fem zabaras finiteelementapproximationsfor2dbvp

8/9/2019 FEM Zabaras FiniteElementApproximationsFor2DBVP

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MAE 4700 FE Analysis for Mechanical & Aerospace DesignN. 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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Finite element approximation of 2D BVP

We approximate the domain with Efinite elements and

Nnodes placing nodes and elements in such a waythat element boundaries coincide as close as possible

to the interface with jump in k.


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Finite element approximation of 2D BVP

We define an N-dimensional subspace Hhof H1(h) by

constructing appropriate global basis functions

using the elements we discussed earlier.

A typical test function in Hh is of the form:

In general the essential BC data (Dirichlet data)

is approximated as:

where the sum is over all nodes on

, 1,2,..,i

N i N =

1

( , )

( , ) ( , )

h j j

N

h j j

jw x y

w x y w N x y=

=

1

u in

( ) ( ( ), ( ))h j jj

u s u N x s y s=

1.

h


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FEM problem statement

Find a function such that

and at the nodes on so the following holds:

The above equation leads to the following system ofalgebraic equations:

,h

hu H

2 2

1

( , ) ( , ) ( , )

0 .

h h h h

h h h hh h h h h h

h

h h h

u w u wk b x y u x y w x y dxdy pu w ds f w dxdy w ds

x x y y

for all w H with w on

+ + + = +

=

1

( , ) ( , )N

h j j

j

u x y u N x y=

=

jju u= 1h

1

, 1, 2,..,N

ij j i

j

K u F i N =

= =

2

( , ) , , 1,...,

h h

j ji iij i j i j

N NN NK k b x y N N dxdy pN N ds i j N

x x y y

= + + + =

2h h

i i iF fN dxdy N ds

= +

Symmetric,banded


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Finite element approximation

Each of the integrals in the stiffness and load vector can becomputed as the sum of contributions from each element inthe mesh. But we need to approach this more carefully!

Let denote a typical finite element. The exact solution uon of our BVP satisfies:

Let and denote the restrictions of the approximations

and to . Then the local approximation of thevariational BVP over is:

e

e

( , ),.

e e e

n

n e

k u w buw dxdy f wdxdy wds

for all admissible functions w x ywhere is the normal component of the flux at

+ =

e

hu

e

hw

hu hw e

e

( )

e e e

e

e e e e e e

h h h h h n h

exact flux on

not known

k u w bu w dxdy f w dxdy w ds

+ =


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Finite element approximation

Since , there will be no contribution in the last

integral from elements with sides that coincide with We already have in place the following approximations:

where are the shape functions in and thenumber of nodes in .

The following linear system is then obtained:

1 .h

10h hw on=

eN

e

e

( )e e e

e

e e e e e e

h h h h h n h

exact flux onnot known

k u w bu w dxdy f w dxdy w ds

+ =

1 1

( ) ( , ), ( ) ( , )e eN N

e e e e e e

h i i h j j

i j

w x w N x y u x u N x y= =

= =

( , )

e

jN x y


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Assembly process

The global system of equations is obtained by summing

over all elements Ein the mesh.

We expand the element stiffness to a matrix NxNwith zeros everywhere except those rows and columnscorresponding to nodes within and and

will be expanded to Nx1 vectors and with nonzeroentries only in those rows corresponding to nodes in

e

e

eF e

ef e

ek

1 1

, , 1, 2,...,

e

E Ej j ei i

i j ij

e e

N NN Nk bN N dxdy K i j N

x x y y= =

+ + = =

1 1

, 1,2,...,

e

E Ee

i i

e e

fN dxdy F i N = =

= =

( )1

0, 1, 2,...,E

e e e

ij j i i

e

K u F i N

=

+ = =

eK


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Boundary conditions

Note that the contributions to Kijand Fifrom boundary

conditions must enter the problem through the terms .

