07.12. The integrals in terms of degrees of freedom - continued

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The Finite Element Method for Problems in Physics

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University of Michigan

The Finite Element Method for Problems in Physics

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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.

Na lição

7

In this unit, we develop the finite element method for three-dimensional scalar problems, such as the heat conduction or mass diffusion problems.

07.01. The strong form of steady state heat conduction and mass diffusion - I 18:24

07.02. The strong form of steady state heat conduction and mass diffusion - II 19:00

07.02q. Response to a question 1:27

07.03. The strong form, continued 19:26

07.03c. In-Video Correction 0:42

07.04. The weak form 24:33

07.05. The finite-dimensional weak form - I 12:35

07.06. The finite-dimensional weak form - II 15:54

07.07. Three-dimensional hexahedral finite elements 21:33

07.08. Aside: Insight to the basis functions by considering the two-dimensional case 17:36

07.08c In-Video Correction 0:44

07.09. Field derivatives. The Jacobian - I 12:38

07.10. Field derivatives. The Jacobian - II 14:20

07.11. The integrals in terms of degrees of freedom 16:25

07.12. The integrals in terms of degrees of freedom - continued 20:55

07.13. The matrix-vector weak form - I 17:19

07.14. The matrix-vector weak form II 11:19

07.15.The matrix-vector weak form, continued - I 17:21

07.15c. In-Video Correction 1:00

07.16. The matrix-vector weak form, continued - II 16:08

07.17. The matrix vector weak form, continued further - I 17:39

07.17c. In-Video Correction 0:47

07.18. The matrix-vector weak form, continued further - II 20:36

07.18c. In-Video Correction 3:12

Conheça os instrutores

Krishna Garikipati, Ph.D.
Professor of Mechanical Engineering, College of Engineering - Professor of Mathematics, College of Literature, Science and the Arts

Okay, so we have everything that we now need to write our integrals in terms of

coordinates in the parent subdomain.

So let's just go ahead and write that.

And in particular the integral that we are working with is the following, right?

So, so actually let me title this.

Integrals in

the finite dimensional

Weak form.

And in particular we're working with this integral

All right. And

where we are is a realization that this integral can now be written as minus.

And that minus comes from the constitutive relation.

We've agreed to pull the sum out, and write it as a sum over A and B,

and for brevity we are not writing the limits that A and B run over.

Right? In the general case,

they run over one to number of nodes in the element, each of them.

All right.

We have an integral now.

Okay?

And inside this integral we have terms of the form,

out here we have C A B, right?

We don't want to forget that.

Okay. The integrand is made up of terms of

the following form.

N A comma i C

capital I

comma little i kappa i, j, N B

comma capital J C Capital j comma little j.

And now, previously we had d, v, being the,

the elemental volume in the, physical subdomain.

And we just agreed at the end of the last segment, to write it as

in terms of an elemental volume in the parent sub domain.

But that does involve our including the determinant of j here.

And let's recall that it depends upon c.

Okay? D V, c.

The integral here now can properly be written as an integral over omicron c.

We recall that the, this is the integral which we enclose in parenthesis.

And we have here, d, d, d.

Okay?

Alright now, if we stare at this integrand,

it looks a little, a little daunting, doubtless, but we observe that there is

a The Einstein summation convention is at work.

And so, really what we have left is going to be an expression which

takes on values depending upon the specific

element degrees of freedom, capital E and capital B.

Okay, the, the capital I, capital G, little I and little G are going to be so

to speak contracted out, to use the terminology of densers and vectors.

Okay right.

So, so when we carry out this, this integral and

let's assume that we do know how to carry it out.

First of all observe that this integral can, can be is,

is, is easy to carry, is relatively easy to carry on because our domain omicron

c is a nice regular sub domain.

It's a nice regular domain, it's a bi unit domain.

So let me just write that little bit here.

So the integral of omega C is the one thing that I want to make a difference

with the way I wrote, so, so this is now a triple integral, okay?

Where, C, 1 equals minus 1 to 1, C2 equals minus 1 to 1.

And C3 also equals minus 1 to 1.

Okay.

Everything that we have in here, I'm not going to explicitly write them as

depending upon c, but we do recognize that indeed,

n a comma i will depend on c, because n a is trilinear.

All right, and the xis.

Okay, so derivative with respect to any one of those coordinates is going to leave

us with something that's bilinear in general, it'll depend on two of the xis.

All right?

If you look at xi i comma i,

and you recall how we are going to compute it It too will depend upon C.

And why's that?

Right.

That's because we are going to compute it or we are going to identify it as

being the components of the inverse of the tangent variable.

Right? And the tangent map,

consists of first derivatives, right?

So, this thing, this, too, will be a function of c, right?

Kappa ij. Well, it depends.

If we have something with, uniform conductivity or uniform diffusivity,

it would be a constant, right?

But that does not res-, we are not restricted to that.

We could have something that's a function of, this thing could.

Potentially be a function of, position.

Right?

Could be a function of position and because we know how to write

the physical position as a function of z that too would be a function of z.

All right, the kappa Okay.

And then we have NB, which of course, we've all ready discussed, right?

We know how to write its derivatives.

It's going to depend upon c as well the derivative of c itself with respect to

little x little j, right?

And then again, we have the determinate here.

Okay? And this again is an integral

over c 1, c 2, c 3.

We close our parenthesis.

Right, in the places multiplying d B element e.

Okay? So

everything in that integrand is properly a function of the xi 1, xi 2, and xi 3.

That integral can be carried out.

Alright, for simple cases this integral is not that

difficult to carry out, analytically, and by simple cases

I mean things where situations where Kappa is indeed independent of position.

