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A semi-analytical approach for A semi-analytical approach for engineering problems with circular engineering problems with circular boundariesboundaries
J. T. Chen Ph.D. Taiwan Ocean University
Keelung, Taiwan
Dec.23, 14:00-14:30, 2006南台科技大學 教學研究大樓 T0111 演講廳
Diff2006.ppt
National Taiwan Ocean University
MSVLABDepartment of Harbor and River
Engineering
1u =
1u =-1u =
1u =-
1u = 1u =-
第十五屆微分方程研討會 15th Diff. Equations
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Research collaborators Research collaborators
Dr. I. L. Chen Dr. K. H. ChenDr. I. L. Chen Dr. K. H. Chen Dr. S. Y. Leu Dr. W. M. LeeDr. S. Y. Leu Dr. W. M. Lee Mr. H. Z. Liao Mr. G. N. KehrMr. H. Z. Liao Mr. G. N. Kehr Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. HsiaoMr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. A. C. Wu Mr. P. Y. ChenMr. A. C. Wu Mr. P. Y. Chen Mr. Y. T. Lee Mr. Y. T. Lee
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Top 25 scholars on BEM/BIEM since 2001Top 25 scholars on BEM/BIEM since 2001
北京清華姚振漢教授提供 (Nov., 2006)
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4URL: http://ind.ntou.edu.tw/~msvlab E-mail: [email protected] 海洋大學工學院河工所力學聲響振動實驗室 ``
Elasticity & Crack Problem李應德
Laplace Equation
Research topics of NTOU / MSV LAB on null-field BIE (2003-2006)
Navier Equation
Null-field BIEM
Biharmonic Equation
Previous research and project
Current work
蕭嘉俊 05’(Plate with circulr holes)
(Stokes’ flow)
BiHelmholtz EquationHelmholtz Equation
沈文成 05’(Potential flow)
(Torsion)(Anti-plane shear)(Degenerate scale)
吳安傑 06’(Inclusion)
(Piezoleectricity)
陳柏源 06’(Beam bending)
李應德Torsion bar(Inclusion)
廖奐禎、李應德Image method
(Green function)
柯佳男Green function of half plane
(Hole and inclusion)
陳佳聰 05’(Interior and exterior
Acoustics)
陳柏源 05’SH wave (exterior acoustics)
(Inclusions)
李為民、李應德(Plate vibration)
ASMEMRC,CMESEABE
ASMEJoM
EABE
CMAME revision
SDEE revision
JCA revision
NUMPDE revision
JSV revision
柯佳男SH wave
Impinging hill
廖奐禎
Degenerate kernel for ellipse
ICOME 2006
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哲人日已遠 典型在宿昔哲人日已遠 典型在宿昔C B Ling (1909-1993)C B Ling (1909-1993)
省立中興大學第一任校長
林致平校長( 民國五十年 ~ 民國五十二年 )
林致平所長 ( 中研院數學所 )
林致平院士 ( 中研院 )
PS: short visit (J T Chen) of Academia Sinica 2006 summer `
林致平院士 ( 數學力學家 )C B Ling (mathematician and expert in mechanics)
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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 ConclusionsConclusions
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MotivationMotivation
Numerical methods for engineering problemsNumerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless methodFDM / FEM / BEM / BIEM / Meshless method
BEM / BIEM (mesh required)BEM / BIEM (mesh required)
Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity
Boundary-layer Boundary-layer effecteffect
Ill-posed modelIll-posed modelConvergence Convergence raterate
Mesh free for circular boundaries ?Mesh free for circular boundaries ?
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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 Singular and hypersingularhypersingular
RegulRegularar
Improper Improper integralintegral
CPV and CPV and HPVHPV
Ill-Ill-posedposed
FictitiFictitious ous
bounboundarydary
CollocatCollocation ion
pointpoint
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Present approachPresent approach
1.1.No principal No principal valuevalue 2. Well-posed2. Well-posed
3. No boundary-laye3. No boundary-layer effectr effect
4. Exponetial converg4. Exponetial convergenceence
(s, x)eK
(s, x)iK
Advantages of Advantages of degenerate kerneldegenerate kernel
(x) (s, x) (s) (s)B
K 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 sO O
- -
(x) (s)a b
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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)
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Motivation and literature reviewMotivation and literature review
Analytical methods for solving Laplace problems
with circular 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
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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
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Contribution and goalContribution and goal
However, they didn’t employ the However, they didn’t employ the null-field integral equationnull-field integral equation and and degenerate kernelsdegenerate kernels to fully to fully capture the circular boundary, capture the circular boundary, although they all employed although they all employed Fourier Fourier series expansionseries expansion..
