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Weighted Residual Methods
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Weighted Residual Methods
Mohammad Tawfik
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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.
Weighted Residual Methods
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General Weighted Residual
Method
Weighted Residual Methods
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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!
Weighted Residual Methods
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Evaluating the Residue
xRxgxLan
i
ii 1
xRxgxLaxLaxLa nn ...2211
n unknown variables
01
Domain
n
i
ii
Domain
dxxgxLadxxR
One equation!!!
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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Collocation Method
0jxR
01
j
n
i
jii
j
xFxLa
xR
Weighted Residual Methods
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Example Problem
Weighted Residual Methods
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The bar tensile problem
0/
00
'
02
2
dxdulx
ux
sBC
xFx
uEA
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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Notes on the trial functions
• They should be at least twice
differentiable!
• They should satisfy all boundary
conditions!
• Those are called the “Admissibility
Conditions”.
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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In Matrix Form
11
2
2 j
j
j
j
x
x
i
x
x
i dxxFadxdx
xdEA
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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The Galerkin Method
0Domain
j dxxxR
01
Domain
j
n
i Domain
iji dxxgxdxxLxa
Weighted Residual Methods
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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
Weighted Residual Methods
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In Matrix Form
Domain
ji
Domain
ij dxxFxadx
dx
xdxEA
2
2
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
Weighted Residual Methods
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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
Weighted Residual Methods
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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
Weighted Residual Methods
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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!
Weighted Residual Methods
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What did we gain?
• The functions are required to be less
differentiable
• Not all boundary conditions need to be
satisfied
• The matrix became symmetric!
Weighted Residual Methods
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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.