In this course we are going to look at the
computational methods in design and manufacturing. As you can see the word computational methods
indicates that we are going to look at the way in which problems in design and manufacture
will be solved using numerical as well as computer methods. About 40 to 50 years back people were solving
complex problems or at least trying to solve complex problem using what are called as analytical
methods. But as the material development took place,
as the complexity, both geometry as well as shape of the components, became quite complex,
people were not able to use the analytical methods and they started using numerical methods. We will go into the details of the numerical
methods in this course. The three most popular numerical methods being
what is called as final difference method, finite element method and what is called as
the boundary element methods. Of these three methods, today finite element
method is extremely popular and in this course we are going to concentrate on finite element
method. In fact, if you look at the number of softwares
that are available today on finite element method you would see or at least you would
hear about numbers ranging from about 400 to 500. In other words, if you include all the commercial
softwares and softwares that are available in universities you would count even up to
500; it is a phenomenal number. In other words, it has just exploded over
the past 10 years. If you look at very popular commercial softwares
then at least there are about 10 to 15 of them which are used in the industry and in
the educational institutions and research institutions. There are number of reasons why this has happened. One of the major reasons is that the industry
is today tuned to the fact that finite element methods can be used to solve lot of practical
problems. That has given an impetus for this kind of
explosion in the market place of finite element method. The other reason, more important reason is
that computers have become now very powerful and are affordable. Lot of companies today can buy high end workstations
to solve very complex finite element problems which will be very useful for them to develop
their product. That is the second and more important reason
why finite element method has become very useful. Though finite element method started about
40 years back, 1950’s rather 56-57 and Boeing Aircraft Company really launched a project
to determine the stresses in their air craft wings, the method actually took off in the
70’s; late 70’s, early 80’s. Initially up to about 60’s or 60-61, there
were about 15 papers that were published in this field of finite elemental method. Today we have about more than 20 to 25000
papers that have been published. That shows the explosion in the research front
as well. Very complex problems can be solved using
finite element method. Both the theory as well as the implementation
has developed considerably over the past 10 years. The method as such is not very difficult to
understand but the theory is more difficult. In other words the crux of the whole issue
is very simple; it can be explained in a matter of 10 minutes and the whole philosophy of
finite element method rests on what is called as the divide and conquer. Now let us look at a simple
problem and let us see how we can do this problem. Let us say that I have a sheet. My whole idea is to find out what is the area
of this sheet? What is the area of this sheet? How do you think you can do this problem? How do you think you can do this problem,
the area of a very complex sheet? Exactly. We divide this into known geometrical shapes. For example I can say that I will divide this
into 1, 2, 3 and 4. In other words this property called area for
some standard shapes are already known to you. In fact the calculus behind it did not come
to your mind that I have to integrate and so on. But what came to your mind is that there is
a relationship between this shape and formula which I know already. For example you know the formula for a square
or a rectangle, a triangle and semicircle and so on. So, the first thing is divide the given area
into a number of smaller geometrical pieces. What is the second step? The second step is to calculate the area of
these individual geometric pieces. The first one is divide into geometrical shapes
which automatically means that into shapes whose area you know. The second step is to calculate the individual
areas of the shapes. Let me call this as Ai’s, where A1, A2,
A3, A4 indicates the areas of this individual geometric pieces. What is the third step, what is the third
step? Assemble them; it is not add, but assemble. Why do I use the word assemble? Because 1, 2 and 3 can be added and fourth
is a hole and so it has to be subtracted. So, the third step is to assemble these areas. Once I do this or do the addition I get the
complete area of this particular complex sheet. The problem of finite element method is very
simple. If I wanted to calculate the complete area
in one go it would have been difficult for me. I am not saying it is impossible, it would
have been difficult for me. There are number of ways in which I would
have done the same problem. I could have divided this into many more number
of squares by putting some sort of grids here and approximating the grids such that I ultimately
cover the entire area. That is one technique but another technique
is what we did here. Another technique, more difficult technique
as I said, is to use more theorems of calculus and calculate independent or the total areas. The finite element method relies on techniques
or the concepts which we used here. It is not of course used to calculate areas,
