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Jacobian Matrix 2-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: Determine, Jacobian, Matrix, Quadrilateral, Element, Function, Natural, Coordinates, Derivative

Typology: Exercises

2011/2012

Uploaded on 07/08/2012

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Assignment#04
Finite Element Methods
Problem 4.4
Determine the jacobian matrix for the quadrilateral element defined in Problem
4.3. Evaluate the jacobian matrix at the point (r, s) = (0.5.0.5).
Solution
We know that if
is a function of the natural coordinate’s r & s, its derivative
w.r.t x & y coordinates can be obtained by jacobian matrix, so
Or
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pf2

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Download Jacobian Matrix 2-Finite Element Method-Assignment Solution and more Exercises Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

Assignment#

Finite Element Methods

Problem 4.

Determine the jacobian matrix for the quadrilateral element defined in Problem

4.3. Evaluate the jacobian matrix at the point (r, s) = (0.5.0.5).

Solution

We know that if is a function of the natural coordinate’s r & s, its derivative

w.r.t x & y coordinates can be obtained by jacobian matrix, so

Or

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 

       

       

1 1

2 2

3 3

4 4

1 1 1 1 1

4 1 1 1 1

x y

s s s s x y

J

x y r r r r

x y

 

 

       

 

  

           

 

 

Substituting values of x 1

to x 4

and y 1

to y 4

we get

0 1

1 1 1 1 5 3 1

4 2 4 1 1 1 1

2 3

s s s s

J

r r r r

  

 

       

 

  

       

   

 

 

Solving we get

s s s s s s s

J

r r r r r r r

Or

s s
J
r r

Putting the given points (r,s)=(0.5,0.5) we get

J

Answer

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