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KAREN WILLCOX: All right.

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So let's see.

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I have the graded projects
here which maybe you guys

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can grab after class.

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00:00:35,306 --> 00:00:38,600
Or you can grab them now if you
want but it can take a while.

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Vikram, maybe you want to
give the projects back.

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Just quietly ask people
what their names are.

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OK.

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00:00:47,590 --> 00:00:52,220
So today, we're going to
talk about finite elements.

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00:00:52,220 --> 00:00:54,530
You guys have read a
little bit in the reading.

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I'm going to talk about a
couple things first, just

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go through a couple of
the very basic ideas.

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And then we're going
to do it interactively.

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You're going to actually
implement the one 1D

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00:01:03,370 --> 00:01:04,819
finite element codes.

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00:01:04,819 --> 00:01:08,570
We'll do it piece
by piece together.

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00:01:08,570 --> 00:01:14,810
So let me just put the
topics up over here.

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00:01:20,310 --> 00:01:24,940
So again we're going to talk
about 1D finite elements.

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We're going to start
off just with basically

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00:01:27,937 --> 00:01:29,145
thinking about the key ideas.

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00:01:32,970 --> 00:01:40,890
We'll just look at a very brief
history of finite elements.

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00:01:40,890 --> 00:01:46,160
We'll think about what 1D
linear elements look like.

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We'll talk about
the nodal bases.

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00:01:58,740 --> 00:02:04,230
And then we will spend
most of the session doing

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00:02:04,230 --> 00:02:08,133
the derivation and
implementation for the 1D

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00:02:08,133 --> 00:02:08,924
diffusion equation.

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00:02:18,430 --> 00:02:21,120
Same derivation
and implementation

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00:02:21,120 --> 00:02:24,120
for 1D diffusion.

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00:02:24,120 --> 00:02:24,620
OK.

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00:02:24,620 --> 00:02:27,070
So by the end of today,
you'll more or less

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have a basic shell
of a code that

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00:02:30,620 --> 00:02:34,070
implements at least the meaty
part of the finite element.

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00:02:34,070 --> 00:02:35,970
I'm not quite sure how
far we'll get today.

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00:02:35,970 --> 00:02:37,490
Then on Monday, I
am unfortunately

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not going to be here.

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But Vikram is going to
run the class session.

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And you guys will continue
working with that code.

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00:02:41,920 --> 00:02:44,880
Probably Monday you'll
implement the boundary condition

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00:02:44,880 --> 00:02:48,410
and implement some
quadrature schemes in it.

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00:02:48,410 --> 00:02:50,250
And so then, by the
end of class on Monday,

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00:02:50,250 --> 00:02:52,120
you should have a
working 1D finite element

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00:02:52,120 --> 00:02:53,990
code which will
then hopefully be

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00:02:53,990 --> 00:02:57,710
of help to do the 2D that you
have to do for the project.

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00:02:57,710 --> 00:02:59,510
And on Wednesday of
next week, Vikram

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will talk a little bit
about 2D finite elements.

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00:03:03,444 --> 00:03:07,370
So let's just start
off with key ideas.

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00:03:07,370 --> 00:03:09,360
I'm just going to write
over there for a bit

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00:03:09,360 --> 00:03:13,392
because I want to put something
up on the screen in a second.

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So remember on Monday, we
talked about the method

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of weighted residuals.

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And again, just
to sort of go over

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what the steps were in the
method of weighted residuals.

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Kind of the fundamental
idea is take the solution

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and to approximate
it in a basis.

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Right?

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Remember we took the
intradimensional PDE

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and we discretized it.

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And this is already
different to what

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we do in finite difference
or finite volume.

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We discretized it by
saying let's assume

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that a solution t of x lives in
this finite dimensional space

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described as a weighted
sum of basis functions.

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Then we said how
do we figure out

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the coefficient of
the basis function?

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And we went through
the derivation

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of choosing weighting functions
to be the same functions

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that we used in the basis
to approximate the solution.

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What was that called?

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What's it called when
you choose the weighting

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functions to be the same
functions that you think

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[INAUDIBLE]?

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Galerkin.

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Yeah.

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It's the Galerkin method.

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When we define the residual,
multiplying by the weighting

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function using Galerkin.

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Integrating over the domain,
setting that equal to zero.

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That's the weighted residual.

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Those gave us the
equations we solve

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to find the approximation.

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So we're more or less going
to do the same thing now

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with finite elements.

