Download - Introduction to Meshfree Method
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Introduction to Meshfree Method
Speaker : Yu-Ling ChenDate : 2013/06/13
National Taiwan Ocean UniversityDepartment of Systems Engineering & Naval Architecture
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Discrete Element Method(DEM) first employed MLS in the construction of “meshfree” discrete eq.
Consider 1-D domain
Moving Least-squares Approximation(MLS)
a a
x=s
xIx a Ix a
Ix
( )a Ix x
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Discrete Reprod (RK) Approximation_
1
1
1
1
( ) ( ; )
( ; ) ( )
( ; ) ( ) ( )
[ ( ; ) ] ( ) ( )
NPR
I II a
NP
I a I II
NPT
I a I II
NPT
I I a II
u x x x x u
C x x x x x u
H x x x b x x x u
H x x x u b x x x
2 3
3
| |where z=
2 14 4 ; 0 z
3 24 1
( ) (1 ) ; z 13 20 ; 1 z
I
a
x x
a
z z
z z
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Exact Reproduction of basis function• • •
•
2 2 2
1: ( ( ) 1) ( ) 1
: ( ( ) ) ( ) [ ( ) ( )] ( ) 0
: ( ( ) 1) ( ) [ ( ) ( ) ] ( ) 0
: ( ( ) ( ) ) ( ) 0
TI I a
I
T TI I I I a I I a
I I
T TI I I a I I a
I I
n T nI I I I a
I
u H x x b x
u x H x x x b x x H x x x x b x
u x H x x b x x H x x x x b x
u x H x x x x b x
[ ( ) ( ) ( )] ( ) (0)
( ) ( ) (0)
TI I a I
I
H x x H x x x x b x H
M x b x H
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1
1
1
( ) (0) ( ) ( ; ) ( )
= ( )
where ( ) (0) ( ) ( ; ) ( )
NPR T T
I a I II
I II
T TI I a I
u x H M x H x x x x x u
x u
x H M x H x x x x x
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Use the Reproducing Kernel Approximation to approximate the following functions• (1)• (2)
3( ) 2 1, [1,0] with 1 ,2 ,3 shape functionsst nd rdf x x x x order RK
( ) sin , [0,2 ]with 1 ,2 ,3 shape functionsst nd rdf x x x order RK
use 1 ,2 ,3 shape functions with 11,21and 31 RK nodesst nd rd order RK
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•
3( ) 2 1, [1,0]f x x x x
RK shape function-discrete point=11 support size=2
1st
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• RK shape function-discrete point=21 support size=2
1st
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• RK shape function-discrete point=31 support size=2
1st
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• RK shape function-discrete point=11 support size=2.001
2nd
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• RK shape function-discrete point=21 support size=2.001
2nd
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• RK shape function-discrete point=31 support size=2.01
2nd
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• RK shape function-discrete point=11 support size=3.001
3rd
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• RK shape function-discrete point=21 support size=43rd
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• RK shape function-discrete point=31 support size=43rd
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( ) sin , [0,2 ]f x x x
1st RK shape function-discrete point=11 support size=2
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• 1st RK shape function-discrete point=21 support size=2
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• 1st RK shape function-discrete point=31 support size=2
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• 2st RK shape function-discrete point=11 support size=2.001
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• 2st RK shape function-discrete point=21 support size=2.001
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• 2st RK shape function-discrete point=31 support size=2.001
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• RK shape function-discrete point=11 support size=3.0013rd
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• RK shape function-discrete point=21 support size=3.0013rd
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• 3rd RK shape function-discrete point=31 support size=3.001
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結論 :由以上圖表得知 support size 在 1,2階時,設定為大於等於 2 就可以貼近解析解。但在第 3 階時, support size 為 2 時反而離散,如果大於 2時就可以貼近解析解。因此由以上得知階數越高並不代表精度可以更高。
References
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