Solved Example | Finite Element Method | Part#1

Summary

In 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

00:00

📚 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.

05:01

🔍 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.

Keywords

💡Finite Element Method (FEM)

The Finite Element Method is a numerical technique used for solving differential equations that have a wide range of engineering applications. In the context of this video, FEM is the primary method being discussed for solving a specific differential equation related to structural analysis. The script mentions that the equation is a special case of those typically solved using FEM, highlighting the method's versatility and application in various scenarios.

💡Differential Equation

A differential equation is an equation that involves derivatives of a function with respect to one or more variables. In the video, a specific differential equation is presented, which is central to the problem being solved. The script specifies that the equation has a particular form with 'a' as -1 and 'f' as -x^2, indicating the unique characteristics of the problem at hand.

💡Boundary Conditions

Boundary conditions are constraints applied to the solution of a differential equation at the limits of the domain. In this video, the boundary conditions are that the displacement at both ends of the domain is zero, indicating a fixed bar with no movement at either end. These conditions are essential for setting up the problem correctly in the FEM framework.

💡Displacement Vectors

Displacement vectors represent the movement or deformation of a structure under load. The script mentions finding the displacement vectors as the target of the analysis, which is a critical output in structural problems solved using FEM. These vectors provide insight into how the structure behaves under the given conditions.

💡Shape Functions

Shape functions are mathematical functions used in FEM to describe the shape of an element within a mesh. They are essential for constructing the stiffness matrix and for assembling the global system of equations. The script refers to these functions as 'psi' and discusses their role in the local coordinate system of each element.

💡Stiffness Matrix

The stiffness matrix is a key component in FEM that relates the internal forces within an element to its deformation. The script explains how to construct this matrix for each element, emphasizing its importance in the analysis. The matrix is derived from the shape functions and the properties of the material, such as the coefficient 'a' in the differential equation.

💡Local Coordinate System

A local coordinate system is used to describe the properties of an element within a FEM model independently of the global system. The script discusses how the stiffness matrix is initially calculated in the local system and then assembled into the global system. This approach simplifies the process of solving the equations for each element.

💡Global Coordinate System

The global coordinate system is the overall reference frame for the entire FEM model. After calculating the properties of each element in their local systems, these are assembled into the global system to solve for the overall behavior of the structure. The script mentions the process of elevating the local stiffness matrices into the global system.

💡Shear Functions

Shear functions, in the context of FEM, are related to the deformation of an element due to shear forces. The script mentions these functions in relation to the shape functions, indicating their role in calculating the derivatives needed for the stiffness matrix.

💡Element Length

The length of each element is a critical parameter in FEM as it affects the calculation of the stiffness matrix and the force vector. The script specifies that the bar is divided into four linear elements, each with a length of 0.25 units, which is used to define the local properties of each element.

💡Force Vector

The force vector represents the external forces acting on the structure within the FEM model. The script indicates that the force vector will be discussed in a subsequent video, suggesting that it is a separate but equally important aspect of the FEM analysis process.

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