2 d Finite Elements

7/29/2019 2 d Finite Elements

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CCOORRNNEELLLLU N I V E R S I T Y 1MAE 4700

FE Analys is for Mechanical & Aerosp ace Design

N. Zabaras (10/18/2012)

Fini te Element Analysis forMechanical and Aerospace Design

Prof. Nicholas Zabaras

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

101 Frank H. T. Rhodes Hall

Cornell University

Ithaca, NY 14853-3801

Email: [email protected]

URL: http://mpdc.mae.cornell.edu/
mailto:[email protected]://mpdc.mae.cornell.edu/http://mpdc.mae.cornell.edu/mailto:[email protected]


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N. Zabaras (10/18/2012)

Isoparametr ic f in i te elements

The basis functions used in the definition of the

mapping Te, do not have to be the same as those

used for the approximation of functions.

Let Mbe the number of basis functions used to define

Te and let Ne be the number of basis functions

(nodes) used in the approximation of functions. Polynomials used to define geometry can be of higher

order (M>Ne), equal (M=Ne) or lower (M


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N. Zabaras (10/18/2012)

Sub -, iso - and super-parametr ic f in i te elements


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N. Zabaras (10/18/2012)

Quadri lateral elements : B i-l inear

We use tensor product of

polynomials as discussed in

1D (Lagrange family)

1

2

3

4

1( , ) (1 )(1 )

4

1( , ) (1 )(1 )4

1( , ) (1 )(1 )

4

1( , ) (1 )(1 )

4

N

N

N

N


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N. Zabaras (10/18/2012)

Quadr i lateral elemen ts: B i-quadrat ic

We use tensor product of

polynomials as discussed in

1D (Lagrange family)

Note that here we have one

internal node (9).

2 2 2 21 5

2 2 2 22 6

2 2 2 23 7

2 2 2 24 8

2 29

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

4 2

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

4 21 1

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

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

4 2

( , ) (1 )(1 )

N N

N N

N N

N N

N


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N. Zabaras (10/18/2012)

Quadrat ic eigh t node element

These type of elements

are not derived from

tensor product of 1D

polynomials. They are

called serendipity

elements.

To derive the shape

function for node 1, we

need a polynomial that

vanishes on thefollowing lines:

1 ,1 ,1

21 5

22 6

23 7

24 8

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

4 2

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

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

4 2

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

4 2

N N

N N

N N

N N


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N. Zabaras (10/18/2012)

Quadratu re ru les

Quadrature rules are defined from the 1D Gauss rules

presented earlier as follows:

Here we re-labeled

(m,n) with a single

index

'

1 11 1

1 1 1

( , ) ( , )

( , ) ( , )i i l

N N N

n m n m l l l m n l

G d d G d d

G w w G w

1l


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N. Zabaras (10/18/2012)

Triangu lar elemen ts

We first consider triangular with straight sides. We

consider the mapping from a right-isosceles master

triangle. By inspection, we can write the basis

functions as:

The coordinate mapping is then defined from:

1

2

3

( , ) 1

( , )

( , )

N

N

N

3

1

3

1

( , )

( , ) ,

jjj

jjj

x x N

y y N


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N. Zabaras (10/18/2012)

Triangu lar elemen ts

Inverting this mapping gives:

Can you recognize these as the linear shape functions of

the 4 node quadrilateral element? (take nodes )

1

2

3

( , ) 1

( , )

( , )

N

N

N

3

1

3

1

( , )

( , ) ,

jjj

jjj

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 yA

y y x x x x y yA

2

3

1

( , )

( , )

1 ( , )

e

e

e

N x y

N x y

N x y

Using these, one can now easily compute

and thus the element stiffness and load

, , , ,| |Jx y x y

3 4


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N. Zabaras (10/18/2012)

Area coo rdinates

The expressions for can

easily be interpreted as ratios of

areas. We will see this interpretation

to be useful in deriving higher order

triangular elements.

Let us join the points and (x,y)with the vertices of the triangles

and respectively. We denote

as the areas of the subtriangles

opposite node in and We define the area coordinates on

as: where is the

area of the master element.

, ,1

( , )

,e

,i ia a

,,e

, 1,2,3,iia i

A 1/ 2A

respectively.

1

2

3

1


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N. Zabaras (10/18/2012)

Area coo rdinates

Since |J| is constant (the ratio of the

areas of and ), the map Te

transforms areas uniformly, thus:

This is only true for triangles withstraight sides.

1

2

3

1

,e ,

, 1,2,3,i i

ie

aai

AA


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N. Zabaras (10/18/2012)

Area coo rdinates

tani cons t

0, 1,2,3.i i

Several interesting properties

of area coordinates are

shown in the figures.

At a given point, the line

is parallel to the

side of the elementopposite node i.

The boundary segments of

the element are defined by

The vertices of the triangle

are (1,0,0), (0,1,0) and

(0,0,1).


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FE Analys is for Mechanical & Aerosp ace DesignN. Zabaras (10/18/2012)

Higher-degree shape functions

The area coordinates on can be used to

determine higher degree shape functionsi ,

( ).i

Quadratic shape

functions

Cubic shape

functions

1 41 1 1 2

2 52 2 2 3

3 63 3 3 1

12 ( ), 4

2

12 ( ), 4

2

12 ( ), 4

2

N N

N N

N N

1 1 1 1

4 1 2 1

5 1 2 2

10 1 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 func t ions on tr iangles using area coord inates

The coordinate transformation is now having the form:

The calculations here defer from those used in

quadrilaterals because of the redundant areacoordinate

Calculation of derivatives proceeds as:

Alternatively, one can use and

proceed exactly as was done before for quadrilaterals.

1 2 3 1 2 31

1 2 3 1 2 31

( , ) ( , , ) ( , , )

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

e

e

N

jjj

N

jjj

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 integrat ion form ulas for tr iang les

We use particular quadrature rules appropriate for the

area coordinates introduced earlier.int

1 2 3 2 3 1 2 3 1 2 31

intint s in

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

l l l l l

egrationquadratureweightspo

G d d note G w

Linear Quadratic CubicPolynomials

up to thisdegree are

integrated

exactly

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2 d Finite Elements