Scattering of sound from axisymetric sources by multiple circular cylinders using addition
theorem and superposition technique
The 32nd National Conference on Theoreticaland Applied Mechanics
Authors : Yi-Jhou Lin, Ying-Te Lee , I-Lin Chen and Jeng-Tzong ChenDate: November 28-29, 2008 Place: National Chung Cheng University, Chia-Yi
Reporter : Yi-Jhou Lin
National Taiwan Ocean UniversityDepartment of Harbor and River Engineering
The 32nd National Conference on Theoretical and Applied Mechanics
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Outlines
Introduction Problem statement Method of solution Mathematical Equivalence Mathematical equivalence between the solution of Green’s third
identity and that of superposition technique
Numerical examples Concluding remarks
Introduction
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Motivation
Numerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless method
BEM / BIEM
Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity
Boundary-layer Boundary-layer effecteffect
Ill-posed modelIll-posed modelConvergence Convergence raterate
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MotivationBEM / BIEMBEM / BIEM
Improper integralImproper integral
Singularity & hypersingularitySingularity & hypersingularity RegularityRegularity
Bump contourBump contour Limit processLimit process Fictitious Fictitious boundaryboundary
Collocation Collocation pointpoint
Fictitious BEMFictitious BEM
Null-field approachNull-field approach
CPV and HPVCPV and HPVIll-posedIll-posed
Guiggiani (1995)Guiggiani (1995) Gray and Manne (199Gray and Manne (1993)3)
Waterman (1965)Waterman (1965)
Achenbach Achenbach et al.et al. (1988) (1988)
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Present approach)()()( sdBsx
B ),( xsK
),( xsK e
Fundamental solutionFundamental solution
No principal valueNo principal value
Advantages of present approach1. mesh-free generation2. well-posed model3. principal value free4. elimination of boundary-layer effect5. exponential convergence
Degenerate kernelDegenerate kernel
CPV and HPVCPV and HPV
xsxsK
xsxsKe
i
),,(
),,(
),( xsK i
4)()1(
0 kriH
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Green’s third identityGreen’s third identity
BIE for Green’s functionBIE for Green’s function
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Outlines
Introduction Problem statement Method of solution Mathematical Equivalence Mathematical equivalence between the solution of Green’s third
identity and that of superposition technique
Numerical examples Concluding remarks
Problem statement
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Problem statementOriginal Problem
Free field
Radiation field (typical BVP)
DxxxGk ),(),()( 22
.1
H
jjBB
BxxG ,0),( (soft)
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FlowchartOriginal problem
Decompose two parts
Free field Radiation field
Expansion
Fourier series of boundary densities
Degenerate kemelFor fundamental solution
Collocate of the real boundary
Linear algebraic system
Calculation of the unknown Fourier
BIE for the domain point
Superposing the solution of two parts
Total field
The 32nd National Conference on Theoretical and Applied Mechanics
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Outlines
Introduction Problem statement Method of solution Mathematical Equivalence Mathematical equivalence between the solution of Green’s third
identity and that of superposition technique
Numerical examples Concluding remarks
Method of solution
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Method of solutionBoundary integral equation and null-field integral equation
Interior case Exterior case
cD
D D
x
xx
xcD
x x
Degenerate (separate) formDegenerate (separate) form
DxsdBstxsUsdBsuxsTxuBB
),()(),()()(),()(2
BxsdBstxsUVPRsdBsuxsTVPCxuBB
),()(),(...)()(),(...)(
Bc
BBDxsdBstxsUsdBsuxsT ),()(),()()(),(0
B
s
s
n
ss
n
xsUxsT
kriHxsU
)()(
),(),(
4
)(),(
)1(0
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Degenerate kernel and Fourier series
,,,2,1,,)sincos()(1
0 NkBsnbnaas kkn
kn
kn
