nn
u a a n b n B
t p p n q n B
q q
q q
=
=
= + + Î
= + + Î
å
å
14
Separable form of fundamental Separable form of fundamental solution (1D)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 x2(s, x)1, x s
2
T
ìïï >ïïï=íï -ï >ïïïî
15-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
Separable form of fundamental Separable form of fundamental solution (2D)solution (2D)
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
Ro
s ( , )R q=
x ( , )r f=
iU
eU
r
1
1
1( , ; , ) ln ( ) cos ( ),
(s, x)1
( , ; , ) ln ( ) cos ( ),
i m
m
e m
m
U R R m Rm R
UR
U R m Rm
rq r f q f r
q r f r q f rr
¥
=
¥
=
ìïï = - - ³ïïïï=íïï = - - >ïïïïî
å
å
x ( , )r f=
16
Boundary density discretizationBoundary density discretization
Fourier Fourier seriesseries
Ex . constant Ex . constant elementelement
Present Present methodmethod
Conventional Conventional BEMBEM
17
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
18
Adaptive observer systemAdaptive observer system
( , )r f
collocation collocation pointpoint
19
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
20
Vector decomposition technique for Vector decomposition technique for potential gradientpotential gradient
zx
z x-
(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 Special case (concentric case) :(concentric case) :
z x=
(s, x)(s, x)
ULr r
¶=
¶(s, x)
(s, x)T
M r r¶
=¶
Non-Non-concentric concentric
case:case:
(x)2 (s, x) (s) (s) (s, x) (s) (s), x
(x)2 (s, x) (s) (s) (s, x) (s) (s), x
B B
B B
uM u dB L t dB D
uM u dB L t dB D
r r
ff
p
p
¶= - Î
¶¶
= - ζ
ò ò
ò ò
n
t
nt
t
n
True normal True normal directiondirection
21
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
22
{ }
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
t
t
t t
t
M
Linear algebraic equationLinear algebraic equation
[ ]{ } [ ]{ }U t T u=
[ ]
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
whwhereere
Column vector of Column vector of Fourier coefficientsFourier coefficients(Nth routing circle)(Nth routing circle)
0B1B
Index of Index of collocation collocation
circlecircle
Index of Index of routing circle routing circle
23
Explicit form of each submatrix [Explicit form of each submatrix [UUpkpk] an] and vector {d vector {ttkk}}
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 Mc Mspk pk pk pk pkc c s Mc Mspk pk pk pk pkc c s Mc Mspk pk pk pk pk
pk
c c s Mc Mspk M pk M pk M pk M pk
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
( )
( ) ( ) ( ) ( ) ( )M
c c s Mc Mspk M pk M pk M pk M pk MU 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 M Mp p q p q=t L
1f
2f
3f
2Mf
2 1Mf +
Fourier Fourier coefficientscoefficients
Truncated Truncated terms of terms of
Fourier seriesFourier series
Number of Number of collocation pointscollocation points
24
Flowchart of present methodFlowchart of present method
0 [ (s, x) (s) (s, x) (s)] (s)BT u U t dB= -ò
Potential Potential of domain of domain
pointpointAnalytiAnalyticalcal
NumeriNumericalcal
Adaptive Adaptive observer observer systemsystem
DegeneratDegenerate kernele kernel
Fourier Fourier seriesseries
Linear algebraic Linear algebraic equation equation
Collocation point and Collocation point and matching B.C.matching B.C.
