regularized meshless method for boundary value problems with multiply-connected domain
DESCRIPTION
Regularized meshless method for boundary value problems with multiply-connected domain. Jeng-Hung Kao Advisor: Jeng-Tzong Chen, Kue-Hung Chen 6, 29, 2006 HRE2-307. Outlines. Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-domain problems - PowerPoint PPT PresentationTRANSCRIPT
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Regularized meshless method for boundary value problems with m
ultiply-connected domain
Jeng-Hung KaoAdvisor: Jeng-Tzong Chen, Kue-Hung Chen
6, 29, 2006HRE2-307
海洋大學力學聲響振動實驗室 htt
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Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-
domain problems Application on multiply-connected-domain
problems Conclusions Further research
海洋大學力學聲響振動實驗室 htt
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Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-
domain problems Application on multiply-connected-domain
problems Conclusions Further research
海洋大學力學聲響振動實驗室 htt
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Motivation and literature review
Numerical Methods
Numerical Methods
Mesh MethodsMesh Methods
Finite Difference Method
Finite Difference Method
Meshless Methods
Meshless Methods
Finite Element Method
Finite Element Method
Boundary Element Method
Boundary Element Method
(MFS) (RMM)
Motivation
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Meshless methodsMeshless methods
FEMFEM BEM BEM
Chen et al.2002JSV
Chen et al.2002JSV
Continuous moving least
square
Continuous moving least
squareContinuous
Kernel
Continuous Kernel
Boundary node method
Boundary node method
Boundary collocation
method
Boundary collocation
method Belyschko et al. 1994
Belyschko et al. 1994
Monagh 1982Liu et al. 1995
Monagh 1982Liu et al. 1995
Mukherjee, Huang,
Chen & Kang 2002
EABE, IJNME
Mukherjee, Huang,
Chen & Kang 2002
EABE, IJNME
RMMRMMMFSMFS
Kupradze 1964 CMMP
Kupradze 1964 CMMP
Young and Chen
2005
JCP, JASA
Young and Chen
2005
JCP, JASA
Nonsingular kernel
Nonsingular kernel
Singular kernel
Singular kernel
Motivation and literature review literature review
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Motivation and literature review
literature review
x
y
B1 and B2
(Physical boundary)
Dc
Collocation pointCollocation point
),( ix ),( ix
B‘
R
d
B1 and B2
(D( Interested Domain)
Dc
Source pointSource point),( ix ),( ix
B‘
R
d
x
D
De
( , )js r
( , )ix
x
y
D
De
( , )js r
( , )ix
x
y
Helmholtz problem
Laplace problem
d=0
Chen,Tanaka BKM onsingular general solution 2002
JT Chen BEM imaginary-part 2002
Young and Chen RMM 2005
SW Kang NDIF imaginary-part 2002
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Motivation and literature review
