regularized meshless method for solving laplace equation with multiple holes speaker: kuo-lun wu...
Post on 19-Dec-2015
222 views
TRANSCRIPT
![Page 1: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/1.jpg)
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
![Page 2: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/2.jpg)
2
Outlines
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
![Page 3: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/3.jpg)
3
Outlines
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
![Page 4: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/4.jpg)
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)
![Page 5: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/5.jpg)
5
Outlines
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
![Page 6: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/6.jpg)
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
![Page 7: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/7.jpg)
7
Outlines
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
![Page 8: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/8.jpg)
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
![Page 9: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/9.jpg)
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.
![Page 10: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/10.jpg)
10
Outlines
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
![Page 11: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/11.jpg)
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
![Page 12: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/12.jpg)
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
![Page 13: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/13.jpg)
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
![Page 14: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/14.jpg)
14
Outlines
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation with multiple holes Numerical examples Conclusions
![Page 15: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/15.jpg)
15
Formulation with multiple holes
Source point Collocation point— Physical boundary
inner holes = m-1
outer hole = m th
![Page 16: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/16.jpg)
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
![Page 17: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/17.jpg)
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
![Page 18: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/18.jpg)
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
![Page 19: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/19.jpg)
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
![Page 20: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/20.jpg)
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
![Page 21: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/21.jpg)
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
![Page 22: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/22.jpg)
22
Outlines
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
![Page 23: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/23.jpg)
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
![Page 24: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/24.jpg)
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)
![Page 25: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/25.jpg)
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)
![Page 26: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/26.jpg)
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)
![Page 27: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/27.jpg)
27
Outlines
Motivation Statement of problem Method of fundamental solutions Regularized meshless method Formulation for multiple holes Numerical examples Conclusions
![Page 28: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/28.jpg)
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.
![Page 29: Regularized meshless method for solving Laplace equation with multiple holes Speaker: Kuo-Lun Wu Coworker : Jeng-Hong Kao 、 Kue-Hong Chen and Jeng-Tzong](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d3e5503460f94a17b3e/html5/thumbnails/29.jpg)
29
The end
Thanks for your attention.
Your comment is much appreciated.