Fourier Fourier coefficientscoefficients
Vector Vector decompodecompo
sitionsition
Potential Potential gradientgradient
25
Comparisons of conventional BEM Comparisons of conventional BEM and present methodand present method
BoundaryBoundarydensitydensity
discretizatdiscretizationion
AuxiliaryAuxiliarysystemsystem
FormulatFormulationion
ObserObserverver
systesystemm
SingularSingularityity
ConventiConventionalonal
BEMBEM
Constant,Constant,Linear,Linear,
QurdraturQurdrature…e…
FundameFundamentalntal
solutionsolution
BoundarBoundaryy
integralintegral
equationequation
FixedFixed
obserobserverver
systesystemm
CPV, RPCPV, RPVV
and HPVand HPV
PresentPresentmethodmethod
FourierFourier
seriesseries
expansioexpansionn
DegeneraDegeneratete
kernelkernel
Null-Null-fieldfield
integralintegral
equationequation
AdaptiAdaptiveve
obserobserverver
systesystemm
NoNo
principprincipalal
valuevalue
26
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
27
Numerical examplesNumerical examples
Steady state heat conduction problemsSteady state heat conduction problems Electrostatic potential of wiresElectrostatic potential of wires Flow of an ideal fluid pass cylindersFlow of an ideal fluid pass cylinders A circular bar under torqueA circular bar under torque An infinite medium under antiplane sheAn infinite medium under antiplane she
arar Half-plane problemsHalf-plane problems
28
Numerical examplesNumerical examples
Steady state heat conduction problemsSteady state heat conduction problems Electrostatic potential of wiresElectrostatic potential of wires Flow of an ideal fluid pass cylindersFlow of an ideal fluid pass cylinders A circular bar under torqueA circular bar under torque An infinite medium under antiplane sheAn infinite medium under antiplane she
arar Half-plane problemsHalf-plane problems
29
Steady state heat conduction Steady state heat conduction problemsproblems
Case Case 11
Case Case 22
1u=
0u=
1 2.5a =2 1.0a =
1u=
1u=
0u=
0 2.0R =
a
a
30
Steady state heat conduction Steady state heat conduction problemsproblems
Case Case 33
Case Case 44
0u
n
¶=
¶
1u=
0u=
0u
n
¶=
¶
0 2.0R =
a
a
a
1u=
0u=
0u
n
¶=
¶
1u=
0 2.0R =
a
a
a
31
Case 1: Isothermal lineCase 1: Isothermal line
Exact Exact solutionsolution
(Carrier and (Carrier and Pearson)Pearson)
BEM-BEPO2DBEM-BEPO2D(N=21)(N=21)
FEM-ABAQUSFEM-ABAQUS(1854 (1854
elements)elements)
Present Present methodmethod(M=10)(M=10)-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
32
0 90 180 270 360
Degr ee ( )
0
1
2
3
Rel
ativ
e er
ror
of f
lux
on t
he
sm
all
circ
le (
%)
B E M -B E P O2 D (N = 2 1 )
P r es ent met hod (M = 1 0 )
Tr efft z met hod (N T= 2 1 )
M FS (N M = 2 1 ) (a1 '= 3 .0 , a2 '= 0 .7 )
Relative error of flux on the small Relative error of flux on the small circlecircle
33
Convergence test - Parseval’s sum for Convergence test - Parseval’s sum for Fourier coefficientsFourier coefficients
0 4 8 12 16 20
Ter ms of Four ier s er ies (M )
1 0
1 1
1 2
1 3
1 4
1 5
Par
sev
al's
sum
0 4 8 12 16 20
Ter ms of Four ier s er ies (M )
2
2.4
2.8
3.2
3.6
Par
sev
al's
sum
22 2 2 2
00
1
( ) 2 ( )M
n nn
f d a a bp
q q p p=
+ +åò B&Parseval’s Parseval’s
sumsum
34
Case 2: Isothermal lineCase 2: Isothermal line
Caulk’s data (1983)Caulk’s data (1983)IMA Journal of Applied MatheIMA Journal of Applied Mathematicsmatics
Present Present method method (M=10)(M=10)
FEM-ABAQUSFEM-ABAQUS(6502 (6502
elements)elements)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
35
Case 3: Isothermal lineCase 3: Isothermal line
FEM-ABAQUSFEM-ABAQUS(8050 (8050
elements)elements)Present Present method method (M=10)(M=10)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Caulk’s data (1983)Caulk’s data (1983)IMA Journal of Applied MatheIMA Journal of Applied Mathematicsmatics
36
Case 4: Isothermal lineCase 4: Isothermal line
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
FEM-ABAQUSFEM-ABAQUS(8050 (8050
elements)elements)Present Present method method (M=10)(M=10)
Caulk’s data (1983)Caulk’s data (1983)IMA Journal of Applied MatheIMA Journal of Applied Mathematicsmatics
37
Numerical examplesNumerical examples
Steady state heat conduction problemsSteady state heat conduction problems Electrostatic potential of wiresElectrostatic potential of wires Flow of an ideal fluid pass cylindersFlow of an ideal fluid pass cylinders A circular bar under torqueA circular bar under torque An infinite medium under antiplane sheAn infinite medium under antiplane she
arar Half-plane problemsHalf-plane problems
38
Electrostatic potential of wiresElectrostatic potential of wires
Hexagonal Hexagonal electrostatic electrostatic
potentialpotential
Two parallel cylinders Two parallel cylinders held positive and held positive and
negative potentialsnegative potentials
1u=- 1u=
2l
aa1u=
1u=-1u=
1u=-
1u= 1u=-
39
Contour plot of potentialContour plot of potential
Exact solution (LebeExact solution (Lebedev et al.)dev et al.)