Exact solution
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Motivation and literature review
d=0.1
d=1.0
Convention MFS
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Motivation and literature review
RMM
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Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-
domain problems Application on multiply-connected-domain
problems Conclusions Further research
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D
Dc
( , )js r q
),( ix
Source pointCollocation point
B1 and B2
R
d x
y
D
Dc
( , )js r q
),( ix
Source pointCollocation point
B1 and B2
R
d
D
Dc
( , )js r q
),( ix
Source pointCollocation pointSource pointCollocation point
B1 and B2
R
d x
y
N
jjiji xsBxt
1
),()( Neumann problem
N
jjiji xsAxu
1
),()( Dirichlet problem
Relation between MFS and RMM
x
y
B1 and B2
(Physical boundary)
Dc
Collocation pointCollocation point
),( ix ),( ix
B‘
R
d
B1 and B2
(D( Interested Domain)
Dc
Source pointSource point),( ix ),( ix
B‘
R
d
x
Interior problem Exterior
problem
Kernel functions
Introduction of MFS
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)(2
),( )1(0 ijij xskH
ixsU
Relation between MFS and RMM
( , ) ln | |j i j iU s x s x
),( ij xsL ),( ij xsM
),( ij xsTsn
xn
sn
xnSingle-layer
Potentials
Double-layerPotentials
Laplace problemHelmholtz problem
Introduction of MFS
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Relation between MFS and RMM
N
jj
Oi
Oji xsTxu
1
),()(
x
y
B1 and B2
(Physical boundary)
Dc
Collocation pointCollocation point
),( ix ),( ix
B‘
R
d
B1 and B2
(D( Interested Domain)
Dc
Source pointSource point),( ix ),( ix
B‘
R
d
x
D
De
( , )js r
( , )ix
x
y
D
De
( , )js r
( , )ix
x
y
N
ij
Oi
Oj
Oi
Oi
i
j
Oi
Oj xsTxsTxsT
1
1
1
),(),(),(
N
jj
Oi
Oji xsMxt
1
),()(
N
ij
Oi
Oj
Oi
Oi
i
j
Oi
Oj xsMxsMxsM
1
1
1
),(),(),(
d=0
Introduction of MFSConvention MFS RMM
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Relation between MFS and RMM
D
De
( , )js r
( , )ix
x
y
D
De
( , )js r
( , )ix
x
y
N
jj
Oi
Oji xsTxu
1
),()(
N
ji
Ii
Ij xsT
1
),(
N
ijj
Oi
Oji
Oi
Oi
i
jj
Oi
Oj xsTxsTxsT
1
1
1
),(),(),(
N
iji
Ii
Iji
Ii
Ii
i
ji
Ii
Ij xsTxsTxsT
1
1
1
),(),(),(
iOi
Oi
N
j
Ii
Ij
N
ijj
Oi
Oj
i
jj
Oi
Oji xsTxsTxsTxsTxu )],(),([),(),()(
11
1
1
iOi
Oi
N
j
Ii
Ij
N
ijj
Oi
Oj
i
jj
Oi
Oji xsMxsMxsMxsMxt )],(),([),(),()(
11
1
1
Introduction of RMM
jixsTxsT
jixsTxsTOi
Oj
Ii
Ij
Oi
Oj
Ii
Ij
),,(),(
),,(),(
jixsMxsM
jixsMxsMOi
Oj
Ii
Ij
Oi
Oj
Ii
Ij
),,(),(
),,(),(
=0
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Introduction of Method of Fundamental Solutions
D
De
( , )js r
( , )ix
x
y
D
De
( , )js r
( , )ix
x
y
)],(),([),(
),()],(),([
11
1111
1
ON
ON
N
j
IN
Ij
ON
O
OON
OON
j
IIj
xsTxsTxsT
xsTxsTxsT
u
)],(),([),(
),()],(),([
