FEM Zabaras 2DFiniteElements

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CC OO RR NN EE LL LL U N I V E R S I T Y 1

MAE 4700 FE Analysis for Mechanical & Aerospace Design N. Zabaras (10/06/2009)

MAE4700/5700Finite Element Analysis for

Mechanical and Aerospace DesignCornell University, Fall 2009

Nicholas ZabarasMaterials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering101 Rhodes Hall

Cornell UniversityIthaca, NY 14853-3801
http://mpdc.mae.cornell.edu/Courses/MAE4700/MAE4700.htmlhttp://mpdc.mae.cornell.edu/http://mpdc.mae.cornell.edu/http://mpdc.mae.cornell.edu/Courses/MAE4700/MAE4700.html


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Isoparametric finite elements

The basis functions used in the definition of themapping T e , do not have to be the same as thoseused for the approximation of functions.

Let M be the number of basis functions used to defineT e and let N e be the number of basis functions

(nodes) used in the approximation of functions.

Polynomials used to define geometry can be of higherorder ( M>N e ), equal ( M=N e ) or lower ( M


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Sub-, iso- and super-parametric finite elements


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Quadrilateral elements: Bi-quadratic

We use tensor product ofpolynomials as discussed in1D (Lagrange family)

Note that here we have oneinternal node (9).

2 2 2 21 5

2 2 2 22 6

2 2 2 2

3 7

2 2 2 24 8

2 29

1 1( , ) ( )( ), ( , ) (1 )( )4 21 1

( , ) ( )( ), ( , ) ( )(1 )4 21 1

( , ) ( )( ), ( , ) (1 )( )4 21 1

( , ) ( )( ), ( , ) ( )(1 )4 2

( , ) (1 )(1 )

N N

N N

N N

N N

N

= =

= + = +

= + + = +

= + =

=


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Quadratic eight node element

These type of elementsare not derived fromtensor product of 1Dpolynomials. They arecalled serendipityelements.

To derive the shapefunction for node 1, weneed a polynomial that

vanishes on thefollowing lines:1 ,1 ,1 + +

21 5

22 6

23 7

2

4 8

1 1( , ) (1 )(1 )( 1 ), ( , ) (1 )(1 )

4 2

1 1( , ) (1 )(1 )( 1 ), ( , ) (1 )(1 )4 21 1

( , ) (1 )(1 )( 1 ), ( , ) (1 )(1 )4 21 1

( , ) (1 )(1 )( 1 ), ( , ) (1 )(1 )4 2

N N

N N

N N

N N

= =

= + + = +

= + + + + = +

= + + =


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Quadrature rules

Quadrature rules are defined from the 1D Gauss rulespresented earlier as follows:

Here we re-labeled(m,n) with a singleindex

( )'

1 11 1

1 1 1

( , ) ( , )

( , ) ( , )i i l N N N

n m n m l l lm n l

G d d G d d

G w w G w

= = =

=

= =

1l=


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Triangular elements

We first consider triangular with straight sides . Weconsider the mapping from a right-isosceles mastertriangle. By inspection, we can write the basisfunctions as:

The coordinate mapping is then defined from:

1

2

3

( , ) 1

( , )

( , )

N

N

N

=

=

=

3

1

3

1

( , )

( , ) ,

j j j

j j

j

x x N

y y N

=

=

=

=


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Triangular elements

Inverting this mapping gives:

Can you recognize these as the linear shape functions ofthe 4 node quadrilateral element? (take nodes )

1

2

3

( , ) 1( , )

( , )

N N

N

= =

=

3

1

3

1

( , )

( , ) ,

j j j

j j j

x x N

y y N

=

=

=

=

{ }

{ }

3 1 1 3 1 1

2 1 1 2 1 1

1( )( ) ( )( )

2

1 ( )( ) ( )( )2

e

e

e

areaof

y y x x x x y y A

y y x x x x y y A

=

= +

2

3

1

( , )

( , )

1 ( , )

e

e

e

N x y

N x y

N x y

=

=

=

Using these, one can now easily computeand thus the element stiffness and load

, , , ,| | J x y x y

3 4


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Area coordinates

The expressions for caneasily be interpreted as ratios ofareas . We will see this interpretationto be useful in deriving higher ordertriangular elements.

Let us join the points and (x,y)

with the vertices of the trianglesand respectively. We denoteas the areas of the subtriangles

opposite node in and We define the area coordinates onas: where is the

area of the master element.

, ,1

( , )

,e,i ia a

,,e

, 1,2,3,iia i= = 1/ 2=

respectively.1

2

3

1

= ==


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Area coordinates

Since |J| is constant (the ratio of theareas of and ), the map T e transforms areas uniformly, thus:

This is only true for triangles with

straight sides.

1

2

3

1

= ==

,e ,

, 1,2,3,i iie

aa i A A

= = =


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Area coordinates

tani cons t =

0, 1,2,3.i i= =

Several interesting propertiesof area coordinates areshown in the figures.

At a given point, the lineis parallel to theside of the element

opposite node i. The boundary segments ofthe element are defined by

The vertices of the triangleare (1,0,0), (0,1,0) and

(0,0,1).


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Higher-degree shape functions

The area coordinates on can be used todetermine higher degree shape functionsi

,( ).i

Quadratic shapefunctions

Cubic shapefunctions

1 41 1 1 2

2 52 2 2 3

3 63 3 3 1

12 ( ), 42

12 ( ), 4

21

2 ( ), 4

2

N N

N N

N N

= =

= =

= =

1 1 1 1

4 1 2 1

3 1 2 2

101 2 3

9 2 1( )( )

2 3 3

27 1( )2 3

27 1( )

2 3

27

N

N

N

N

=

=

=

=


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Shape functions on triangles using area coordinates

The coordinate transformation is now having the form:

The calculations here defer from those used inquadrilaterals because of the redundant areacoordinate

Calculation of derivatives proceeds as:

Alternatively, one can use andproceed exactly as was done before for quadrilaterals.

1 2 3 1 2 31

1 2 3 1 2 31

( , ) ( , , ) ( , , )

( , ) ( , , ) ( , , ),

e

e

N j j

j

N j j

j

x x x N

y y y N

=

=

=

=

1 2 31 . =

1 1 1 1 31 2

1 2 3

N N N N x x x x

= + +

1 2 3 2 31 , , , =


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Quadrature integration formulas for triangles

We use particular quadrature rules appropriate for thearea coordinates introduced earlier.

int

1 2 3 2 3 1 2 3 1 2 31

int

int s in

( , , ) ( 1 ) ( , , ) N

l l l ll

egrationquadratureweights po

G d d note G w =

= =

Linear Quadratic Cubic Polynomialsup to thisdegree are

integratedexactly

FEM Zabaras 2DFiniteElements

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