SlideShare une entreprise Scribd logo
1  sur  37
Télécharger pour lire hors ligne
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Weighted Residual Methods
Mohammad Tawfik
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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 xaxf
1

Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Basic Concepts
• Applying the differential operator on the approximate
solution, you get:
        
     0
1
1












xgxLa
xgxaLxgxfL
n
i
ii
n
i
ii


      xRxgxLa
n
i
ii 1

Residue
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
General Weighted Residual
Method
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Evaluating the Residue
      xRxgxLa
n
i
ii 1

            xRxgxLaxLaxLa nn   ...2211
n unknown variables
       0
1






   Domain
n
i
ii
Domain
dxxgxLadxxR 
One equation!!!
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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.
           0
1






   Domain
n
i
iij
Domain
j dxxgxLaxwdxxRxw 
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Collocation Method
  0jxR
 
     0
1



j
n
i
jii
j
xFxLa
xR

Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Example Problem
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The bar tensile problem
 
0/
00
'
02
2





dxdulx
ux
sBC
xF
x
u
EA
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Bar application
  02
2



xF
x
u
EA
   

n
i
ii xaxu
1

     xRxF
dx
xd
aEA
n
i
i
i 1
2
2

Applying the collocation method
    0
1
2
2

j
n
i
ji
i xF
dx
xd
aEA

Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
i
ij
dx
xd
EAk

 2
2

Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
x
Sinx
2


Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
x
Sin
l
EA
dx
xd
EA
22
2
2
2

   faSin
l
EA 




















 1
2
42

Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
f
a
2
2
2
21 57.0
24
42


  






l
x
Sin
EA
fl
xu
2
57.0
2

   5.057.0
2
 exact
EA
fl
lu
   0.19.00  exact
EA
lf
ux
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Bar application
  02
2



xF
x
u
EA
   

n
i
ii xaxu
1

     xRxF
dx
xd
aEA
n
i
i
i 1
2
2

Applying the subdomain method
    



11
1
2
2 j
j
j
j
x
x
n
i
x
x
i
i dxxFdx
dx
xd
aEA

Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
In Matrix Form
     


















 11
2
2 j
j
j
j
x
x
i
x
x
i
dxxFadx
dx
xd
EA

Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
x
Sinx
2


Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Using the function into the DE:
• Since we only have one term in the series,
we will select one subdomain!
 













l
x
Sin
l
EA
dx
xd
EA
22
2
2
2

 




























ll
fdxadx
l
x
Sin
l
EA
0
1
0
2
22

Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
fl
a
22
1 637.0
2
2


  






l
x
Sin
EA
fl
xu
2
637.0
2

   5.0637.0
2
 exact
EA
fl
lu
   0.10.10  exact
EA
lf
ux
   fla
l
x
Cos
l
EA
l



























1
0
22

Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
The Galerkin Method
    0Domain
j dxxxR 
         0
1
   Domain
j
n
i Domain
iji dxxgxdxxLxa 
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Bar application
  02
2



xF
x
u
EA
   

n
i
ii xaxu
1

     xRxF
dx
xd
aEA
n
i
i
i 1
2
2

Applying Galerkin method
         
 Domain
j
n
i Domain
i
ji dxxFxdx
dx
xd
xaEA 


1
2
2
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
In Matrix Form
         

















 Domain
ji
Domain
i
j dxxFxadx
dx
xd
xEA 

 2
2
Solve the above system for the “generalized
coordinates” ai to get the solution for u(x)
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
x
Sinx
2


Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Substituting with the approximate
solution:
         
 Domain
j
n
i Domain
i
ji dxxFxdx
dx
xd
xaEA 


1
2
2




























l
l
fdx
l
x
Sin
dx
l
x
Sin
l
x
Sina
l
EA
0
0
1
2
2
222



 ll
a
l
EA
2
22
1
2







EA
fll
EA
f
a
2
3
2
1 52.0
16


Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
Substituting with the approximate
solution: (Int. by Parts)
         
 Domain
j
n
i Domain
i
ji dxxFxdx
dx
xd
xaEA 


1
2
2

 ll
a
l
EA
2
22
1
2







EA
fll
EA
f
a
2
3
2
1 52.0
16


   
       



Domain
ij
l
i
j
Domain
i
j
dx
dx
xd
dx
xd
dx
xd
x
dx
dx
xd
x




0
2
2
Zero!
Weighted Residual Methods
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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
Mohammad Tawfik
#WikiCourses
http://WikiCourses.WikiSpaces.com
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.