11
1111
1
ON
ON
N
j
IN
Ij
ON
O
OON
OON
j
IIj
xsMxsMxsM
xsMxsMxsM
t
Introduction of RMM
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Relation between MFS and RMM
Source points Collocation points Kernel functions
MFS
RMM
fictitious
boundary
Real boundary
Real boundary
Real boundary
Single-layer potentialsDouble-layer potentialsDouble-layer potentials
Compared RMM with MFS
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Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-
domain problems Application on multiply-connected-domain
problems Conclusions Further research
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p
P
NN
NNjj
Ii
Ij
N
jj
Ii
Ij
Ii xsTxsTxu
1
11
1
11
),(),()(
N
NNjj
Ii
Oj
NN
NNjj
Ii
Ij
m
m
m
xsTxsT11 11
11
21
),(),(
p
P
NN
NNji
Ii
Ij xsT
1
11 1
),(
RMM for solving multiply-connected-domain problems
Source point Collocation point
1
11 11
1
),(),()(i
NNjj
Ii
Ij
N
jj
Ii
Ij
Ii
p
xsTxsTxu
11
21
1
11
),(),(m
m
p NN
NNjj
Ii
Ij
NN
ijj
Ii
Ij xsTxsT
iIi
Ii
NN
NNj
Ii
Ij
N
NNjj
Ii
Oj xsTxsTxsT
p
Pm
),(),(),(1
1111 11
p
p
NN
ijj
Ii
Iji
Ii
Ii
i
NNjj
Ii
Ij xsTxsTxsT
1
11 1
1
1
),(),(),(
Laplace problem
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11
21
21
1
1
111
),(),(),()(m
m
NN
NNjj
Oi
Ij
NN
Njj
Oi
Ij
N
jj
Oi
Ij
Oi xsTxsTxsTxu
N
ijj
Oi
Oj
i
NNjj
Oi
Oj xsTxsT
m 1
1
1
),(),(11
iOi
Oi
N
NNj
Ii
Ij xsTxsT
m
),(),(111
RMM for solving multiply-connected-domain problems
p
P
NN
NNji
Ii
Ij xsT
1
11 1
),(
Source point Collocation point
11
21
21
1
1
111
),(),(),()(m
m
NN
NNjj
Oi
Ijj
NN
Nj
Oi
Ij
N
jj
Oi
Ij
Oi xsTxsTxsTxu
N
NNjj
Oi
Oj
m
xsT111
),(
N
ijj
Oi
Oji
Oi
Oi
i
NNjj
Oi
Oj xsTxsTxsT
m 1
1
1
),(),(),(11
Laplace problem
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1 11 1 2
1 1 21 1 1
( ) ( , ) ( , ) ( , )m
m
N NN N NO I O I O I Oi j i j j i j j i j
j j N j N N
t x M s x M s x M s x
N
ijj
Oi
Oj
i
NNjj
Oi
Oj xsMxsM
m 1
1
1
),(),(11
iOi
Oi
N
NNj
Ii
Ij xsMxsM
m
),(),(111
11
21
1
11
),(),(m
m
p NN
NNjj
Ii
Ij
NN
ijj
Ii
Ij xsMxsM
1
11 11
1
),(),()(i
NNjj
Ii
Ij
N
jj
Ii
Ij
Ii
p
xsMxsMxt
iIi
Ii
NN
NNj
Ii
Ij
N
NNjj
Ii
Oj xsMxsMxsM
p
Pm
),(),(),(1
1111 11
RMM for solving multiply-connected-domain problems
Source point Collocation point
Laplace problem
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RMM for solving multiply-connected-domain problems
Construction of influence matrices
1 11 1 1
1 2 1 1 2 11
1 1
11 1
11 1
1
m
m mm m m
mN NN N N N
m mmN N N N N NN N N NN N
N NNN
u
T Tu
T Tu
u
1
11
11
1
111
1
1
1
1
),(),(),(),(
),(),(),(),(
),(),(),(),(
121
2221
221
112111
1
11
NN
IN
IN
N
j
IN
Ij
IN
IIN
I
IIN
IIN
j
IIj
II
IIN
IIIIN
j
IIj
xsTxsTxsTxsT
xsTxsTxsTxsT
xsTxsTxsTxsT
T
mmm
mm
mm
NN
IN
ON
IN
ONN
IN
ONN
ION
IONN
IONN
ION
IONN
IONN
m
xsTxsTxsT
xsTxsTxsT
xsTxsTxsT
T
11111111
1111