kk
,,,2,1,,)sincos()(1
0 NkBsnqnpps kkn
kn
kn
kk
s
Ox
R
kth circularboundary
cosnθ, sinnθboundary distributions
eU
x
iU
Expand fundamental solution by using degenerate kernel
Expand boundary densities by using Fourier series
,),()(),(
,),()(),(
),(
0
0
sxsBxAxsU
sxxBsAxsU
xsU
jjj
E
jjj
I
The 32nd National Conference on Theoretical and Applied Mechanics
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Degenerate kernels
xn
xn
sn
U(s,x)U(s,x) T(s,x)T(s,x)
L(s,x)L(s,x) M(s,x)M(s,x)sn
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Degenerate kernels
,)],(cos[)()(4
),(
,)],(cos[)()(4
),(),(
0
)1(
0
)1(
RmkHkRJki
xsL
RmkRHkJki
xsLxsL
mmmm
e
mmmm
i
,)],(cos[)()(4
),(
,)],(cos[)()(4
),(),(
0
)1(
2
0
)1(
2
RmkHkRJik
xsM
RmkRHkJik
xsMxsM
mmmm
e
mmmm
i
,)],(cos[)()(4
),(
,)],(cos[)()(4
),(),(
0
)1(
0
)1(
RmkHkRJki
xsT
RmkRHkJki
xsTxsT
mmmm
e
mmmm
i
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Adaptive observer system
Source pointSource point
Collocation pointCollocation point
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Linear algebraic system
B¥
1B2B
NB
x
y
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Outlines
Introduction Problem statement Method of solution Mathematical Equivalence Mathematical equivalence between the solution of Green’s third
identity and that of superposition technique
Numerical examples Concluding remarks
Mathematical Equivalence
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Mathematical equivalence between the solution of Green’s third identity and that of superposition technique
+=( , )rG x
( , )G x
( , )fG x
Green’s third identity
Superposition technique
The 32nd National Conference on Theoretical and Applied Mechanics
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Outlines
Introduction Problem statement Method of solution Mathematical Equivalence Mathematical equivalence between the solution of Green’s third
identity and that of superposition technique
Numerical examples Concluding remarks
Numerical examples
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An infinite plane with two equal circular
cylinders subject to a point sound source.
2 0 313
2
.; ;b a
k k
( )(1)0 ( )
, , where4
iH krU s x r s x
-= º -
2 2 k G x x x D ,,
Governing equation:
Dirichlet Boundary condition:
(soft)
Fundamental solution:
0,Bx
G x
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Distribution potential on the artificial
boundaries in the free field
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Case 1 parameter use size and cylinder
B1
B2
b
by
Probe(soft)
(soft)
parameter Probe
Case 1-1 2
b
2
k
0 313.
ak
240 y
Case 1-2 2
b
2
k
1 253.
ak
180 y
Case 1-3 2
b
2
k
2
ak
150 y
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Distribution potential on the artificial boundaries in the free field versus polar angle.
0 60 120 180 240 300 360
angle
-0 .015
-0.01
-0.005
0
0.005
0.01
uM = 2 0
rea l p a rt (se rie s -fo rm )
im ag . p a rt (se rie s -fo rm )
rea l p a rt ( c lo sed -fo rm )
im ag . p a rt ( c lo sed -fo rm )
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Relative amplitude of total field versus the probe location y (M=20).
0 4 8 12 16 20 24
P r o b e p o s itio n y (c m )
0
0.2
0.4
0.6
0.8
1
1.2
Rel
ativ
e to
tal s
catt
ered
fie
ld
k a = 0 .3 1 3T H E O
E X P
P resen t m eth o d
Total field
Free field
B1
B2
b
bProbe
(soft)
(soft)
B1
B2
b
bProbe
(soft)
(soft)
Versus
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Relative amplitude of total field versus the probe location (M=20).
0 2 4 6 8 10 12 14 16 18
P r o b e p o s itio n y (c m )
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Rel
ativ
e to
tal s
catt
ered
fie
ldk a = 1 .2 5 3
E X P E R IM E N T A L
G A U S S -S E ID E L A P P R O X
E X A C T S O L U T IO N
IN D E P E N D E N T
P R E S E N T M E T H O D
Total field
Free field
B1
B2
b
b
(soft)
(soft)
B1
B2
b
b
(soft)
(soft)
Probe
Probe
Versus
The 32nd National Conference on Theoretical and Applied Mechanics
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Relative amplitude of total field versus
(M=20).