Present Present method method (M=10)(M=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 10-10
-8
-6
-4
-2
0
2
4
6
8
10
40
Contour plot of potentialContour plot of potential
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Onishi’s data Onishi’s data (1991)(1991)
Present Present method method (M=10)(M=10)
41
Numerical examplesNumerical examples
Steady state heat conduction problemsSteady state heat conduction problems Electrostatic potential of wiresElectrostatic potential of wires Flow of an ideal fluid pass cylindersFlow of an ideal fluid pass cylinders A circular bar under torqueA circular bar under torque An infinite medium under antiplane sheAn infinite medium under antiplane she
arar Half-plane problemsHalf-plane problems
42
Flow of an ideal fluid pass two Flow of an ideal fluid pass two parallel cylindersparallel cylinders
is the velocity of flow far is the velocity of flow far from the cylindersfrom the cylinders
is the incident angleis the incident angle
v¥
g
v¥
g
2l
a a
43
Velocity field in different incident Velocity field in different incident angleangle
-14 -12 -10 -8 -6 -4 -2 0 2 4-10
-8
-6
-4
-2
0
2
4
6
8
10
-14 -12 -10 -8 -6 -4 -2 0 2 4-10
-8
-6
-4
-2
0
2
4
6
8
10
Present Present method method (M=10)(M=10)
180g= o
Present Present method method (M=10)(M=10)
135g= o
44
Numerical examplesNumerical examples
Steady state heat conduction problemsSteady state heat conduction problems Electrostatic potential of wiresElectrostatic potential of wires Flow of an ideal fluid pass cylindersFlow of an ideal fluid pass cylinders A circular bar under torqueA circular bar under torque An infinite medium under antiplane sheAn infinite medium under antiplane she
arar Half-plane problemsHalf-plane problems
45
Torsion bar with circular holes Torsion bar with circular holes removedremoved
The warping The warping functionfunction
Boundary conditionBoundary condition
wherewhere
2 ( ) 0,x x DjÑ = Î
j
sin cosk k k kx yn
jq q
¶= -
¶ kB
2 2cos , sini i
i ix b y b
N N
p p= =
2 k
N
p
a
a
ab q
R
oonn
TorqTorqueue
46
Axial displacement with two circular Axial displacement with two circular holesholes
Present Present method method (M=10)(M=10)
Caulk’s data (1983)Caulk’s data (1983)ASME Journal of Applied MechASME Journal of Applied Mechanicsanics
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2-1.5-1-0.500.511.52
Dashed line: exact Dashed line: exact solutionsolution
Solid line: first-order Solid line: first-order solutionsolution
47
Axial displacement with three Axial displacement with three circular holescircular holes
Present Present method method (M=10)(M=10)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Caulk’s data (1983)Caulk’s data (1983)ASME Journal of Applied MechASME Journal of Applied Mechanicsanics
Dashed line: exact Dashed line: exact solutionsolution
Solid line: first-order Solid line: first-order solutionsolution
48
Axial displacement with four circular Axial displacement with four circular holesholes
Present Present method method (M=10)(M=10)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Caulk’s data (1983)Caulk’s data (1983)ASME Journal of Applied MechASME Journal of Applied Mechanicsanics
Dashed line: exact Dashed line: exact solutionsolution
Solid line: first-order Solid line: first-order solutionsolution
49
Numerical examplesNumerical examples
Steady state heat conduction problemsSteady state heat conduction problems Electrostatic potential of wiresElectrostatic potential of wires Flow of an ideal fluid pass cylindersFlow of an ideal fluid pass cylinders A circular bar under torqueA circular bar under torque An infinite medium under antiplane sheAn infinite medium under antiplane she
arar Half-plane problemsHalf-plane problems
50
Infinite medium under antiplane shearInfinite medium under antiplane shear
The displacementThe displacement
Boundary conditionBoundary condition
Total displacementTotal displacement
t
m
sw2 ( ) 0,sw x x DÑ = Î
( )sin
sw x
n
tq
m¶
=¶
sw w w¥= +
oonn
kB
51