To develop a To develop a systematic approachsystematic approach for solving Laplace problems with for solving Laplace problems with multiple holesmultiple holes is our goal. is our goal.
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Outlines (Direct problem)Outlines (Direct problem)
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 ConclusionsConclusions
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Boundary integral equation Boundary integral equation and null-field integral equationand null-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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• R.P.V. (Riemann principal value)
• C.P.V.(Cauchy principal value)
• H.P.V.(Hadamard principal value)
Definitions of R.P.V., C.P.V. and H.P.V.using bump approach
NTUCE
R P V x dx x x x. . . ln ( ln )z
1
1
- = -2 x=-1x=1
C P Vx
dxx
dx. . . lim
z zz 1
1 1
1
1 1 = 0
H P Vx
dxx
dx. . . lim
z zz 1
1
2
1
12
1 1 2 = -2
0
0
x
ln x
x
x
1 / x
1 / x 2
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Principal value in who’s sensePrincipal value in who’s sense
Riemann sense (Common sense)Riemann sense (Common sense) Lebesgue senseLebesgue sense Cauchy senseCauchy sense Hadamard sense (elasticity)Hadamard sense (elasticity) Mangler sense (aerodynamics)Mangler sense (aerodynamics) Liggett and Liu’s senseLiggett and Liu’s sense The singularity that occur when the The singularity that occur when the base point and field point coincide are not base point and field point coincide are not
integrable. (1983)integrable. (1983)
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Two approaches to understand Two approaches to understand HPVHPV
1
2 2-10
1lim =-2 y
dxx y
H P Vx
dxx
dx. . . lim
z zz 1
1
2
1
12
1 1 2 = -2
(Limit and integral operator can not be commuted)
(Leibnitz rule should be considered)
1
01
1{ } 2
t
dCPV dx
dt x t
Differential first and then trace operator
Trace first and then differential operator
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Bump contribution (2-D)Bump contribution (2-D)
( )u x
( )u xx
sT
x
sU
x
sL
x
sM
0
0
1( )
2t x
1 2( ) ( )
2t x u x
1( )
2t x
1 2( ) ( )
2t x u x
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Bump contribution (3-D)Bump contribution (3-D)
2 ( )u x
2 ( )u x
2( )
3t x
4 2( ) ( )
3t x u x
2( )
3t x
4 2( ) ( )
3t x u x
x
xx
x
s
s s
s0`
0
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Outlines (Direct problem)Outlines (Direct problem)
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
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Gain of introducing the degenerate Gain of introducing the degenerate kernelkernel
(x) (s, x) (s) (s)B
K dBj f=ò
Degenerate kernel Fundamental solution
CPV and HPV
No principal value?
0
(x) (s)(x) (s) (s)jB
jja dBbj f
¥
=
= åò
0
0
(s,x) (s) (x), x s
(s,x)
(s,x) (x) (s), x s
ij j
j
ej j
j
K a b
K
K a b
¥
=
¥
=
ìïï = <ïïïï=íïï = >ïïïïî
å
åinterior
exterior
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How to separate the regionHow to separate the region
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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
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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 x
2(s, x)1
, x s2
T
ìïï >ïïï=íï -ï >ïïïî
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26-20 -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=
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Boundary density discretizationBoundary density discretization
Fourier Fourier seriesseries
Ex . constant Ex . constant elementelement
Present Present methodmethod
Conventional Conventional BEMBEM
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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 ConclusionsConclusions
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Adaptive observer systemAdaptive observer system
( , )r f
collocation collocation pointpoint
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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 ConclusionsConclusions
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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
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32
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 ConclusionsConclusions
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{ }
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
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Physical meaning of influence Physical meaning of influence coefficientcoefficient
kth circularboundary
xmmth collocation point
on the jth circular boundary
jth circular boundary
Physical meaning of the influence coefficient )( mncjkU
cosnθ, sinnθboundary distributions
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Flowchart of present methodFlowchart of present method
0 [ (s, x) (s) (s, x) (s)] (s)B
T 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
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36
