analysis of circular torsion bar with circular holes using null-field approach
Post on 22-Jan-2016
44 Views
Preview:
DESCRIPTION
TRANSCRIPT
九十四年電子計算機於土木水利工程應用研討會
1
Analysis of circular torsion bar with Analysis of circular torsion bar with circular holes using null-field circular holes using null-field approachapproach
研究生:沈文成指導教授:陳正宗 教授時間: 15:10 ~ 15:25地點: Room 47450-3
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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples 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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples 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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples 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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples 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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples 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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples ConclusionsConclusions
20
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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples ConclusionsConclusions
21
{ }
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
22
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
23
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
24
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
25
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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples ConclusionsConclusions
26
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
27
Torsion problemsTorsion problems
TorqTorqueue
TorqTorqueue
ab
qR
Eccentric Eccentric casecase
Two holes Two holes (N=2)(N=2)
a
b
q
R
a
28
Torsion problemsTorsion problems
a
a
ab q
R
TorqTorqueue
TorqTorqueuea
a
ab q
R
a
Three Three holes holes (N=3)(N=3)
Four holes Four holes (N=4)(N=4)
29
Torsional rigidity (Eccentric case)Torsional rigidity (Eccentric case)
Present Present methodmethod
(M=10)(M=10)
Exact Exact solutionsolution
BIEBIE
formulatioformulationn
0.200.20 0.978720.97872 0.978720.97872 0.978720.97872
0.400.40 0.951370.95137 0.951370.95137 0.951370.95137
0.600.60 0.903120.90312 0.903120.90312 0.903160.90316
0.800.80 0.824730.82473 0.824730.82473 0.824970.82497
0.900.90 0.761680.76168 0.761680.76168 0.762520.76252
0.980.98 0.667050.66705 0.665550.66555 0.667320.66732
42 /G Rmpb
R a-
30
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
31
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
32
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
33
Torsional rigidity (N=2,3,4)Torsional rigidity (N=2,3,4)
Number of Number of holesholes
Present Present methodmethod
(M=10)(M=10)
BIEBIE
formulatioformulationn
First-orderFirst-order
solutionsolution
22 0.86570.8657 0.86570.8657 0.86610.8661
33 0.82140.8214 0.82140.8214 0.82240.8224
44 0.78930.7893 0.78930.7893 0.79340.7934
42 /G Rmp
34
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 Linear algebraic equationLinear algebraic equation
Numerical examplesNumerical examples ConclusionsConclusions
35
ConclusionsConclusions
A systematic approach using A systematic approach using degenerdegenerate kernelsate kernels, , Fourier seriesFourier series and and null-fielnull-field integral equationd integral equation has been successf has been successfully proposed to solve torsion probleully proposed to solve torsion problems with circular boundaries.ms with circular boundaries.
Numerical results Numerical results agree wellagree well with avai with available exact solutions and Caulk’s datlable exact solutions and Caulk’s data for a for only few terms of Fourier seriesonly few terms of Fourier series..
36
ConclusionsConclusions
Engineering problemsEngineering problems with with circular bocircular boundariesundaries which satisfy the which satisfy the Laplace eqLaplace equationuation can be solved by using the pro can be solved by using the proposed approach in a posed approach in a more efficient anmore efficient and accurate mannerd accurate manner..
37
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.
38
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
top related