but things like displacements, temperatures, velocities, stresses and so on. What we will do is before we even go through
what finite element means we will look at a few problems in order to understand its
application areas or in other words let us now understand why we are going to study this
course? Once we know that, it would be possible to
appreciate the importance of finite element analysis as well as understand the things
that we should learn in this course in order to apply whatever you have learnt here in
the practical as well as in the research fronts. I know there are lot of research students
also here. What I am going to do is to give you a few
examples now, go back to the theory of finite elements, explain the theory of finite element
analysis, come back to these examples and show you why we have done these problems in
this fashion; then go back to the theory and expand the theory in such a fashion that it
would be possible for you to attack many of the complex industrial problems as well as
research problems. I am going to now summarize some of the work
that we have done over the past 2 to 3 years and these projects have been sponsored by
different companies. You can also read the companies that have
sponsored these projects which clearly show the practical significance of this technique. Let us look at the first problem and let us
now see what we have exactly done with this problem. The first problem is the stress analysis of
a LP rotor, low pressure side rotor done for NTPC, New Delhi. You can see that our major aim is to determine
the stresses in this LP rotor and the next slide will show you that this LP rotor can
be looked at as a solid model. You can also see a small marking in the LP
rotor. Actually what happened was that during the
manufacture of this LP rotor, there was a small mistake that was done during CNC programming
and instead of machining LP rotor as it is, the LP rotor was machined inappropriately
or not correctly. So, a small hole was created which you can
see in that particular slide which is been marked by a small ring. You can see that in the next slide in a much
more close up fashion. You can see that there is a small hole there. The problem is that this small hole or defect
which has been created due to improper manufacturing, it so happens that it is at a place where
the stresses are also high. But this rotor being very expensive, it is
about 10 crores and the lead time to manufacture this rotor is nearly 6 months which means
that a project worth about 3000 crores; a particular thermal power project worth about
3000 crores is going to come to a stand still, a screeching halt for 6 months which means
that it is not acceptable. The whole question here is that can we use
a predictive technique a computational method in order to find out whether this mistake
can be tolerated. If not can we modify this rotor which has
a defect in such a fashion that we do not compromise on the safety aspect of the system. So, that is the question. If we can use finite element analysis in order
to pass this rotor, then we are going to save very valuable time. We did this and we can see from the next slide
how what is called as a finite element model looks like. We took into account the centrifugal forces
which are very, very important; in fact the rotor is designed for centrifugal forces and
you can see those arrow marks there which indicates the position at which the blades
are attached. In the next slide you can see the stresses. This is for an existing good LP rotor. The whole idea here is that the good LP rotor
is analyzed first so that the stresses are determined and when we do for a new or this
modified rotor the stress levels that you are going to see here are going to be the
same in that particular modified rotor as well. If we do that then we are not compromising
on the quality or the safety features. Imagine that it is impossible for you to make
this kind of modifications in the actual piece and try it out. In fact we did about 4 or 5 modifications
before we came to that modification which is going to be helpful or which is going to
be implemented. It is impossible for you to do. As I said it is 6 months lead time and hence
computational method is extremely useful in order that such a technique can be used. You can see that in the next slide. The modified model is better seen in the zoomed
up view of the next slide. You can see that a small portion has been
removed axially, completely; it is an axial piece; axially and again stress analysis has
been conducted. The next slide shows the results of these
stresses. Though The stresses are not very significant
for you at this point of time. Because we have not talked what are called
as Mises stresses, you may not understand right now what are Mises stresses and we are
going to explain those things in the course, which we are going to see now. Nevertheless, if you are able to see the numbers,
you can see that the numbers are exactly the same, are almost the same which means we have
modified it with absolutely total control over quality. So that is a very important example. We were able to do the whole thing in a matter
of about 5 days. What would have taken not even months but
years we were able to do that. We did about 4 iterations; we were able to
come to this conclusion in matter of about a week maximum. That is the power of computational methods
and that is the power of finite element method. Let us look at the next problem which has
been done to another company. Though the first problem was critical, was
very, very critical problem, because as I told you it involves 3000 crores, the problem
as such is not difficult because the problem involves only a linear elastic analysis; one