Shear stress Shear stress σz around the hole of radiu around the hole of radius as a11 (x axis) (x axis)
0 1 2 3 4 5 6 (in r adians )
- 2
0
2
4
6
8
z
/
(aro
un
d h
ole
wit
h r
ad
ius
a1
)
d/a1 = 0 .0 1
d/a1 = 0 .1
d/a1 = 2 .0
s ingle hole
Present Present method method (M=20)(M=20)
Honein’s data (1Honein’s data (1992)992)Quarterly of Applied MathQuarterly of Applied Mathematicsematics
52
Shear stress Shear stress σz around the hole of radiu around the hole of radius as a11 (y axis) (y axis)
0 1 2 3 4 5 6 (in r adians )
- 2
0
2
4
6
8
z
/
(aro
un
d h
ole
wit
h r
adiu
s a
1)
d/a1= 0 .0 1
d/a1= 0 .1
d/a1= 2 .0
Present Present method method (M=20)(M=20)
Honein’s data (1Honein’s data (1992)992)Quarterly of Applied MathQuarterly of Applied Mathematicsematics
53
Shear stress Shear stress σz around the hole of radiu around the hole of radius as a11 (45 degrees) (45 degrees)
0 1 2 3 4 5 6 (in r adians )
- 2
0
2
4
6
8
1 0
z
/
(aro
und
hole
wit
h ra
dius
a1
)
d/a1= 0 .0 1
d/a1= 0 .1
d/a1= 2 .0
Present Present method method (M=20)(M=20)
Honein’s data (1Honein’s data (1992)992)Quarterly of Applied MathQuarterly of Applied Mathematicsematics
54
Shear stress Shear stress σz around the hole of radiu around the hole of radius as a11 (Touching) (Touching)
0 1 2 3 4 5 6 (in r adians )
- 2
0
2
4
6
8
1 0
z
/
(aro
un
d h
ole
wit
h r
ad
ius
a1
)
M = 1 0M = 2 0M = 3 0M = 4 0
Present Present methodmethod
discontidiscontinuousnuous
discontidiscontinuousnuous
1a 2a
Honein’s data (1Honein’s data (1992)992)Quarterly of Applied MathQuarterly of Applied Mathematicsematics
Gibb’s Gibb’s phenomenophenomeno
nn
55
Two equivalent approachesTwo equivalent approaches
0 sinw R q=
0R
d
2a
1a
sinw
nq
¶=
¶
0R
d
2a
1a
Displacement Displacement approachapproach
Stress Stress approachapproach
Present Present methodmethod
Bird and Steele Bird and Steele (1992)(1992)
ASME Journal of Applied ASME Journal of Applied MechanicsMechanics
56
Shear stress Shear stress σz around the hole of radiu around the hole of radius as a11
0 90 180 270 360
-2
0
2
4
R 0= 7 .5
R 0= 1 5 .0
R 0= 3 0 .0
0 90 180 270 360
-2
0
2
4
R 0= 7 .5
R 0= 1 5 .0
R 0= 3 0 .0
Present Present method method (M=20)(M=20)
Present Present method method (M=20)(M=20)
Steele’s data Steele’s data (1992)(1992)
Stress Stress approachapproach
Displacement Displacement approachapproachHonein’s data Honein’s data
(1992)(1992)5.35.34848
5.35.34949
4.64.64747
5.35.34545
13.1313.13%%
0.020.02%%
AnalytiAnalyticalcal
0.060.06%%
57
Convergence of stress σzat =45 degrees versus R0
0 30 60 90 120 150
R adius R 0
0
2
4
6
8
z
at
=
45
deg
rees
Equivalent dis placement appr oach
Equivalent s t r es s appr oach
0 sinw R q=
sinw
nq
¶=
¶
0t=
0t=
0R
58
Three circular holes with centers on Three circular holes with centers on the x axisthe x axis
0 1 2 3 4 5 6 (in r adians )
- 2
0
2
4
6
8
z
/
(aro
und
hol
e w
ith
radi
us
a1
)
d/a1= 2 .0
d/a1= 0 .1
d/a1= 0 .0 1
1a2a3a
y
x
t
m
dd
59
Three circular holes with centers on Three circular holes with centers on the y axisthe y axis
0 1 2 3 4 5 6 (in r adians )
- 2
- 1
0
1
2
z
/
(aro
un
d h
ole
wit
h r
ad
ius
a1)
d/a1= 2 .0
d/a1= 0 .1
d/a1= 0 .0 1
x
y
1a
2a
3a
t
m
d
d
60
Three circular holes with centers on Three circular holes with centers on the line making 45 degreesthe line making 45 degrees
0 1 2 3 4 5 6 (in r adians )
- 2
0
2
4
6
8
1 0
z
/
(aro
und
hole
wit
h ra
dius
a1)
d/a1= 2 .0
d/a1= 0 .1
d/a1= 0 .0 1
1a
2a
3a
x
y
t
m d
d
61
Numerical examplesNumerical examples
Steady state heat conduction problemsSteady state heat conduction problems Electrostatic potential of wiresElectrostatic potential of wires Flow of an ideal fluid pass cylindersFlow of an ideal fluid pass cylinders A circular bar under torqueA circular bar under torque An infinite medium under antiplane sheAn infinite medium under antiplane she
arar Half-plane problemsHalf-plane problems
62
Half-plane problemsHalf-plane problems
Dirichlet boundary cDirichlet boundary conditionondition(Lebedev et al.)(Lebedev et al.)