1111
),(),(),(
),(),(),(
),(),(),(
21
22221
11211
1
11
1111111
1111111
),(),(),(
),(),(),(
),(),(),(
21
22221
11211
1
NN
ON
IN
ON
ION
I
ONN
IN
ONN
IONN
I
ONN
IN
ONN
IONN
I
m
m
mmm
mmm
xsTxsTxsT
xsTxsTxsT
xsTxsTxsT
T
mmm
m
m
mmm
m
m
NN
N
NNj
ON
ON
IN
Ij
ON
ONN
ONN
ON
ONN
ON
ONN
ONN
N
NNj
INN
Ij
mm
xsTxsTxsT
xsT
xsTxsTxsT
T
11
2
1111
1
11
11
11
111111
11
11
),(),(),(
),(
),(),(),(
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RMM for solving multiply-connected-domain problems
)sin()2cos(2 rru
Test cases
Neumann problem
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RMM for solving multiply-connected-domain problems Test cases
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RMM for solving multiply-connected-domain problems
)cos(yeu x
Arbitrary-shape problem
Test cases
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RMM for solving multiply-connected-domain problems Test cases
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RMM for solving multiply-connected-domain problems
ij
kkijij r
ynkrH
kixsT )(
2),( )1(
1
2),(),(lim
ij
kkijij
sx r
ynxsTxsT
ji
})()({2
),( )1(12
)1(2
ij
kkij
ij
lklkijij r
nnkrH
r
nnyykrkH
kixsM
ik
r
nn
r
nnyyi
kxsMxsM
ij
kk
ij
lklkijij
sx ji 4)2(
4),(),(lim
2
24
2
Source point Collocation point
Helmholtz problem
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RMM for solving multiply-connected-domain problems
p
P
NN
NNjj
Ii
Ij
N
jj
Ii
Ij
Ii xsTxsTxu
1
11
1
11
),(),()(
N
NNjj
Ii
Oj
NN
NNjj
Ii
Ij
m
m
m
xsTxsT11 11
11
21
),(),(
p
P
NN
NNji
Ii
Ij xsT
1
11 1
),(
1
11 11
1
),(),()(i
NNjj
Ii
Ij
N
jj
Ii
Ij
Ii
p
xsTxsTxu
11
21
1
11
),(),(m
m
p NN
NNjj
Ii
Ij
NN
ijj
Ii
Ij xsTxsT
iIi
Ii
NN
NNj
Ii
Ij
N
NNjj
Ii
Oj xsTxsTxsT
p
Pm
),(),(),(1
1111 11
Source point Collocation point
N
ijj
Ii
Iji
Ii
Ii
i
NNjj
Ii
Ij xsTxsTxsT
m 1
1
1
),(),(),(11
Helmholtz problem
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RMM for solving multiply-connected-domain problems
N
NNji
Ii
Ij
m
xsT111
),(
11
21
21
1
1
111
),(),(),()(m
m
NN
NNjj
Oi
Ij
NN
Njj
Oi
Ij
N
jj
Oi
Ij
Oi xsTxsTxsTxu
N
ijj
Oi
Oj
i
NNjj
Oi
Oj xsTxsT
m 1
1
1
),(),(11
iOi
Oi
N
NNj
Ii
Ij xsTxsT
m
),(),(111
11
21
21
1
1
111
),(),(),()(m
m
NN
NNjj
Oi
Ijj
NN
Nj
Oi
Ij
N
jj
Oi
Ij
Oi xsTxsTxsTxu
N
NNjj
Oi
Oj
m
xsT111
),(
Source point Collocation point
N
ijj
Oi
Oji
Oi
Oi
i
NNjj
Oi
Oj xsTxsTxsT
m 1
1
1
),(),(),(11
Helmholtz problem
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1
11 11
1
),(),()(i
NNjj
Ii
Ij
N
jj
Ii
Ij
Ii
p
xsMxsMxt
11
21
1
11
),(),(m
m
p NN
NNjj
Ii
Ij
NN
ijj
Ii
Ij xsMxsM
iIi
Ii
NN
NNj
Ii
Ij
N
NNjj
Ii
Oj xsMxsMxsM
p
Pm
),(),(),(1
1111 11
11
21
21
1
1
111
),(),(),()(m
m
NN
NNjj
Oi
Ijj
NN
Nj
Oi
Ij
N
jj
Oi
Ij
Oi xsMxsMxsMxt
N
ijj
Oi
Oj
i
NNjj
Oi
Oj xsMxsM
m 1
1
1
),(),(11
iOi
Oi
N
NNj
Ii
Ij xsMxsM
