Deriving Shape Functions for Hexahedron Element by Lagrange Functions and Verified

(1)

Deriving Shape Functions for Hexahedron

Element by Lagrange Functions and Verified

P. Reddaiah#1

#_{Professor of Mathematics, Global College of Engineering and Technology, kadapa, Andhra Pradesh, India.}

Abstract — In this paper, I derived shape functions for hexahedron element by lagrange functions and also I verified two verification conditions for shape functions. First verification condition is sum of all the shape functions is equal to one and second verification condition is each shape function has a value of one at its own node and zero at the other nodes. For computational purpose I used Mathematica 9 Software [2].

Keywords - Hexahedron element, Lagrange functions, Shape functions.

I.INTRODUCTION

Interpolation functions derived using the dependent unknown-not its derivatives-at the nodes i.e., interpolation functions with C0 continuity are called the Lagrange family of interpolation functions or Shape functions[3].

II. GEOMETRICAL DESCREPTION

Hexahedron element with eight nodes is shown in figure.1 with axis   , , respectively.

Figure.1 Typical hexahedron element

III. DERIVING SHAPE FUNCTIONS FOR HEXAHEDRON ELEMENT

The typical element is shown in Figure.1

1(1,-1,-1), 2(1,1,-1), 3(1,1,1), 4(1,-1,1), 5(-1,-1,-1),6(-1,1,-1), 7(-1,1,1), 8(-1,-1,1)

0

the C continuity element in three dimensions

For

     

(1)

i i i i

N L  L  L 

i

L refers to Lagrangian function at node i.

where

Figure.1 there are 3 nodes in each direction. Hence n=3 in Lagrange function.

In

polynomial in three dimension is defined by Lagrange

























Node i, -axis node Node i, -axis node Node i, -axis node

Node i, -axis node Node i, -axis node

Node i, -axis node

( , , ) .

.

(2) For

i

i For

For

i For

For

i For

N 

 

 



    

   

     

 

 



1 1 1

, , Node 1(1,-1,-1)

For

  

i=1

5 5 5

2 2 2

4 4 4

, ,

Node 5(-1,-1,-1) and , ,

Node 2(1,1,-1) and , ,

axis

   

   

   



(2)



 

_



_













5 1

1 5

2 4

1 2 1 4

(2) , , .

. (3) N     

     

   



 



 







_

_{ }

 

_



_





_















 





 



1

1 1

1, 1, 1 . .

1 1

1 1 1 1

. .

1 1 1 1 2 2

N

 

   

  

  

   

   

 

    







1

1 1 1

(4) 8

N     

2 2 2

, , Node 2(1,1,-1)

For

  

i=2

6 6 6

1 1 1

3 3 3

, ,

Node 6(-1,1,-1) and , ,

Node 1(1,-1,-1) and , ,

Node 3(1,1,1) axis

axis

   

   

   



 



 

_



_













6 1

2 6

1 3

2 1 2 3

2 , , .

.

N     

     

   



 



 







_

_{ }

 

_





_

_{ }

 

_

















 

 



2

1 1

1,1, 1 . .

1 1 1 1

. .

1 1 1 1 1 1 2

N

 

   

   

 

   

   

 

    







2

1 1 1

(5) 8

N    



3 3 3

, , Node 3(1,1,1) For

  

i=3

7 7 7

4 4 4

2 2 2

, , Node 7(-1,1,1) and , ,

Node 4(1,-1,1) and , ,

Node 2(1,1,-1) axis

axis

   

   

   



 



 

_



_













7 3

3 7

4 2

3 4 3 2

2 , , .

.

N     

     

   



 



 







_

_{ }

 

_





_

_{ }

 

_



 





 

















3

1 1

1,1,1 . .

1 1 1 1

. .

1 1 1 1 1 1

1 1

N

 

   

   



   

    

 

  

 







3

1 1 1

(6) 8

N    

4 4 4

, , Node 4(1,-1,1) For

  

i=4

8 8 8

3 3 3

1 1 1

, ,

Node 8(-1,-1,1) and , ,

Node 3(1,1,1) and , ,

Node 1(1,-1,-1) axis

axis

   

   

   



(3)

 



 

_



_













8 4

4 8

3 1

4 3 4 1

2 , , .

.

N     

     

   



 



 







_

_{ }

 

_



_





_

 





 











 







4

1 1

1, 1, 1 . .

1 1

1 1 1 1

. .

1 1 2 1 1

1 1

N

 

   

  

  

   

    

 

  

 







4

1 1 1

(7) 8

N     



5 5 5

, , Node 5(-1,-1,-1) For

  

i=5

1 1 1

6 6 6

8 8 8

, ,

Node 1(1,-1,-1) and , ,

Node 6(-1,1,-1) and , ,

Node 8(-1,-1,1) axis

axis

   

   

   



 



 

_



_













1 5

5 1

6 8

5 6 5 8

2 , , .

