meshless method moving least-square method and differential reproducing kernel method

2009/06/18(JWLee)高結期末報告

Meshless methodmoving least-square method

anddifferential reproducing kernel method

Reporter: Jai-Wei Lee

Teacher: Shyh-Rong Kuo

Date: June, 18, 2009

Time: 7:00pm


Outline

• Introduction• MLS• DRKM• Weight functions• Shape function• Conclusions


Introduction

1

(( ) ( ))i

N

ii

u x u xx

FEM (finite element method)

LSM (least square method)

MLS (moving least square method)

MLS-RKM (moving least square reproducing kernel method )

DRKM (differential reproducing kernel method)

mesh

meshless

node

element

ix Nx1x 2x

1 21 2( )) ) ) ((( ( )u xx u x u xx 1x 2x


MLS

2

1 1

( )( )) (N M

j i j ii j

i iL p x a uw x x x

( 1) ,( 2,) 1,jjp x j Mx

1 1

2 ( ) ( ) ( ) ( ) 0N M

i i j i j i k ii jk

Lw x x p x a u x p x

a

1

( ) ( ) ( )M T

j j j jj

u x p x a p x a

1 1 1

( ) ( ) ( ) ( ) ( ) ( )M N N

i i j i k i j i i k i ij i i

w x x p x p x a w x x p x u x

kj j ki iA a B u 1

( ) ( ) ( )N

kj i i j i k ii

A w x x p x p x

1

( ) ( )N

ki i i k ii

B w x x p x

1

j kj ki ia A B u

1

1( )

( )

( )

( ) ( )

( )

T

j kj ki

T

i

T

j j

i

i

N

iii

p x A B

x

x

u x p x a

u

u

u x

base function

Weight functions


MLS (differential)

1

1 1kj

kj kj kj

d AA A A

dx

1( )( ) ( )i

TT

i j kj ki ix p x Au x uBu

1

1

1 1

( )

( )

( )

( ) ( )

kkj

k

T

j

T

j T T

j

i

kiki k

T

i

i

j

kj kjj

A

d AA

B

d BB B

p x

d p xp x p x

d

d x d

x

dxx

A

d

dxdx

(( )

)T

i

iu x ud x

dx

( )???

l

l

T

id xcomplex

dx


DRKM

( )( )) ( iii w x xx c x x

1

( ( ) ( ))M T

j i ji j i jj

p x x ac p x ax x x

2 2

1

2 2 2 2 2

1 1 1 1

2 ( )

( ) ( ) ( ) 2 ( ) ( ) 2 0

N

i ii

N N N N

i i i i i i ii i i i

k x x x

x x x x x x x x x x x x x

1

0 ( ) 1N

ii

k x

1

1 1 1

1 ( )

( ) ( ) ( ) ( ) 0

N

i ii

N N N

i i i i ii i i

k x x x

x x x x x x x x x

1

( ) ( )( ) ( ) i

TN

i i ii

u x u xx x u

1

( ) , 0,1, 2, 1N

k ki i

i

x x x k M

1

1, 0( ) ( )

0, 0

Nk

i ii

kx x x

k

reproducing condition1

( ) ( ) ( )M

i j i jj

i w x x p xx x a

correction function


DRKM

1 1 1

( ) ( ) ( ) ( ) ( ) (0)N N M

k i i k i i j i j ki i j

p x x x p x x w x x p x x a b

1

( ) ( ) ( )N

kj k i i j ii

A p x x w x x p x x

1( ) ( ) (0) )(

T

i j i kj ki w x x p x x Ax b

1

( ) ( ) ( )) (( )M T

i j i j i j i jj

i w x x p x x a w x x xx p x a

1

1, 0( ) ( ) (0)

0, 0

Nk

i i ki

kx x x b

k

(0) , 1, 2, , 1, 2,kj j kA a b k M j M

1(0)j kj ka A b


DRKM (differential)

1

) )( ( ( )M T

j i j j ij

i jp x x p x xc x x a a

2

1

2 2 2

1 1 1 1

( )2 2

( ) ( ) ( ) ( )( ) 2 0 2 2 0

Ni

ii

N N N Ni i i i

i i ii i i i

d xk x x

dx

d x d x d x d xx x x x x x x x

dx dx dx dx

1

( ) ( ) ( )M

i j ii jj

x w x p x x ax

1

( ) ( )( )( )N

i i

T

i

i

id xd x

dx dxu x u x u

1

( )0 0

Ni

i

d xk

dx

1

1 1 1

( )1 1

( ) ( ) ( )( ) 0 1 1

Ni

ii

N N Ni i i

i ii i i

d xk x

dx

d x d x d xx x x x

dx dx dx

( 1)

