null field integral approach for engineering problems with circular boundaries
DESCRIPTION
National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering. Null field integral approach for engineering problems with circular boundaries. J. T. Chen Ph.D. 終身特聘教授 Taiwan Ocean University Keelung, Taiwan Dec.26, 15:30-17:00, 2007 交通大學 機械系 邀請演講 交大機械 2007.ppt. - PowerPoint PPT PresentationTRANSCRIPT
1
Null field integral approach for Null field integral approach for engineering problems with circular engineering problems with circular boundariesboundaries
J. T. Chen Ph.D.終身特聘教授
Taiwan Ocean UniversityKeelung, Taiwan
Dec.26, 15:30-17:00, 2007交通大學 機械系 邀請演講
交大機械 2007.ppt
National Taiwan Ocean University
MSVLABDepartment of Harbor and River
Engineering
1u =
1u =-1u =
1u =-
1u = 1u =-
2
Overview of numerical Overview of numerical methodsmethods
Finite Difference M ethod Finite Element M ethod Boundary Element M ethod
M esh M ethods M eshless M ethods
Numerical M ethods
2
PDE- variational IEDE
Domain
BoundaryMFS,Trefftz method MLS, EFG
開刀 把脈
針灸
3
Number of Papers of FEM, BEM and Number of Papers of FEM, BEM and FDMFDM
(Data form Prof. Cheng A. H. D.)
126
4
Growth of BEM/BIEM Growth of BEM/BIEM paperspapers
(data from Prof. Cheng A.H.D.)
5
BEMBEM USA USA (3569)(3569) , China , China (1372) (1372) , UK, , UK,
JapanJapan, Germany, France, , Germany, France, Taiwan Taiwan (580)(580), Canada, South Korea, Italy , Canada, South Korea, Italy (No.7)(No.7)
Dual BEM (Made in Taiwan)Dual BEM (Made in Taiwan) UK UK (124)(124), USA , USA (98)(98), , Taiwan (70)Taiwan (70), ,
China China (46)(46), Germany, France, , Germany, France, JapanJapan, , Australia, Brazil, Slovenia (No.3) Australia, Brazil, Slovenia (No.3)
(ISI information Apr.20, 2007)(ISI information Apr.20, 2007)
Top ten countries of BEM and dual BEMTop ten countries of BEM and dual BEM
6
FEMFEM USA(9946), USA(9946), China(4107)China(4107), , Japan(3519)Japan(3519), France, Germany,, France, Germany,
England, South Korea, Canada, England, South Korea, Canada, Taiwan(1383), Taiwan(1383), ItalyItaly Meshless methodsMeshless methods USA(297), USA(297), China(179)China(179), Singapore(105), , Singapore(105), Japan(49)Japan(49), Spain,, Spain,
Germany, Slovakia, England, Portugal, Germany, Slovakia, England, Portugal, Taiwan(28)Taiwan(28) FDMFDM USA(4041), USA(4041), Japan(1478)Japan(1478), , China(1411)China(1411), England, France, , England, France,
Canada, Germany, Canada, Germany, Taiwan (559)Taiwan (559), South Korea, India, South Korea, India (ISI information Apr.20, 2007)(ISI information Apr.20, 2007)
Top ten countries of FEM, FDM and Meshless methodsTop ten countries of FEM, FDM and Meshless methods
7
