Solved Example | Finite Element Method | Part#1
Summary
TLDRIn this educational video from E Academy, the presenter, DA, delves into solving a differential equation using the finite element method (FEM). The video focuses on a specific case where the differential equation involves a force function of -x squared. The domain is from 0 to 1, with both ends fixed, indicating no displacement. The objective is to find the displacement vectors across five nodes, divided into four linear elements, each of length 0.25. The video outlines the process of calculating the stiffness matrix for each element, highlighting the importance of the element's length and the shape functions in the local coordinate system. The summary sets the stage for the next video, which will address finding the force vector and its integration into the global system.
Takeaways
- 📚 The video is a tutorial on solving a differential equation using the finite element method, focusing on the application of general steps previously discussed.
- 🔍 The differential equation in question is a special case with 'a' specified as -1 and the force 'f' as -x^2, contrasting with the general form.
- 📐 The domain for the problem is from 0 to 1, with both boundary conditions set to zero, indicating no displacement at either end of the domain.
- 🛠 The target is to find the displacement vectors 'u' for an unspecified number of elements in the bar, which is later clarified to be four linear elements.
- 🔑 The bar is divided into four linear elements, and the displacement at each node needs to be determined.
- 📏 Each element's length is crucial for constructing the stiffness matrix, with each element having a length of 0.25 units.
- 📝 The script outlines the general structure for the stiffness matrix at the element level, using shape functions and local coordinates.
- 🔄 The stiffness matrix for each element is the same due to identical element lengths and material properties, allowing for a symmetric approach.
- 📉 The shape functions, crucial for the stiffness matrix, are derived from the shear function and involve taking derivatives to integrate.
- 🔍 The script mentions that the force vector will be discussed in a subsequent video, indicating a continuation of the tutorial series.
- 📺 The video concludes with an invitation to subscribe for more educational content and a farewell until the next video.
Q & A
What is the main topic of the video?
-The main topic of the video is solving a differential equation using the finite element method.
What is the differential equation in the video a special case of?
-The differential equation in the video is a special case of the general differential equation discussed in the finite element method steps, with 'a' specified as -1.
What is the force function 'f' in the differential equation?
-The force function 'f' in the differential equation is minus x squared (-f(x) = -x^2).
What are the boundary conditions for the differential equation discussed in the video?
-The boundary conditions are that the displacement at both ends of the domain is equal to zero, implying that the bar is fixed at both ends.
How many elements are used to divide the bar in the finite element method solution?
-The bar is divided into four linear elements for the solution.
What is the length of each element when the bar is divided into four equal parts?
-The length of each element is 0.25 units when the bar is divided into four equal parts.
What is the general structure of the stiffness matrix for an element level in the finite element method?
-The general structure of the stiffness matrix for an element level involves the shape functions (psi_i and psi_j) and is dependent on the local coordinates x_a and x_b.
What is the significance of the shape functions in the context of the stiffness matrix?
-The shape functions are crucial for determining the local stiffness matrix as they relate to the displacement and force within each element.
Why is it beneficial to solve the stiffness matrix at the local level?
-Solving the stiffness matrix at the local level is beneficial because of the symmetry and uniformity in the matrix structure for all elements, allowing for easier assembly into the global system.
What is the next step after finding the stiffness matrix for each element?
-The next step after finding the stiffness matrix for each element is to find the force vector for each element, which will be discussed in the next video.
How can the force vector be approached in terms of the local coordinate system?
-The approach to finding the force vector may be similar to the stiffness matrix by working in the local coordinate system, but the specifics will be discussed in the next video.
Outlines
📚 Introduction to Solving Differential Equations with FEM
This paragraph introduces the video's focus on solving a specific differential equation using the finite element method (FEM). The equation is a special case with 'a' specified as -1 and force 'f' as -x^2. The domain is from 0 to 1 with boundary conditions set to zero displacement at both ends, indicating a fixed bar. The goal is to find the displacement vectors 'u' without knowing the number of elements in the bar. The bar is divided into four linear elements, and the process involves understanding the general steps of FEM and comparing them with the general differential equation discussed in previous videos.
🔍 Detailed Explanation of Stiffness Matrix and Element Length
The second paragraph delves into the importance of element length in calculating the stiffness matrix for FEM. It explains that the length of each of the four linear elements is 0.25 units, which is crucial for constructing the stiffness matrix at the element level. The paragraph outlines the general structure of the stiffness matrix using the shape functions 'psi' and their derivatives, which are integral to finding the matrix for each element. The uniformity in element length and the linear nature of the elements allow for a symmetric and simplified approach to calculating the stiffness matrix for all elements.
Mindmap
Keywords
💡Finite Element Method (FEM)
💡Differential Equation
💡Boundary Conditions
💡Displacement Vectors
💡Shape Functions
💡Stiffness Matrix
💡Local Coordinate System
💡Global Coordinate System
💡Shear Functions
💡Element Length
💡Force Vector
Highlights
Introduction to solving a differential equation using the finite element method.
The differential equation is a special case with specific values for 'a' and 'f'.
Comparison of the given differential equation with the general form used in finite element method steps.
Domain of the problem is from zero to one with both ends having zero displacement.
The bar is fixed at both ends, implying no displacement.
Objective is to find the displacement vectors 'u' for an unknown number of elements.
Division of the bar into four linear elements for analysis.
Identification of nodes and elements in the system.
Importance of element length 'h' in calculating the stiffness matrix.
General structure of the stiffness matrix at the element level.
All elements are linear with the same stiffness matrix structure due to uniform length and properties.
