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PROFESSOR STRANG: OK, so.
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This is a big day.
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Part One of the course is
completed, and I have your
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quizzes for you, and that was
a very successful result,
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00:00:31 --> 00:00:33
I'm very pleased.
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00:00:33 --> 00:00:35
I hope you are, too.
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Quiz average of 85, that's on
the first part of the course.
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And then the second part so
this is, Chapter 3 now.
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Starts in one dimension with an
equation of a type that we've
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already seen a little bit.
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So there's some more things to
say about the equation and the
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framework, but then we get to
make a start on the finite
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element approach to solving it.
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We could of course in 1-D
finite differences are probably
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the way to go, actually.
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In one dimension the special
success of finite elements
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doesn't really show up that
much because finite elements
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have been, I mean one great
reason for their success
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00:01:30 --> 00:01:33
is that they handle
different geometries.
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They're flexible; you could
have regions in the plane,
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three dimensional bodies
of different shapes.
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Finite differences doesn't
really know what to do on
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a curved boundary
in in 2- or 3-D.
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00:01:47 --> 00:01:50
Finite elements
cope much better.
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So, we'll make a start today,
more Friday on one dimensional
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finite elements and then, a
couple of weeks later will be
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the real thing, 2-D and 3-D.
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00:02:05 --> 00:02:09
OK, so, ready to
go on Chapter 3?
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00:02:09 --> 00:02:12
So, that's our equation and
everybody sees right away
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00:02:12 --> 00:02:17
what's the framework that
that's A transpose in some way;
39
00:02:17 --> 00:02:21
this is A transpose C A, but
what's new of course is
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that we're dealing with
functions, not vectors.
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00:02:25 --> 00:02:28
So we're dealing with, you
could say, operators,
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00:02:28 --> 00:02:30
not matrices.
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00:02:30 --> 00:02:35
And nevertheless, the big
picture is still as it was.
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00:02:35 --> 00:02:40
So let me take u(x) to be
the displacements again.
45
00:02:40 --> 00:02:43
So I'm thinking more
of mechanics than
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electronics here.
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00:02:45 --> 00:02:52
Displacements, and then we have
du, the e(x) will be du/dx,
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that'll be the stretching, the
elongation and, of course at
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that step you already see the
big new item, the fact that the
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A, the one that gets us from u
to du/dx, instead of being a
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difference matrix which it has
been, our matrix A is
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now a derivative.
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00:03:18 --> 00:03:19
A is d/dx.
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So maybe I'll just
take out that arrow.
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So A is d/dx.
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00:03:29 --> 00:03:37
OK, but if we dealt OK
with difference matrices,
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we're going to deal
OK with derivatives.
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Then, of course, this is the
C part, that produces w(x).
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And it's a multiplication
by this possibly varying,
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possibly jumping,
stiffness constancy of x.
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So w(x) is c(x) c(x),
that's our old w=Ce,
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this is Hooke's Law.
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I'll put Hooke's Law,
but that's, or who's
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ever law it is.
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It's like a diagonal matrix;
I hope you see that it's
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like a diagonal matrix.
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This function u is kind of like
of a vector but a continuum
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vector instead of just a fixed,
finite number of values.
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Then at each value we used to
multiply by c_i, now our
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values are continuous with x,
so we multiply by c(x).
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And then you're going to expect
that, going up here, there's
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going to be an A transpose w f,
and of course that A transpose
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we have to identify and
that's the first point
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of the lecture, really.
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To identify what
is A transpose.
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00:05:01 --> 00:05:05
What do I mean by A transpose?
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And I've got to say right away
that I'm a little, the
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notation, writing a transpose
of a derivative is like,
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00:05:16 --> 00:05:18
that's not legal.
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00:05:18 --> 00:05:21
Because we think of the
transpose of a matrix; you sort
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of flip it over the main
diagonal, but obviously
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it's got to be something
more to it than that.
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And so that's a central math
part of this lecture is
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what's really going on
when you transpose?