Contenu connexe

Tendances

Method of weighted residuals
Method of weighted residualsMethod of weighted residuals
Method of weighted residualsJasim Almuhandis
 
Classification of clutches, torque transmission capacity, considerations for ...
Classification of clutches, torque transmission capacity, considerations for ...Classification of clutches, torque transmission capacity, considerations for ...
Classification of clutches, torque transmission capacity, considerations for ...vaibhav tailor
 
Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4propaul
 
Axisymmetric
Axisymmetric Axisymmetric
Axisymmetric Raj Kumar
 
Lecture 9 shear force and bending moment in beams
Lecture 9 shear force and bending moment in beamsLecture 9 shear force and bending moment in beams
Lecture 9 shear force and bending moment in beamsDeepak Agarwal
 
Analysis of Thin Plates
Analysis of  Thin PlatesAnalysis of  Thin Plates
Analysis of Thin PlatesJaya Teja
 
Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5propaul
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANKASHOK KUMAR RAJENDRAN
 
Introduction to finite element analysis
Introduction to finite element analysisIntroduction to finite element analysis
Introduction to finite element analysisTarun Gehlot
 
COMPUTER AIDED ENGINEERING - INTRODUCTION
COMPUTER AIDED ENGINEERING - INTRODUCTIONCOMPUTER AIDED ENGINEERING - INTRODUCTION
COMPUTER AIDED ENGINEERING - INTRODUCTIONISAAC SAMUEL RAJA T
 
01. steps involved, merits, demerits & limitations of fem
01. steps involved, merits, demerits & limitations of fem01. steps involved, merits, demerits & limitations of fem
01. steps involved, merits, demerits & limitations of femSura Venkata Mahesh
 

Tendances (20)

Method of weighted residuals
Method of weighted residualsMethod of weighted residuals
Method of weighted residuals
 
Classification of clutches, torque transmission capacity, considerations for ...
Classification of clutches, torque transmission capacity, considerations for ...Classification of clutches, torque transmission capacity, considerations for ...
Classification of clutches, torque transmission capacity, considerations for ...
 
Finite Element Method
Finite Element MethodFinite Element Method
Finite Element Method
 
Axis symmetric
Axis symmetricAxis symmetric
Axis symmetric
 
Introduction to Engineering Mechanics
Introduction to Engineering MechanicsIntroduction to Engineering Mechanics
Introduction to Engineering Mechanics
 
Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4
 
Axisymmetric
Axisymmetric Axisymmetric
Axisymmetric
 
FEA Using Ansys
FEA Using AnsysFEA Using Ansys
FEA Using Ansys
 
Finite Element Methods
Finite Element  MethodsFinite Element  Methods
Finite Element Methods
 
Lecture 9 shear force and bending moment in beams
Lecture 9 shear force and bending moment in beamsLecture 9 shear force and bending moment in beams
Lecture 9 shear force and bending moment in beams
 
ME6603 - FINITE ELEMENT ANALYSIS
ME6603 - FINITE ELEMENT ANALYSIS ME6603 - FINITE ELEMENT ANALYSIS
ME6603 - FINITE ELEMENT ANALYSIS
 
Analysis of Thin Plates
Analysis of  Thin PlatesAnalysis of  Thin Plates
Analysis of Thin Plates
 
Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5
 
FEM
FEMFEM
FEM
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
 
Introduction to finite element analysis
Introduction to finite element analysisIntroduction to finite element analysis
Introduction to finite element analysis
 
Lect14
Lect14Lect14
Lect14
 
Bending stresses in beams
Bending stresses in beamsBending stresses in beams
Bending stresses in beams
 
COMPUTER AIDED ENGINEERING - INTRODUCTION
COMPUTER AIDED ENGINEERING - INTRODUCTIONCOMPUTER AIDED ENGINEERING - INTRODUCTION
COMPUTER AIDED ENGINEERING - INTRODUCTION
 
01. steps involved, merits, demerits & limitations of fem
01. steps involved, merits, demerits & limitations of fem01. steps involved, merits, demerits & limitations of fem
01. steps involved, merits, demerits & limitations of fem
 

Similaire à FEM: Introduction and Weighted Residual Methods

Multiple Degree of Freedom (MDOF) Systems
Multiple Degree of Freedom (MDOF) SystemsMultiple Degree of Freedom (MDOF) Systems
Multiple Degree of Freedom (MDOF) SystemsMohammad Tawfik
 
