1 null-field integral equations and engineering applications i. l. chen ph.d. department of naval...
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1
Null-field integral equations Null-field integral equations and engineering and engineering applicationsapplications
I. L. Chen Ph.D. Department of Naval Architecture,
National Kaohsiung Marine University
Mar. 11, 2010
2
Research collaborators Research collaborators
Prof. J. T. ChenProf. J. T. Chen Dr. K. H. ChenDr. K. H. Chen Dr. S. Y. Leu Dr. W. M. LeeDr. S. Y. Leu Dr. W. M. Lee Dr. Y. T. LeeDr. Y. T. Lee 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. J. N. Ke Mr. H. Z. Liao Mr. J. N. Ke Mr. H. Z. Liao Mr. Y. J. LinMr. Y. J. Lin Mr. C. F. Wu Mr. J. W. LeeMr. C. F. Wu Mr. J. W. Lee
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 nullsystem2007.ppt`
Elasticity & Crack Problem
Laplace Equation
Research topics of NTOU / MSV LAB on null-field BIE (2003-2010)
Navier Equation
Null-field BIEM
Biharmonic Equation
Previous research and project
Current work
(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
ASMEJoM
EABE
CMAME 2007
SDEE
JCA
NUMPDE revision
JSV
SH wave
Impinging canyonsDegenerate kernel for ellipse
ICOME 2006
Added mass
李應德Water wave impinging circul
ar cylinders
Screw dislocation
Green function foran annular plate
SH wave
Impinging hillGreen function of`circular inc
lusion (special case:staic)
Effective conductivity
CMC
(Stokes flow)
(Free vibration of plate) Direct BIEM
(Flexural wave of plate)
AOR 2009
5
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
6
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 ?
7
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
8
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
5. Meshless 5. Meshless
(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 sO O
- -
(x) (s)a b
9
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)
10
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
11
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
12
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.
13
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
14
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
15
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
16
Gain of introducing the degenerate Gain of introducing the degenerate kernelkernel
(x) (s, x) (s) (s)BK dBj f=ò
Degenerate kernel Fundamental solution
CPV and HPV
No principal value?
0
(x) (s)(x) (s) (s)jBj
ja 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
18
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
19
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
20-20 -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=
21
Boundary density discretizationBoundary density discretization
Fourier Fourier seriesseries
Ex . constant Ex . constant elementelement
Present Present methodmethod
Conventional Conventional BEMBEM
22
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
24
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
25
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
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 ConclusionsConclusions
27
{ }
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
28
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
29
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
30
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
31
Numerical examplesNumerical examples
Laplace equation Laplace equation (EABE 2005, EABE 2007) (EABE 2005, EABE 2007) (CMES 2005, ASME 2007, JoM2007)(CMES 2005, ASME 2007, JoM2007) (MRC 2007, NUMPDE 2010)(MRC 2007, NUMPDE 2010) Biharmonic equation Biharmonic equation (JAM, ASME 2006(JAM, ASME 2006)) Plate eigenproblem Plate eigenproblem (JSV )(JSV ) Membrane eigenproblem Membrane eigenproblem (JCA)(JCA) Exterior acoustics Exterior acoustics (CMAME, SDEE (CMAME, SDEE )) Water waveWater wave (AOR 2009) (AOR 2009)
32
Laplace equationLaplace equation
A circular bar under torqueA circular bar under torque
(free of mesh generation)(free of mesh generation)
33
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
34
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
36
Numerical examplesNumerical examples
Biharmonic equationBiharmonic equation (exponential convergence)(exponential convergence)
37
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)
38
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)
39
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)
40
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
42
Distribution of dynamic moment concentration factors Distribution of dynamic moment concentration factors
by using the present method and FEM( by using the present method and FEM( L/a L/a = = 2.12.1))
43
Membrane eigenproblemMembrane eigenproblem ((Nonuniqueness problems ))
A confocal elliptical annulusA confocal elliptical annulus
44
A confocal elliptical annulusA confocal elliptical annulus
2 2( ) ( ) 0,k u D x x
0 1( ) 0,u B B x x
G. E.:
B. Cs.:
1
1
1 11
1
2 21 1
0 1
0 0
0 0
1
0.5
tanh
2
cosh( )
sinh( )
a
b
ab
c a b
a c
b c
45
True and spurious eigenvaluesTrue and spurious eigenvalues
Note: the data inside parentheses denote the spurious eigenvalue.
(42)
(11)
Eigenvalues of an elliptical membrane
UT equation Spurious
BEM mesh FEM mesh
True 1
1
( , ) 0
( , ) 0m
m
Je q
Jo q
47
Water waveWater wave ((Nonuniqueness problems ))
Interaction of water waves Interaction of water waves with vertical cylinderswith vertical cylinders
48
Trapped modeTrapped mode (( nonuniqueness in physics nonuniqueness in physics ))
M.S. Longuet-higgins
JFM, 1967.
A.E.H. Love
1966.
Williams & Li
OE, 2000.