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Because then we can copy what's
going on and it's quite
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00:05:41 --> 00:05:43
important to get it.
87
00:05:43 --> 00:05:47
Because the transpose, well,
other notations and other words
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00:05:47 --> 00:05:51
for it would be notation
might be a star.
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00:05:51 --> 00:05:56
Star would be way more common
than transpose, I'll just stay
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00:05:56 --> 00:06:00
with transpose because I want
to keep pressing the parallel
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00:06:00 --> 00:06:04
with A transpose C A.
92
00:06:04 --> 00:06:08
And the name for it
would be the adjoint.
93
00:06:08 --> 00:06:11
And the adjoint method,
and adjoint operator,
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00:06:11 --> 00:06:13
those appear a lot.
95
00:06:13 --> 00:06:16
And you'll see them up
here in finite elements.
96
00:06:16 --> 00:06:24
So this is a good
thing to catch on to.
97
00:06:24 --> 00:06:25
Why?
98
00:06:25 --> 00:06:30
Why should the transpose or
the adjoint of the derivative
99
00:06:30 --> 00:06:32
be minus the derivative?
100
00:06:32 --> 00:06:37
And by the way, just while
we're fixing this, this is a
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key factor which is certainly,
we have a very strong hint from
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00:06:45 --> 00:06:47
center difference, right?
103
00:06:47 --> 00:06:51
If I think of derivatives,
if I associate them with
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00:06:51 --> 00:06:54
differences, the center
difference matrix, so
105
00:06:54 --> 00:06:59
the a matrix may be
centered, would be.
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00:06:59 --> 00:07:03
Just to remind us, it's a
center difference has onws
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00:07:03 --> 00:07:08
and minus one, one, zeroes
on the diagonal, right?
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00:07:08 --> 00:07:10
Minus one, one.
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Takes that difference
at every row.
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00:07:16 --> 00:07:18
Except possibly boundary rows.
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00:07:18 --> 00:07:21
And of course as soon as you
look at that matrix you see,
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00:07:21 --> 00:07:23
yeah, it's anti symmetric.
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00:07:23 --> 00:07:26
It's an anti symmetric matrix.
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00:07:26 --> 00:07:32
So a transpose is minus a for
center differences and
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00:07:32 --> 00:07:36
therefore we're not so
surprised to see a minus sign
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00:07:36 --> 00:07:39
up here when we go to the
continuous case,
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00:07:39 --> 00:07:40
the derivative.
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00:07:40 --> 00:07:45
But, we still have to
say what it means.
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00:07:45 --> 00:07:47
So that's what
I'll do next, OK?
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00:07:47 --> 00:07:49
So this is a good
thing to know.
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00:07:49 --> 00:07:52
And I was just going to
comment, what would be
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00:07:52 --> 00:07:55
the transpose of the
second derivative?
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00:07:55 --> 00:07:57
I won't even write this down.
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If the derivative transpose
sort of flips its sign to
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00:08:02 --> 00:08:07
minus, what would you guess for
this x transpose of second
126
00:08:07 --> 00:08:12
derivative, our more familiar
d second by dx squared?
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Well we'll have
two minus signs.
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So it'll come out fine.
129
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So second derivatives, even
order derivatives are sort
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of like symmetric guys.
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00:08:24 --> 00:08:27
Odd order derivatives, first
and third and fifth
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00:08:27 --> 00:08:30
derivatives, well, God forbid
we ever meet a fifth
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00:08:30 --> 00:08:34
derivative, but first
derivative anyway,
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00:08:34 --> 00:08:36
is anti symmetric.
135
00:08:36 --> 00:08:38
Except for boundary conditions.
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00:08:38 --> 00:08:42
So I really have to
emphasize that the boundary
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00:08:42 --> 00:08:44
conditions come in.
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00:08:44 --> 00:08:45
And you'll see them come in.
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00:08:45 --> 00:08:47
They have to come in.