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00:04:37,454 --> 00:04:38,870
So what you are
going to see today

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is that a finite element
as we're going to view it

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is based on the method
of weighted residuals.

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But what is different
is the first step.

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Before we start with the
approximation of the solution,

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the very first thing
we're going to do

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is to discretize the
domain into small cells.

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00:05:07,330 --> 00:05:07,830
OK?

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00:05:07,830 --> 00:05:09,470
So we're going to
divide up our domain--

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00:05:09,470 --> 00:05:11,990
rather than thinking about the
approximation of the solution

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00:05:11,990 --> 00:05:13,850
globally over the
whole domain, we're

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00:05:13,850 --> 00:05:19,970
going to discretize our domain
into small cells, elements.

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That's the element of the
finite element method.

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They've been called-- I
think you guys called them

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00:05:24,285 --> 00:05:26,201
cells when you talked
about finite difference.

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Is that right?

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Nodes and cells?

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00:05:28,970 --> 00:05:30,790
And for finite
elements, the cells

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are referred to as elements.

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Then once we've discretized
the domain into small cells,

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00:05:36,607 --> 00:05:38,440
then we're just going
to go and basically do

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what we did with negative
weighted residuals.

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00:05:40,315 --> 00:05:43,420
So now we're going to
think about approximating

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the solution in each element.

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00:05:50,980 --> 00:05:52,830
So in each element,
we're going to say

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00:05:52,830 --> 00:05:55,740
let's assume that
the solution can

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00:05:55,740 --> 00:05:59,280
be approximated as a linear
combination of basis functions.

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00:05:59,280 --> 00:06:01,550
And we're going
to use polynomials

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just like we did on Monday.

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00:06:03,217 --> 00:06:04,800
But now the polynomials
are just going

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to be defined over the little
element, the little piece

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00:06:07,677 --> 00:06:08,260
of the domain.

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00:06:08,260 --> 00:06:09,220
Not the whole domain.

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00:06:12,500 --> 00:06:13,000
E.g.

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Here with polynomial functions.

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Again, just like we
did on Monday but now

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00:06:18,850 --> 00:06:23,109
to serve a small element
polynomial function.

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00:06:23,109 --> 00:06:25,150
And then again, it's going
to be the same as what

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00:06:25,150 --> 00:06:26,090
we did on Monday.

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00:06:26,090 --> 00:06:28,350
Once we've done
the approximation,

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00:06:28,350 --> 00:06:32,260
we're going to evaluate weighted
residuals for each element.

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00:06:38,330 --> 00:06:43,300
We're going to use the
Galerkin method to do that,

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to define the weighted residual.

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That's going to give us
a system of equations.

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00:06:50,280 --> 00:06:56,011
And we're going to solve
to determine the weighting

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coefficient.

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00:07:06,880 --> 00:07:08,340
Weighting coefficients.

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00:07:08,340 --> 00:07:11,180
So basically exactly
what we did on Monday,

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except that first step
which is discretize

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00:07:13,570 --> 00:07:15,405
the domain to small cells.

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00:07:15,405 --> 00:07:16,780
And then what
you're going to see

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00:07:16,780 --> 00:07:20,780
is that idea of breaking the
domain up into small cells,

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we're going to make
a very special choice

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for the polynomial
basis functions.

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It's going to be really
neat because when

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we do the weighted residuals
and we do all the integrals--

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00:07:28,160 --> 00:07:29,090
like we mentioned a
little bit, I think,

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00:07:29,090 --> 00:07:30,920
in response to one of
[INAUDIBLE] questions--

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00:07:30,920 --> 00:07:32,961
a whole bunch of integrals
are going to fall out.

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00:07:32,961 --> 00:07:34,500
And it'll turn out
that everything

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has got a very special and
really efficient and kind

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of neat structure to it.

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AUDIENCE: Did you
pick the same basis

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functions for the entire problem
or for each individual cell?

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KAREN WILLCOX: Yeah.

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So the question is do you
pick the same basic functions

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for the entire problem, or
for each individual cell?

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You're going to see
it in just a second.

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There are a lot
of different ways

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to choose the basic function.

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Today we're going to talk
about a special choice that's

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00:07:58,642 --> 00:08:00,340
referred to as the nodal basis.

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00:08:00,340 --> 00:08:02,280
And you'll see
exactly what they are.

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Before I start,
are there any kind

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00:08:07,530 --> 00:08:11,410
of residual questions from
method of weighted residuals?