of the simplest kinds, not very difficult, geometry very manageable. So, the problem is a delight to do, it is
well behaved. On other hand, look at this problem. We are now going to talk about a radial tyre. This is the section of a tyre. It is very, very complex component, extremely
complex due to so many reasons. What are the reasons? The geometry itself is very complex; in a
minute we are going to see how complex it is. When we did the solid model of this piece
we had to put nearly 21,000 surfaces in order to define this particular piece. Number one geometry is extremely complex;
not only just external geometry, geometry is also defined by certain internal modifications. You can see that there are some reinforcements
that are given. This tyre is called a radial tyre. You can see that there are some steel belts
or reinforcements that are given in this tyre in order to withstand the extreme conditions
that this tyre would be subjected to, as it is in a vehicle. The second point that is important is not
only the external geometry but the internal geometry or the internal arrangement of material. The third point that is important is that
the material that is used to make this tyre, the material is a non-linear elastic material;
it is a rubber, it is a non-linear elastic material. The behavior of the material is very complex. Again towards the end of the course we will
also study how to tackle such difficult problems, but right now all of you I am sure know that
rubber is a non-linear material or it has a non-linear material behavior, so that is
the third complexity. The fourth is that the tyre is in contact
with a number of components. It is in contact with the rim, it is in contact
with the ground and so on. Contact makes life really miserable when you
look at the computational aspects and rubber or this tyre rather which is a rubber component
has contact as well. It has all the complexity; name it, it has
it. On top of it, it has to be looked at both
as 2D as well as three dimensional problems. Why is that we do it? We will understand looking at the results. Let us look at the first step that we have
done with the next slide. This is the solid model of the complete tyre. You can see that, as I told you, it has nearly
21,000 surfaces. It has been meticulously done and now we know
as to how or what is the procedure that has to be followed in order to create this particular
tyre. Though we took about a month’s time to do
this work, if you ask me today, matter of about 2 to 3 days it will be possible to produce
such a tyre. Let us look at the next slide and see what
we have done there. What we have done is what is called as axi-symmetric
analysis in order to predict the deformation during inflation; during inflation. What do I mean by that? I mean to say that it is important to realize
or understand how the tyre is going to behave due to the inflation pressure. This particular optimization of shape is very
important for the tyre industry. Let us look at the next slide and see what
we can infer from this slide. You can see those red marks there. I hope that is very clear. There is one big red area, a circle which
is seen to your left as well as a red streak of line that you see. That red area, that circle is called as a
bead; that is what is called as bead. You can see it here. You can see here the bead and the red streak
is actually a reinforcement that you can see. That is the reinforcement that is given and
what it shows? What does this result show? The result clearly shows that the stresses
are quite nicely taken by these reinforcements during inflation so that the rubber does not
take that kind of stresses but this reinforcement takes the stresses. As all of you know the stress levels to which
a steel bar can be raised to, is much higher than that of the rubber. But we need rubber due to various reasons. You can see also from that particular slide,
a small area near this place, near this place where the stresses are higher than other locations. Again you can also see in these locations
there are reinforcements. There are reinforcements here as well. These reinforcements are again steel wires,
very small steel wires which run at around 22 degrees or 20-22 degrees. The whole exercise in this particular example
is to find out what is the optimum angle of these steel belts such that the inflation
pressure can be very nicely handled; such that the pressure that is developed due to
contact of this tyre on to the ground is very nicely dispersed and so on. That is the first part of the story on tyre. Now let us look at the second part of the
story. Let us see the additional things that can
be done. This is the three dimensional model, a complete
three dimensional model of the tyre. You can see that the tyre, complete tyre is
not analyzed but only half the tyre is analyzed; only half the tyre is analyzed. Why because we are taking care of correct
symmetry. So, only one half is analyzed. We put what are called as boundary conditions
in such a fashion that we take into account exactly; there is no approximation. Please note that there is no approximation
but we put boundary conditions in such a fashion that we exactly, theoretically take into account
the complete piece. Let us go back and see in a more, say closer
fashion how exactly this model looks like. Let us see that again. You can see that when we look at this model,
we can look at all the grooves in it. In fact this particular slide gives the result
due to contact and so on. In fact it would be better again to go back
to the previous slide and to look at it much more closely. There are some very interesting things that
you can observe. You see a small rectangular piece at the bottom
