Shape Functions 3-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Download Shape Functions 3-Finite Element Method-Assignment Solution and more Exercises Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

ASSIGNMENT # 4

FINITE ELEMENT METHOD

CHAPTER 4

Question No 4.

Consider the shape functions described in Eq. (4.10) for a one-dimensional cubic

element. Show that the shape function corresponding to a particular node i, N, (x),

has a value of one at node i and zero at the other three nodes j, k, and 1. Repeat

the procedure for the shape functions Nj(x), Nk(x), and N1 (x).

Solution:

The Shape Function for one dimensional cubic element at node (^) i is given by

i

x x x N x l l l

At x  0 value of shape function is

(at node ) (1 0)(1 0)(1 0) 1 i

N i     

At

3

l x  ;

3 3(^3 ) 3

(at node ) (1 3 )(1 )(1 ) 0 2

l l l Ni j l l l

At

l x  ;

2 2 2 3 3(^3 ) 3 (at node ) (1 3 )(1 )(1 ) 0 2

l l l Ni k l l l

At x  l ;

(at node ) (1 3 )(1 )(1 ) 0 2

i

l l l N l l l l

The Shape Function for one dimensional cubic element at node k is given by

At x  0 ;

(at node ) ( )(1 0)(1 0) 0 2

N k i l

At

3

l x  ;

(at node ) ( )(1 )(1 ) 0 2

l l l

Nk j l l l

At

l x  ;

2 2 2 9 3 3( 3 ) 3 (at node ) ( )(1 )(1 ) 1 2

l l l

Nk k l l l

At x  l ;

(at node ) ( )(1 )(1 ) 0 2

k

l l l N l l l l

0 at Node ( =0)

0 at Node ( = ) 3

1 at Node ( = ) 3

0 at Node ( = )

k

i x

l j x

N

l k x

l x l

The Shape Function for one dimensional cubic element at node l is given by

l

x x x N x l l l

At x  0 ;

Nl (at node ) i ( )(1 0)(1 0) 0 l

At

3

l x  ;

3 3(^3 )^ 3(^3 )

(at node ) ( )(1 )(1 ) 0 2

l l l

Nl j l l l

At

l x  ;

2 2 2 3 3(^3 )^ 3(^3 ) (at node ) ( )(1 )(1 ) 0 2

l l l

Nl k l l l

At x  l ;

(at node ) ( )(1 )(1 ) 1 2

l

l l l N l l l l

0 at Node ( =0)

0 at Node ( = ) 3

0 at Node ( = ) 3

1 at Node ( = )

l

i x

l j x

N

l k x

l x l