This course is an introduction to the finite element method as applicable to a range of problems in physics and engineering sciences. The treatment is mathematical, but only for the purpose of clarifying the formulation. The emphasis is on coding up the formulations in a modern, open-source environment that can be expanded to other applications, subsequently. The course includes about 45 hours of lectures covering the material I normally teach in an introductory graduate class at University of Michigan. The treatment is mathematical, which is natural for a topic whose roots lie deep in functional analysis and variational calculus. It is not formal, however, because the main goal of these lectures is to turn the viewer into a competent developer of finite element code. We do spend time in rudimentary functional analysis, and variational calculus, but this is only to highlight the mathematical basis for the methods, which in turn explains why they work so well. Much of the success of the Finite Element Method as a computational framework lies in the rigor of its mathematical foundation, and this needs to be appreciated, even if only in the elementary manner presented here. A background in PDEs and, more importantly, linear algebra, is assumed, although the viewer will find that we develop all the relevant ideas that are needed. The development itself focuses on the classical forms of partial differential equations (PDEs): elliptic, parabolic and hyperbolic. At each stage, however, we make numerous connections to the physical phenomena represented by the PDEs. For clarity we begin with elliptic PDEs in one dimension (linearized elasticity, steady state heat conduction and mass diffusion). We then move on to three dimensional elliptic PDEs in scalar unknowns (heat conduction and mass diffusion), before ending the treatment of elliptic PDEs with three dimensional problems in vector unknowns (linearized elasticity). Parabolic PDEs in three dimensions come next (unsteady heat conduction and mass diffusion), and the lectures end with hyperbolic PDEs in three dimensions (linear elastodynamics). Interspersed among the lectures are responses to questions that arose from a small group of graduate students and post-doctoral scholars who followed the lectures live. At suitable points in the lectures, we interrupt the mathematical development to lay out the code framework, which is entirely open source, and C++ based. Books: There are many books on finite element methods. This class does not have a required textbook. However, we do recommend the following books for more detailed and broader treatments than can be provided in any form of class: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, T.J.R. Hughes, Dover Publications, 2000. The Finite Element Method: Its Basis and Fundamentals, O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu, Butterworth-Heinemann, 2005. A First Course in Finite Elements, J. Fish and T. Belytschko, Wiley, 2007. Resources: You can download the deal.ii library at dealii.org. The lectures include coding tutorials where we list other resources that you can use if you are unable to install deal.ii on your own computer. You will need cmake to run deal.ii. It is available at cmake.org.
01.04. Constitutive relations
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From the course by University of Michigan
The Finite Element Method for Problems in Physics
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From the lesson
1
This unit is an introduction to a simple one-dimensional problem that can be solved by the finite element method.
Meet the Instructors
Krishna Garikipati, Ph.D.Professor of Mechanical Engineering, College of Engineering - Professor of Mathematics, College of Literature, Science and the Arts
Okay, we will continue with laying out the elements of our
linear elliptic differential equation in one dimension.
Okay?
Let's, let's look at our
differential equation again, so.
We've written it in the form d sigma
dx plus f function of x as we discussed,
is equal to 0 in 0, L.
Right.
Well, one thing you note of course is that I'm using total derivatives here.
I'm not using partial derivatives, that's because when we're doing one dimension,
doing things in one dimension, we don't even have time.
There are, there are no other variables, right?
So we're free do to this.
Of course when we go into multi,
multi-dimensional problems we will have to use proper partial derivatives.
Right, so though I've been calling it a partial differential equation sometimes
and a differential equation, really they are one and the same thing for
this one dimensional problem in a single variable.
Okay?
Just a few more things I need to state here.
One thing you will note is that I've been careful to say that
we have an open interval over which the differential equation is specified.
Okay?
So, that is an open interval.
Right, and you recall what this means?
It says that we are interested in looking at this differential equation
over the range 0 to L, right.
So, we have our domain here.
That is 0, that is L.
This is the domain over which we are interested in looking at this differential
equation.
But importantly we are seeing that the differential equation does
not apply at the point x equals 0 and x equals f, right.
So open interval remember that
this implies the domain between.
X equals 0 and x equals L, but
importantly excluding the end points.
X equals 0 and x equals L themselves, right?
This is important.
And what I want you to do is think of why that maybe, why is it that we specify
the differential equation, the partial differential equation or
just the differential equation, in this case excluding the point 0 and L.
And to help you think about the answer,
recall that we do need to know something about 0 and L.
If we don't have the differential equation specified at those points we
must have something else.
All right?
Well, you probably get the answer.
The an, the, the answer is that we do
already have boundary conditions specified at x equals 0 and x equals L.
If we were to insist that the differential equation also held at the boundary points,
we would actually have a, an over-constrained system.
Okay and this will often manifest itself in the mathematical or
numerical solution of problems where you by, by,
by essentially developing instabilities in your solution mattered
or, or, or situations where you just can't find a solution, okay?
So let me just state that.
This is because we have boundary conditions.
Right.
And those boundary conditions
are either on the primal field itself, or
on effectively on u, x, okay, at.
The boundary points.
Okay? And this is why the partial differential
equation is always specified over an open interval.
This, by the way, is the case for any differential equation, right?
And any, any, any, any partial differential equation,
is not specified at its boundary points.
Likewise, if you had an ordinary differential equation it would not
be specified at the end points of the interval of interest.
Okay? This is an important thing to note and,
and to recognize that it's not purely, matter of pace.
It really means something.
Okay? It means something mathematically and
physically as well.
Okay.
That's one thing I wanted to state.
The other thing I also want to state is I also,
I also want to talk about is the constitutive relations, right.
Constitutive relation.
And in this problem, we specify the constitutive
relation as being sigma equals E times u, x.