Comparisons of conventional BEM and present Comparisons of conventional BEM and present
methodmethod
BoundaryBoundarydensitydensity
discretizatiodiscretizationn
AuxiliaryAuxiliarysystemsystem
FormulatiFormulationon
ObservObserverer
systemsystem
SingulariSingularityty
ConvergenConvergencece
BoundarBoundaryy
layerlayereffecteffect
ConventionConventionalal
BEMBEM
Constant,Constant,linear,linear,
quadratic…quadratic…elementselements
FundamenFundamentaltal
solutionsolution
BoundaryBoundaryintegralintegralequationequation
FixedFixedobservobserv
erersystemsystem
CPV, RPVCPV, RPVand HPVand HPV LinearLinear AppearAppear
PresentPresentmethodmethod
FourierFourierseriesseries
expansionexpansion
DegeneratDegeneratee
kernelkernel
Null-fieldNull-fieldintegralintegralequationequation
AdaptivAdaptivee
observobserverer
systemsystem
DisappeaDisappearr
ExponentiaExponentiall
EliminatEliminatee
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37
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 ConclusionsConclusions
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38
Numerical examplesNumerical examples
Laplace equation Laplace equation (EABE 2005, EABE 2007) (EABE 2005, EABE 2007) (CMES 2005, ASME 2007, JoM200(CMES 2005, ASME 2007, JoM200
7)7) (MRC 2007, NUMPDE revision)(MRC 2007, NUMPDE revision) Eigen problem Eigen problem (JCA revision)(JCA revision) Exterior acoustics Exterior acoustics (CMAME, SDEE revision(CMAME, SDEE revision)) Biharmonic equation Biharmonic equation (JAM, ASME 2006(JAM, ASME 2006)) Plate vibration Plate vibration (JSV revision)(JSV revision)
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39
Laplace equationLaplace equation
Steady state heat conduction Steady state heat conduction problemsproblems
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 An infinite medium under antiplane
shearshear Half-plane problemsHalf-plane problems
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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
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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
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Laplace equationLaplace equation
Steady state heat conduction Steady state heat conduction problemsproblems
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 An infinite medium under antiplane
shearshear Half-plane problemsHalf-plane problems
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43
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 =-
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44
Laplace equationLaplace equation
Steady state heat conduction Steady state heat conduction problemsproblems
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 An infinite medium under antiplane
shearshear Half-plane problemsHalf-plane problems
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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
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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
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Torsional rigidityTorsional rigidity
?
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48
Laplace equationLaplace equation
Steady state heat conduction Steady state heat conduction problemsproblems
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 An infinite medium under antiplane
shearshear Half-plane problemsHalf-plane problems
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49
Numerical examplesNumerical examples
Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation
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Problem statementProblem statement
Doubly-connected domain
Multiply-connected domain
Simply-connected domain
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Example 1Example 1
2 2( ) ( ) 0,k u x x D
2 2.0r
1B0u
2B
0u
1 0.5r
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kk11 kk22 kk33 kk44 kk55
FEMFEM(ABAQUS)(ABAQUS)
2.032.03 2.202.20 2.622.62 3.153.15 3.713.71
BEMBEM(Burton & Miller)(Burton & Miller)
2.062.06 2.232.23 2.672.67 3.223.22 3.813.81
BEMBEM(CHIEF)(CHIEF)
2.052.05 2.232.23 2.672.67 3.223.22 3.813.81
BEMBEM(null-field)(null-field)
2.042.04 2.202.20 2.652.65 3.213.21 3.803.80
BEMBEM(fictitious)(fictitious)
2.042.04 2.212.21 2.662.66 3.213.21 3.803.80
Present methodPresent method 2.052.05 2.222.22 2.662.66 3.213.21 3.803.80
Analytical Analytical solution[19]solution[19]
2.052.05 2.232.23 2.662.66 3.213.21 3.803.80
The former five true eigenvalues The former five true eigenvalues by usinby using different approachesg different approaches
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The former five eigenmodes by using prThe former five eigenmodes by using present method, FEM and BEMesent method, FEM and BEM
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54
Numerical examplesNumerical examples
Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation
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u=0 u=0
u=0
u=0 u=0
. .
.
.
.