Okay, however we're not going to get involved right now with evaluating it

analytically, because we already have a way to evaluate this, right?

Do you remember how we're going to evaluate in, in the general case?

Right.

Using numerical integration.

Okay.

So we'll come back to that later We'll come back es

essentially to how to do that in three dimensions, okay.

But the point I want to make now is that if one looks at

the integral here, okay?

That can be evaluated.

It's going to give us a scalar value, right?

Because all the little i's, capital Is, the little js and

capital Js are going to be contracted away.

It's going to give us a scalar value, however.

That is going to be indexed by the, by the capital A and capital B

corresponding to the local degrees of freedom, okay?

So I am going to denote this K, AB,

sub e, okay?

The AB is obvious, why it needs to be indexed by AB.

It's also obvious why it needs to be indexed by little e.

Okay, because this is for a specific element, okay?

And so we have it.

Our integral on

the left is now just

minus sum over A comma B,

c Ae KAB, for element e,

dBe, all right?

And now, we take a step that we took in the one d case as well.

We are going to well actually, what are we going to do?

Do you recall what we do next in, in setting up our finite element equations?

We get rid of our some, our explicit sum over a and

d by going to matrix vector notation.

Okay, let's do that on the next slide.

So what we're doing is using matrix vector notation.

We're using matrix vector notation for

local degrees of freedom.

In order to keep track of local degrees of freedom,

we're going to use matrix vector notation, right, and really for their numbering.

Okay?

And what this means is that for

element e, if we have c 1 e, c 2 e,

up to c number of nodes in the element e, all right?

These are the degrees of freedom.

As you recall that are used to interpolate the weighting

function using our basis functions, all right?

Okay, so using this, right, and

of course the same thing for d's.

D 1 e, d 2 e, all the way

up to d number of nodes in

the element e, all right?

Using this, what we observe is that

we essentially are able to write our our integral, right?

This implies that we can write integral over omega e.

As now, minus we're going to write the c

vector in a, as a row vector, right?

And you recall how we did this in the one d case as well.

Okay?

We get a big matrix here.

Which consists of entries K 1 1 e,

all the way up to K 1 number of

nodes in the element, e, okay?

And, the last entry here is k number of nodes in the element time number of nodes

in the element, e, okay?

That is the matrix that we get.

And this is multiplied by a column vector using the notation that

is sometimes employed in linear algebra.

Okay.

All right.

And, for, even for the gravity we

could write this as minus c e, right?

Where ce, or ce transpose is that row vector.

All right, this, Ke, right,

Ke is the matrix, de, okay?

And by, by reference to what was done in the one d case,

we would call this the element conductivity.

Or diffusivity matrix, right, depending upon what problem we were solving.

Depending upon what physical problem we were solving, okay?

All right, clear enough hopefully.

We can now follow exactly that same approach to look at the first integral on

the right-hand side of our finite-dimensional weak form.

Okay, so now consider

integral over omega e w h f d v.

With everything we've gathered together this should be relatively quick,

all right?

We're going to write this as integral over omega e.

W h, you recall, we will right as a sum over A N A.

There are no derivatives here, right?

So, we have N A, C A e.

I put parentheses on this to remind us that this represents w h, multiplying f,

d v, okay?

We can take several steps all at once now.

Let's do that.

Let's, pull the summation out, c, A e.

We have an integral over omega e, but right, but

let's plan to write that with a change of variable as an integral over omega c.

All right.

Inside here we continue to have N a because it does depend on position.

Multiplying f and let's recall that f could be a function of position but

through our mapping we have that.

Okay, and any, of course, is the function of position.

Okay.

We're writing everything as a function of coordinates in the parent sub domain.

That leaves us with just d v, right, but we know how to handle that.

We know that it's a determinant of

the jovian of our mapping.

D, V, C.

Okay.

Sorry that's a little to squeezed in there.

So let me write it out here more clearly.

Okay.

That's what we have.

Right.

And, of course, this is now, again, we can take several steps at once.

So we can write this now as we can abandon our explicit

Writing of the, of the sum over A.

Right.

And we know how to do that.

That base, that can be done by going to this rho vector notation.

With a c degrees of freedom.

The integral now explicitly is an, a triple integral.

C 1 equals minus 1 to 1.

C 2 equals minus 1 to 1.

And z 3 equals minus 1 to 1.

Right?

Our integrand here consists of a column vector of the shape fun,

oh sorry, the basis functions.

It's all, they're often called shape functions, but

a term that I've been avoiding.

Okay, we have the basis functions.

All right, we have this.

We have f, right, which could be a function,

which could depend upon c through x.

We know how that dependents comes about.

There's an abuse of notation in, in using the same symbol f, but I will,

just dock that abuse of notation.

All right.

And then, we have the determinant of j,

function of back position.

Okay.

We have this and and yes, and we observe that this integrant is essentially,

integral, sorry, it's integral is over D C 1.

D C 2.

D C 3.

All right.

Okay, we can compute this integral in general.

Again, we will look at how we handle these

types of integrals using numerical quadrature, numerical integration.

And we observe that we essentially

get what I will write out as being c 1 e, c 2 e,

c number of nodes in the element e.

Times, the column vector which I will

denote as F internal.

Okay?

Degree of freedom one in element e.

F internal, degree of freedom 2, element e.

Coming down as far as f,

internal Degree of freedom,

in an E, element E.

Okay. And finally, we could

write that as C, E,

transpose F Internal for element e.

Okay?

All very nice and concise.

All right, and the, the advantages that having done it with,

with the, in excruciating detail for the,

1D case we know you don't immediately have the sense to work on it.

All right, we'll stop this segment here.

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