1

MANE 4240 & CIVL 4240

Introduction to Finite Elements

Higher order elements

Reading assignment:

Lecture notes

Summary:

• Properties of shape functions

• Higher order elements in 1D

• Higher order triangular elements (using area coordinates)

• Higher order rectangular elements

Lagrange family

Serendipity family

Recall that the finite element shape functions need to satisfy the

following properties

1. Kronecker delta property

⎩

⎨

⎧

=nodesotherallat

inodeat

Ni0

1

...

2211 ++= uNuNu

Inside an element

At node 1, N1=1, N2=N3=…=0, hence

1

1uu node =

Facilitates the imposition of boundary conditions

2. Polynomial completeness

yyN

xxN

N

i

ii

i

ii

ii

=

=

=

∑

∑

∑

1

yxuIf 321

α

α

α

+

+

≈

Then

Docsity.com

2

Higher order elements in 1D

2-noded (linear) element:

12

x2

x1

12

1

2

21

2

1

xx

xx

N

xx

xx

N

−

−

=

−

−

=

In “local” coordinate system (shifted to center of element)

12

x

aa a

xa

N

a

xa

N

2

2

2

1

+

=

−

=

x

3-noded (quadratic) element:

12

x2

x1

()

(

)

()()

()()

()()

()()

()()

2313

21

3

3212

31

2

3121

32

1

xxxx

xxxx

N

xxxx

xxxx

N

xxxx

xxxx

N

−−

−−

=

−−

−−

=

−−

−

−

=

In “local” coordinate system (shifted to center of element)

12

x

aa

()

()

2

22

3

2

2

2

1

2

2

a

xa

N

a

xax

N

a

xax

N

−

=

+

=

−

−=

3

x3x

axxax

=

=−= 321 ;0;

3

4-noded (cubic) element:

12

x2

x1

()()()

()()()

()()()

()()()

()()()

()()()

()()()

()()()

342414

321

4

432313

421

3

423212

431

2

413121

432

1

xxxxxx

xxxxxx

N

xxxxxx

xxxxxx

N

xxxxxx

xxxxxx

N

xxxxxx

xxxxxx

N

−−−

−−−

=

−−−

−−−

=

−−−

−−−

=

−−−

−−−

=

In “local” coordinate system (shifted to center of element)

12

2a/3 x

aa

)3/)(3/)((

16

27

))(3/)((

16

27

)3/)(3/)((

16

9

)3/)(3/)((

16

9

3

4

3

3

3

2

3

1

axaxax

a

N

xaxaxa

a

N

axaxax

a

N

axaxax

a

N

+−−−=

+−−=

+−+=

+−−−=

3

x3x

x4

4

34

2a/3

2a/3

Polynomial completeness

#

4

3

2

1

x

x

x

x2 node; k=1; p=2

3 node; k=2; p=3

4 node; k=3; p=4

Convergence

rate (displacement)

1;

0+=≤− kpChuu p

h

Recall that the convergence in displacements

k=order of complete polynomial

Docsity.com

##### Document information

Uploaded by:
anuhya

Views: 1140

Downloads :
0

Address:
Engineering

University:
Shree Ram Swarup College of Engineering & Management

Subject:
Finite Element Method

Upload date:
07/05/2013