regularized meshless method for solving laplace equation with multiple holes speaker: kuo-lun wu...
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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
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
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
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
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
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
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
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
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
Source point Collocation point— Physical boundary
inner holes = m-1
outer hole = m th
16
inner holes = m-1
outer hole = m th
Source point Collocation point— Physical boundary
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
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, 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
Source point Collocation point— Physical boundary
18
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
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
Source point Collocation point— Physical boundary
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
Source point Collocation point— Physical boundary
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
Source point Collocation point— Physical boundary
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
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
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
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
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.