We note that the sum of the contour integrals can bewritten as:

( )1

0, 1, 2,...,

E

e e eij j i i

e

K u F i N =

+ = =

ei

(0) (1) ( 2)

1

, 1, 2,...,E

e

i i i i

e

S S S i N =

= + + =(0)

1e h

E

i n i

eS N ds

= = 1(1)

1h

E

i n i

eS N ds

= = 2

(2)

1h

E

i n ie

S N ds=

=

e h Portion of the boundaryof not on

(interelement boundaries)

e

e h


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Boundary conditions

This vector is defined only at interior nodes. Consider

the patch of 4 elements sharing node 1.

From the conservation lawacross an interface

where no point or line sources

are applied. Thus if fis smooth

in the patch shown

(0)

1e h

E

i n ie

S N ds=

=

1 2 3 4

(0)

1 1 1 1 1

1e

E

i n n n n n

e

S N ds N ds N ds N ds N ds=

= = + +

0n =

(0)0

iS =


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Boundary conditions

When the source fcontains a line source or

concentrated point source, then is equal to theintensity of the line source.

We can include point sources by writing f(x,y) as

We assume that the mesh is constructed so that there

is a node at the source location.

(0)

1e h

E

i n ie

S N ds=

=

n

int ( , )

( , ) ( , ) ( , )

i i h

i i

smooth part po source at x y

f x y f x y f x x y y

= +


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Boundary conditions

For source at node 1 in the figure, we have:

(0)

1e h

E

i n ie

S N ds= =

int ( , )

( , ) ( , ) ( , )

i i h

i ismooth part po source at x y

f x y f x y f x x y y

= +

4 4(0)

1

1 1

1

e h m

i n i n

e m

weighted average of the jumpsat node

S N ds N ds= =

= =

(0)i

S f=

The presence of sources

leads to very singularsolutions u.


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Boundary conditions

The values of uh

are prescribed at . Since isnot known on , cannot be described here.

However, once all nodal u1,u2,uNare computed, we

can evaluate from

1

(1)

1h

E

i n ie

S N ds= =

( )1

0, 1, 2,...,E

e e e

ij j i i

e

K u F i N =

+ = =

1h

n

1h (1)

iS

(1)

iS


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Boundary conditions

On the natural boundary condition is prescribed.

There we set

2

(2)

1h

E

i n ieS N ds=

=

2 h

( ) ( ) ( ) ( )n hs p s u s s=

2

(2)

1 1 1h

E N N

i j j i ij j i

e j j

S p u N N ds P u

= = =

= =

2 2

1 1eh h

E Ee

i i i i

e e

N ds N ds= =

= = =

2 2

int sec 2

1 1eh h

eportion of

er ting h

E Ee

ij i j i j ij

e e

P pN N ds pN N ds P

= =

= = =


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Final algebraic equations

The final system of equations is given as:

Once boundary conditions are imposed on we can

proceed solving the system of equations for the

unknown nodal values.

(1)

1

1 1

, 1, 2,...,

( ), ( )

N

ij j i i

j

E Ee e e e

ij ij ij i i i

e e

K u F S i N

K K P F F

=

= =

= =

= + = +

1h


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Example

Consider the FEM

solution of the followingproblem:

( , ) ( , )u x y f x y in =

410u on=

12 25, 67 74,0 , ,u onn

=

56

u

u onn

+ =

2 12 25 56 67 74, =

1 41 =

1 1h =

2 2h =


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Example

Element 1:

2 12 25 56 67 74, =

1 41 =

11

12

13

1

00

0

0

f

f

fF

=

1 1 111 12 13

1 1 121 22 23

1 1 131 32 33

1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

k k k

k k k

k k k

K

=

1 1h =

2 2h =


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Example

Element 2:

2 12 25 56 67 74, =

1 41 =

2

1

22

22

3

0

0

0

0

f

f

Ff

=

2 2 211 12 13

2 2 221 22 23

22 2 231 32 33

0 0 0 0

0 0 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

k k k

k k k

K k k k

=

1 1h =

2 2h =


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Example

Element 6:

