Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong.

Regularized meshless method for solving Laplace equation

with multiple holes

Speaker: Kuo-Lun WuCoworker : Jeng-Hong Kao 、 Kue-Hong Chen

and Jeng-Tzong Chen

以正規化無網格法求解含多孔洞拉普拉斯方程式

工學院 2005/04/01

2

Outlines

Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions

3

Outlines


4

MotivationNumerical 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)

5

Outlines


6

Statement of problem Laplace equation with multiple holes :

potential flow around

cylinders

electrostatic field of wires

torsion bar with holes

21 2( , ) 0u x x MZ

7

Outlines


8

Method of fundamental solutions (MFS)

Method of fundamental solutions (MFS) :

Source point Collocation point— Physical boundary-- Off-set boundary

d = off-set distance

d

Double-layer

potential approach

Single-layer

Potential approach

Dirichlet problem

Neumann problem

Dirichlet problem

Neumann problem

Distributed type

1

( ) ( , )N

i j i jj

u x U s x

1

( ) ( , )N

i j i jj

t x L s x

1

( ) ( , )N

i j i jj

u x T s x

1

( ) ( , )N

i j i jj

t x M s x

( , ) ln | |j i j iU s x s x

( , )( , ) j i

j is

U s xT s x

n

9

The artificial boundary (off-set boundary) distance is debatable.

The diagonal coefficients of influence matrices are singular when the source point coincides the collocation point.

10

Outlines


11

Regularized meshless method (RMM)

Source point Collocation point— Physical boundary

Regularized meshless method (RMM)

Double-layer

potential approach

Dirichlet problem

Neumann problem

where

( ) ( )

1 1

( ) ( , ) ( , )N N

I Oi j i j j i i

j j

u x T s x T s x

( ) ( )

1 1

( ) ( , ) ( , )N N

I Oi j i j j i i

j j

t x M s x M s x

( )

1

( , ) 0,N

Oj i

j

T s x

( )

1

( , ) 0N

Oj i

j

M s x

ixis

1s

2s

3s4s

Ns

I = Inward normal vectorO = Outward normal vector

12

( ) ( )

1 1

( ) ( , ) ( , )N N

I Oi j i j j i i

j j

u x T s x T s x

1( ) ( ) ( ) ( )

1 1 1

( , ) ( , ) ( , ) ( , ) ,i N N

I I I Ij i j j i j m i i i i i

j j i m

T s x T s x T s x T s x x B

1( ) ( ) ( ) ( )

1 1 1

( , ) ( , ) ( , ) ( , )i N N

I I I Oj i j i i i j i j j i i

j j i j

T s x T s x T s x T s x

In a similar way, 1

( ) ( ) ( ) ( )

1 1 1

( ) ( , ) ( , ) ( , ) ( , ) ,i N N

I I I Ii j i j j i j m i i i i

j j i m

t x M s x M s x M s x M s x

ix B

jixsTxsT

jixsTxsTOi

Oj

Ii

Ij

Oi

Oj

Ii

Ij

),,(),(

),,(),(

( , ) ( , ),

( , ) ( , ),

I I O Oj i j iI I O Oj i j i

M s x M s x i j

M s x M s x i j

13

1, 1,1 1,2 1,1

2,1 2, 2,2 2,1

,1 ,2 , ,1

,

N

m Nm

N

m Ni jm

N

N N N m N Nm

T T T T

T T T Tu

T T T T

1, 1,1 1,2 1,1

2,1 2, 2,2 2,1

,1 ,2 , ,1

( )

( ).