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00:08:11,410 --> 00:08:12,435
Residual questions.

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00:08:12,435 --> 00:08:13,810
Are there any
things about method

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00:08:13,810 --> 00:08:19,212
of weighted residuals that is
mysterious or uncomfortable?

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00:08:19,212 --> 00:08:20,670
It's pretty
straightforward, right?

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So just before we start, I
want to give you a little bit

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of a-- I hate this WebEx thing.

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Because when you
try to grab things,

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it goes down-- little bit
of a historical perspective.

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And actually I went to
Wikipedia as you do,

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which actually doesn't have
a bad-- this is the Wikipedia

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00:08:56,240 --> 00:09:00,796
entry for finite element method,
which actually surprised me.

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00:09:00,796 --> 00:09:02,170
Look how many
sections there are.

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Vikram, have you
contributing to Wikipedia?

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00:09:04,522 --> 00:09:05,935
AUDIENCE: No.

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00:09:05,935 --> 00:09:07,348
[LAUGHTER]

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00:09:07,348 --> 00:09:10,180
KAREN WILLCOX: Aha.

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00:09:10,180 --> 00:09:12,345
There's actually
quite a lot here.

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00:09:12,345 --> 00:09:13,220
I didn't read it all.

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00:09:13,220 --> 00:09:15,515
So I can't verify
how good it is.

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Although Wikipedia
appears to be pretty good,

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00:09:17,390 --> 00:09:20,994
there's enough of a
community of people.

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00:09:20,994 --> 00:09:22,870
So I was going to show
you guys the history

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about technical discussion,
general principles

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00:09:26,530 --> 00:09:28,614
of weak formulation,
discretization.

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Then you can see all
the different kinds

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00:09:30,280 --> 00:09:35,040
of finite element
methods that exist.

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00:09:35,040 --> 00:09:37,190
So that should give you
a sense of just how big

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00:09:37,190 --> 00:09:38,670
of a field of study this is.

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We're going to be just
a little tiny bit of it.

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00:09:40,930 --> 00:09:43,360
[INAUDIBLE] for finite
difference method application.

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00:09:43,360 --> 00:09:47,910
I just wanted to mention
a little bit just

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about the history.

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So like it says here,
it's difficult to quote

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00:09:51,310 --> 00:09:54,840
the date of the invention of
the finite element method.

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00:09:54,840 --> 00:09:58,650
More or less, its
origins are actually

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00:09:58,650 --> 00:10:02,290
from structures in both civil
and aerospace aeronautical

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00:10:02,290 --> 00:10:02,790
engineering.

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00:10:02,790 --> 00:10:06,490
So it has roots in
aeronautical engineering.

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00:10:06,490 --> 00:10:13,000
And there were three kind of
groups where if you trace back,

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00:10:13,000 --> 00:10:16,280
that's where things seem
to have really originated.

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00:10:16,280 --> 00:10:20,320
Courant is often the person
whose is cited as being,

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00:10:20,320 --> 00:10:22,344
I guess, the father
of a lot of theories

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00:10:22,344 --> 00:10:23,510
that led to finite elements.

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00:10:23,510 --> 00:10:28,980
So he was at NYU and then
established an Applied Math

209
00:10:28,980 --> 00:10:31,470
institute, the Courant Institute
which still exists today.

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00:10:31,470 --> 00:10:33,469
It is one of the best
Applied Math computational

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00:10:33,469 --> 00:10:35,760
places in the world.

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00:10:35,760 --> 00:10:41,130
And as you can see here, I think
this is a German mathematician.

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00:10:41,130 --> 00:10:45,720
And also things were going
on in China at the same time.

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00:10:45,720 --> 00:10:47,790
So there's a little bit here.

215
00:10:47,790 --> 00:10:50,009
You can see there Rayleigh,
Ritz, and Galerkin.

216
00:10:50,009 --> 00:10:51,550
Those are all names
of mathematicians

217
00:10:51,550 --> 00:10:55,120
that you guys have seen
in various methods, things

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00:10:55,120 --> 00:10:56,880
that came up.

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00:10:56,880 --> 00:10:59,020
Really the way the
finite element method

220
00:10:59,020 --> 00:11:01,120
started to really
sort of become like

221
00:11:01,120 --> 00:11:04,200
a computational
powerhouse, I think,

222
00:11:04,200 --> 00:11:05,980
was here in the 60s and 70s.