All right, in constitutive relation as,
the term suggests, it tells us something more about
the constitution of the domain of the problem, okay?
In this particular case, we started out with a primal field u,
but then we, introduced another field sigma.
Okay?
And, what we are seeing,
is that we know something more about the domain of interest,
and in our case the domain of interest is this one.
Sorry. That is our domain of interest, right?
We're seeing that, sorry,
it's not x here, it's 0, that is x, okay.
We are saying we know something more about the domain of this bar,
over the domain 0 and L, by specifying this particular relation.
Now we are trying to set this problem up using elasticity as a,
as an economical physical phenomenon, right?
In that context sigma is the stress okay and.
And we rewrite it here.
Sigma is the stress.
And it is related, not to the displacement itself,
but to the gradient of the displacement,
which in this setting of linear or
linearized elasticity is the strain, okay?
The strain is sometime given its own symbol.
And sometimes, or I guess pretty often
is denoted by epsilon, which is u comma x.
Okay, so in the context of this particular problem that we're looking at,
the elasticity problem, we are seeing that there is a relation between the stress and
this which is this field signal and the strain, which is the displacement reading.
Okay?
And that relation as you can see here is a linear one.
Okay, so when we see sigma
equals E times u comma x what
we have here is linear,
linearized elasticity.
Right, and it's obvious why we say it's linear or
linearized because that's a linear relation between the stress and strain.
In this setting, E is a physical constant,
that often gets called the Young's modulus,
or you may simply call it the modulus in a one dimensional setting.
Right, in, in a one dimensional setting, there is only one modulus,
might as well call it the Young's modulus.
Okay?
So this is the other bit that I wanted to state about this,
about the physical phenomenon that we're using for
now as canonical setting for the problem we're considering.
The probably very last thing I will do is restate essentially the same problem
but in the context of a different physical phenomenon.
Okay?
So let me state it here the 1D scalar,
because remember we said that in 1D
everything is essentially scalar.
So the 1D scalar 1D scalar
linear elliptic problem.
Also, models are heat or
diffusive mass transport.
Okay?
Let's suppose we are talking about heat transport, okay?
In that case we would be, we would state the problem as following.
We would say find u function of X,
which is a mapping, again, from this domain 0, L, open interval
to the one dimensional real space.
Okay?
Given, I, I will attempt to use different symbols here, okay?
And I, I, I'm going to use different symbols.
But because we're talking about the heat condition of,
of the heat transport problem we, I, I will also state here that what
we're talking about when we talk about u, the primary field, we,
we have the temperature in mind, okay, for, for heat transport.
Okay, so define the temperature U,
which is mapping from 0,L open
interval to R1, given the following,
okay, given, u0 and ug or.
Now, I want to reserve t for traction.
So for the other sort of boundary condition
that we could have in the context of heat
conduction, I'm going to use j, j bar.
Okay?
We're given these.
We have our forcing function,
f, and the constitutive relation.
Now in this case the constitutive relation that I want to use is going to be for
the heat flux.
And want to use a different symbol instead of sigma here.
Okay. I'm going to use the, the symbol j for
the heat flux.
J equals minus kappa u comma x.
Okay, all right?
Such that.
We have the following, okay?
Minus d,
dx of j equals f
of x in (0,L),
okay?
All right.
Now let, let me put down the boundary conditions.
And the boundary conditions are.
'Kay, u at zero
equals u naught and
either u at L equals ug.
Right? Now both these, u at 0 being u naught, and
u at L being ug would be given temperature conditions, all right?
Or at X equals L, we may
also have the condition
that j at L, okay,
equals minus j bar.
Okay?
All right.
Right, so this is how we may specify the heat conduction problem.
Now in doing this observe that what we have here is the,
is to be be thought of as the divergence of, of, of the heat flux.
Okay, and in fact G.
In this case,
would be the heat flux.
Okay?
What I've circled here would
be the divergence of the heat flux.
Okay?
In the con, in the context of heat conduction, what we would be seeing
would be that the, so, sorry this would be the negative diversions of the heat flux.
Okay. So the minus diversions will be heat flux.
What we would be seeing is that the negative of the diversion of the heat flux
in the context of heat conduction is the amount of heat you're supplying to
a particular little volume element in, in this domain of interest.
Okay?
I already said that you, you, that the conditions on you would be,
would be temperature boundary conditions.
So let me specify that also, okay?
So the Dirichlet boundary
conditions would imply in this case,
temperature on great conditions.
Okay?
And this, and finally,
the Neumann boundary condition
would imply a heat flux on recondition.
Okay?
In particular, this condition
that j at L equals minus j bar.
Okay?
This would have the following
significance that j bar,
as I've written it here would
be the heat influx at x equals L.
Okay?
So, if you look at this problem that we've just described here rather quickly and
you just make substitutions, right?
You just make substitutions instead of you know, instead of j,
you go back to sigma instead of kappa, you go back to e.
And instead of j bar, you go back to t.
Okay?
You will see that the problems are in fact very similar.
Mathematically, it is exactly the same problem.
But we know that this is a problem of steady state heat transport or
diffusive mass transport.
So, so, so this is important to, to note that there
is a multitude of problems that are described by the same
differential equation or the same partial differential equation.
It's just a matter of redefining our fields of interest boundary conditions
of interest and
giving them different ascribing to them different physical interpretations.
Okay?
And because we're interested in developing in this class
finite element methods that apply to a range of problems.
We will tend to focus more upon the underlying differential
equation of interest.
And, of, we will repeatedly make connections with the physical problems but
we will also bear in mind that rather than thinking of the method as
a problem to be developed as a problem applicable.
Only to structure mechanics and only to heat conduction,
it truly is a lot more general than that.
All right?
Okay.
So this is where we were and this is where we are going to stop for this segment.
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