FEM: Introduction and Weighted Residual Methods

Weighted Residual Methods

Mohammad Tawfik #WikiCourses

http://WikiCourses.WikiSpaces.com


Mohammad Tawfik




Objectives

• In this section we will be introduced to the general classification of approximate methods

• Special attention will be paid for the weighted residual method

• Derivation of a system of linear equations to approximate the solution of an ODE will be presented using different techniques




Classification of Approximate

Solutions of D.E.’s

• Discrete Coordinate Method

– Finite difference Methods

– Stepwise integration methods

• Euler method

• Runge-Kutta methods

• Etc…

• Distributed Coordinate Method




Distributed Coordinate Methods

• Weighted Residual Methods – Interior Residual

• Collocation

• Galrekin

• Finite Element

– Boundary Residual • Boundary Element Method

• Stationary Functional Methods – Reyligh-Ritz methods

– Finite Element method




Basic Concepts

• A linear differential equation may be written in the form:

xgxfL

• Where L(.) is a linear differential operator.

• An approximate solution maybe of the form:

n

i

ii xaxf1




Basic Concepts • Applying the differential operator on the approximate

solution, you get:

01

1

xgxLa

xgxaLxgxfL

n

i

ii

n

i

ii

xRxgxLan

i

ii 1

Residue




Handling the Residue

• The weighted residual methods are all

based on minimizing the value of the

residue.

• Since the residue can not be zero over the

whole domain, different techniques were

introduced.




General Weighted Residual

Method




Objective of WRM

• As any other numerical method, the

objective is to obtain of algebraic

equations, that, when solved, produce a

result with an acceptable accuracy.

• If we are seeking the values of ai that

would reduce the Residue (R(x)) allover

the domain, we may integrate the residue

over the domain and evaluate it!




Evaluating the Residue

xRxgxLan

i

ii 1

xRxgxLaxLaxLa nn ...2211

n unknown variables

01

Domain

n

i

ii

Domain

dxxgxLadxxR

One equation!!!




Using Weighting Functions

• If you can select n different weighting

functions, you will produce n equations!

• You will end up with n equations in n

variables.

01

Domain

n

i

iij

Domain

j dxxgxLaxwdxxRxw




Collocation Method

• The idea behind the collocation method is

similar to that behind the buttons of your

shirt!

• Assume a solution, then force the residue

to be zero at the collocation points




Collocation Method

0jxR

01

j

n

i

jii

j

xFxLa

xR




Example Problem




The bar tensile problem

0/

00

'

02

2

dxdulx

ux

sBC

xFx

uEA




Bar application

02

2

xF

x

uEA

n

i

ii xaxu1

xRxF

dx

xdaEA

n

i

ii

12

2Applying the collocation method

0

12

2

j

n

i

ji

i xFdx

xdaEA




In Matrix Form

nnnnnn

n

n

xF

xF

xF

a

a

a

kkk

kkk

kkk

2

1

2

1

21

22212

12111

...

...

...

Solve the above system for the “generalized

coordinates” ai to get the solution for u(x)

jxx

iij

dx

xdEAk

2

2




Notes on the trial functions

• They should be at least twice

differentiable!

• They should satisfy all boundary

conditions!

• Those are called the “Admissibility

Conditions”.




Using Admissible Functions

• For a constant forcing function, F(x)=f

• The strain at the free end of the bar should

be zero (slope of displacement is zero).

We may use:

l

xSinx

2




Using the function into the DE:

• Since we only have one term in the series,

we will select one collocation point!

• The midpoint is a reasonable choice!

l

xSin

lEA

dx

xdEA

22

2

2

2

faSinl

EA

1

2

42




Solving:

• Then, the approximate

solution for this problem is:

• Which gives the maximum

displacement to be:

• And maximum strain to be:

EA

fl

EA

fl

SinlEA

fa

2

2

2

21 57.024

42

l

xSin

EA

flxu

257.0

2

5.057.02

exactEA

fllu

0.19.00 exactEA

lfux




The Subdomain Method

• The idea behind the

subdomain method is

to force the integral

of the residue to be

equal to zero on a

subinterval of the

domain




The Subdomain Method

0

1

j

j

x

x

dxxR

0

11

1

j

j

j

j

x

x

n

i

x

x

ii dxxgdxxLa




Bar application

02

2

xF

x

uEA

n

i

ii xaxu1

xRxF

dx

xdaEA

n

i

ii

12

2Applying the subdomain method

11

12

2 j

j

j

j

x

x

n

i

x

x

ii dxxFdx

dx

xdaEA




In Matrix Form

11

2

2 j

j

j

j

x

x

i

x

x

i dxxFadxdx

xdEA






Using Admissible Functions

• For a constant forcing function, F(x)=f

• The strain at the free end of the bar should

be zero (slope of displacement is zero).

We may use:

l

xSinx

2




Using the function into the DE:

• Since we only have one term in the series,

we will select one subdomain!

l

xSin

lEA

dx

xdEA

22

2

2

2

ll

fdxadxl

xSin

lEA

0

1

0

2

22




Solving:

• Then, the approximate

solution for this problem is:

• Which gives the maximum

displacement to be:

• And maximum strain to be:

EA

fl

EA

fl

lEA

fla

22

1 637.02

2

l

xSin

EA

flxu

2637.0

2

5.0637.02

exactEA

fllu

0.10.10 exactEA

lfux

flal

xCos

lEA

l

1

022




The Galerkin Method

• Galerkin suggested that the residue

should be multiplied by a weighting

function that is a part of the suggested

solution then the integration is performed

over the whole domain!!!

• Actually, it turned out to be a VERY

GOOD idea




The Galerkin Method

0Domain

j dxxxR

01

Domain

j

n

i Domain

iji dxxgxdxxLxa




Bar application

02

2

xF

x

uEA

n

i

ii xaxu1

xRxF

dx

xdaEA

n

i

ii

12

2Applying Galerkin method

Domain

j

n

i Domain

iji dxxFxdx

dx

xdxaEA

12

2




In Matrix Form

Domain

ji

Domain

ij dxxFxadx

dx

xdxEA

2

2






Same conditions on the functions

are applied

• They should be at least twice

differentiable!

• They should satisfy all boundary

conditions!

• Let’s use the same function as in the

collocation method:

l

xSinx

2




Substituting with the approximate

solution:

Domain

j

n

i Domain

iji dxxFxdx

dx

xdxaEA

12

2

l

l

fdxl

xSin

dxl

xSin

l

xSina

lEA

0

0

1

2

2

222

lla

lEA

2

221

2

EA

fll

EA

fa

2

3

2

1 52.016




Substituting with the approximate

solution: (Int. by Parts)

Domain

j

n

i Domain

iji dxxFxdx

dx

xdxaEA

12

2

lla

lEA

2

221

2

EA

fll

EA

fa

2

3

2

1 52.016

Domain

ij

l

ij

Domain

ij

dxdx

xd

dx

xd

dx

xdx

dxdx

xdx

0

2

2

Zero!




What did we gain?

• The functions are required to be less

differentiable

• Not all boundary conditions need to be

satisfied

• The matrix became symmetric!




Summary

• We may solve differential equations using a

series of functions with different weights.

• When those functions are used, Residue

appears in the differential equation

• The weights of the functions may be determined

to minimize the residue by different techniques

• One very important technique is the Galerkin

method.

Education

FEM: Introduction and Weighted Residual Methods