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Analysis of two-spheres radiation problems by using the
null-field integral equation approach
The 32nd Conference of Theoretical and Applied Mechanics
Ying-Te Lee ( 李應德 ) and Jeng-Tzong Chen ( 陳正宗 )
學 校 : 國立臺灣海洋大學科 系 : 河海工程學系時 間 : 2008 年 11 月 28-29日地 點 : 國立中正大學
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Outline
Introduction1.
2.
3.
Problem statement
Method of solution
Numerical examples4.
5. Concluding remarks
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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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 hy
persingularitypersingularity
Boundary-layer Boundary-layer effecteffect
Ill-posed modelIll-posed modelConvergence Convergence raterate
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Advantages of BEM
1. Mesh reduction
2. Solve infinite problem
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
FEM BEM
Area (2D) Line (1D)2-D problem: Volume (3D) Surface (2D)3-D problem:
Only boundary discretization is needed and without the DtN map.
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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BEM FEM
5
DtN interface
BEM and FEM
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Singular and hypersingular integrals
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
x
sx
s
Conventional approach to calculate singular and hypersingular integral (Bump contour)
Present approach x
s
x
s
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Fictitious frequency
(1) CHIEF method (Schenck, JASA , 1968)
(2) Burton and Miller method (Burton and Miller, PRS , 1971)
(3) SVD updating term technique (Chen et al., JSV, 2002)
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
tL
Uu
M
T
uMk
iTtL
k
iU
uTt
U
Additional constraint (CHIEF point)
Non-unique solution:
0 2 4 6 8
-2
-1
0
1
2UT m ethod
LM method
Burton & Miller method
t(a,0)
1),( au0),( au
Drruk ),( ,0),()( 22
9
1),( au0),( au
Drruk ),( ,0),()( 22
9
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Successful experiences in 2-D problems using the present approach
)()()( sdBsxB ),( xsK
),( xsK e
Fundamental solutionFundamental solution
Advantages of present approach:1. No principal value2. Well-posed model3. Exponential convergence4. Free of mesh generation
Degenerate kernelDegenerate kernel
xsxsK
xsxsKe
i
),,(
),,(
),( xsK i
sx ln
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
The proposed approach will be extended to deal with 3-D problem.
)()1(0 sxkH
(Laplace)
(Helmholtz)
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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3-D radiation problem
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Dxxuk ,0)()( 22
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Interior case Exterior case
cD
D D
x
xx
xcD
x
Degenerate (separate) formDegenerate (separate) form
DxsdBstxsUsdBsuxsTxuBB
),()(),()()(),()(
BxsdBstxsUVPRsdBsuxsTVPCxuBB
),()(),(...)()(),(...)(2
1
Bc
BBDxsdBstxsUsdBsuxsT ),()(),()()(),(0
B
Boundary integral equation and null-field integral equation
s
s
ikr
n
sust
n
xsUxsT
r
exsU
)()(
),(),(
4),(
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
x
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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),,( UE
x),,( UI x
Expansions
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Expand fundamental solution by the using degenerate kernel
Expand boundary densities by using the spherical harmonics
,),()(),(
,),()(),(
),(
0
0
sxsBxAxsU
sxxBsAxsU
xsU
jjj
E
jjj
I
,,)cos()(cos)(0 0
kv
v
w
wv
kvwk BswPBst
,,)cos()(cos)(0 0
kv
v
w
wv
kvwk BswPAsu
M term in the real implementation
),,( s
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Degenerate kernels
0 0
)2( )(cos)(cos)(cos)()()!(
)!()12(
4
cossin
44),(
n m
mn
mnnnm
ikr
mPPkhkjmn
mnn
ik
kr
krikrik
r
exsU
0 0
)2( )(cos)(cos)(cos)()()!(
)!()12(
4),(
n m
mn
mnnnm mPPkhkj
mn
mnn
ikxsU
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
,)(cos)()()12(sin
0nnnn Pkjkjn
kr
kr
n
m
mn
mnnnn mPP
mn
mnPPP
1
)(cos)(cos)(cos)!(
)!(2)(cos)(cos)(cos
,)(cos)()()12(cos
0nnnn Pkykjn
kr
krAddition theorem
),,(
),,(
x
s
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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U(s,x)
T(s,x)
L(s,x) M(s,x)
sn
sn xn
xn
Relationship of kernel functions
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
DxsdBstxsUsdBsuxsTxuBB
