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.
02.09. The finite dimensional weak form as a sum over element subdomains - I
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來自 University of Michigan 的課程
The Finite Element Method for Problems in Physics
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從本節課中
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In this unit you will be introduced to the approximate, or finite-dimensional, weak form for the one-dimensional problem.
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Krishna Garikipati, Ph.D.Professor of Mechanical Engineering, College of Engineering - Professor of Mathematics, College of Literature, Science and the Arts
Okay, let's continue now.
You recall that at the end of the last segment
we've reached as far as observing that
in order to proceed with our with,
with computing the terms needed in our weak form,
we needed to figure out a way to write out uh,x,
which is sum over A.
NA,xi, xi,x,
dAe, and
the same for wh,x.
Okay?
And, and in particular the question is how do we do this term, right?
Simply because this one is easy, right?
This this one is easy.
Right, you already know how to compute the derivatives with respect
to xi of the basis functions, 'kay?
So let's look at how we compute xi,x.
And in order to do that, let's just recall that what we've set up for
ourselves is a description where omega e,
right, and some arbitrary point x in omega e,
where xe and xe plus 1 are the nodal positions, right?
Some arbitrary point in x is always thought of as being
obtained as a mapping from this by unit domain.
Okay, now if we could figure out this mapping
we may have a chance of figuring out what xi,x is.
And what we're going to do now is do exactly that.
We're going to define this mapping, okay?
So the, we are going to define the mapping.
To omega e from
omega xi.
It's an extremely straightforward manner in which we are going to do it, 'kay?
What we are going to do here is the following.
We're going to say that any point,
x paramaterized by xi over element e is obtained as a,
is obtained by representing it in a basis, okay?
And I'm not going to complete the limits on that summation sign just yet, 'kay?
We're going to write it as some basis function.
Right, here, and I'm going to leave myself a blank spot for it
times the nodal values, okay?
So we're going to write the nodal values here as a let me think, okay,
so I'm going to write these nodal values as
xeA, okay?
Now, we're going to have the sum running
over the number of nodes in each element, right?
So that is A equals 1 to Nne.
When, when I write xAe,
right, what I mean here is for
A equals 1 I'll get x1e and
I get x2e, okay?
These are equal to
my global nodes,
xe and xe plus 1, okay?
The coordinate x1e is the same as the coordinate xe.
Right, but when I write xe, I'm viewing that as a global node,
right, the coordinate is the same.
Likewise x2e simply is the second node of element e.
Well, globally that is just global node number e plus 1, okay?
All right, because of the fact that we need to write this
as a summation over the nodes in each element and there are, in this case,
two nodes in each element which we're writing generally as N, sub n sub e,
'kay, we need to go to this other sort of numbering for local nodes, okay?
So what I have in the brace brackets
here are local nodes, and
here, the same things appear
as global nodes, okay?
Those coordinates are the same.
It was just that in one case, when we're just viewing a single element,
we are regarding those nodes as local nodes, nodes local to that element and
numbered 1 and 2 in that element, okay?
When we look at them globally, we recognize that they're one of the many
nodes in the global domain, and we number them as xe and xe plus 1, okay?
So just below this, let me also write the local nodes.
So that would be x1 for element e, that would x2 for element e, okay?
So the idea there, is that we just want to take our nodes,
our two local nodes, describe to them the actual coordinate values,
right, that they each possess and essentially interpolate.
Now, this blank spot that I've left here in the parenthesis, right, the,
the, the parenthesis that I've left unfilled is where we're going to put in
our basis function in order to do this interpolation of the geometry.
Now, we're free choose whatever basis functions we would want here.
It would just mean that we're interpolating the geometry in some way.
What is the most straightforward way to do it,
the simplest way to do it, one that would probably make our lives the most easy?
Yeah, it is to choose the same basis functions as we chose to inter,
to, to represent our finite dimensional functions, right?
So we'll just use NA, a, A being 1 and 2, right,
the corresponding two in fact the same linear
Lagrange polynomials that we already know, okay?
So, what we are doing here,
using the same basis functions.
As for representing.
And wh,
okay?
When we do this, when we use the same basis functions to represent our
finite dimensional functions as well as to interpolate our geometry,
we have what is called an isoparametric formulation, 'kay,
isoparametric meaning same sort of parameterization.
Okay, so we have
here an what
we have here is
an isoparametric
formulation, okay?
Isoparametic, but because we're using the same parameterization for
our finite dimensional functions as well as for the geometry, okay?
Again, it's not necessary, it is just a convenient way to do it.
There may be special cases where we don't actually do the same thing, okay,
where we don't have an isoparam, parametric formulation.
One can design methods of this kind also, right?
Finite element methods.
Okay, but we will be taking simple approach here,
which is to go isoparametric, okay?
All right, what that lets us do then, is the following,
it lets us say that x,xi is easy to compute, right?
x,xi for
element e is,
now, sum over A,
Na,xi xAe,
all right?
And explicitly, this is just N1,xi, x1 for
element e, where x1e is the global node number,
right, it's sorry, it's, it's, it's so,
I'm sorry, x1e is the local node, right?
But the coordinate x1e is the same
as xe viewed globally, okay,
plus N2,xi x2e, 'kay?
And just to make that clear let me state that,
that that is just equal to xe viewed globally.
That is just equal to xe plus 1, viewed globally.
Right, so the coordinates are the same.
Okay again, that's easy, easy enough to do.
In this particular case, we know exactly what N1 and N2 are.
That's easy to compute.
So that is just d/d xi of N1,
which is 1 minus xi divided
by 2 times x1e plus
d/d xi of 1 plus xi divided
by 2 x to e, okay?
Carrying it out,
we basically get x
2e minus x1e divided by 2,
which is also xe plus 1 minus
xe divided by 2, okay?
Now, what we will do is
observe that xe plus 1 minus xe.
For element e is simply the length of that element, right?
So we will write this
as he over 2, okay?
Where 'kay, this is to be
viewed as the element link.
Importantly what this allows us to do from the ground up, from the,
even from the very simplest of finite element formulations is to have
elements of unequal lengths, right?
So in our partition of the domain omega into subdomains omega e,
there was no requirement and indeed there is no requirement that
the elements all have to be of the same length, okay?
So he can be
different for
each omega e,
'kay?
We don't need a uniform discretization, 'kay?
So this is, allows us to have
a non-uniform discretization.
Okay?
And the straight at, at the very outset,
it's an important property of finite element methods.
Okay let's see how all of this works now.
Now so, so we've done very well.
We have gone ahead and observed that well where did we start this.
Remember we started this calculation all the way up here, okay?
So what we've seen is that the derivative of x with respect to xi,
which can be thought of as the tangent of this mapping, right,
of a tangent of the map that gives us the physical element from this by-unit domain.
[INAUDIBLE]
Right, the essentially what we've figured
out is that we found for this mapping,
we know that you know, every little well, well,
we know that the scaling of the, the element of the physical domain,
relative to the scaling in the parent domain is essentially he over 2, okay?
It's a constant because, why, why is it a constant?
Because for our geometry we're using linear interpolations, right?
We're using linear basis functions, okay?
That's what makes it a constant, that's what makes this,
this tangent map a cons, a constant.
