Jacobian Matrix 1-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

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Assignment of

Chapter#4(Question#25)

FINITE ELEMENT

METHOD

Problem#4.25: Determine the Jacobian matrix for the quadratic isoparametric triangular element shown in Figure 4.

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Solution:

From “Applied Finite Element Analysis for Engineers” by Frank L. Stasa we have;

 

1 2 6 1 2 6

:

x x

x x N y y y

y

  1 2 6 1 2 6

1 2 1 1 1 2 2 2

Where N N N N N N N And N N L L N L L N L L

 

^ ^ 

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(^41) 2 (^52) 1 (^52 1 2 1 ) 2 (^61 1 2 1 ) 1 (^61) 2 6 1 1 2 1 2 1 1 2

j j j

N L

L

N L

L

N L L L L L

L

N L L L L L

L

N L

L

N

x L L L L L L L





6 1 1 2 2 1 2 1 1 2

j j j

N

y L L L L L L L L





 

6 2 1 1 2 1 (^1 ) 2 1 6 1 2 1 1 2 1 2 2 1

1 2 1 2

j j j

j j j

N

x L L L L L L L L N y L L L L L L L L So the required Jacobian Matrix is

L L L L

J









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