Derivation of shape functions and their role in the stiffness matrix calculation.
Integration of shape functions to obtain the stiffness matrix for each element.
Same initial stiffness 'c' and length for all elements, resulting in identical stiffness matrices.
Assembly of element stiffness matrices into the global system.
Upcoming discussion on finding the force vector for each element in the next video.
Invitation to subscribe for more educational content on the finite element method.
Transcripts
hello everyone this is d a from e
academy in this video we will be solving
example
uh related to the finite element method
because previously what we have
seen and what we have discussed is about
the general steps uh that is in the
finite element method so now we will be
taking a differential equation and we
will trying to solve it with the help of
the finite element method uh by
specifying the general and the mandatory
things
so let's start so we want to solve this
differential equation
with the help of the finite element
method and this differential equation is
the special case of the differential
equation that we have seen differential
equation that we have seen in the steps
of the finite element method because
here a the small a that we have seen in
the previous videos is specified as -1
and you can compare it by yourself and
the force
small f here is minus x square because
usually the force was on the other side
of the equation that's why it's minus x
square in this differential equation so
to better understand you have to compare
it
with the general differential equation
that we have
discussed earlier in the finite element
method steps
so here is a differential equation and
the domain is
from zero to one the two boundary
condition that we have the first one is
that the displacement at the first end
is equal to zero
and the other boundary condition is
displacement at the other end is also
equal to zero so displacement at both
ends of the domain is equal to zero
which implies that the bar is
fixed
from both of the ends here there is no
displacement on both of the ends
and the target is to find
uh the displacement vectors u basically
uh and how we can find the displacement
vectors if we don't know how
many elements are in there
uh in this bar so four
and it should be linear as well
so we have to divide the bar in four
elements into four linear elements and
we have to find the displacement on each
of those
ends right so let's
specify this general bar with these
uh four element system
here is our bar and
both ends are fixed which implies that u
at zero is equal to zero and u at one is
equal to one a is equal to zero as well
so this is the first
node this is the second node this is the
third node fourth node and the fifth
node these nodes are in the full
coordinate system right
and how many elements do we have this is
the first element the second element the
third element and the fourth element as
per
the
question
right so this is the question that we
have now we are going to solve it uh to
find the
u to find the displacements on each of
these nodes
right in on all of these five nodes so
let's start the solution
first of all we know that
the length of each element matters a lot
uh because in order to put in the
stiffness matrix you know to find the
stiffness matrix now to find the force
vector it is really important to know
the h the length for each element
as we know this is the zeroth
x and this is
the x is equal to one
so if
there are four
linear elements then the length for each
element should be point
two
five right units
point two five here this is the point
two five right
and this is point five
this is point seven five
and this is one right so the length for
each element is 0.25
and we know the general structure for
the stiffness matrix uh that we have
seen in the previous videos so let's
write it down here for the stiffness
matrix
uh for the element level and x a and x b
right here uh the one the first and then
the second end because this is the x
axis so we have psi
that is also known as the three function
we have discussed in the previous videos
the link of all the previous videos will
be in the card and will be in the
description
here the x bar is showing that we are in
the local area here so this is for the
element level which implies that we are
on the local level psi i and i j are the
share functions
for the element levels for a particular
element so this is how we can find the
stiffness matrix for each of the element
now the now the
point here
that all of these elements
all of these elements are linear and the
stiffness matrix
the structure of the stiffness matrix
would be the same for each of these
elements weak in the structure because
we have the shape function the shape
functions won't change at each end plus
the length of the element is also same
the length of each element is 0.25 and
that there is no difference in the
length of each element that's why the
stiffness matrix for each element
for example if we find the stiffness
matrix for e1
the first element that will be the same
for e2 because
here we are in the local system so in
local level this would be 0 and this
would be 0.25 again for the e3 we are in
the local system this point would be 0
this point would be 0.25 and so on so
that is very positive and the beneficial
point for the stiffest matrix is that we
can solve it
by going into the local level and just
elevate them because there is no
difference there is symmetry in this
difference matrix
so let's solve it for the element level
one
before going into this we have to recall
this psi 1 inside of the shear functions
because the shear function is y minus x
bar by
the length of each element so x 1 minus
x bar divided by 0.25 and here this
would be x bar by zero point two five
right so this is psi one and this is psi
two we just what we are going to do
we're just going to plug it there and
we're going to take the derivative of
each of the side one because if we want
to take the derivative of this
it would be this
it would be 0 minus
[Music]
1 by 0.25 and this would be 1 by 0.25 so
this would be the derivative
for this shape function one and for the
save function too and this would be the
shape function one and two we were we
have just plug the values in here take
the integration
so after plugging the values and taking
the integration the output would be
so that would be the stiffness matrix
for each of the element and this is this
would be the same for all of the
elements because why because all of
these elements are sharing the same a
initial stiffness c that is minus 1 and
and the length for each interval
so this is same this stiffness is same
for the all element so this is the
stiffness matrix for each of the element
again this is in the local coordinate
system we have to assemble them lately
uh
to elevate them into the global system
but this is the main structure for the
element level one
we are just going to write it
on the side in order to use it later now
we are going to find out the force
vector
first thing that we have done here we
have solved the stiffness matrix for
each element level talk about the force
vector in the next video that how we can
find the force vector for each element
and is it possible to
solve the force vector just like the
stiffness matrix on the local coordinate
system or we have to change the
technique so this is for now look for
more such videos then you can subscribe
this channel to watch more upcoming
videos we will meet in the next video
till then take care goodbye
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