m
),(),(111
RMM for solving multiply-connected-domain problems
Source point Collocation point
Helmholtz problem
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RMM for solving multiply-connected-domain problems
Construction of influence matrices
1
1
1
1
111
1
1
1
121
1
1
111
121
1
NN
NNN
N
NNNNmmNNm
NNmNN
N
N
NNN
N
mmmm
m
m TT
TT
u
u
u
u
11
11
1
111
1
1
1
1
),(),(),(),(
),(),(),(),(
),(),(),(),(
121
2221
221
112111
1
11
NN
IN
IN
N
j
IN
Ij
IN
IIN
I
IIN
IIN
j
IIj
II
IIN
IIIIN
j
IIj
xsTxsTxsTxsT
xsTxsTxsTxsT
xsTxsTxsTxsT
T
mmm
mm
mm
NN
IN
ON
IN
ONN
IN
ONN
ION
IONN
IONN
ION
IONN
IONN
m
xsTxsTxsT
xsTxsTxsT
xsTxsTxsT
T
11111111
1111
1111
),(),(),(
),(),(),(
),(),(),(
21
22221
11211
1
11
1111111
1111111
),(),(),(
),(),(),(
),(),(),(
21
22221
11211
1
NN
ON
IN
ON
ION
I
ONN
IN
ONN
IONN
I
ONN
IN
ONN
IONN
I
m
m
mmm
mmm
xsTxsTxsT
xsTxsTxsT
xsTxsTxsT
T
mm
m
m
mmm
m
m
NN
N
NNj
ON
ON
IN
Ij
ON
ONN
ONN
ON
ONN
ONN
N
NNj
INN
Ij
mm
xsTxsTxsT
xsTxsTxsT
T
11
1111
1
11
11
111111
11
11
),(),(),(
),(),(),(
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RMM for solving multiply-connected-domain problems
Extracting out the eigenvalues
T M
Treatments of spurious eigenvalues
0
NN
NN
M
TP
H
M
T
M
T
M
TP
0
0
0
0
0
0
SVD and SVD updating term
HTTT
HMMM
SVD
SVD
Singular values matrix
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RMM for solving multiply-connected-domain problems
22 )/()/( LnLmkmn
0 2 4 6 8 10 12k
0.001
0.01
0.1
1
10
1
4.44 (T)<4.44>
7.02 (T)<7.02>
8.87 (T)<8.88>
9.93 (T)<9.93>
11.33 (T)<11.33>
Test case
海洋大學力學聲響振動實驗室 htt
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Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-
domain problems Application on multiply-connected-domain
problems Conclusions Further research
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
),( yxww ),( yx
0
yxzyzx
0
y
D
x
D yx
xzxzx Eec 1544
yzyzy Eec 1544
xzxx EeD 1115
yzyy EeD 1115
0
02
112
15
215
244
we
ewc
02 w 02 mi ww mzr
izr
mi mr
ir DD
0 0
Antiplane piezoelectricity problem Antiplane shear problem
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
Decomposition of the problem
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
n
w
w
MM
TT
m
i
Ow
Iwm
i
Ow
Iw
Inclusion Matrix
iww
iw
I
Tu
iww
iw
I
Mt
ii I
Tu
ii I
Mt
mww
mw
O
Tu
mww
mw
O
Mt
mm O
Tu
mOmMt
mi ww mzr
izr
mi mr
ir DD
nn
we
nc
e
n
w
w
MMMeMe
Mc
eM
c
eMM
c
c
TT
TT
mm
m
m
m
i
mw
iw
OmIiOw
mIw
i
Om
mI
m
iOw
Iwm
i
OI
Ow
Iw
1115
44
15
11111515
44
15
44
15
44
44
00
00
Antiplane piezoelectricity problems Antiplane shear problems
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
G.E.Continuous conditions
Shear stressElectric displacements
Piezoelectricity problem
Antiplane shear problem
02 w
02
02 w
mi ww mzr
izr
mi mr
ir DD
mi ww mzr
izr