Roots of Nonlinear Equations - Open Methods
Roots of Nonlinear Equations - Open MethodsRoots of Nonlinear Equations - Open Methods
Roots of Nonlinear Equations - Open MethodsMohammad Tawfik
 
System of Initial Value Problems
System of Initial Value ProblemsSystem of Initial Value Problems
System of Initial Value ProblemsMohammad Tawfik
 
Word2vec in Theory Practice with TensorFlow
Word2vec in Theory Practice with TensorFlowWord2vec in Theory Practice with TensorFlow
Word2vec in Theory Practice with TensorFlowBruno Gonçalves
 
FEM: Stationary Functional Approach
FEM: Stationary Functional ApproachFEM: Stationary Functional Approach
FEM: Stationary Functional ApproachMohammad Tawfik
 
Boundary Value Problems - Finite Difference
Boundary Value Problems - Finite DifferenceBoundary Value Problems - Finite Difference
Boundary Value Problems - Finite DifferenceMohammad Tawfik
 
Bounded Model Checking
Bounded Model CheckingBounded Model Checking
Bounded Model CheckingIlham Amezzane
 
super vector machines algorithms using deep
super vector machines algorithms using deepsuper vector machines algorithms using deep
super vector machines algorithms using deepKNaveenKumarECE
 
Bp150513(compiler)
Bp150513(compiler)Bp150513(compiler)
Bp150513(compiler)indhu mathi
 
Distributed algorithms for big data @ GeeCon
Distributed algorithms for big data @ GeeConDistributed algorithms for big data @ GeeCon
Distributed algorithms for big data @ GeeConDuyhai Doan
 

Similaire à FEM: Introduction and Weighted Residual Methods (20)

13 weightedresidual
13 weightedresidual13 weightedresidual
13 weightedresidual
 
Multiple Degree of Freedom (MDOF) Systems
Multiple Degree of Freedom (MDOF) SystemsMultiple Degree of Freedom (MDOF) Systems
Multiple Degree of Freedom (MDOF) Systems
 
Roots of Nonlinear Equations - Open Methods
Roots of Nonlinear Equations - Open MethodsRoots of Nonlinear Equations - Open Methods
Roots of Nonlinear Equations - Open Methods
 
System of Initial Value Problems
System of Initial Value ProblemsSystem of Initial Value Problems
System of Initial Value Problems
 
Numerical Integration
Numerical IntegrationNumerical Integration
Numerical Integration
 
Word2vec in Theory Practice with TensorFlow
Word2vec in Theory Practice with TensorFlowWord2vec in Theory Practice with TensorFlow
Word2vec in Theory Practice with TensorFlow
 
FEM: Element Equations
FEM: Element EquationsFEM: Element Equations
FEM: Element Equations
 
FEM: Stationary Functional Approach
FEM: Stationary Functional ApproachFEM: Stationary Functional Approach
FEM: Stationary Functional Approach
 
september4.ppt
september4.pptseptember4.ppt
september4.ppt
 
03 multipledof
03 multipledof03 multipledof
03 multipledof
 
Boundary Value Problems - Finite Difference
Boundary Value Problems - Finite DifferenceBoundary Value Problems - Finite Difference
Boundary Value Problems - Finite Difference
 
Bracketing Methods
Bracketing MethodsBracketing Methods
Bracketing Methods
 
Bounded Model Checking
Bounded Model CheckingBounded Model Checking
Bounded Model Checking
 
Word2vec and Friends
Word2vec and FriendsWord2vec and Friends
Word2vec and Friends
 
super vector machines algorithms using deep
super vector machines algorithms using deepsuper vector machines algorithms using deep
super vector machines algorithms using deep
 
Interpolation Methods
Interpolation MethodsInterpolation Methods
Interpolation Methods
 
Bp150513(compiler)
Bp150513(compiler)Bp150513(compiler)
Bp150513(compiler)
 
FEM: 2-D Problems
FEM: 2-D ProblemsFEM: 2-D Problems
FEM: 2-D Problems
 
Greedy Algorithms with examples' b-18298
Greedy Algorithms with examples'  b-18298Greedy Algorithms with examples'  b-18298
Greedy Algorithms with examples' b-18298
 
Distributed algorithms for big data @ GeeCon
Distributed algorithms for big data @ GeeConDistributed algorithms for big data @ GeeCon
Distributed algorithms for big data @ GeeCon
 

Plus de Mohammad Tawfik

Supply Chain Management for Engineers - INDE073
Supply Chain Management for Engineers - INDE073Supply Chain Management for Engineers - INDE073
Supply Chain Management for Engineers - INDE073Mohammad Tawfik
 
Supply Chain Management 01 - Introduction
Supply Chain Management 01 - IntroductionSupply Chain Management 01 - Introduction
Supply Chain Management 01 - IntroductionMohammad Tawfik
 
Supply Chain Management 02 - Logistics
Supply Chain Management 02 - LogisticsSupply Chain Management 02 - Logistics
Supply Chain Management 02 - LogisticsMohammad Tawfik
 
Supply Chain Management 03 - Inventory Management
Supply Chain Management 03 - Inventory ManagementSupply Chain Management 03 - Inventory Management
Supply Chain Management 03 - Inventory ManagementMohammad Tawfik
 
Creative problem solving and decision making
Creative problem solving and decision makingCreative problem solving and decision making
Creative problem solving and decision makingMohammad Tawfik
 
Digital content for teaching introduction
Digital content for teaching introductionDigital content for teaching introduction
Digital content for teaching introductionMohammad Tawfik
 
Crisis Management Basics
Crisis Management BasicsCrisis Management Basics
Crisis Management BasicsMohammad Tawfik
 
Effective Delegation Skills
Effective Delegation SkillsEffective Delegation Skills
Effective Delegation SkillsMohammad Tawfik
 
Business Management - Marketing
Business Management - MarketingBusiness Management - Marketing
Business Management - MarketingMohammad Tawfik
 
Project Management (CAPM) - Integration
Project Management (CAPM) - IntegrationProject Management (CAPM) - Integration
Project Management (CAPM) - IntegrationMohammad Tawfik
 
Project Management (CAPM) - The Framework
Project Management (CAPM) - The FrameworkProject Management (CAPM) - The Framework
Project Management (CAPM) - The FrameworkMohammad Tawfik
 
Project Management (CAPM) - Introduction
Project Management (CAPM) - IntroductionProject Management (CAPM) - Introduction
Project Management (CAPM) - IntroductionMohammad Tawfik
 
Introduction to Wind Energy
Introduction to Wind EnergyIntroduction to Wind Energy
Introduction to Wind EnergyMohammad Tawfik
 
Finite Element for Trusses in 2-D
Finite Element for Trusses in 2-DFinite Element for Trusses in 2-D
Finite Element for Trusses in 2-DMohammad Tawfik
 

Plus de Mohammad Tawfik (20)

Supply Chain Management for Engineers - INDE073
Supply Chain Management for Engineers - INDE073Supply Chain Management for Engineers - INDE073
Supply Chain Management for Engineers - INDE073
 
Supply Chain Management 01 - Introduction
Supply Chain Management 01 - IntroductionSupply Chain Management 01 - Introduction
Supply Chain Management 01 - Introduction
 
Supply Chain Management 02 - Logistics
Supply Chain Management 02 - LogisticsSupply Chain Management 02 - Logistics
Supply Chain Management 02 - Logistics
 
Supply Chain Management 03 - Inventory Management
Supply Chain Management 03 - Inventory ManagementSupply Chain Management 03 - Inventory Management
Supply Chain Management 03 - Inventory Management
 
Creative problem solving and decision making
Creative problem solving and decision makingCreative problem solving and decision making
Creative problem solving and decision making
 
Digital content for teaching introduction
Digital content for teaching introductionDigital content for teaching introduction
Digital content for teaching introduction
 
Crisis Management Basics
Crisis Management BasicsCrisis Management Basics
Crisis Management Basics
 
DISC Personality Model
DISC Personality ModelDISC Personality Model
DISC Personality Model
 
Training of Trainers
Training of TrainersTraining of Trainers
Training of Trainers
 
Effective Delegation Skills
Effective Delegation SkillsEffective Delegation Skills
Effective Delegation Skills
 
Train The Trainer
Train The TrainerTrain The Trainer
Train The Trainer
 
Business Management - Marketing
Business Management - MarketingBusiness Management - Marketing
Business Management - Marketing
 
Stress Management
Stress ManagementStress Management
Stress Management
 
Project Management (CAPM) - Integration
Project Management (CAPM) - IntegrationProject Management (CAPM) - Integration
Project Management (CAPM) - Integration
 
Project Management (CAPM) - The Framework
Project Management (CAPM) - The FrameworkProject Management (CAPM) - The Framework
Project Management (CAPM) - The Framework
 
Project Management (CAPM) - Introduction
Project Management (CAPM) - IntroductionProject Management (CAPM) - Introduction
Project Management (CAPM) - Introduction
 
The Creative Individual
The Creative IndividualThe Creative Individual
The Creative Individual
 
Introduction to Wind Energy
Introduction to Wind EnergyIntroduction to Wind Energy
Introduction to Wind Energy
 
Finite Element for Trusses in 2-D
Finite Element for Trusses in 2-DFinite Element for Trusses in 2-D
Finite Element for Trusses in 2-D
 
Future of Drones ITW'16
Future of Drones ITW'16Future of Drones ITW'16
Future of Drones ITW'16
 

Dernier

3.21.24 The Origins of Black Power.pptx
3.21.24  The Origins of Black Power.pptx3.21.24  The Origins of Black Power.pptx
3.21.24 The Origins of Black Power.pptxmary850239
 
How to Print Employee Resume in the Odoo 17
How to Print Employee Resume in the Odoo 17How to Print Employee Resume in the Odoo 17
How to Print Employee Resume in the Odoo 17Celine George
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17Celine George
 
How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17Celine George
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17Celine George
 
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptxmary850239
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxraviapr7
 
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdfMaximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdfTechSoup
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice documentXsasf Sfdfasd
 
How to Filter Blank Lines in Odoo 17 Accounting
How to Filter Blank Lines in Odoo 17 AccountingHow to Filter Blank Lines in Odoo 17 Accounting
How to Filter Blank Lines in Odoo 17 AccountingCeline George
 
General views of Histopathology and step
General views of Histopathology and stepGeneral views of Histopathology and step
General views of Histopathology and stepobaje godwin sunday
 
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxPractical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxKatherine Villaluna
 
CapTechU Doctoral Presentation -March 2024 slides.pptx
CapTechU Doctoral Presentation -March 2024 slides.pptxCapTechU Doctoral Presentation -March 2024 slides.pptx
CapTechU Doctoral Presentation -March 2024 slides.pptxCapitolTechU
 
Presentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphPresentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphNetziValdelomar1
 
How to Add a many2many Relational Field in Odoo 17
How to Add a many2many Relational Field in Odoo 17How to Add a many2many Relational Field in Odoo 17
How to Add a many2many Relational Field in Odoo 17Celine George
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRATanmoy Mishra
 
UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE
 
M-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxM-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxDr. Santhosh Kumar. N
 
Patterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxPatterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxMYDA ANGELICA SUAN
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfMohonDas
 

Dernier (20)

3.21.24 The Origins of Black Power.pptx
3.21.24  The Origins of Black Power.pptx3.21.24  The Origins of Black Power.pptx
3.21.24 The Origins of Black Power.pptx
 
How to Print Employee Resume in the Odoo 17
How to Print Employee Resume in the Odoo 17How to Print Employee Resume in the Odoo 17
How to Print Employee Resume in the Odoo 17
 
How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17How to Add a New Field in Existing Kanban View in Odoo 17
How to Add a New Field in Existing Kanban View in Odoo 17
 
How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17
 
How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17How to Show Error_Warning Messages in Odoo 17
How to Show Error_Warning Messages in Odoo 17
 
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
3.19.24 Urban Uprisings and the Chicago Freedom Movement.pptx
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptx
 
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdfMaximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
Maximizing Impact_ Nonprofit Website Planning, Budgeting, and Design.pdf
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice document
 
How to Filter Blank Lines in Odoo 17 Accounting
How to Filter Blank Lines in Odoo 17 AccountingHow to Filter Blank Lines in Odoo 17 Accounting
How to Filter Blank Lines in Odoo 17 Accounting
 
General views of Histopathology and step
General views of Histopathology and stepGeneral views of Histopathology and step
General views of Histopathology and step
 
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxPractical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
 
CapTechU Doctoral Presentation -March 2024 slides.pptx
CapTechU Doctoral Presentation -March 2024 slides.pptxCapTechU Doctoral Presentation -March 2024 slides.pptx
CapTechU Doctoral Presentation -March 2024 slides.pptx
 
Presentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a ParagraphPresentation on the Basics of Writing. Writing a Paragraph
Presentation on the Basics of Writing. Writing a Paragraph
 
How to Add a many2many Relational Field in Odoo 17
How to Add a many2many Relational Field in Odoo 17How to Add a many2many Relational Field in Odoo 17
How to Add a many2many Relational Field in Odoo 17
 
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRADUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
DUST OF SNOW_BY ROBERT FROST_EDITED BY_ TANMOY MISHRA
 
UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024
 
M-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptxM-2- General Reactions of amino acids.pptx
M-2- General Reactions of amino acids.pptx
 
Patterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxPatterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptx
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdf
 

FEM: Introduction and Weighted Residual Methods

  • 1. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Weighted Residual Methods Mohammad Tawfik
  • 2. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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
  • 3. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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
  • 4. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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
  • 5. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 xaxf 1 
  • 6. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Basic Concepts • Applying the differential operator on the approximate solution, you get:               0 1 1             xgxLa xgxaLxgxfL n i ii n i ii         xRxgxLa n i ii 1  Residue
  • 7. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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.
  • 8. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com General Weighted Residual Method
  • 9. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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!
  • 10. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Evaluating the Residue       xRxgxLa n i ii 1              xRxgxLaxLaxLa nn   ...2211 n unknown variables        0 1          Domain n i ii Domain dxxgxLadxxR  One equation!!!
  • 11. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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.            0 1          Domain n i iij Domain j dxxgxLaxwdxxRxw 
  • 12. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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
  • 13. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Collocation Method   0jxR        0 1    j n i jii j xFxLa xR 
  • 14. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example Problem
  • 15. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The bar tensile problem   0/ 00 ' 02 2      dxdulx ux sBC xF x u EA
  • 16. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Bar application   02 2    xF x u EA      n i ii xaxu 1       xRxF dx xd aEA n i i i 1 2 2  Applying the collocation method     0 1 2 2  j n i ji i xF dx xd aEA 
  • 17. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 i ij dx xd EAk   2 2 
  • 18. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Notes on the trial functions • They should be at least twice differentiable! • They should satisfy all boundary conditions! • Those are called the “Admissibility Conditions”.
  • 19. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 x Sinx 2  
  • 20. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 x Sin l EA dx xd EA 22 2 2 2     faSin l EA                       1 2 42 
  • 21. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 f a 2 2 2 21 57.0 24 42            l x Sin EA fl xu 2 57.0 2     5.057.0 2  exact EA fl lu    0.19.00  exact EA lf ux
  • 22. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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
  • 23. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 
  • 24. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Bar application   02 2    xF x u EA      n i ii xaxu 1       xRxF dx xd aEA n i i i 1 2 2  Applying the subdomain method         11 1 2 2 j j j j x x n i x x i i dxxFdx dx xd aEA 
  • 25. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In Matrix Form                          11 2 2 j j j j x x i x x i dxxFadx dx xd EA  Solve the above system for the “generalized coordinates” ai to get the solution for u(x)
  • 26. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 x Sinx 2  
  • 27. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using the function into the DE: • Since we only have one term in the series, we will select one subdomain!                l x Sin l EA dx xd EA 22 2 2 2                                ll fdxadx l x Sin l EA 0 1 0 2 22 
  • 28. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 fl a 22 1 637.0 2 2            l x Sin EA fl xu 2 637.0 2     5.0637.0 2  exact EA fl lu    0.10.10  exact EA lf ux    fla l x Cos l EA l                            1 0 22 
  • 29. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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
  • 30. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com The Galerkin Method     0Domain j dxxxR           0 1    Domain j n i Domain iji dxxgxdxxLxa 
  • 31. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Bar application   02 2    xF x u EA      n i ii xaxu 1       xRxF dx xd aEA n i i i 1 2 2  Applying Galerkin method            Domain j n i Domain i ji dxxFxdx dx xd xaEA    1 2 2
  • 32. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In Matrix Form                             Domain ji Domain i j dxxFxadx dx xd xEA    2 2 Solve the above system for the “generalized coordinates” ai to get the solution for u(x)
  • 33. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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 x Sinx 2  
  • 34. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Substituting with the approximate solution:            Domain j n i Domain i ji dxxFxdx dx xd xaEA    1 2 2                             l l fdx l x Sin dx l x Sin l x Sina l EA 0 0 1 2 2 222     ll a l EA 2 22 1 2        EA fll EA f a 2 3 2 1 52.0 16  
  • 35. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Substituting with the approximate solution: (Int. by Parts)            Domain j n i Domain i ji dxxFxdx dx xd xaEA    1 2 2   ll a l EA 2 22 1 2        EA fll EA f a 2 3 2 1 52.0 16                  Domain ij l i j Domain i j dx dx xd dx xd dx xd x dx dx xd x     0 2 2 Zero!
  • 36. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com What did we gain? • The functions are required to be less differentiable • Not all boundary conditions need to be satisfied • The matrix became symmetric!
  • 37. Weighted Residual Methods Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com 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.