),()(),()()(),()(
DxsdBstxsLsdBsuxsMxtBB
),()(),()()(),()(
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Degenerate kernels
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
,)],(cos[)()()(cos)(cos)!(
)!()12(
4
,)],(cos[)()()(cos)(cos)!(
)!()12(
4),()2(
0 0
)2(
0 0
mkhkjPPmn
mnn
ikU
mkhkjPPmn
mnn
ikU
xsU
nnmn
mn
n
n
mm
e
nnmn
mn
n
n
mm
i
,)],(cos[)()()(cos)(cos)!(
)!()12(
4
,)],(cos[)()()(cos)(cos)!(
)!()12(
4),()2(
0 0
2
)2(
0 0
2
mkhkjPPmn
mnn
ikT
mkhkjPPmn
mnn
ikT
xsT
nnmn
mn
n
n
mm
e
nnmn
mn
n
n
mm
i
,)],(cos[)()()(cos)(cos)!(
)!()12(
4
,)],(cos[)()()(cos)(cos)!(
)!()12(
4),()2(
0 0
2
)2(
0 0
2
mkhkjPPmn
mnn
ikL
mkhkjPPmn
mnn
ikL
xsL
nnmn
mn
n
n
mm
e
nnmn
mn
n
n
mm
i
,)],(cos[)()()(cos)(cos)!(
)!()12(
4
,)],(cos[)()()(cos)(cos)!(
)!()12(
4),()2(
0 0
3
)2(
0 0
3
mkhkjPPmn
mnn
ikM
mkhkjPPmn
mnn
ikM
xsM
nnmn
mn
n
n
mm
e
nnmn
mn
n
n
mm
i
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Adaptive observer system
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
),,( 111 ),,( 222
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Problem with spherical boundaries
Null-field BIE
Expansion
Degenerate kernel for the fundamental
solution
Spherical harmonics for
boundary density
Collocating the collocation point and matching the boundary conditions
Boundary integration in adaptive observer system
Linear algebraic system
Obtain the unknown spherical harmonics coefficients
Velocity potential
Flowchart
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Sound pressure
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Case 1: A sphere pulsating with uniform radial velocity Case 2: A sphere oscillating with non-uniform radial velocity Case 3: Two spheres vibrating from uniform radial velocity
Numerical examples
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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a
Case 1: A sphere pulsating with uniform radial velocity
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Uniform radial velocity U0 (Seybert et al., JASA, 1985)
Oy
z
x
)(0
0
1aikeU
ika
kaizap
Analytical solution:
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Results
0 0
)cos()(cos)(v
v
w
wvvw wPBst
0
)()( U
n
sust
s
000 UB
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Uniform radial velocity U0
S
i
S
i sdSstxsUsdSsuxsT )()(),()()(),(0 0)2(0
)2(0
00 )(
)(1U
kah
kah
kA
)(
)()(
)2(0
)2(0
00 kah
khUizp
S
e
S
e sdSstxsUsdSsuxsTxu )()(),()()(),()(0)2(
0
)2(0
)(
)(1U
kah
kh
ku
ukizuip 00
000 UB
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Comparison
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
)(0
0
1aikeU
ika
kaizap
Exact solution (Seybert et al.)
Spherical Hankel function of series form:
n
k
kizn
n izknk
kne
z
izh
0
1)2( )2(
)1(!
)!()(
n
k
kkizn
n
k
kiznn
k
kizn
n
zkiknk
kne
z
i
izknk
kne
z
iiz
knk
kne
z
izh
0
)1(1
0
2
02
1)2(
)()2()1(!
)!(
)2()1(!
)!()2(
)1(!
)!()(
)(
)()2(
0
)2(0
00 kah
khUizp
Present approach
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Distribution of collocation points
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarksSound pressure
0 1 2 3 4 5k a
0
0.2
0.4
0.6
0.8
1
Re(
p/z 0
U0)
P resen t a p p ra o chE x a ct so lu tio n
0 1 2 3 4 5
k a
0
0.2
0.4
0.6
Im(p
/z0U
0)
P resen t a p p ro a chE x a ct so lu tio n
Real part of non-dimensional pressure on the surface
Imaginary part of non-dimensional pressure on the surface
ka=π exist a fictitious frequency in the result of Seybert et al.
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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a
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Non-uniform radial velocity U0cosθ (Seybert et al., JASA, 1985)
Oy
z
x
Analytical solution:
Case 2: A sphere oscillating with non-uniform radial velocity
)(022
0
2
cos)1(2
)1( aikeUakika
ikkaizap
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Result
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
0 0
)cos()(cos)(v
v
w
wvvw wPBst
cos)(
)( 0Un
sust
s
000 UB
Uniform radial velocity U0
S
i
S
i sdSstxsUsdSsuxsT )()(),()()(),(0 0)2(1
)2(1
10 )(
)(1U
kah
kah
kA
cos)(
)()( 0)2(
1
)2(1
0 Ukah
khizp
S
e
S
e sdSstxsUsdSsuxsTxu )()(),()()(),()(0)2(
1
)2(1
)(
)(1U
kah
kh
ku
ukizuip 00
010 UB
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Exact solution
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
cos
)(
)(0)2(
1
)2(1
0 Ukah
khizp
)(
0220
2
cos)1(2
)1( aikeUakika
ikkaizap
Exact solution (Seybert et al.)