xzxx EeD 1115
yzyy EeD 1115
zxzx
zyzy
absent
Compared antiplane piezoelectric with antiplane shear problems
zxzx c 44 xEe15
zyzy c 44yEe15
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
Influence matrices
Piezoelectricity problem
Antiplane shear problem
nn
we
nc
e
n
w
w
MMMeMe
Mc
eM
c
eMM
c
c
TT
TT
mm
m
m
m
i
mw
iw
OmIiOw
mIw
i
Om
mI
m
iOw
Iwm
i
OI
Ow
Iw
1115
44
15
11111515
44
15
44
15
44
44
00
00
n
w
w
MM
TT
m
i
Ow
Iwm
i
Ow
Iw
0
Compared antiplane piezoelectric with antiplane shear problems
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
7105 Nm-2
0.1015 ie Cm-2
81111 1051.1 im CV-1m-1
104444 1053.3 im cc Nm-2
Antiplane piezoelectric problems with multiple inclusions
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-6
-4
-2
0
2
4
6m
z/
E = -10 6 V /mana lytica l so lu tion
R M M
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-6
-4
-2
0
2
4
6m
z/
E =0.0 V /mana lytica l so lu tion
R M M
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-6
-4
-2
0
2
4
6m
z/
E =10 6 V /mana lytica l so lution
R M M
32
31
mVE /106 mVE /106
ie15
me15
Case 1: Single inclusion
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-4
-2
0
2
4
6
8
mzr
/
E = -10 6 V /mana lytica l so lu tion
R M M
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-4
-2
0
2
4
6
8
mzr
/
E =0.0 V /mana lytica l so lu tion
R M M
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-4
-2
0
2
4
6
8
mzr
/
E =10 6 V /mana lytica l so lu tion
R M M
32
31
mVE /106 mVE /106
ie15
me15
Case 1: Single inclusion
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
7105 Nm-2
0.1015 ie Cm-2
81111 1051.1 im CV-1m-1
104444 1053.3 im cc Nm-2
Antiplane piezoelectric problems with multiple inclusions
海洋大學力學聲響振動實驗室 htt
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d
31
31
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-6
-4
-2
0
2
4
6
mz
/
E= -10 6 V /manalytica l so lu tion (Pak 1992)
R M M (d /r1=10., =/2)
C hao 's resu lt
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-6
-4
-2
0
2
4
6
mz
/
E=10 6 V /manalysis so lu tion (Pak 1992)
R M M (d /r1=10., =/2)
C hao's resu lt
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-6
-4
-2
0
2
4
6
mz
/
E=0 V /mana lytica l so lution (Pak 1992)
R M M (d /r1=10., =/2) C hao 's result
Application on multiply-connected-domain problems
mVE /106 mVE /106
me15
ie15
Case 2: Two inclusions
海洋大學力學聲響振動實驗室 htt
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31
32
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-60
-40
-20
0
20
40
60
mz
/
=510 7 N /m 2 E=10 6 V /m d /r1=10.0
R M MC hao 's resu lt
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-60
-40
-20
0
20
40
60
mz
/
=510 7 N /m 2 E=10 6 V /m d /r1=1.0
R M M
C hao's resu lt
-10 -8 -6 -4 -2 0 2 4 6 8 10em
15/e i15