Mixed-type boundary coMixed-type boundary conditionndition(Lebedev et al.)(Lebedev et al.)
0u=
1u=
1B
2B
0u=
1u
n
¶=
¶
1B
2B
h ha a
63
Dirichlet problemDirichlet problem
Exact solution (LebeExact solution (Lebedev et al.)dev et al.)
Present Present method method (M=10)(M=10)
IsothermIsothermal lineal line
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 1 0
- 8
- 6
- 4
- 2
0
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 1 0
- 8
- 6
- 4
- 2
0
64
Mixed-type problemMixed-type problem
Exact solution (LebeExact solution (Lebedev et al.)dev et al.)
Present Present method method (M=10)(M=10)
IsothermIsothermal lineal line
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 1 0
- 8
- 6
- 4
- 2
0
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0- 1 0
- 8
- 6
- 4
- 2
0
65
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
66
Numerical instability in BEMNumerical instability in BEM
2r
1r
a Annular Annular casecase
Interior Interior casecase
Max Max errorerror
DegeneratDegenerate scalee scale
u specified= u specified=
International Journal International Journal forfor
Numerical Methods in Numerical Methods in EngineeringEngineering
Engineering Engineering AnalysisAnalysis
with Boundary with Boundary Elements Elements
Matrix Matrix singularsingular
ErroErrorr
SinguSingularlar
valuevalue
67
2 21 1 1 1 1 1 2 2 2 1 2 1
2 2
2 21 1 1 2 1 2 2 2 2 2 2 2
2 2
2 21 1 1 2 1 1 2 1 2 2 2 2 1 2
2 2
2 ln cos sin 2 ln ( )cos ( )sin
2 ln cos sin 2 ln ( )cos ( )sin
2 ln cos sin 2 ln ( ) cos ( )sinM M M
a aa a a a a a a
a aa a a a a a a
a aa a a a a a a
p p f p f p r p f p fr r
p p f p f p r p f p fr r
p p f p f p r p f pr r+ + +
L L
L L
M M M O M M M O
L 2 1
1 11 1 1 1 1 1 2 2 2 1 2 1
1 1
1 11 1 1 2 1 2 2 2 2 2 2 2
1 1
1 11 1 1 2 1 1 2 1 2 2 2 2 1
1 1
2 ln ( )cos ( )sin 2 ln cos sin
2 ln ( )cos ( )sin 2 ln cos sin
2 ln ( ) cos ( )sin 2 ln cos
M
M M M
a a a a a a a aa a
a a a a a a a aa a
a a a a a a aa a
f
r rp p f p f p p f p f
r rp p f p f p p f p f
r rp p f p f p p f
+
+ + +
L
L L
L L
M M M O M M M O
L
1,0
1,1
1,1
1,
1,
2,0
2,1
2,1
2,
2,2 2 1sin
M
M
M
MM
p
p
q
p
q
p
p
q
p
qap f +
é ùê úì üê úï ïï ïê úï ïï ïê úï ïê úï ïï ïê úï ïï ïê úï ïê úï ïï ïê úï ïï ïê úï ïê úï ïï ïê úï ïï ïê úï ïï ïí ýê úï ïï ïê úï ïï ïê úï ïï ïê úï ïê úï ïï ïê úï ïï ïê úï ïê úï ïïê úïïê úïê úïïê úïïî þê úïïê úê úë ûê ú
M
M
L
[ ]
1,0
1,1
1,1
1,
1,
2,0
2,1
2,1
2,
2,
M
M
M
M
a
a
b
a
b
a
a
b
a
b
ì üï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ï= í ýï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïï ï ïî þï ï ïï ï ï
T
M
M
Degenerate scale in the multiply Degenerate scale in the multiply connected problemconnected problem
a1 =1.0, influence matrix [U] is singular
1a
68
Treatments of degenerate scale Treatments of degenerate scale problemproblem
Method of adding a Method of adding a rigid body termrigid body term
CHEEF conceptCHEEF concept
[ ]{ } [ ]{ }=U t T u
(s, x) (s, x)mU U c= +
12 a cp+ 1 12 (ln )a a cpé ù+ê úê úë û
