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FINITE ELEMENT METHOD


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Objective

The purpose of the present course is to describe the use of

Finite Element method for the solution of problems of fluid

flow or seepage through porous media.


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What is FEM

Powerful computational technique for the solution ofdifferential equtions.

too complicated to be solved satisfactorily by classical

analytical methods.

Problems-Solid mechanics, Heat transfer, fluid mechanics,acoustics, electromagnetism, coupled interaction of thesephenomena

Commercial packages


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Course coverage

Finite element Basic concepts

Formulation of Finite elements

Steps in the finite element method

Examples


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WHY FEM ?


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Simple Example

Simple 1-D problem

fdx

du

dx

d=

0 1x on

Boundary conditions

0)0( uu = (1)du qdx

=

( Dirichlet ) ( Neumann )

:

0 :

1 :

Strong form


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Analytical solution

2

02

q f fu u x x

= + +

d duf

dx dx =

Integrating both sides

2

1 22

f xu C x C = + +

Imposing the boundary conditions we get

1 2 0C q f C u= =


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Finite difference formulation

is approximated in the mid-point of the intervaldx

du

1

1/2

l l

l

u udu

dx x

+

+

1

1/2

l l

l

u udu

dx x

1/2 1 1/2 1

2

( ) ( )l l l l l l

l

u u u uf

x

+ + =

Finite difference approximation

llll f

x

uuu=

+ +

2

11 2

For constant

1..l L=

0 0x = 1x lx 1Lx =1lx +x

1 1( , ) ( , )

l l l lx x and x x by+


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Neumann boundary condition

1/2 1 1/2 1( ) ( )1 1

2 2

L L L L L Lu u u u

qx x

+ + + =

Discretisation at the end l=L

1/2 1 1/2 1

2

( ) ( )L L L L L L

L

u u u uf

x

+ + =

Combining

11/2

1

2

L LL L

u uq f x

x

=


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Discrete problem-algebraic equations

KU F=

2 1

1 2 1

...

1 2 11 1

K

=

2

0 1

2

2

2

1

2

2

L

xu f

xf

F

x

f

x xq f

=

M

where,

1

2

1L

L

u

u

U

uu

=

M


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Finite difference -solution

2

01

221

2 1

1

2

xu f

u

u x xq f

=

For a three noded mesh

1u

0u 1u 2u


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Grid is structured .

Equation is discretized.

Proper treatment of boundary conditions is vitally important.

Gives solution at only discrete points in the domain grid

points.


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FEM- Building blocks

Form of the approximate solution is assumed apriori

Approximate solution0

0 1

N N

k k k k

k k

u u u = =

= = +

on substitution leaves a residual

d dur f

dx dx

=

Optimizing criterion- the weighted integral of the residual is zero.

0lw r= 1, 2...,l n=


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Trial solution procedure

Trial solution u

uis undetermined

Optimizing criterion

Determines best values of ui

Approximate solution

Accuracy

Accuracy unacceptable repeat cycle with a different trial function


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Galerkins weighted residual method

weight functions are the basis functions

The optimization criteria transforms the original differential

equation into a set of algebraic equations.

Can be solved to determined the undetermined parameters

Can be applied to any differential equation

l lw =


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Galerkin method

2

0 1 2( )u x u u x u x= + +

Approximating function of the form

are to be determined

Imposing the boundary conditions

'i

u s

0 0 1 22q

u u u u

= =

2

0 2 2

2

0 2 2

( ) 2

2

qu x u u x u x

qu x u x u x

= + +

= + +


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Galerkins method

xx =)(12

2 )(xx

=

2( : ) 2R x u u f=

2

02

q f fu u x x

= + +

From the weighted residual statements we get 2 2u

f

=

2

0 2 2( ) 2

qu x u x u x u x

= + +

0 0

q

u x = +


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Galerkin method-limitations

Approximating field is defined over the entire domain.

Major disadvantage is the construction of approximation

functions that satisfy the boundary conditions of the problem

Approximating fields must be admissible and easy to use.

Only polynomials and sine and cosine functions are simple

enough to be practicable.

The undetermined parameters have no physical meaning.iu


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Finite Element Method

Approximating field is defined in a piecewise fashion bydividing the entire region over subregions

Undetermined parameters are the nodal values of the field

The approximation functions can be generated systematically

over these subregions

FEM is the piecewise (or elementwise )application of the

weighted residual method.

We get different finite element approximations depending on

the choice of the weighted residual method.

iu


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Discretization of the domain into a set of finite elements.

Defining an approximate solution over the element.

Weighted integral formulation of the differential equation.

Substitute the approximate solution and get the algebraicequation

?


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1. Discretization of the domain into a set of finite elements (mesh generation).--- the domain is subdivided into non-overlapping subdomains e calledelements of simple geometrical form.

---for this 1-D problem an obvious choice for an element e is the interval

1e e

x x x

0 0x = 1x 1ex 1Lx =exex


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Step 2: To set up a weak formulation of the differential equation.

(i) Multiply the equation by a weight function and integrate the equation over thedomain e.