BEMBEM Aliabadi M H (UK, Queen Mary College)Aliabadi M H (UK, Queen Mary College) Chen J TChen J T (Taiwan, Ocean Univ.) (Taiwan, Ocean Univ.) 113 SCI papers 479citing113 SCI papers 479citing Mukherjee S (USA, Cornell Univ.)Mukherjee S (USA, Cornell Univ.) Leisnic D(UK, Univ. of Leeds)Leisnic D(UK, Univ. of Leeds) Tanaka M (Japan, Shinshu Univ.)Tanaka M (Japan, Shinshu Univ.) Dual BEM (Made in Taiwan)Dual BEM (Made in Taiwan) Aliabadi M H (UK, Queen Mary Univ. London)Aliabadi M H (UK, Queen Mary Univ. London) Chen J T (Taiwan, Ocean Univ.) Chen J T (Taiwan, Ocean Univ.) Power H (UK, Univ. Nottingham)Power H (UK, Univ. Nottingham) (ISI information Apr.20, 2007)(ISI information Apr.20, 2007)
Active scholars on BEM and dual BEMActive scholars on BEM and dual BEM
8
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)
NTOU/MSVTaiwan
9URL: 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 Frame2007-c.ppt
Elasticity(holes and/or inclusions)
Laplace Equation
Research topics of NTOU / MSV LAB using null-field BIEs (2005-2007) (2008-2011)
Navier Equation
Null-field BIEM
Biharmonic Equation
Previous work (done)
Interested topic
Present project
(Plate with circulr holes)
BiHelmholtz EquationHelmholtz Equation
(Potential flow)(Torsion)
(Anti-plane shear)(Degenerate scale)
(Inclusion)(Piezoleectricity) (Beam bending)
Torsion bar (Inclusion)Imperfect interface
Image method(Green function)
Green function of half plane (Hole and inclusion)
(Interior and exteriorAcoustics)
SH wave (exterior acoustics)(Inclusions)
(Free vibration of plate)Indirect BIEM
ASME JAM 2006MRC,CMESEABE
ASME JoMEABE
CMAME 2007
SDEE
JCA
NUMPDE revision
JSV
SH wave
Impinging canyonsDegenerate kernel for ellipse
ICOME 2006
Added mass
Water wave impinging circul
ar cylinders
Screw dislocation
Green function foran annular plate
SH wave
Impinging hill
Green function of`circular inclusion (special case:static)
Effective conductivity
CMC
(Stokes flow)
(Free vibration of plate) Direct BIEM
(Flexural wave of plate)
10
Research collaborators Research collaborators Dr. I. L. Chen (Dr. I. L. Chen ( 高海大高海大 ) Dr. K. H. Chen () Dr. K. H. Chen ( 宜蘭大學宜蘭大學 )) Dr. S. Y. Leu Dr. W. M. Lee (Dr. S. Y. Leu Dr. W. M. Lee ( 中華技術學院中華技術學院 )) Mr. S. K. Kao Mr. Y. Z. Lin Mr. S. Z. Hsieh Mr. S. K. Kao Mr. Y. Z. Lin Mr. S. Z. Hsieh Mr. H. Z. Liao Mr. G. N. Kehr (2007)Mr. H. Z. Liao Mr. G. N. Kehr (2007) Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao (2006)Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao (2006) Mr. A. C. Wu Mr. P. Y. Chen (2005)Mr. A. C. Wu Mr. P. Y. Chen (2005) Mr. Y. T. Lee Mr. S. K. Kao Mr. Y. T. Lee Mr. S. K. Kao Mr. S. Z. Shieh Mr. Y. Z. LinMr. S. Z. Shieh Mr. Y. Z. Lin
11
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)
12
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)
13
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
14
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 ?除四舊
15
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
16
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
17
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 (Legendre polynomial)polynomial)
(Chebyshev poly(Chebyshev polynomial)nomial)
(Mathieu (Mathieu function)function)
18
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
19
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
20
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.