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00:08:47 --> 00:08:53
OK, so what meaning can I
assign to the transpose, or
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00:08:53 --> 00:08:58
what was the real thing
happening when we flipped the
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matrix across its diagonal?
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00:09:00 --> 00:09:14
I claim that we really define
the transpose by this rule.
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00:09:14 --> 00:09:17
By we know what
inner products are.
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00:09:17 --> 00:09:20
I'll do vectors first, we know
about inner products, dot
146
00:09:20 --> 00:09:23
products, we know what the dot
product of two vectors is.
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00:09:23 --> 00:09:27
So, this is the transpose of A.
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00:09:27 --> 00:09:29
How am I going to define
the transpose of A?
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00:09:29 --> 00:09:36
Well, I look at the dot
product of Au with w.
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00:09:36 --> 00:09:40
I'll use a dot here for once, I
may erase it and replace it.
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00:09:40 --> 00:09:47
If at the dot product of Au
with w, then that equals for
152
00:09:47 --> 00:09:51
all u and w, all vectors u
and w, that equals the dot
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00:09:51 --> 00:09:56
product of u with something.
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00:09:56 --> 00:09:59
Because u is coming.
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00:09:59 --> 00:10:02
If I write out what the dot
product is, I see u_1
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00:10:02 --> 00:10:05
multiplies something, u_2
multiplies something.
157
00:10:05 --> 00:10:13
And what goes in
that little space?
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00:10:13 --> 00:10:15
This is just an identity.
159
00:10:15 --> 00:10:18
I mean, it's like,
you'll say no big deal.
160
00:10:18 --> 00:10:22
But I'm saying there is
at least a small deal.
161
00:10:22 --> 00:10:22
OK.
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00:10:22 --> 00:10:30
So if I write it this way,
you'll tell me right away this
163
00:10:30 --> 00:10:33
should be the same as u
transpose times something.
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00:10:33 --> 00:10:36
And again, so I'm asking
for the same something
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00:10:36 --> 00:10:38
on both lines.
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00:10:38 --> 00:10:41
What is that something?
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00:10:41 --> 00:10:44
A transpose w.
168
00:10:44 --> 00:10:48
Whatever A transpose is,
it's the matrix that
169
00:10:48 --> 00:10:50
makes this right.
170
00:10:50 --> 00:10:51
That's really my message.
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00:10:51 --> 00:10:55
That A transpose is the reason
we flipped the matrix across
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00:10:55 --> 00:11:01
the diagonal, is that it
makes that equation correct.
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00:11:01 --> 00:11:03
And I'm writing the
same thing here.
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00:11:03 --> 00:11:04
OK.
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00:11:04 --> 00:11:10
So again, if we knew what dot
products were, what inner
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product of vectors were, then A
transpose is the matrix that
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00:11:15 --> 00:11:18
makes this identity correct.
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00:11:18 --> 00:11:23
And of course if you write it
all out in terms of i, j every
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00:11:23 --> 00:11:26
component, you find
it is correct.
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00:11:26 --> 00:11:31
So that defines the
transpose of a matrix.
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00:11:31 --> 00:11:34
And of course it coincides with
flipping across the diagonal.
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00:11:34 --> 00:11:39
Now, how about the
transpose of a derivative.
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00:11:39 --> 00:11:43
OK, so I'm going to
follow the same rule.
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00:11:43 --> 00:11:47
Here A is now going to be the
derivative, and A transpose
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00:11:47 --> 00:11:50
is going to be whatever it
takes to make this true.
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00:11:50 --> 00:11:51
But what do I mean?
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00:11:51 --> 00:11:55
Now I have functions, so I have
to think again, what do I mean
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00:11:55 --> 00:11:57
by the inner product,
the dot product?
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So for this to make sense I
need to say, and it's a very
190
00:12:03 --> 00:12:07
important thing anyway, and
it's the right natural choice,
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00:12:07 --> 00:12:14
I need to say the dot product,
or the inner product is a
192
00:12:14 --> 00:12:19
better word, of functions.
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00:12:19 --> 00:12:21
Of two functions.