223
00:11:05,980 --> 00:11:08,964
So Agyris at the
University of Stuttgart.

224
00:11:08,964 --> 00:11:10,380
I think this is
pronounced Clough.

225
00:11:10,380 --> 00:11:12,130
Is that right, Vikram?

226
00:11:12,130 --> 00:11:13,000
Yeah.

227
00:11:13,000 --> 00:11:15,650
Clough at UC Berkeley.

228
00:11:15,650 --> 00:11:17,786
And Zienkiewicz at
University of Swansea.

229
00:11:17,786 --> 00:11:19,160
And in fact
University of Swansea

230
00:11:19,160 --> 00:11:21,880
is where Professor
[INAUDIBLE] did his PhD.

231
00:11:21,880 --> 00:11:24,309
And he was working with
some of the later people

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00:11:24,309 --> 00:11:26,392
there that were at the
forefront of finite element

233
00:11:26,392 --> 00:11:30,400
methods and their development.

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00:11:30,400 --> 00:11:32,240
So it's actually a
really kind of neat story

235
00:11:32,240 --> 00:11:34,100
because it really comes
from applied math.

236
00:11:34,100 --> 00:11:36,430
And you're not going to see
a whole lot of it in class.

237
00:11:36,430 --> 00:11:40,270
But there's a lot of
incredible analysis

238
00:11:40,270 --> 00:11:44,870
and theoretical results that
go with finite element methods,

239
00:11:44,870 --> 00:11:46,760
a lot of guarantees
that can be made.

240
00:11:46,760 --> 00:11:50,000
So there's really a
hardcore math component.

241
00:11:50,000 --> 00:11:54,240
But then it has gone on to
be of such practical use.

242
00:11:54,240 --> 00:11:56,220
So it started off in
structural problems

243
00:11:56,220 --> 00:11:59,680
but now is applied to
fluid problems, acoustics,

244
00:11:59,680 --> 00:12:01,785
all kinds of things.

245
00:12:01,785 --> 00:12:04,410
You can see here NASA sponsored
the original vision of NASTRAN.

246
00:12:04,410 --> 00:12:06,550
I actually didn't know
that until I read this.

247
00:12:06,550 --> 00:12:07,925
So now there's a
lot of software,

248
00:12:07,925 --> 00:12:10,420
things like NASTRAN
ADINA is Professor Bathe

249
00:12:10,420 --> 00:12:13,350
from mechanical
engineering package.

250
00:12:13,350 --> 00:12:15,170
Professor Strang in
the math department

251
00:12:15,170 --> 00:12:17,640
also plays a really
important role

252
00:12:17,640 --> 00:12:21,436
in some of the rigorous
results early on.

253
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So there's also a lot
of connections to MIT.

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So we're going to see
only just a little piece

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of the finite element method.

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But it is certainly,
I think, one

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of the really great examples
of where applied methods made

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enormous impact on real problems
and particularly on engineering

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and engineering design.

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So let's get into talking
about how it works.

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We're going to talk
about 1D linear elements.

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I'm just going to
go over a few things

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that you guys read
about just to make sure

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that the notation is clear
and that the ideas are clear.

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And then we're going
to start together

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deriving and implementing the
methods for the 1D diffusion

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equation.

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So we're at number three.

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00:13:22,980 --> 00:13:24,480
1D linear elements.

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OK.

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00:13:30,330 --> 00:13:34,790
So I'm just going
to draw a picture.

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And again this is the picture
that was in the online reading.

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Board is not very clean.

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All right.

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So we're thinking in 1D.

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So draw this line that's going
to go in both directions.

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So this is the x direction.

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And I'm going to have
nodes just like I

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had in finite difference.

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00:14:03,915 --> 00:14:04,790
So I draw some nodes.

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00:14:08,260 --> 00:14:13,936
This guy here is
going to be node i.

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00:14:13,936 --> 00:14:16,942
And this guy here is going
to be node i plus one.

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00:14:16,942 --> 00:14:21,130
So node i is located
in at position xi.

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00:14:21,130 --> 00:14:25,470
Node i plus one is located
at position xi plus one.

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00:14:25,470 --> 00:14:28,820
i minus one to the
left and so on.

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00:14:28,820 --> 00:14:33,284
And then in between xi and
xi plus one is element i.

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00:14:33,284 --> 00:14:33,784
OK?

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So when we talk about
element i, we're

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talking about the region
of the domain that lies

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between xi and xi plus one.

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And--