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Shape Function of 2D Simplex Element 1-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Aligarh Muslim UniversityMathematical Methods for Numerical Analysis and Optimization

This assignment solution was submitted to Amar Sharma for Finite Element Method course at Aligarh Muslim University. It includes: Determination, Shape, Function, Simplex, Element, Interpolation, Model, Node, Adjoint, Coefficicents

Typology: Exercises

2011/2012

Uploaded on 07/08/2012

ramu
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Problem No 3.15
Finite Element Methods
problem no 3.15
determinatin of shape function of 2D simplex element .
A 2D simplex element is a triangle for which the interpolation model is given by:
1 2 3
,)
1 x y
1 x y
(At node i) 1 x y
(At node j) 1 x y
(At node k)
Tii
Tjj
kk
T
x y x y
















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Download Shape Function of 2D Simplex Element 1-Finite Element Method-Assignment Solution and more Exercises Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

Problem No 3.

Finite Element Methods

problem no 3.

determinatin of shape function of 2D simplex element.

A 2D simplex element is a triangle for which the interpolation model is given by:

 ^  

1 x y

1 x y (At node i) 1 x y (At node j) 1 x y (At node k)

T i i

T (^) j j

T k^ k

x y   xy

 

  ^ 

  ^ 

 ^  ^ 

  ^ 

  ^ 

  ^ 

1

1

Now we find out as follow:

1 x y 1 x y

1 x y 1 1 x y 2 or A= 1 x y 2 1 x y

For adjoint first

i i i i

j j j j

k k k k

Adj

A

  ^ 

 ^ ^  ^ 

11 12

13 21

22 23

find coefficients of

x y^1 y det , det x y (^1) y

1 x (^) x y det , det 1 x x^ y

1 y det 1 y

i i^ j i i k k (^) k

j (^) i i i j k k^ k

i j k

A a A b

A c A a

A b A

  ^ 

  ^ 

31 32

33

1 x det 1 x

x y 1 y det det x y 1 y

1 x det 1 x

i j k

i i i k k j j j

i k j

c

A a A b

A c

j k

j k 1

c c

c c

i

i j k

i j k

i

i j k

i j k

a a a

Adj b b b

c

a a a

b b b A c