where this tyre rests. That actually depicts the ground; that rectangular
piece there depicts the ground. There is one center dot; dot at the center. Are you able to see that clearly? That dot at the center actually depicts the
rim. When compared to the tyre whose deformations
are quite high, the rim as well as the contact surface is considered to be rigid. In other words it is important to understand
that in finite element analysis it is possible to combine a very rigid piece like a rim,
treat it as rigid, as well as a highly deformable piece like rubber. It does not mean that I cannot look at contact
between two deforming surfaces. But it is advantageous many times to make
such engineering assumptions and say that look any way the rim deformation will be very
small when compared to the rubber deformation and I am interested in the rubber deformation
and hence I will make the rim deformation to be negligible when compared to this and
study the problem. That is what makes a model handleable. The question that you may ask is why do you
want to make such assumptions? Why is that you want this rim to be rigid? Why not you make that also as deformable? Yes, I can make that deformable. But the problem becomes huge, handling becomes
difficult in the sense that this has to be handled in a computer and the computer resources
that are required are quite enormous for such a problem. Hence we make some assumptions like that. You will notice as we go along this course
that computer resources, though we have very powerful computers no doubt about it, computer
resources are going to be one of the major questions that have to be answered for this
kind of high end problems. This particular problem took nearly 40 hours
in a very powerful system. Yes, this was possible today with 40 hours;
may be 5 years hence this may be possible within 4 minutes and 5 years before I would
not have been able to do this problem. It is a question of a compromise today with
the powerful systems that are available; yes, they are powerful. Can we solve very, very complex problems and
what are the types of problems that can be solved? Many rubber companies or tyre companies rather
still use super computers to solve many of those problems. Since we do not possess so many super computers
in India we have to make some engineering judgment; we have to make some assumptions
in order to do such kind of problems. Why is it that this problem becomes important? Let us look at the result to understand why
we have to do this problem and that is shown in the next slide. This is the contact pressure; this is the
contact pressure distribution. You can see an area which has a different
colour when compared to the rest of the areas and this colour changes. You can see green, blue, sorry green, yellow
which merges to red. This area is actually in contact with the
ground and the contact pressure distribution or the foot print, as it is called in the
tyre industry, is very important for them to understand how or why wear takes place
or how the design of the tyre can be modified and how the pressure can be much more uniformly
distributed. I do not know how many of you have really
seen radial tyres. If you go and look at new cars say for example
a Zen car and look at the tyre, a layman would always feel that the air is not enough. You would see the deformation to be more. You would always have a lurking fear is it
ok? Is there a problem? You go and check up the air; you would see
the inflation pressure to be perfect but still you would feel that the deformations are higher
but you need not worry about it. The point there is that the pressure has to
be much more uniformly distributed and you know what happens if I just have a line contact. If the inflation is high or if the pressure
is concentrated in one or two areas, small area, then obviously the pressure will be
high. Why? Anyway I am going to take the total load of
the vehicle. Hence pressure distribution is a very important
input for the design of the tyre. In today’s competitive world it is almost
impossible for me to first design a tyre, test it out, make a mould and test it out. You have to make the manufacturing of this
tyre and in order to do that you have to look at or you have to look at ways and means in
which you can design, develop, test moulds and so on. It is a question of about 4 to 5 months. I cannot afford to have about six different
varieties of moulds, then test them, then make the tyre, then again go back and test
this prototype, go and modify them, remake another tyre and so on. You know it is impossible. By that time the car model itself would have
changed. Computational methods are very important in
order to optimize such products which are very, very complex and this is not the only
output that you can get. The previous slide shows a very interesting
output as well. Let us look at that. It is clear for you that the treads, as it
is called, you can see that there are tread patterns here. The tread patterns are shown in this particular
tyre. It is very important to know how this tread
pattern behaves during the contact, when it is in contact with the ground. Let us go back and see that. You can see that the picture clearly shows
that a small gap develops towards the top of the tread when compared to the bottom. In other words the bottom of the tread gets
closed or buckles whereas there is a small opening at the top. This may lead to problems when the vehicle
goes over water. The water may get in. There can be some sort of a hydrodynamic lubrication
and so the vehicle may start skidding. There should be ways and means by which you