( )

N

m Nm

N

m Ni jm

N

N N N m N Nm

M M M M

M M M Mt

M M M M

14

Outlines

Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation with multiple holes Numerical examples Conclusions

15

Formulation with multiple holes


inner holes = m-1

outer hole = m th

16

inner holes = m-1

outer hole = m th


1

1 1

1

1 1

1 2

1 1

1

1 1

1

1

1

( ) ( , ) ( , )

( , )

( , )

( , )

(

p

p

m

m

m

N iI I I I Ii j i j j i j

j j N N

N NI Ij i j

j i

N NI Ij i j

j N N

NO Ij i j

j N N

u x T s x T s x

T s x

T s x

T s x

T s

1

1 1 1

, ) ( , ) ,

, 1

p

P

N NI I I Ij i i i i

j N N

Ii p

x T s x

x B p

P=1

17

1

1 1

1

1 1

1 2

1 1

1

1 1

1

1

1

( ) ( , ) ( , )

( , )

( , )

( , )

(

p

p

m

m

m


j j N N

N NI Ij i j

j i

N NI Ij i j

j N N

NO Ij i j

j N N

u x T s x T s x

T s x

T s x

T s x

T s

1

1 1 1

, ) ( , ) ,

, 1, 2, 3, , 1

p

P

N NI I I Ij i i i i

j N N

Ii p

x T s x

x B p m

inner holes = m-1

outer hole = m th


18

1

1 1

1

1 1

1 2

1 1

1

1 1

1

1

1

( ) ( , ) ( , )

( , )

( , )

( , )

( ,

p

p

m

m

m


j j N N

N NI Ij i j

j i

N NI Ij i j

j N N

NO Ij i j

j N N

Ij

t x M s x M s x

M s x

M s x

M s x

M s

1

1 1 1

) ( , ) ,

, 1, 2, 3, , 1

p

P

N NI I Ii i i i

j N N

Ii p

x M s x

x B p m

inner holes = m-1

outer hole = m th


19

1 1 2

1

1 1

1 2 1 1

1 1

1 1

1

1 1

1

( ) ( , ) ( , )

( , ) ( , )

( , )

( , )

m

m m

m

N N NO I O I Oi j i j j i j

j j N

N N iI O O Oj i j j i j

j N N j N N

NO Oj i j

j i

I Ij i

j N N

u x T s x T s x

T s x T s x

T s x

T s x

1

( , ) ,

,

NO Oi i i

Oi p

T s x

x B p m

inner holes = m-1

outer hole = m th


P=m

20

1 1 2

1

1 1

1 2 1 1

1 1

1 1

1

1 1

1

1

( ) ( , ) ( , )

( , ) ( , )

( , )

( , )

m

m m

m

N N NO I O I Oi j i j j i j

j j N

N N iI O O Oj i j j i j

j N N j N N

NO Oj i j

j i

NI Ij i

j N N

t x M s x M s x

M s x M s x

M s x

M s x

( , ) ,

,

O Oi i i

Oi p

M s x

x B p m

inner holes = m-1

outer hole = m th


P=m

21

The linear algebraic systems

1 1 1

1

11 11 1

1

m

m m m

mN N N N

m mm NN N N NNN N

T Tu

T Tu

1 1 1

1

11 11 1

1

m

m m m

mN N N N

m mm NN N N NNN N

M Mt

M Mt

s

s

x

x

22

Outlines


23

Numerical examples

2 2.0r 1 1.0r

u

u2 0u

y

x

y

x1.0a

1.0a

1r

1r

1r2 0u

0r

t

u

u

t

0

1

2.0

0.25

r

r

Case 1 Dirichlet B.C. Case 2 Mixed-type B.C.

1( , ) cos( )u r

r

3 cos(3 )u r

24

Contour of potential (case 1)

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

Exact solution RMM (360 points)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

BEM (360 elements)

25

Contour of potential (case 2)

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

Exact solution RMM (400 points)-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

BEM (800 elements)

260 200 400 600

N um ber of nodes (N )

1 .0E-005

1.0E-004

1.0E-003

1.0E-002

1.0E-001

1.0E+000

1.0E+001

1.0E+002

1.0E+003N

orm

err

or

Error convergence (case 2)

27

Outlines


28

Conclusions

Only boundary nodes on the real boundary are required.

Singularity of kernels is desingularized.

The present results for multiply-hole cases were well compared with exact solutions and BEM.

29

The end

Thanks for your attention.

Your comment is much appreciated.

Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong.

Documents