-60
-40
-20
0
20
40
60
mz
/
=510 7 N /m 2 E=10 6 V /m d /r1=0.1
R M MC hao's resu lt
Application on multiply-connected-domain problems
me15
d=10d=1d=0.1
mVE /106
ie15
Case 2: Two inclusions
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
0 1 2 3 4 5 6 (in rad iu s)
-12
-8
-4
0
4
8
12
E/
E
=510 7 N /m 2 E=10 6 V /m em
15/e i15= -5
d/r1=0.02
R M M
C hao's resu lt
0 1 2 3 4 5 6 ( in rad iu s)
-12
-8
-4
0
4
8
12
E/
E
=510 7 N /m 2 E=10 6 V /m em
15/e i15= -5
d/r1=10.0
R M M C hao's resu lt
0 1 2 3 4 5 6 ( in rad iu s)
-12
-8
-4
0
4
8
12
E/
E
=510 7 N /m 2 E=10 6 V /m e m
15/e i15= -5
d /r1=1.0
R M M C hao 's resu lt
0 1 2 3 4 5 6 ( in rad iu s)
-12
-8
-4
0
4
8
12
E/
E
=510 7 N /m 2 E=10 6 V /m e m
15/e i15= -5
d /r1=0.1
R M M
C hao's resu lt
d=1
d=10d=0.1d=0.01
d=0.02
0 1 2 3 4 5 6 (in rad iu s)
-12
-8
-4
0
4
8
12
E/
E
=510 7 N /m 2 E=10 6 V /m em
15/e i15= -5
d/r1=0.01
R M M
C hao 's resu lt
610E
31
32
Case 2: Two inclusions
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
12 2rr
11.0 rd
0.1 Nm-2
0.10
01 3
2
02 7
13
Antiplane shear problems with multiple inclusions
海洋大學力學聲響振動實驗室 htt
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31
32
0 1 2 3 4 5 6
( in rad iu s)
-2
0
2
4
6
8
10
Str
esse
s ar
ound
incl
usio
n of
rad
ius
r 1
R M M
m zr/H onein et a l.
Application on multiply-connected-domain problems
0 1 2 3 4 5 6
( in ra d iu s)
-2
0
2
4
6
8
10
Str
esse
s ar
ound
incl
usio
n of
rad
ius
r 1
R M M
iz r /H onein et a l.
Case 1: Two inclusions
海洋大學力學聲響振動實驗室 htt
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31
32
0 1 2 3 4 5 6
( in ra d iu s)
-2
0
2
4
6
8
10
Str
esse
s ar
ound
incl
usio
n of
rad
ius
r 1
R M M
m z/H onein e t a l.
Application on multiply-connected-domain problems
0 1 2 3 4 5 6
( in ra d iu s)
-2
0
2
4
6
8
10
Str
esse
s ar
ound
incl
usio
n of
rad
ius
r 1
R M M
iz/H o n e in e t a l.
Case 1: Two inclusions
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
12rd
Antiplane shear problems with multiple inclusions
海洋大學力學聲響振動實驗室 htt
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31
32
Application on multiply-connected-domain problems
0 0.2 0.4 0.6 0.8 1
/
-2 .5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Str
ess
Con
cent
rati
on F
acto
r m
z/ R M M
1/0=2/0=3/0= 0 .0G o n g e t a l.
0 0.2 0.4 0.6 0.8 1
/
-2 .5
-2
-1 .5
-1
-0 .5
0
0.5
1
1.5
2
2.5
Str
ess
Con
cent
rati
on F
acto
r m
z/ R M M
1/0=2/0=3/0=0 .5G o n g e t a l.
10 01
02
03
21
22
23
51
52
53
5.01
5.02
5.03
0 0.2 0.4 0.6 0 .8 1
/
-2 .5
-2
-1 .5
-1
-0 .5
0
0 .5
1
1.5
2
2.5
Str
ess
Con
cent
rati
on F
acto
r m
z/ R M M
1/0=2/0=3/0=2 .0G o n g e t a l.
0 0.2 0.4 0.6 0 .8 1
/
-2 .5
-2
-1 .5
-1
-0 .5
0
0 .5
1
1.5
2
2.5
Str
ess
Con
cent
rati
on F
acto
r m
z/ R M M
1/0=2/0=3/0=5 .0G o n g e t a l.