can eliminate this water which is sitting inside. This is called as aqua planning. You can get very important input from this
kind of deformations that you see in this particular output. From a very simple component to a very complex
component like tyre, it is possible for us to look at the computational technique in
order to optimize our design. Let us go and look at some other example. We will see what are the other things that
can be done with finite element analysis? We are going to look at a much more complex
component geometrically, very complex component geometrically and what we are going to do
here or at least what we are going to see right now, we will come back to this example
towards the later part of the course, but what we are going to see right now is that
a complex piece like a railway coach can be done in a computer or can be modeled in a
computer. The whole of manufacturing process which is
used to make this particular coach can be simulated in the computer beforehand and we
can learn so many things by doing the simulation. What is that we are going to learn? Let us look at the first slide and see what
it teaches. Let us now see the model. This is what is called as a computer aided
design model of the entire coach. See how realistic it is. Many people have asked me if it is a photograph
or is it really a model. It is so very realistic. There are about 250 major components which
go to make this coach, which actually takes the stress and strain of travel, of payload
and so on. Our model is put into this particular model. You can see that again and you see that the
whole coach is divided into three different areas; the roof which is shown as white, the
side wall which is, of course all of you know it looks, red and the under carriage on which
all of us sit and travel in the train. One of the problems that our Indian coaches
have is the lack of very good aesthetic appearance. All of us have traveled by train and maybe
you would have all noticed some undulations on the sides of the coach. Have you all seen it? Especially when you go to the station and
in the evening times or night times when there is light, it is very clear that there are
undulations on the sides. Though if you look at the performance of the
coach this kind of undulations are not going to affect but aesthetically does not give
a very nice look. If especially when you want to import or sorry
export this kind of coach people are not going to accept with this kind of undulations. We have to take care of these undulations. These undulations are unfortunately part of
the manufacturing process or it comes out because of the way coaches are manufactured. It is not very easy to change that. I can say that it is because of the way in
which it is manufactured; it does not mean that I can give you a complete alternate way
of manufacturing basically because you should understand that these sheets which have undulations
have a thickness of around 2 mm; see how thin it is, 2 mm thickness. Basic manufacturing process is a welding process. Due to assembly and welding these undulations
creep in during manufacturing. Yes, the first part is understood; but there
are so many intricate details that can be looked at during manufacturing and which have
to be understood and which have to be segregated in order to understand why this kind of undulation
is coming or other words is it possible to pinpoint out of some 10 reasons what is the
most important reason which causes this kind of undulations? That is not very easy to do in a shop floor. We have in fact tried it out and it is not
very easy to do it in the shop floor. On the other hand it is possible to do that
in a computer using finite element method. Let us go and have a look at that and see
whether it is possible to predict this kind of manufacturing process as well as the undulations. Let us look at that in the next slide. This is the result. We are not looking at how to do it; that we
will do it later, but this is the result of the analysis. You can very clearly see the undulation step. In fact the next slide shows it in a much
more closer view. You can see the zoomed up view of that in
the next slide. Look at that; though it is a magnified view
you can see that the undulations are very clear; undulations are very clear. These undulations are due to the assembly
procedure itself. As I told you it is impossible to change that. But there are ways and means by which you
can change the design and reduce these undulations. More importantly it is possible to predict
such very, very intricate details that you come across during manufacturing in computational
methods or by using finite element method. Let us now look at the next problem which
is again a very interesting problem. The next problem that we are going to look
at is what is called as the contact analysis of a rail wheel. All of us know railway vehicle coaches and
all these things very well and all of us know that the rail wheel is subjected to lot of
stresses as bad or as good, however you wish to call it, as that of the tyre. Though, geometrically the railway wheel does
not seem to be as difficult as that of a rubber tyre, there are other difficult things that
you would also notice in the analysis of a railway wheel. The analysis of the railway wheel is very
well documented; the analytical procedures are very well documented. There are standards which are used to design
the rail wheel. But in spite of all these things, there are
quite a few problems. If you follow the newspapers probably you
would have heard that there were some accidents, railway accidents last year in Euorpe. In Germany there was a railway accident and
probably if you keep your ears open you would have heard about some rail wheel failure in