.

N     

     

   



 



 



 

_



_

_





_











 



5

1 1

1, 1, 1 . .

1 1 1 1

. .

1 1 2 2 2

N

 

   

 

   

   

 

    







5

1 1 1

(8) 8

N    



6 6 6

, , Node 6(-1,1,-1) For

  

i=6

2 2 2

5 5 5

7 7 7

, , Node 2(1,1,-1) and , ,

Node 5(-1,-1,-1) and , ,

Node 7(-1,1,1) axis

axis

   

   

   



 



 

_



_













2 6

6 2

5 7

6 5 6 7

2 , , .

.

N     

     

   



 



 



 

_



_



_

_{ }

 

_













 



6

1 1

1, 1,1 . .

1 1 1 1

. .

1 1 2 2 2

N

 

   

  

  

   

   

 

   







6

1 1 1

(9) 8

N     

7 7 7

, , Node 7(-1,1,1) For

  

i=7

3 3 3

8 8 8

6 6 6

, ,

Node 3(1,1,1) and , ,

Node 8(-1,-1,1) and , ,

Node 6(-1,1,-1) axis

axis

   

   

   



(4)

 



 

_



_













3 7

7 3

8 6

7 8 8 6

2 , , .

.

N     

     

   



 



 



 

_



_



_

_{ }

 

_



 





 







 



7

6

1 1

1,1, 1 . .

1 1 1 1

. .

2 2 2

1 1

N

 

   

  

  

   

    

 

  







7

1 1 1

(10) 8

N     



8 8 8

, , Node 8(-1,-1,1) For

  

i=8

4 4 4

7 7 7

5 5 5

, ,

Node 4(1,-1,1) and , ,

Node 7(-1,1,1) and , ,

Node 5(-1,-1,-1) axis

axis

   

   

   



 



 

_



_













4 8

8 4

7 5

8 7 8 5

2 , , .

.

N     

     

   



 



 



 

_



_

_





_

 





 







 



8

1 1

1,1,1 . .

1 1 1 1

. .

2 2 2

1 1

N

 

   

 

 

   

    

 

 







8

1 1 1

(11) 8

N     

1 , 2 , 3 , 4 , 5 , 6 , 7 , 8

are called shape functions

N N N N N N N N

IV. VERIFICATION

(I) 1st Conditon

Sum of all the shape functions is equal to one 1+N +N +N + N +N +N +N =2 3 4 5 6 7 8

(5)+(6)+(7)+(8)+(9)+(10)+(11)+(12) N

Output

1+N +N + N +N + N +N +N =12 3 4 5 6 7 8

N

( ) 2 Condition

Each shape function has a value of one at its own node and zero at the other nodes.

nd II

Node 1 (1,-1,-1) :=1, :=-1, :=-1 At

  

1 2 3 N

N

4 5 6 N

N

7 8 N

N

Output 1 0 0 0 0 0 0 0

Node 2 (1,1,-1) :=1, :=1, :=-1 At

  

1 2 3 N

N

4 5 6 N

N

7 8 N

N

Output 0 1 0 0 0 0 0 0

Node 3 (1,1,1) :=1, :=1, :=1 At

  

1 2 3 N

N

4 5 6 N

N

Node 4 (1,-1,1) :=1, :=-1, :=1 At

  

1 2 3 N

N

4 5 6 N

N

(5)

7 8 N

N

Output 0 0 1 0 0 0 0 0

7 8 N

N

Output 0 0 0 1 0 0 0 0

Node 5 (-1,-1,-1) :=-1, :=-1, :=-1 At

  

1 2 3 N

N

4 5 6 N

N

7 8 N

N

Output 0 0 0 0 1 0 0 0

Node 6 (-1,1,-1) :=-1, :=1, :=-1 At

  

1 2 3 N

N

4 5 6 N

N

7 8 N

N

Output 0 0 0 0 0 1 0 0

Node 7 (-1,1,1) :=-1, :=1, :=1 At

  

1 2 3 N

N

4 5 N

N

Node 8 (-1,-1,1) :=-1, :=-1, :=1 At

  

1 2 3 N

N

7 8 N

N

Output 0 0 0 0 0 0 1 0

4 5 6 N

N

7 8 N

N

Output 0 0 0 0 0 0 0 1

V. CONCLUSIONS

1. Derived Shape functions for hexahedron element. 2. Verified sum of all the shape functions is equal to one

3. Verified each shape function has a value of one at its own node and zero at the other nodes.

REFERENCES

[1] S.S. Bhavikatti, Finite Element Analysis, New Age International (P) Limited, Publishers, 2

nd

Edition, 2010. [2] Mathematica 9 Software, Wolfram Research, Version

number 9.0.0.0, 1988-2012.