L
M O1 12 lna apé ù
ê úê úë û
L
M O
SinguSingularlar
[ ]{ } [ ]{ }=U t T u
SinguSingularlar
Auxiliary Auxiliary constraint constraint { } { }=w t v u
[ ]{ } [ ]{ }=U t T u[ ]
{ }[ ]
{ }é ù é ùê ú ê ú=ê ú ê úê ú ê úë û ë û
U Tt u
w vNonsingulaNonsingula
rr
1a
CHEEF CHEEF pointpoint
Promote Promote rankrank
69
0.5 1 1.5 2 2.5 3
R adius a1
0
0.1
0.2
0.3
0.4
0.5
1
P r es ent met hod
A dding a CH EE F poing (5 .0 ,5 .0 )
A dding a r igid body t er m (c= 1 .0 )
The minimum singular value versus The minimum singular value versus radius aradius a11
DegeneratDegenerate scalee scale
1a
Numerical Numerical failurefailure
70
OutlinesOutlines
Motivation and literature reviewMotivation and literature review Mathematical formulationMathematical formulation
Expansions of fundamental solutionExpansions of fundamental solution and boundary densityand boundary density
Adaptive observer systemAdaptive observer system Vector decomposition techniqueVector decomposition technique Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples Degenerate scaleDegenerate scale ConclusionsConclusions
71
ConclusionsConclusions
A systematic approach using A systematic approach using degenerate degenerate kernelskernels, , Fourier seriesFourier series and and null-field integnull-field integral equationral equation has been successfully propo has been successfully proposed to solve Laplace problems with circulsed to solve Laplace problems with circular boundaries.ar boundaries.
Numerical results Numerical results agree wellagree well with availabl with available exact solutions, Caulk’s data, Onishi’e exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for s data and FEM (ABAQUS) for only few teronly few terms of Fourier seriesms of Fourier series..
72
ConclusionsConclusions
Method of adding a rigid body termMethod of adding a rigid body term and and CHEEF aCHEEF approachpproach have been successfully adopted to over have been successfully adopted to overcome the come the degenerate scale for multiply connectdegenerate scale for multiply connected problemed problem..
The The stress concentrationstress concentration due to due to different orientdifferent orientationsations was discussed by using present method. was discussed by using present method.
Engineering problemsEngineering problems with with circular boundariescircular boundaries which satisfy the which satisfy the Laplace equationLaplace equation can be solve can be solved by using the proposed approach in a d by using the proposed approach in a more effimore efficient and accurate mannercient and accurate manner..
73
The endThe end
Thanks for your kind attentions.Thanks for your kind attentions.
Your comments will be highly apprYour comments will be highly appreciated.eciated.
74
Further researchFurther research
Expansion to general boundary, e.g. elExpansion to general boundary, e.g. elliptic, straight, degenerate.liptic, straight, degenerate.