0 4 8 12 16
P r o b e p o s itio n y (c m )
0
0.4
0.8
1.2
1.6
Rel
ativ
e to
tal s
catt
ered
fie
ld
k a = 2 .0C O U P L E D
IN D E P E N D E N T
T H E O R
P R E S E N T M E T H O D
B1
B2
b
bProbe
(soft)
(soft)
B1
B2
b
bProbe
(soft)
(soft)
Versus
Total field
Free field
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Convergence test of Parseval’s sum for (real part).
0 2 4 6 8 10 12 14 16 18 20
T erm s o f F o u rie r se rie s (M )
0.00264
0.00272
0.0028
0.00288
0.00296
Pars
eval
's s
um o
f r
eal p
art s
olut
ion
xnxG ),(
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Convergence test of Parseval’s sum for
(imaginary part).
0 2 4 6 8 10 12 14 16 18 20
T erm s o f F o u rie r se rie s (M )
0.0093
0.009302
0.009304
0.009306
0.009308
0.00931P
arse
val's
sum
of
imag
inar
y pa
rt s
olut
ion
xnxG ),(
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parameter
Case 2-1 2
k
42
~b
1 253.a
k
Case 2-2 2
k
42
~b
1 5.a
k
Case 2-3 2
k
42
~b
2a
k
Case 2-4 2
k
42
~b
3a
k
Case 2 parameter use cylinder center-to-center
B1
B2
b
bProbe
(soft)
(soft)
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Relative amplitude of total field versus
(M=20).
0 2 4 6 8
S p a c in g b etw een cen ters o f cy lin d ers 2 b /
0
0.4
0.8
1.2
1.6
2
Rel
ativ
e to
tal s
catt
ered
fie
ld a
t p
rob
e
k a = 1 .2 53C O U P L E D
IN D E P E N D E N T
P R E S E N T M E T H O D
B1
B2
b
bProbe
(soft)
(soft)
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Relative amplitude of total field versus
(M=20).
0 1 2 3 4 5 6 7 8
S p a c in g b etw een cen ters o f cy lin d ers 2 b /
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Rel
ativ
e to
tal s
catt
ered
fie
ld a
t p
rob
e
k a = 1 .5 0IN D E P E N D E N T
C O U P L E D
P R E S E N T M E T H O D
D IA G O N A L T E E M S O N L Y
B1
B2
b
bProbe
(soft)
(soft)
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Relative amplitude of total field versus
(M=20).
0 1 2 3 4 5 6 7 8
S p a c in g b etw een cen ters o f cy lin d ers 2 b /
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Rel
ativ
e to
tal s
catt
ered
fie
ld a
t p
rob
e
k a = 2 .0IN D E P E N D E N T
C O U P L E D
D IA G O N A L T E E M S O N L Y
P R E S E N T M E T H O D
B1
B2
b
bProbe
(soft)
(soft)
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Relative amplitude of total field versus
(M=20).
0 1 2 3 4 5 6 7 8
S pa c in g b e tw e e n c e nter s o f c y lin d e rs 2 b /
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Rel
ativ
e to
tal s
catt
ered
fie
ld a
t p
rob
e
k a = 3 .0C O U P L E D
IN D E P E N D E N T
D IA G O N A L T E E M S O N L Y
P R E S E N T M E T H O D
b
bProbe
(soft)
(soft)
B1
B2
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Outlines
Introduction Problem statement Method of solution Mathematical Equivalence Mathematical equivalence between the solution of Green’s third
identity and that of superposition technique
Numerical examples Concluding remarksConcluding remarks
The 32nd National Conference on Theoretical and Applied Mechanics
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Concluding remarks
A general-purpose program for solving the problems with arbitrary number, size and various locations of circular cavities was developed.
We have proposed a BIEM formulation by using degenerate kernels, null-field integral equation and Fourier series in companion with adaptive observer system.