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Weighted-Residual Method,

Galerkin Variational form and

Piece-wise Continuous TrialFunctions

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Residual Method

The Weighted-Residual (WR method is

a !owerful wa" of finding a!!ro#imatesolutions to differential e$uations%

&n !articular, the Galerkin Weighted-Residual formulation is the most !o!ular

from the finite element !oint of 'iew%

Piece-wise trial function a!!ro#imationof the weak form of the Galerkin

weighted residual techni$ue forms the

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Residual Method

For finding a!!ro#imate solution to

differential e$uations)

(i *ssume a trial solution, like

(ii +ustitute the trial function and a!!l"oundar" conditions into the to

make its .residual/ form

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Residual Method

(iii etermine the unknown !arameters

(C0, C1, C2,3 in the assumed trialfunction in such a wa" as to make these

residuals as low as !ossile%

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Residual Method

4ar under uniform a#ial load, is

, BCs:

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+olution)

(i *ssume trial function

a!!l" 4Cs, we get

(ii Find the domain residual

Residual Method

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+olution)

(iii Minimise the residual (i%e%, Rd50

+olution is)

Residual Method

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Residual Method

Cantile'er ean under 67, is

, BCs:

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+olution)

(i *ssume trial function

a!!l" 4Cs, we get

and

Residual Method

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(ii Find the domain residual

(iii Minimise the residual (i%e%, Rd50

+olution is

Residual Method

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Residual Method

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Residual Method

Residual is 'ar"ing with #,

Collocation method

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Residual Method

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Residual Method

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Weighted-Residual Method

* weighting function Wi(# is used

minimise the residual o'er the entiredomain as)

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Weighted-Residual Method

W-R methods

(i Collocation Method

(ii 7east +$uare Method

(iii Galerkin Method (most !o!ular

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Galerkin Weighted-Residual Method

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Galerkin Weighted-Residual Method

+olution)

(i *ssume trial function

(ii omain residual

(iii Galerkin method

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Galerkin Weighted-Residual Method

or

we get,

The solution is

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Galerkin Weighted-Residual Method

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Galerkin Weighted-Residual Method

4ar under uniform a#ial load $5a#, is

, BCs:

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Galerkin Weighted-Residual Method

+olution)

(i *ssume trial function

*!!l" 4C,

l k h d d l h d

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Galerkin Weighted-Residual Method

(ii Find domain residual

(iii Minimise the residual

l ki i h d id l h d

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Galerkin Weighted-Residual Method

we get,

+olution is,

k ( i i l f

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Consider the a#ial ar !rolem,

Residual form is)

Weighted-Residual form is)

Weak (Variational form W-R

, BCs:

W k (V i i l f W R

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or

&ntegration " !arts formula,

here,

Weak (Variational form W-R

W k (V i i l f W R

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7et,

Weak form of W-R is)

&t is referred to as the weak form ecauseof the weaker continuit" demand on the

solution%

Weak (Variational form W-R

W k (V i i l f W R

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*d'antages of Weak form)

(i The continuit" demanded on trialfunction is down%

(ii 8atural (or Force oundar" conditions

(i%e% Po or P7 ha'e een e#!licitl"

rought out in the WR statement itself%

(iii The trial function assumed need onl"satisf" the ssential (or Geometric

oundar" condition at # 5 0, i%e % u(0 5

Weak (Variational form W-R

W k (V i ti l f W R

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4oundar" conditions)

(i ssential oundar" conditions normall"in'ol'e deflection and slo!e

(ii 8atural oundar" conditions normall"in'ol'e force and ending moments%

Weak (Variational form W-R

W k (V i ti l f W R

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Weak (Variational form W-R

W k (V i ti l f W R

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Consider the a#ial ar !rolem,

+olution)*ssume trial function)

Weak (Variational form W-R

, BCs:

W k (V i ti l f W R

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4Cs)

Weak form w%r%t% W1

Weak (Variational form W-R

W k (V i ti l f W R

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Weak form w%r%t% W2

Rearranging,

or

Weak (Variational form W-R

W k (V i ti l f W R

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+olution is)

Weak (Variational form W-R

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Weak (Variational form W R

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Weak (Variational form W-R

Weak (Variational form W R

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Weak (Variational form W-R

+olution)

W-R form

Weak (Variational form W R

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Weak (Variational form W-R

&ntegrating " !arts again,

*!!l" 8atural 4Cs,

Weak (Variational form W R

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Weak (Variational form W-R

*!!l" ssential 4Cs

we get, weak form as

+u9ect to

Weak (Variational form W R

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Weak (Variational form W-R

+olution)

*ssume $uadratic trial function satisf"ingthe essential 4Cs)

Weighting fn is)For sim!licit", let

or

Weak (Variational form W R

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Weak (Variational form W-R

Thus, the $uadratic, weak W-R solution is)

*nd, the e#act solution is)

W-R sinusoidal solution is)

Galerkin Weighted Residual Method

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Galerkin Weighted-Residual Method

Galerkin Weighted Residual Method

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Galerkin Weighted-Residual Method

Galerkin Weighted Residual Method

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Galerkin Weighted-Residual Method

Piece-wise Continuous Trial Function

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Piece-wise Continuous Trial Function

+olution of the Weak Form

Piece-wise Continuous Trial Function

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Piece-wise Continuous Trial Function

+olution of the Weak Form

7inear &nter!olation Function

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7inear &nter!olation Function

:ne-dimensional 4ar Finite lement

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:ne-dimensional 4ar Finite lement

4ar under uniform a#ial load $5a#, is

, BCs:

:ne-dimensional 4ar Finite lement

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:ne-dimensional 4ar Finite lement

:ne-dimensional 4ar Finite lement

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:ne-dimensional 4ar Finite lement

:ne-dimensional 4ar Finite lement

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:ne-dimensional 4ar Finite lement

Let u(x) within each element be given by the

interpolation as

:ne-dimensional 4ar Finite lement

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:ne dimensional 4ar Finite lement

Galerkin weighting function = Shape function

:ne-dimensional 4ar Finite lement

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:ne dimensional 4ar Finite lement

:ne-dimensional 4ar Finite lement

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:ne dimensional 4ar Finite lement

Weak (Variational form W-R

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Weak (Variational form W R

+olution)

:ne-dimensional 4ar Finite lement

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:ne dimensional 4ar Finite lement

:ne-dimensional 4ar Finite lement

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:ne dimensional 4ar Finite lement