海洋大學力學聲響振動實驗室 1 Regularized meshless method for boundary value problems with multiply-connected domain Jeng-Hung Kao Advisor:

海洋大學力學聲響振動實驗室http://

msvlab.hre.ntou.edu.tw 1

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

p://msvlab.hre.ntou.edu.tw 2

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








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



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




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




Exact solution




d=0.1

d=1.0

Convention MFS




RMM








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


),( ix ),( ix

B‘

R

d

B1 and B2


Dc


B‘

R

d

x

Interior problem Exterior

problem

Kernel functions

Introduction of MFS



)(2

),( )1(0 ijij xskH

ixsU


( , ) 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




N

jj

Oi

Oji xsTxu

1

),()(

x

y

B1 and B2

(Physical boundary)

Dc


),( ix ),( ix

B‘

R

d

B1 and B2


Dc


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




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



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




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








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



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


p

P

NN

NNji

Ii

Ij xsT

1

11 1

),(


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



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



Laplace problem




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

),(),(),(

),(

),(),(),(




)sin()2cos(2 rru

Test cases

Neumann problem



RMM for solving multiply-connected-domain problems Test cases




)cos(yeu x

Arbitrary-shape problem

Test cases



RMM for solving multiply-connected-domain problems Test cases




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


Helmholtz problem




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


N

ijj

Ii

Iji

Ii

Ii

i

NNjj

Ii

Ij xsTxsTxsT

m 1

1

1

),(),(),(11

Helmholtz problem




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

),(


N

ijj

Oi

Oji

Oi

Oi

i

NNjj

Oi

Oj xsTxsTxsT

m 1

1

1

),(),(),(11

Helmholtz problem



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



Helmholtz problem




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

),(),(),(

),(),(),(




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




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








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




Decomposition of the problem




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




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




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




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




-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




-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




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



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


mVE /106 mVE /106

me15

ie15

Case 2: Two inclusions



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


me15

d=10d=1d=0.1

mVE /106

ie15





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





12 2rr

11.0 rd

0.1 Nm-2

0.10

01 3

2

02 7

13

Antiplane shear problems with multiple inclusions



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.


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.




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.


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.





12rd

Antiplane shear problems with multiple inclusions



31

32


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




0)()]()()()([ 11221 krJkrYkrJkrYkrJ nnnnn

0)()]()()()([ 11221 krJkrYkrJkrYkrJ nnnnn

True eigenequations

Dirichlet type

Neumann type

Spurious eigenequation

Acoustic 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




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




0 1 2 3 4 5k

0.1

1

10

100 1 3.68

(S)<3.68>

Case 1: RMM+SVD updating term approached




Case 2: A circular domain with two equal holes Case 3: A circular domain with four equal holes

Acoustic problems



Application on multiply-connected-domain problems Case 2: A circular domain with two equal holes




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



Application on multiply-connected-domain problems Case 3: A circular domain with four equal holes




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








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.



Conclusions

The RMM successfully are applied on three engineering problems. (antiplane, piezoelectricity, acoustics)








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.



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



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

)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




Formulation

0 ( , ) ( ) ( ) ( , ) ( ) ( )B BT s x u s dB s U s x t s dB s 1, 0Rigid body test u t

( , )x ( , )s R

1

0 ( , ) ( ) ( , )N

j i jBj

T s x dB s T s x l

1

( , ) 0N

j ij

T s x

2 0u

Documents

海洋大學力學聲響振動實驗室 1 Regularized meshless method for boundary value problems with multiply-connected domain Jeng-Hung Kao Advisor: