What are weighted residual methods?
How to apply Galerkin Method to the finite element model?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/TopicX+Approximate+Methods+-+Weighted+Residual+Methods
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
A Presentation About The Introduction Of Finite Element Analysis (With Example Problem) ... (Download It To Get More Out Of It: Animations Don't Work In Preview) ... !
General steps of the finite element methodmahesh gaikwad
General Steps used to solve FEA/ FEM Problems. Steps Involves involves dividing the body into a finite elements with associated nodes and choosing the most appropriate element type for the model.
What is a multiple dgree of freedom (MDOF) system?
How to calculate the natural frequencies?
What is a mode shape?
What is the dynamic stiffness matrix approach?
#WikiCourses
https://wikicourses.wikispaces.com/Lect04+Multiple+Degree+of+Freedom+Systems
https://eau-esa.wikispaces.com/Topic+Multiple+Degree+of+Freedom+%28MDOF%29+Systems
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
A Presentation About The Introduction Of Finite Element Analysis (With Example Problem) ... (Download It To Get More Out Of It: Animations Don't Work In Preview) ... !
General steps of the finite element methodmahesh gaikwad
General Steps used to solve FEA/ FEM Problems. Steps Involves involves dividing the body into a finite elements with associated nodes and choosing the most appropriate element type for the model.
What is a multiple dgree of freedom (MDOF) system?
How to calculate the natural frequencies?
What is a mode shape?
What is the dynamic stiffness matrix approach?
#WikiCourses
https://wikicourses.wikispaces.com/Lect04+Multiple+Degree+of+Freedom+Systems
https://eau-esa.wikispaces.com/Topic+Multiple+Degree+of+Freedom+%28MDOF%29+Systems
How to find the roots of Nonlinear Equations?
Newton-Raphson method is not the only way!
How about a system of nonlinear equations?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Roots+of+Nonlinear+Equations
How to handle a system of initial value problems using Runge-Kutta method?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
Neural networks for word embeddings have received a lot of attention since some Googlers published word2vec in 2013. They showed that the internal state (embeddings) that the neural network learned by "reading" a large corpus of text preserved semantic relations between words.
As a result, this type of embedding started being studied in more detail and applied to more serious Natural Language Processing + NLP and IR tasks such as summarization, query expansion, etc...
In this talk we will cover the intuitions and algorithms underlying word2vec family of algorithms. On the second half of the presentation we will quickly review than basics of tensorflow and analyze in detail the tensorflow reference implementation of word2vec
How to create and solve finite element models?
Application to 2nd Order Differential Equations!
#WikiCourses #FEM
https://wikicourses.wikispaces.com/TopicX+Element+Equations
How to derive the finite element model using the stationary functional approach?
Application to bars and beams!
#WikiCourses
https://wikicourses.wikispaces.com/TopicX+Stationary+Functional+Approach
https://eau-esa.wikispaces.com/Topic+Stationary+Functional+Approach
What is numerical differentiation?
What is finite difference?
How to apply that to boundary value problems?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/Topic+Boundary+Value+Problems+-+Finite+Difference
Solve nonlinear equations using bracketing methods: Bisection and False Position
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Roots+of+Nonlinear+Equations
Word embeddings have received a lot of attention since some Tomas Mikolov published word2vec in 2013 and showed that the embeddings that the neural network learned by “reading” a large corpus of text preserved semantic relations between words. As a result, this type of embedding started being studied in more detail and applied to more serious NLP and IR tasks such as summarization, query expansion, etc… More recently, researchers and practitioners alike have come to appreciate the power of this type of approach and have started a cottage industry of modifying Mikolov’s original approach to many different areas.
In this talk we will cover the implementation and mathematical details underlying tools like word2vec and some of the applications word embeddings have found in various areas. Starting from an intuitive overview of the main concepts and algorithms underlying the neural network architecture used in word2vec we will proceed to discussing the implementation details of the word2vec reference implementation in tensorflow. Finally, we will provide a birds eye view of the emerging field of “2vec" (dna2vec, node2vec, etc...) methods that use variations of the word2vec neural network architecture.
This (long) version of the Tutorial was presented at #O'Reilly AI 2017 in San Francisco. See https://bmtgoncalves.github.io/word2vec-and-friends/ for further details.
What is interpolation?
How to interpolate a polynomial through a given set of data?
General approach, Newton method, Lagrange method
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Interpolation
Why would a company hire a trainer? To produce a change. The trainer by default is
an agent for change. Regardless of any results a trainer may accomplish, the bottom line is a
measurable change in employees’ performance.
What is marketing?
How to find out about customers?
How to reach them?
How to get them to know about you?
What is a product life cycle?
How about Marketing strategies?
Learn more ...
http://AcademyOfKnowledge.org
Brief description of current state of drones and some future challenges.
The presentation is prepared for delivery in the "Interact with Today's World" conference held in Bibliotica Alexandria 5-6 August 2016
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
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
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.
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
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
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.