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A, e(x), and w. if I have two
functions, what do I mean
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00:12:28 --> 00:12:30
by their inner products?
196
00:12:30 --> 00:12:35
Well, really I just think back
what did we mean in the finite
197
00:12:35 --> 00:12:39
dimensional case, I multiplied
each e by a w, each component
198
00:12:39 --> 00:12:43
of e by w, and I added, so
what am I going to do here?
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00:12:43 --> 00:12:50
Maybe my notation should be
parentheses with a comma
200
00:12:50 --> 00:12:54
would be better than
a dot, for function.
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00:12:54 --> 00:12:55
So I have a function.
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00:12:55 --> 00:12:59
I'm in function space now.
203
00:12:59 --> 00:13:04
We moved out of our n,
today, into function space.
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00:13:04 --> 00:13:06
Our vectors have
become functions.
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00:13:06 --> 00:13:09
And now what's the dot
product of two vectors?
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00:13:09 --> 00:13:13
Well, what am I going to do?
207
00:13:13 --> 00:13:15
I'm going to do
what I have to do.
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00:13:15 --> 00:13:21
I'm going to multiply each e by
its corresponding w, and now
209
00:13:21 --> 00:13:26
they depend on this continuous
variable x, so that's
210
00:13:26 --> 00:13:27
e(x) times w(x).
211
00:13:28 --> 00:13:32
And what do I do now?
212
00:13:32 --> 00:13:33
Integrate.
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00:13:33 --> 00:13:38
Here, I added e_i
times w_i, of course.
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00:13:38 --> 00:13:41
Over here I have functions.
215
00:13:41 --> 00:13:42
I integrate dx.
216
00:13:43 --> 00:13:47
Over whatever the region
of the problem is.
217
00:13:47 --> 00:13:52
And then our example's
in 1-D, be zero to one.
218
00:13:52 --> 00:13:55
If these are functions of two
variables I'd be integrating
219
00:13:55 --> 00:13:58
over some 2-D region,
but we're in 1-D today.
220
00:13:58 --> 00:14:05
OK, so you see that I'm
prepared to say this
221
00:14:05 --> 00:14:15
now makes sense.
222
00:14:15 --> 00:14:20
I now want to say, I'm going to
let A be the derivative, and
223
00:14:20 --> 00:14:24
I'm going to figure out what
A transpose has to be.
224
00:14:24 --> 00:14:29
So if A is the derivative,
so now is this key step.
225
00:14:29 --> 00:14:32
y is the transpose x, OK?
226
00:14:32 --> 00:14:41
So I look at the derivative,
du/dx, with w, so that's this
227
00:14:41 --> 00:14:52
integral, zero to one
of du/dx*w(x)dx, so
228
00:14:52 --> 00:14:54
that's my left side.
229
00:14:54 --> 00:14:58
Now I want to get u by itself.
230
00:14:58 --> 00:15:01
I want to get the dot product,
so I want to get another
231
00:15:01 --> 00:15:07
integral here that
has u(x) by itself.
232
00:15:07 --> 00:15:09
Times something, and
that something is
233
00:15:09 --> 00:15:10
what I'm looking for.
234
00:15:10 --> 00:15:15
That something will
be A transpose w.
235
00:15:15 --> 00:15:17
Right?
236
00:15:17 --> 00:15:19
Do you see what I'm doing?
237
00:15:19 --> 00:15:25
This is is the dot product,
this Auw, so I've written
238
00:15:25 --> 00:15:28
out what a u in a
product with w is.
239
00:15:28 --> 00:15:32
And now I want to get u out by
itself and what it multiplies
240
00:15:32 --> 00:15:38
here will be the a transpose w,
and my rule will be extended to
241
00:15:38 --> 00:15:42
the function case and
I'll be ready to go.
242
00:15:42 --> 00:15:49
Now do you recognize, this is a
basic calculus step, what rule
243
00:15:49 --> 00:15:51
of calculus am I going to use?
244
00:15:51 --> 00:15:53
We're back to 18.01.