1 1

( )( ) , 0,1, 2, 1

N Nk k k ki

i i ii i

d xx x x x kx k M

dx

1

1, 1( )( )

0, 1

Nk i

ii

kd xx x

kdx

reproducing condition

1

( ) ( )( ) ( ) i

TN

i i ii

u x u xx x u

( )( )) ( iii w x x c x xx


DRKM (differential)

1 1 1

( ) ( ) ( ) ( ) ((

0)

)N N M

k i k i i jj ii i j

kip x x p x x

d xa bw x x p x

dxx

1(0)j kj ka bA

1(0( )

))

(( )

T

i j i ki

kj

d x

dxw x x p x x A b

1

( )( ) ( ) ( ) ( )

M T

i j i j i ji

j ij

d x

dxw x x p x x w x x p xa ax

1

1, 1( )( )

0,(

10)k

Nk i

ii

bkd x

x xkdx

, 1, 2, ,( 10) , 2,kj kjA a k M jb M

1( ) ( (0)

))

( T

i j i kj

llkl

i w x x p xd x

dxA bx


Weight functions

ix xr

d

2 3

2 3

2 14 4 ,

3 24 4 1

( ) 4 4 , 13 3 20 , 1

r r r

w r r r r r

r

2 3 41 6 8 3 , 1( )

0 , 1

r r r rw r

r

exponential:

cubic spline:

quartic spline:

2( ), 1( )

0 , 1

re rw r

r

d is the radius of the support


Shape function

5N number of nodes

1x 3x 4x2x 5x

1 3 4 52

base functions2

2( ) 1, ( )

( ) 1, , ( )

, ( ) ( )j i i

j

ip x x x

p x

x x

x x MLS

x DRKM

3M

weight functions2 3 41 6 8 3 , 1

( )0 , 1

r r r rw r

r

ix x

rd

quartic spline:


Shape function

ixi 1 2 3 4 5

0 1 2 3 4

1 2 3 4

0.2

0.4

0.6

0.8

1

1 2 3 4

0.2

0.4

0.6

0.8

1

MLS DRKM

21 543, , ( )( ,( ) ( (,) )) xx xx x 2d


Shape function

1

( ) ( )N

i ii

x u x

ix

( ) 1iu x

i 1 2 3 4 5

0 1 2 3 4

1 1 1 1 1

0.5 1 1.5 2 2.5 3 3.5 4

0.25

0.5

0.75

1

1.25

1.5

1.75

2

( ) 1f x

MLS DRKM0.5 1 1.5 2 2.5 3 3.5 4

0.25

0.5

0.75

1

1.25

1.5

1.75

2


Shape function

1

( ) ( )N

i ii

x u x

ix

( )i iu x x

i 1 2 3 4 5

0 1 2 3 4

0 1 2 3 4

( )f x x

MLS DRKM0.5 1 1.5 2 2.5 3 3.5 4

0.5

1

1.5

2

2.5

3

3.5

4

0.5 1 1.5 2 2.5 3 3.5 4

0.5

1

1.5

2

2.5

3

3.5

4


Shape function

1

( ) ( )N

i ii

x u x

ix2( )i iu x x

i 1 2 3 4 5

0 1 2 3 4

0 1 4 9 16

2( )f x x

MLS DRKM0.5 1 1.5 2 2.5 3 3.5 4

2

4

6

8

10

12

14

16

0.5 1 1.5 2 2.5 3 3.5 4

2

4

6

8

10

12

14

16


Shape function

1

( ) ( )N

i ii

x u x

ix3( )i iu x x

i 1 2 3 4 5

0 1 2 3 4

0 1 8 27 67

3( )f x x

MLS DRKM0.5 1 1.5 2 2.5 3 3.5 4

10

20

30

40

50

60

0.5 1 1.5 2 2.5 3 3.5 4

10

20

30

40

50

60


Shape function (differential)

ixi 1 2 3 4 5

0 1 2 3 4

MLS DRKM

541 2 3 ( ( )( ) ( ), , , ,

) )( d xd d x

dx

d x

dx dx x

x

dx

d x

d

1 2 3 4

-2

-1

1

2

1 2 3 4

-2

-1

1

2


Shape function (2)

ixi 1 2 3 4 5

0 2 3 6 8

MLS DRKM

21 543, , ( )( ,( ) ( (,) )) xx xx x 5d

2 4 6 8

-0.2

0.2

0.4

0.6

0.8

1

2 4 6 8

-0.2

0.2

0.4

0.6

0.8

1


Conclusions

• The differential shape function can be easily obtained by using the DRKM.

• At least, there are M nodes in the influence domain.


The end

Thanks for your kind attentions

meshless method moving least-square method and differential reproducing kernel method

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