50
Numerical and physical resonanceNumerical and physical resonancePhysical resonance Fictitious frequency
(BEM/BIEM)
t(a,0)
0 2 4 6 8
-2
-1
0
1
2UT method
LM method
Burton & Miller method
1),( au0),( au
Drruk ),( ,0),()( 22
9
1),( au0),( au
Drruk ),( ,0),()( 22
9
Present
e i t
51
Water wave interaction with surface-piercing Water wave interaction with surface-piercing cylinderscylinders
.),,(,0);,,(2 Dzyxtzyx
( , , , ) Re{ ( , ) ( ) }i tx y z t u x y f z e
Governing equation:
Separation variable :
).,(,0 yxhzn
Seabed boundary conditions :
52
Problem statementProblem statement
2 2( ) ( ) 0,k u x x D
,
.
Dispersion relationship:2
tanhk khg
Dynamic pressure:cosh ( )
( , )cosh
i tk z hp gA u x y e
t kh
Force:2
0
costanh ( , )
sinjjj
jj
gAaX kh u x y d
k
Original problem
inc
Governing equation:
1
2
3
j
54
Physical phenomenon and fictitious frequencyPhysical phenomenon and fictitious frequency
Near-trapped mode
Fictitious frequency
Near-trapped mode
Fictitious frequency
Near-trapped mode
Fictitious frequency
Near-trapped mode
Near-trapped mode
2.43.8
5.15.5
6.4ka4.08482
55
Mechanism of fictitious frequencyMechanism of fictitious frequency
i i =1=1 i i == 2 2
N=1N=1 2.40422.4042 5.52015.5201
N=2N=2 3.83173.8317 7.01567.0156
N=3N=3 5.13565.1356 8.41728.4172
N=4N=4 6.38026.3802 9.76109.7610
N=5N=5 7.58837.5883 11.064711.0647
N=6N=6 8.77158.7715 12.338612.3386
0N N iJ k a a
N
ikNi
56
Near-trapped mode for the four cylinders at Near-trapped mode for the four cylinders at kaka=4.08482 (=4.08482 (a/d=a/d=0.8)0.8)
(a) Contour by the present method (M=20)
57
0 1 2 3 4 5 6 7k a
0
0.5
1
1.5
2
2.5
3C y lin d e r 1
C y lin d e r 2
C y lin d e r 3
E v an s & P o rte r (C y lin d er 1 )
E v an s & P o rte r (C y lin d er 2 )
E v an s & P o rte r (C y lin d er 3 )
C y lin d e r 1 : 5 4 .0 7 8C y lin d e r 2 : 1 .0 0 0 0C y lin d e r 3 : 5 4 .1 1 1
Near-trapped mode for the four cylinders at Near-trapped mode for the four cylinders at kaka=4.08482 (=4.08482 (aa//dd=0.8)=0.8)
(c) Horizontal force on the four cylinders against wavenumber(b) Free-surface elevations by the present method (M=20)
54
12
34
59
By perturbing the radius of one cylinder By perturbing the radius of one cylinder (a1/d≠0.8) to destroy the periodical setup(a1/d≠0.8) to destroy the periodical setup
a1/d=0.82
Evans and Porter, JEM ,1999. Present method
0 2 4 6ka
0
1
2
3
4
|X j||F |
a1/d=0.82
C ylinder 1
C ylinder 2 , 4
C ylinder 3
61
1 /a d dai /ii=2,3,4=2,3,4
Cylinder 1Cylinder 1 Cylinder 3Cylinder 3
ForceForce ForceForce
0.860.86 0.80.8 1.151.15 0.250.25
0.840.84 0.80.8 1.201.20 0.250.25
0.820.82 0.80.8 1.301.30 0.270.27
0.80.8 0.80.8 54.154.1 54.154.1
0.780.78 0.80.8 1.021.02 0.340.34
0.760.76 0.80.8 1.131.13 0.300.30
0.740.74 0.80.8 1.191.19 0.300.30
'/1 da dai /ii=2,3,4=2,3,4
Cylinder 1Cylinder 1 Cylinder 3Cylinder 3
ForceForce ForceForce
0.860.86 0.80.8 1.151.15 0.290.29
0.840.84 0.80.8 1.201.20 0.280.28
0.820.82 0.80.8 1.271.27 0.270.27
0.80.8 0.80.8 54.154.1 54.154.1
0.780.78 0.80.8 1.121.12 0.270.27
0.760.76 0.80.8 1.171.17 0.260.26
0.740.74 0.80.8 1.161.16 0.260.26
Changing radius Moving the center of one cylinder
ka=4.08482ka=4.08482
Disorder of the periodical patternDisorder of the periodical pattern
62
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 kdegenerate kernelsernels, , Fourier seriesFourier series and and null-field integranull-field integral equationl equation has been successfully proposed has been successfully proposed to solve Laplace Helmholtz and Biharminito solve Laplace Helmholtz and Biharminic problems with circular boundaries.c problems with circular boundaries.
Numerical results Numerical results agree wellagree well with available with available exact solutions, Caulk’s data, Onishi’s dexact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for ata and FEM (ABAQUS) for only few terms only few terms of Fourier seriesof Fourier series..
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ConclusionsConclusions
Physical phenomena of Physical phenomena of near-trapped near-trapped mode as well as the numerical instability mode as well as the numerical instability due to due to fictitious frequencyfictitious frequency in BIEM were in BIEM were both observed.both observed.
Fictitious frequency appears and is Fictitious frequency appears and is suppressed in sacrifice of suppressed in sacrifice of higher number higher number of Fourier termsof Fourier terms..
The effect of incident angle and disorder The effect of incident angle and disorder on the near-trapped mode was examined.on the near-trapped mode was examined.
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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