21
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
22
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
23
• 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
24
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)
25
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
26
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
27
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
28
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
29
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
30
How to separate the regionHow to separate the region
31
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
32
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
33-20 -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=
34
Boundary density discretizationBoundary density discretization
Fourier Fourier seriesseries
Ex . constant Ex . constant elementelement
Present Present methodmethod
Conventional Conventional BEMBEM
35
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
36
Adaptive observer systemAdaptive observer system
( , )r f
collocation collocation pointpoint
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
38
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
39
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
40
{ }
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
41
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
42
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
43
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
44
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
45
Numerical examplesNumerical examples
Laplace equation Laplace equation (EABE 2006, EABE2006,EABE 2007) (EABE 2006, EABE2006,EABE 2007) (CMES 2006, ASME 2007, JoM2007, CMC)(CMES 2006, ASME 2007, JoM2007, CMC) (MRC 2007, NUMPDE 2007,ASME 2007)(MRC 2007, NUMPDE 2007,ASME 2007) Eigen problem Eigen problem (JCA)(JCA) Exterior acoustics Exterior acoustics (CMAME 2007, SDEE 2008(CMAME 2007, SDEE 2008)) Biharmonic equation Biharmonic equation (JAM, ASME 2006, IJNME(JAM, ASME 2006, IJNME)) Plate vibration Plate vibration (JSV 2007)(JSV 2007)
46
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
47
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
48
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
49
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
50
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
51
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
52
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 =-
53
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
54
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)
55
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
56
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
57
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
58
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
59
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
60
Axial displacement with two circular Axial displacement with two circular holesholes
Present Present method method (M=10)(M=10)
Caulk’s data Caulk’s data (1983)(1983)
ASME Journal of Applied ASME Journal of Applied MechanicsMechanics -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
61
Torsional rigidityTorsional rigidity
?
62
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
63
Infinite medium under antiplane Infinite medium under antiplane shearshear
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
64
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
65
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%%
66
Extension to inclusionExtension to inclusion
Anti-plane piezoelectricity problemsAnti-plane piezoelectricity problems In-plane electrostatics problemsIn-plane electrostatics problems Anti-plane elasticity problemsAnti-plane elasticity problems
67
Two circular inclusions with Two circular inclusions with centers on the centers 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
68
Convergence test and Convergence test and boundary-layer effectboundary-layer effect analysisanalysis
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
boundary-layer effectboundary-layer effect
69
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 o
yE 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)
70
Three identical inclusions Three identical inclusions forming an equilateral triangleforming an equilateral triangle
12d r=x
yt ¥
2r
1r
2m
3m3r
0m
30o1m
30o
71
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=
72
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
73
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
74
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
75
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
76
Numerical examplesNumerical examples
Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation
77
Problem statementProblem statement
Doubly-connected domain
Multiply-connected domain
Simply-connected domain
78
Example 1Example 1
2 2( ) ( ) 0,k u x x D
2 2.0r
1B0u
2B
0u
1 0.5r
79
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
80
The former five eigenmodes by using prThe former five eigenmodes by using present method, FEM and BEMesent method, FEM and BEM
81
Numerical examplesNumerical examples
Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Water waveWater wave Biharmonic equationBiharmonic equation
82
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
83
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
84
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)
85
Fictitious frequenciesFictitious frequencies
0 2 4 6 8
-8
-4
0
4
8Present m ethod (M =20)
BEM (N =60)
86
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
87
Water wave impinging four Water wave impinging four cylinderscylinders
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
88
Maximum free-surface elevationMaximum free-surface elevation
Perrey-Debain et al. (JSV 2003 ) Present method (2007)
89
Numerical examplesNumerical examples
Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation
90
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)
91
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)
92
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)
93
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
94
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
95
Numerical examplesNumerical examples
Laplace equationLaplace equation Eigen problemEigen problem Exterior acousticsExterior acoustics Biharmonic equationBiharmonic equation Plate vibrationPlate vibration
96
Free vibration of plateFree vibration of plate
97
Comparisons with FEMComparisons with FEM
98
Disclaimer (commercial Disclaimer (commercial code)code)
The concepts, methods, and examples usThe concepts, methods, and examples using our software are for illustrative and eing our software are for illustrative and educational purposes only. Our cooperatioducational purposes only. Our cooperation assumes no liability or responsibility to n assumes no liability or responsibility to any person or company for direct or indirany person or company for direct or indirect damages resulting from the use of anect damages resulting from the use of any information contained here.y information contained here.
inherent weakness ?inherent weakness ? mismisinterpretation ? User interpretation ? User 當自強當自強
99
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
100
Numerical phenomenaNumerical phenomena(Degenerate scale)(Degenerate scale)
Error (%)of
torsionalrigidity
a
0
5
125
da
T
da
Commercial ode output ?
101
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
102
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
103
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
104
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
105
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..
106
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
107
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//
108
109