Deriving Shape Functions for
12-noded quartic serendipity element and verified two verification conditions
Reddaiah P
Professor of Mathematics, Global College
of Engineering and Technology, Kadapa, Andhra Pradesh, India.
*Corresponding Author
E-mail: reddaiah123@yahoo.co.in
ABSTRACT:
In this paper, I derived shape functions
for 12-noded quartic serendipity element by using natural Co-ordinate system
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 we use
Mathematica 4 Software [3].
KEY WORDS: Serendipity
quartic element, Natural Co-ordinate system, Shape functions, 12-noded, Mathematica
4 Software.
INTRODUCTION:
Serendiptiy element means when nodes are
located on the boundary only. In Heat and Mass Transfer problems serendipity
element is very very essential to analyze the behavior of heat and mass
transfer on the top of the boundary. Reddaiah et al analyzed Deriving vertices,
shape functions for Elliptic Duct Geometry and verified two verification
conditions [1].
GEOMETRICAL DESCREPTION:
Figure.1 12-noded
quartic rectangular serendipity element
There are 12 nodal values for u and 12
for v. Hence the displacement function is to be selected with only 12
constants. The polynomial has to maintain geometric isotropy also. This may be
obtained by dropping 
terms
from complete 4th order polynomial.
Thus
DERIVING SHAPE FUNCTIONS FOR 12-NODED
QUARTIC SERENDIPITY ELEMENT:
By using Mathematica Software calculate
Inverse of A.
Inverse of [A]//Matrix Form
Where
VERIFICATION:
(I) 1stConditon
Sum of all the shape functions is equal
to one
Output
(II) 2ndConditon
Each shape function has a value of one
at its own node and zero at the other nodes
CONCLUSIONS:
1. Derived Shape functions for quartic
12-noded rectangular serendipity 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.
Reddaiah
P, Prasada Rao DRV. Deriving vertices, shape functions for Elliptic Duct
Geometry and verified two verification conditions. International Journal of
Scientific & Engineering Research. May-2017; Volume 8: Issue 5.
2.
Bhavikatti
S.S. Finite Element Analysis. New Age International (P) Limited, Publishers.
2010; 2nd Edition.
3.
Mathematica
4 Software. Wolfram Research. 1988-2000; Version number 4.1.0.0.