Spherical Hankel function of series form:
n
k
kizn
n izknk
kne
z
izh
0
1)2( )2(
)1(!
)!()(
n
k
kkizn
n
k
kiznn
k
kizn
n
zkiknk
kne
z
i
izknk
kne
z
iiz
knk
kne
z
izh
0
)1(1
0
2
02
1)2(
)()2()1(!
)!(
)2()1(!
)!()2(
)1(!
)!()(
Present approach
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Case 3: Two spheres vibrating from uniform radial velocity
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Uniform radial velocity U0 (Dokumaci, JSV, 1995)
2a 2a
Oy
z
x
aa
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Distribution of collocation points
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Contour of sound pressure
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
z=0 and ka=1
Present approach
(Dokumaci and Sarigül, JSV, 1995)
Surface Helmholtz Integral Equation (SHIE)
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarksContour of sound pressure
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
Present approach
SHIE (Dokumaci and Sarigül)
z=0 and ka=2
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Contour of sound pressure
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Present approach
SHIE (Dokumaci and Sarigül)
z=0 and ka=0.1
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Potential of the nearest point
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
SHIE SHIE+CHIEF
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Potential of the nearest point
0 2 4 6 8 10k a
- 6
- 4
- 2
0
2
4
Re(
p/z 0U
0)
N ea res t p o in tR ea l p a rt (M = 1 0 )
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
Present approach Burton & Miller approach0 2 4 6 8 10
k a
- 2
- 1
0
1
2
Re(
p/z 0
U0)
N ea rest p o in tR ea l p a rt (M = 1 0 )
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Potential of the furthest point
SHIE SHIE+CHIEF
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Potential of the furthest point
0 2 4 6 8 10k a
- 6
- 4
- 2
0
2
4
6
Re(
p/z 0U
0)
F u rth es t p o in tR ea l p a rt (M = 1 0 )
Present approach Burton & Miller approach
0 2 4 6 8 10k a
- 2
- 1
0
1
2
Re(
p/z 0U
0)
F u rth es t p o in tR ea l p a rt (M = 1 0 )
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Concluding remarks
A systematic approach, null-field integral equation in conjunction with degenerate kernel and spherical harmonics, was successfully proposed to deal with the three-dimensional radiation problem.
1.
3.
The present approach can be seen as one kind of semi-analytical approach, since error comes from the number of truncated term of spherical harmonics.
1. Introduction2. Problem statement3. Method of solution4. Numerical examples5. Concluding remarks
4.
A general-purpose program for multiple radiators of various number, radii, and arbitrary positions was developed.
2. Only boundary nodes were needed in the present approach.
5.
The Burton and Miller approach was successfully used to remedy the fictitious frequency.
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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The End
Thanks for your kind attention
http://ind.ntou.edu.tw/~msvlabWelcome to visit the web site of MSVLAB/NTOU
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
37
Potential of the nearest point
0 2 4 6 8 10k a
- 1
0
1
2
3
4
5
Re(
p/z 0U
0)
N ea res t p o in tR ea l p a rt (M = 1 6 )
Im ag in a ry p a rt (M = 1 6 )
4 .49 5 .7 6 6 .9 8
7 .7 3
8 .2 2 9 .0 9
9 .3 6
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Elliptic coordinates
01
000
01
0000
],sinsinsinhcoscos[cosh4
)4
ln(2
],sinsinsinhcoscos[cosh4
)4
ln(2),(
0
n
n
n
n
nnnnnnen
a
nnnnnnen
a
xsU
Degenerate kernel in the 2-D Laplace problems
Degenerate kernel in the 2-D Helmholtz problems
01
00
00
0
01
00
00
0
,)cosh,()cosh,()cos,()(
)cos,(
)cosh,()cosh,()cos,()(
)cos,(
,)cosh,()cosh,()cos,()(
)cos,(
)cosh,()cosh,()cos,()(
)cos,(
),(
nmmmo
m
m
nmmme
m
m
nmmmo
m
m
nmmme
m
m
hHohJohSohM
hSo
hHehJehSehM
hSe
hHohJohSohM
hSo
hHehJehSehM
hSe
xsU
P. M. Morse and H. Feshbach, Methods of theoretical physics, 1953.
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
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Motivation
BEM / 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)
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
40
-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
BEM (64 elements) FEM (2791 elements)
Non-uniform radiation problem (2D)– Dirichlet BC)2( ka
Successful experience
The 32nd Conference of Theoretical and Applied MechanicsNovember 28-29, 2008, National Chung Cheng University, Chia-Yi
41
-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
BEM (63 elements) FEM (7816 elements))4( ka
Successful experience
Non-uniform radiation problem (2D)– Neumann BC