0e

e

l

d duw f dx

dx dx

=

(ii) Move the differentiation to the weight function by by doing integration by parts

1

0

e

e e

e

xe e

ll l

x

dwdu duw dx w fdx

dx dx dx

=

1

e

e e

e

xe e

ll l

x

dw du dudx w fdx wdx dx dx

= + Weak Form


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1

1

e e

e e

x x

du duq qdx dx

= =

1

1 1( ). ( ).

e

e

xe

l l e e l e e

x

duw w x q w x q

dx

=

Weak Form

1 1( ). ( ).e e

e

ll l e e l e e

dw dudx w f dx w x q w x q

dx dx

= +

Equivalent to both the governing differential equation and the associated natural boundary condition


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Steps in the development of weak form

Multiply the equation by a weight function and integrate the

equation over the domain e.

Distribute the differentiation among the weight function w

and ue

by doing integration by parts

Use the definition of the natural boundary condition in the

weak form

ew


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Step 3:FEM model from the weak form

Approximation over a finite elemente

2

1

( )e e e

j j

j

u x u =

=

1

e e

e

x xx

=

where

12

e e

e

x xx

=

From the weak form

l lw =

2

1 1

1

( ). ( ).e e

e

jl e

ele l e ej l

j

ddw u d w x q w x qx w fdxdx dx

=

= +


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The choice 1 1 2 2e e

w and w gives = =2

11

1

1 1 1 1( ) ( )e e

eeje e

j

e e

e e e e

j

ddu dx x q x qfdx

dx dx

=

= +

Now,

22

2

1

2 2 1 1( ) ( )e e

eeje e

j

e e

e e e e

j

ddu dx x q x qfdxdx dx

=

= +

1 1 1 1 1 1( ) ( )e e e

e e e e eq x q q q = =

2 2 1 1 2( ) ( )e e e

e e e e ex q x q q q = =


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The choice 1 1 2 2e e

w and w gives = =2

11 1

1e e

eeje e e

j

j

ddu dx fdx q

dx dx

=

= +

e e e e

ij j i iK u f q= +

which takes the form

e

eeje i

ij

ddK

dx dx

= ee ei if fdx=

1

2

ee

i e

qq

q

=

22

2 21

e e

eeje e e

jj

dd

u dx fdx qdx dx

=

= +


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The element equations are

1 1

2 2

1 1 2

1 12

ee e

e eee

xf

u q

xx u qf

= +

Diagram here

2

e

u

1

1

eu

2

2

eq1eq


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Results

Draw a diagram for assembly here

1 1

1 1

1 1

2 2

21 1 0

1 1 1 02

0 0 0 0 00

e

e

e

x f

u qx

forelement u f qx

= +

First we rewrite the element matrices in terms of the global numbering scheme

2

1

2

2

2

1

2

2

00 0 0 0

2 0 1 1

1

2

0

20 1

e

e

e

xfor element f q

xu

xf

u

q

= +

1

1u2

1u12u

2

2u

1 2 3

1 2 2 3

1 2

1 2


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Step 4:Assemble the element equations to obtain the globalsystem

1 1

1 1 1

1 2

2 2 1

2 2

3 2 2

1 1 0 2

1 2 1 0

0 1 1

e

e

e

e

x fu q q

u x f q qx

u x q qf

= + + =


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STEP 5:Imposition of boundary conditions

Dirichlet condition

1, , , 0

s s

ss s s i i is s is si

u u

k f u f f k u k k

=

= = = = =

Neumann condition

Replace the corresponding component of the right hand column bythe specified value.


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STEP 5:Imposition of boundary conditions

( )

( )

( )

2

121 1

2 0

23

21 0 0

0 2 1 0

0 1 1 0

2

e

e

e

e

xf

u qx

u f u

q xu

xf

= + +

+


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Step 6: Solution of the algebraic system of equations

This is a standard matrix equation can be solved by direct or

indirect(iterative) method.

Step 7: Postprocessing

This final operation displays the solution to system equations in

tabular graphical or pictorial form. Other meaningful quantities

may be derived from the solution and also displayed


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Results

1.51.51.5u3(@ x=1.0)

0.6250.6250.625u2

(@ x=0.5)

000

u1

(@ x=0)

Finite

Difference

Finite

ElementExact


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Results

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2

spatial co-ordinate, x

displacement,u

Exact Solution

Finite Element solution

Finite Element solution


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FEM and FDM

FDM is a direct method- replacesdifferential equations by a differenceapproximations

Based on rectangular discretisation

Treatment of boundary conditions

needs extra care

Making higher order approximation istedious

FEM is an indirect method- works on aweak form

Arbitrary geometry can be modelled

(can be rectangular, quadrilateral,

triangle)

Boundary conditions is more

systematic

Making higher order approximation iseasy


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FEM and FDM

No closed form solution that permits analytical study of the

effects of changing various parameters.

Good engineering judgement is required

- type of element- type of difference approximation

Many input data are required and voluminous output must

be sorted and understood

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