certain metros and so on. In other words rail wheel failures are still
common and in India, there have existed some failures in metro cities and the whole idea
here is to check up, to find out what is the reason for such failures. Can we see or can we really predict them by
using finite element analysis and take corrective action? Though surely the railway wheel are very,
very safe but today we no more talk about percentage, we talk about parts per million;
failures in terms of parts per million. In other words, we do not even tolerate one
failure out of million. We have to be very safe especially in the
case of air craft; in the case of railways and all these things we have to be extremely
safe. You need not worry about that. When I say that there have been failures,
it does not mean when you go and sit in a vehicle you have always a lurking fear that
whether the wheel will fail or whether the coach will fail and so on; not like that. But nevertheless, as engineers, we have to
make this kind of public transport systems as safe as possible and hence it is better
that we understand what exactly happens in this kind of wheels? Let us look at that in this particular slide. This is actually the temperature distribution
during breaking. In many of the vehicles or emu’s or electric
coaches which run in metros, you would have noticed that there are number of stations
and that braking in these vehicles are quite often carried out; all of you know that. From one station to the other, the guy just
accelerates and then he has to decelerate and apply brakes. In such circumstances, it is very important
to understand the temperature distribution during braking and this can be solved using
finite element analysis. It is not only the stress, we saw that stress
can be determined; not only deformation which we saw in the previous experiment or previous
example that deformations of the coach can be determined. The stresses are obtained from deformations. We are going to see that in this course; deformation
is the first step towards going to the stress but deformation per say are important in certain
problems and now you see that temperatures can also be predicted. You see in this particular slide that the
temperatures can be predicted. Let us worry about how to do it later but
let us only look at the result. You obviously see that the outer side of the
wheel has maximum temperature but it does not mean that the outer surface has maximum
stress. We will discuss this problem again and look
at what are the complexities that are involved in this problem. But at this point of time it is important
to understand that from this temperature it is possible for me to predict what the stresses
are or in other words using finite element analysis I can also look at coupled problems;
temperature giving rise to stresses and vice versa; stresses giving rise to temperatures
as well. These kind of coupled problems can also be
looked at using finite element analysis. Let us look at the next slide for this railway
wheel. You can see again that the contact analysis
of the railway wheel becomes important. Apart from temperature, the contact analysis
or the contact or the stresses, the strains that exist because of contact of the railway
wheel with the rail also becomes important and that can be seen in a very close up view
in the next slide. Look at the plastic strain that exist very
near the place where the wheel is in contact. For this particular problem not only are we
determining the displacement, we are determining the temperatures. We are also determining the contact plastic
strains which may become very important to look at there and so on. So, many, many important inputs, important
from a design point of view can be obtained by using techniques like finite element analysis. We will come back to this example and further
examples in the next class. But, before that let us see the types of inputs
that we give in order to do these problems. Can someone come out and say what are the
type of inputs that are required in order to solve this kinds of problems? Boundary condition; yes, boundary condition
is important but before that, no; we are not looking at finite element input; input from
a designers point of view. Dimension that means geometry; that is number
one, the geometry becomes important. That is the first input. The second input is the type of force that
is acting on this geometry. What is the type of force, how much and so
on. The forces are important. Thirdly as you correctly said what are the
boundary conditions? Usually we call this as BC or boundary conditions. These are the major inputs. It is easily said than done, there are some
complexities in each of them. As we go along we will see that when we look
at a complete model, we have to properly take into account each of these things. We are going to make some assumptions. As I clearly said that the assumptions are
important in certain problems as that of the tyre. As we go along we will see how we can take
into account these things to complete the problem. Yes, the other problems being what the type
of mesh is, how big it is and all that. But we have not yet talked about the mesh. We have only looked at what is a problem,
what is the type of solution or output that you can get using finite element analysis
in order that it will be useful to the design; that is all we have got. We have not yet talked about the correct or
the complete technique or correct way of doing this problem; that we have not talked about. That requires lot of input from this course
and that is what we are going to do in this course. We will stop at this point in this class and
we will continue on certain other examples in the next class.