Case 2: Three inclusions
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Application on multiply-connected-domain problems
0)()]()()()([ 11221 krJkrYkrJkrYkrJ nnnnn
0)()]()()()([ 11221 krJkrYkrJkrYkrJ nnnnn
True eigenequations
Dirichlet type
Neumann type
Spurious eigenequation
Acoustic problems
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
0 1 2 3 4 5k
0.01
0.1
1
1
2.05 (T)<2.05>
2.22 (T)<2.22>
2.66 (T)<2.66>
3.21 (T)<3.21>
3.80 (T)<3.80>
4.27 (T)<4.27>
4.39 (T)<4.39>
4.57 (T)<4.57>
4.97 (T)<5.03>
3.68 (S)<3.68>
4.16 (T)<4.16>
(T): True eigenvalue(S): Spurious eigenvalue< >:Analytical soluition
Case 1: Dirichlet BC
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
0 1 2 3 4 5k
1
10
100
1
0.83(T)<0.82>
1.52(T)<1.50>
2.12(T)<2.10>
2.25(T)<2.22>
2.52(T)<2.50>
2.68(T)<2.66>
3.20(T)<3.18>
3.24(T)<3.21>
3.76(S)<3.68>
3.95(T)<3.93>
(T): True eigenvalue(S): Spurious eigenvalue< >:Analytical soluition
Case 1: Neumann BC
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Application on multiply-connected-domain problems
0 1 2 3 4 5k
0.1
1
10
100 1 3.68
(S)<3.68>
Case 1: RMM+SVD updating term approached
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
Case 2: A circular domain with two equal holes Case 3: A circular domain with four equal holes
Acoustic problems
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems Case 2: A circular domain with two equal holes
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems
RMM
BEM
Mode 1
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Mode 2 Mode 3 Mode 4 Mode 5
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Case 2: A circular domain with two equal holes
海洋大學力學聲響振動實驗室 htt
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Application on multiply-connected-domain problems Case 3: A circular domain with four equal holes
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Application on multiply-connected-domain problems
RMM
BEM
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0 .8 -0 .6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Case 3: A circular domain with four equal holes
海洋大學力學聲響振動實驗室 htt
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Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-
domain problems Application on multiply-connected-domain
problems Conclusions Further research
海洋大學力學聲響振動實驗室 htt
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Conclusions
Only the boundary nodes on the physical boundary are required by using proposed method.
The proposed method can regularize singularity by using subtracting and adding-back technique.
A systematic approach to solve the Laplace and eigenproblems with multiply-connected domain was proposed successfully by using the regularized meshless method.
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Conclusions
The RMM successfully are applied on three engineering problems. (antiplane, piezoelectricity, acoustics)
海洋大學力學聲響振動實驗室 htt
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Outlines Motivation and literature review Relation between MFS and RMM RMM for solving multiply-connected-
domain problems Application on multiply-connected-domain
problems Conclusions Further research
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Further research
Three-dimensional problems with inclusions.
Plane problems with multiple inclusions in an anisotropic medium.
Piezoelectric inclusions subject to an incident wave and a harmonic inplane electric field.
Multiple scattering problems.
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The end
Thanks for your attentions.
Your comment is much appreciated.
You can get more information on our website.http://msvlab.hre.ntou.edu.tw
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Regularized meshless method for Helmholtz problems with multiply-connected domain
ij
kkijij r
ynkrH
kixsT )(
2),( )1(
1
2),(),(lim
ij
kkijij
sx r
ynxsTxsT
ji
where
ikr
krkrH
ij
ijij
rij 2
2)(lim )1(
10
ikr
krkrH
ij
ijij
rij2
2)1(
20 )(
4
8
)()(lim
})()({2
),( )1(12
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24
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Source point Collocation point
海洋大學力學聲響振動實驗室 htt
p://msvlab.hre.ntou.edu.tw 67
Formulation
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1
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j ij
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