x
y
cos( )8
ikre
2 2( ) ( ) 0,k u x x D
Sketch of the scattering problem (DirichSketch of the scattering problem (Dirichlet condition) for five cylinderslet condition) for five cylinders
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k
-3 -2 -1 0 1 2 3-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
(a) Present method (M=20) (b) Multiple DtN method (N=50)
The contour plot of the real-part The contour plot of the real-part solutions of total field forsolutions of total field for
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The contour plot of the real-part The contour plot of the real-part solutions of total field forsolutions of total field for 8k
-3 -2 -1 0 1 2 3-2 .5
-2
-1 .5
-1
-0 .5
0
0 .5
1
1 .5
2
2 .5
(a) Present method (M=20) (b) Multiple DtN method (N=50)
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Fictitious frequenciesFictitious frequencies
0 2 4 6 8
-8
-4
0
4
8Present m ethod (M =20)
BEM (N =60)
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Present methodPresent method
Soft-basin effect Soft-basin effect
/ 2 /3, 2I M
/ 1/ 2I Mc c / 1/3I Mc c / 2I Mc c
/x a /x a /x a
14 18
3
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Numerical examplesNumerical examples
Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation
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Plate problemsPlate problems
1B
4B
3B
2B1O
4O
3O
2O
Geometric data:
1 20;R 2 5;R
( ) 0u s 1B( ) 0s
1 (0,0),O 2 ( 14,0),O
3 (5,3),O 4 (5,10),O 3 2;R 4 4.R
( ) sinu s
( ) 1u s
( ) 1u s
( ) 0s
( ) 0s
( ) 0s
2B
3B
4B
and
and
and
and
on
on
on
on
Essential boundary conditions:
(Bird & Steele, 1991)
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Contour plot of displacementContour plot of displacement
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
Present method (N=101)
Bird and Steele (1991)
FEM (ABAQUS)FEM mesh
(No. of nodes=3,462, No. of elements=6,606)
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Stokes flow problemStokes flow problem
1
2 1R
e
1 0.5R
1B
Governing equation:
4 ( ) 0,u x x
Boundary conditions:
1( )u s u and ( ) 0.5s on 1B
( ) 0u s and ( ) 0s on 2B
2 1( )
e
R R
Eccentricity:
Angular velocity:
1 1
2B
(Stationary)
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0 80 160 240 320 400 480 560 640
0.0736
0.074
0.0744
0.0748
0 80 160 240 320
Comparison forComparison for 0.5
DOF of BIE (Kelmanson)
DOF of present method
BIE (Kelmanson) Present method Analytical solution
(160)
(320)(640)
u1
(28)
(36)
(44)(∞)
Algebraic convergence
Exponential convergence
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Contour plot of Streamline forContour plot of Streamline for
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Present method (N=81)
Kelmanson (Q=0.0740, n=160)
Kamal (Q=0.0738)
e
Q/2
Q
Q/5
Q/20-Q/90
-Q/30
0.5
0
Q/2
Q
Q/5
Q/20-Q/90
-Q/30
0
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66
Numerical examplesNumerical examples
Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation Plate vibrationPlate vibration
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Free vibration of plateFree vibration of plate
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Comparisons with FEMComparisons with FEM
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BEM trap ?BEM trap ?Why engineers should learn Why engineers should learn
mathematics ?mathematics ? Well-posed ?Well-posed ? Existence ?Existence ? Unique ?Unique ?
Mathematics versus Mathematics versus Computation Computation
Some examplesSome examples
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Numerical phenomenaNumerical phenomena(Degenerate scale)(Degenerate scale)
Error (%)of
torsionalrigidity
a
0
5
125
da
Previous approach : Try and error on aPresent approach : Only one trial
T
da
Commercial ode output ?
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Numerical and physical resonanceNumerical and physical resonance
a
m
k
e i t
incident wave
e i t e i t
radiation
Physical resonance Numerical resonance
, if ufinite
( )
2 2
, if u finite lim00
k
m
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Numerical phenomenaNumerical phenomena(Fictitious frequency)(Fictitious frequency)
0 2 4 6 8
-2
-1
0
1
2UT method
LM method
Burton & Miller method
t(a,0)
1),( au0),( au
Drruk ),( ,0),()( 22
9
1),( au0),( au
Drruk ),( ,0),()( 22
9
A story of NTU Ph.D. students
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Numerical phenomenaNumerical phenomena(Spurious eigensolution)(Spurious eigensolution)
0 1 2 3 4 5 6 7 8 9 10 11 12fre q u e n cy p a ra m e te r
1E-080
1E-060
1E-040
1E-020
de
t|SM
|
C -C annular p la teu, com plex-vauled form ulation
T<9.447>
T: T rue e igenvalues
T<10.370>
T<10.940>
T<9.499>
T<9.660>
T<9.945>
S<9.222>
S<6.392>
S<11.810>
S : Spurious e igenvalues
ma 1
mb 5.0
1B
2B
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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 ConclusionsConclusions
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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 BVPs with sed to solve BVPs with arbitraryarbitrary circular circular holes and/or inclusions.holes and/or inclusions.
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..
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ConclusionsConclusions
Free of boundary-layer effectFree of boundary-layer effect Free of singular integralsFree of singular integrals Well posedWell posed Exponetial convergenceExponetial convergence Mesh-free approachMesh-free approach
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The EndThe End
Thanks for your kind attentions.Thanks for your kind attentions.Your comments will be highly apprYour comments will be highly appr
eciated.eciated.
URL: URL: http://http://msvlab.hre.ntou.edu.twmsvlab.hre.ntou.edu.tw//
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