Antiplane problem with rigid inclusioAntiplane problem with rigid inclusionn
Expansion to three-dimensional problExpansion to three-dimensional problemem
Bi-center expansion techniqueBi-center expansion technique
75
Derivation of degenerate kernelDerivation of degenerate kernel
Graf’s addition theoremGraf’s addition theorem Complex variableComplex variable
s xs ( , ) , x ( , )R z zq r f= = = =
x sln x s ln z z- = - Real Real partpart
x x xs x s s s
1s s s
1ln( ) ln[( )(1 )] ln( ) ln(1 ) ln( ) ( )m
m
z z zz z z z z
z z m z
¥
=
- = - = + - = - å
( )x
1 1 1 1s
1 1 1 1( ) ( ) ( ) [ ] ( ) cos ( )
im m m i m m
im m m m
z ee m
m z m Re m R m R
ff q
q
r r rq f
¥ ¥ ¥ ¥-
= = = =
= = = -å å å å
Real Real partpart
IfIf s xz z-
1
1
1( , ; , ) ln ( ) cos ( ),
(s, x)1
( , ; , ) ln ( ) cos ( ),
i m
m
e m
m
U R R m Rm R
UR
U R m Rm
rq r f q f r
q r f r q f rr
¥
=
¥
=
ìïï = - - ³ïïïï=íïï = - - >ïïïïî
å
å
ln R
2
2 3
1
1ln(1 ) (1 )
11 1
( )2 31 m
m
x dx x x dxx
x x x
xm
¥
=
- =- =- + + +-
=- + + +
=-
ò ò
å
L
L
0k ®
Bessel’s Bessel’s functionfunction
76
Non-unique solutionsNon-unique solutions
Non-unique Non-unique solutionssolutions
Rigid body Rigid body solutionsolution
for Neumann for Neumann problemsproblems
Critical size of thCritical size of theedomain in plane domain in plane BVPs BVPs
Hypersingular formulatiHypersingular formulationonfor multiply connected for multiply connected problemsproblems
uspecified
n
¶=
¶u specified=1a= 1a=
MathematicalMathematically andly and
physically physically realizablerealizable
Mathematically Mathematically realizablerealizable
Mathematically Mathematically realizablerealizable
[ ]{ } [ ]{ }=U t T u[ ]{ } [ ]{ }[ ]{ } [ ]{ }
=
=
U t T u
L t M u
[ ]{ } [ ]{ }[ ]{ } [ ]{ }
=
=
U t T u
L t M u
DegeneratDegenerate scalee scaleNon-Non-
uniquenesuniquenesss
2 0uÑ =2 0uÑ =
2 0uÑ =
77
Non-unique solutions in direct BEMNon-unique solutions in direct BEM
Domain of Domain of interestinterest
SingularSingular
formulationformulationHypersinHypersin
gulargularformulatiformulati
onon
SimplySimply
connecconnectedted
domaidomainn
InteriInterioror
casecase
a=1.0a=1.0 NANA
ExterExteriorior
casecase
a=1.0a=1.0 a is a is arbitrararbitrar
yy
MultiplMultiplyy
connecconnectedted
domaidomainn
AnnulAnnularar
casecase
a=1.0a=1.0 a is a is arbitrararbitrar
yy
EccenEccentrictric
casecase
a=1.0a=1.0 a is a is arbitrararbitrar
yy
a
a
a
a
78
Solutions of half-plane problemSolutions of half-plane problem
1u=-
1u= 1u=
Half-plane Half-plane problemproblem
Infinite Infinite problemproblem
Image Image conceptconcept
Anti-symmetry Anti-symmetry propertyproperty
(s; x, x ) ln x s ln x sU ¢ ¢= - - -
1s(s; x, x ) 0BU ΢ =0u=
1u=
1B
2B 2B
1B
1B
x¢
s
xr
r¢
2B
79
FormulationFormulation
2 (x) (s; x, x ) (s) (s) (s; x, x ) (s) (s), xB B
u T u dB U t dB Dp ¢ ¢= - Îò ò0 (s; x, x ) (s) (s) (s; x, x ) (s) (s), x c
B BT u dB U t dB D¢ ¢= - Îò ò
1
1
1
1
1( , ; , , , ) ln ( ) cos ( )
1ln ( ) cos ( ),
(s; x, x )1
( , ; , , , ) ln ( ) cos ( )
1ln ( ) cos ( ),
i m
m
m
m
e m
m
m
m
U R R mm R
Rm R
mU
RU R m
m
Rm R
m
rq r f r f q f
r q f r rr
q r f r f r q fr
r q f r rr
¥
=
¥
=
¥
=
¥
=
ìïï ¢ ¢= - -ïïïïïïï ¢ ¢ ¢- + - > ³ïï ¢ïï¢=ïíïï ¢ ¢= - -ïïïïïïï ¢ ¢ ¢- + - > >ïï ¢ïïîï
å
å
å
å