Introduction to Meshfree Method

Speaker ： Yu-Ling ChenDate ： 2013/06/13

National Taiwan Ocean UniversityDepartment of Systems Engineering & Naval Architecture

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

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

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

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

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

•

3( ) 2 1, [1,0]f x x x x

RK shape function-discrete point=11 support size=2

1st

• RK shape function-discrete point=21 support size=2

1st

• RK shape function-discrete point=31 support size=2

1st

• RK shape function-discrete point=11 support size=2.001

2nd

• RK shape function-discrete point=21 support size=2.001

2nd

• RK shape function-discrete point=31 support size=2.01

2nd

• RK shape function-discrete point=11 support size=3.001

3rd

• RK shape function-discrete point=21 support size=43rd

• RK shape function-discrete point=31 support size=43rd

( ) sin , [0,2 ]f x x x

1st RK shape function-discrete point=11 support size=2

• 1st RK shape function-discrete point=21 support size=2

• 1st RK shape function-discrete point=31 support size=2

• 2st RK shape function-discrete point=11 support size=2.001

• 2st RK shape function-discrete point=21 support size=2.001

• 2st RK shape function-discrete point=31 support size=2.001

• RK shape function-discrete point=11 support size=3.0013rd

• RK shape function-discrete point=21 support size=3.0013rd

• 3rd RK shape function-discrete point=31 support size=3.001

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