2 12 25 56 67 74, =

1 41 =

61

1

62

63

00

0

0

f

Ff

f

=

6 6 611 12 13

6

6 6 621 22 23

6 6 631 32 33

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0 0

k k k

K

k k k

k k k

=

1 1h =

2 2h =


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Example problem: Assembly

1 1

2

3

14

4

5 5

66

7

0

0

0

FF

F

FF

F

F

F

=

11 12 13 14

21 22 23 25

31 32 33 34 35 36 37

41 43 44 47

55 5652 53

65 6663 67

73 74 76 77

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

K K K K

K K K K

K K K K K K K

K K K K K

K K K K

K K K K

K K K K

=

1 1h =

2 2h =

The stiffness terms with the symbolwill be modified once natural BC are applied.

K


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Natural boundary conditions

5

6

0

0

0

0

0

=

55 56

65 66

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

P

P P

P P

=


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Revisiting the FEM formulation for 2D scalar field problems

In the earlier part of this lecture, we used the stiffness

matrix and load vectors in component forms.

We will next discuss a similar example (notation is

used from heat conduction) repeating the samecalculations but in a matrix form. These calculationshave been programmed in the 2dBVP MatLab

software. Let us consider the following

2D BVP: Compute T(x,y):

( ) ( , )

T

q

D T f x y

T T on

q D T n q on

=

=

= =

D H d i k f


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2D Heat conduction weak form

The weak statement for this problem takes the form:

We denote here

The weak form is written in a matrix form ready for

FEM discretization, e.g.

Find a function with such that:

{ } { }

0

[ ]

( , )

q

T T Tw D T d w f d w qd

for all w x y U

=

1( , ) ( ),T x y H ( ) ( ), TT s T s s=

1

0: ( ) 0

TU w H with w on =

{ } { }

{ }

, , [ ] , [ ]T x xx xy

y yx yyConductivityheatmatrixflux

T q k kw wxT w q D T D

T q k kx y

y

= = = = =


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2D H d i FE i l i


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2D Heat conduction FE interpolation

The gradient fields of Teand weare obtained as:

Similarly for the gradient of we:

{ }

11 2 1 2

1 2

1 2 1 2

1 2

... ...

... ...

e

e ee e e eee e enen nen

nen

e

e e ee e e ee e enen nen

nen

The B matrix

TN N N N N N TT T T

x x x x x x xT

T N N N N N N T T T

y y y y y y y

+ + + = = =

+ + +

{ }

{ }

{ }

2( , )

...

e

e

e

e e e e

globalscatternodale matrixtemperaturesnen

d

T B x y d B L d

T

= =

{ } { }{ }

{ }

1 1

2 2

1 2( , ) ..... ...Te

e e

e e

e eT T T T TTe e e e e e e e

nen

scatterw matrix

e e

nen nen

N N

x y

N Nw w

w w B x y w w w w L Bx yx y

N N

x y

= = = =

Here, nen: number element nodes

2D H t d ti k f


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We can substitute these expressions into the weak

form

Note as we did earlier for the 1D BVP, the vector wwas

partitioned as follows:

{ } { }1 1 1

[ ]T T

e e eq

nel nel nelT

e e e e e

e e e

w D T d w f d w qd = = =

=

{ } { } { }1

[ ] 0e e e

q

nelT T T T T e e e e e e e

F

e

w L B D B d L d N f d N qd d w=

+ =

{ } { } 0 0,E E

F FF

wdd ww wd

= = = =

2D H t d ti k f


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From this equation, we can easily identify:

We now can write the weak form as:

{ } { } { }1

[ ] 0e e e

q

nelT T T T T e e e e e e e

F

ew L B D B d L d N f d N qd w

=

+ =

[ ]e

Te e e e

K B D B d

= { }

{ } { }

e eq

ee

T Te e e

f f

f N f d N qd

=

{ } { }

{ }

{ }1 1

Re

0nel nel

T TT e e e e e

F

e e

sidual r

w L K L d L f w= =

=


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fem zabaras finiteelementapproximationsfor2dbvp

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