245
00:15:53 --> 00:15:56
I have the integral of a
derivative times w, and
246
00:15:56 --> 00:15:57
what do I want to do?
247
00:15:57 --> 00:16:01
I want to get the
derivative off of u.
248
00:16:01 --> 00:16:02
What happens?
249
00:16:02 --> 00:16:04
What's it called?
250
00:16:04 --> 00:16:06
Integration by part.
251
00:16:06 --> 00:16:08
Very important thing.
252
00:16:08 --> 00:16:09
Very important.
253
00:16:09 --> 00:16:13
If you miss it's
important in calculus.
254
00:16:13 --> 00:16:17
It gets sometimes introduced as
a rule, or a trick to find some
255
00:16:17 --> 00:16:21
goofy integral, but it's
really the real thing.
256
00:16:21 --> 00:16:23
So what is integration
by parts?
257
00:16:23 --> 00:16:24
What's the rule?
258
00:16:24 --> 00:16:29
You take the derivative off of
u, you put it on to the other
259
00:16:29 --> 00:16:34
one just what we hope for, and
then you also have to remember
260
00:16:34 --> 00:16:40
that there is a minus.
261
00:16:40 --> 00:16:42
Integration by
parts has a minus.
262
00:16:42 --> 00:16:45
And usually you'd see it out
there but here I've left
263
00:16:45 --> 00:16:49
more room for it there.
264
00:16:49 --> 00:16:54
So I have identified
now, A transpose w.
265
00:16:54 --> 00:17:00
A transpose w has, if this is
Au, in a product with w, then
266
00:17:00 --> 00:17:03
this is u in a product with A
transpose w, it had
267
00:17:03 --> 00:17:04
to be what was.
268
00:17:04 --> 00:17:08
And so that one integration by
parts brought out a minus sign.
269
00:17:08 --> 00:17:11
If I was looking at second
derivatives there would
270
00:17:11 --> 00:17:14
probably be somewhere two
integration by parts;
271
00:17:14 --> 00:17:17
I'd have minus twice,
I'd be back to plus.
272
00:17:17 --> 00:17:20
And you're going to ask
about boundary conditions.
273
00:17:20 --> 00:17:22
And you're right to ask
about boundary conditions.
274
00:17:22 --> 00:17:26
I even circled that, because
that is so important.
275
00:17:26 --> 00:17:33
So what we've done so far
is to get the interior
276
00:17:33 --> 00:17:35
of the interval right.
277
00:17:35 --> 00:17:39
Between zero and one, if A
is the derivative, then A
278
00:17:39 --> 00:17:42
transpose is minus
the derivative.
279
00:17:42 --> 00:17:43
That's all we've done.
280
00:17:43 --> 00:17:46
We have not got the
boundary conditions yet.
281
00:17:46 --> 00:17:50
And we can't go
on without that.
282
00:17:50 --> 00:17:54
OK, so I'm ready now to
say something about
283
00:17:54 --> 00:17:57
boundary conditions.
284
00:17:57 --> 00:18:00
And it will bring up this
square versus rectangular
285
00:18:00 --> 00:18:07
also, so we're getting the
rules straight before we
286
00:18:07 --> 00:18:09
tackle finite elements.
287
00:18:09 --> 00:18:15
OK, let me take an example of
a matrix and its transpose.
288
00:18:15 --> 00:18:18
Just so you see how
boundary conditions.
289
00:18:18 --> 00:18:23
Suppose I have a
free fixed problem.
290
00:18:23 --> 00:18:26
Suppose I have a free
fixed line of springs.
291
00:18:26 --> 00:18:30
What's the matrix A for that?
292
00:18:30 --> 00:18:32
Well - question?
293
00:18:32 --> 00:18:33
Yes.
294
00:18:33 --> 00:18:36
AUDIENCE: [INAUDIBLE]
295
00:18:36 --> 00:18:40
PROFESSOR STRANG:
Yeah. that's Yes.
296
00:18:40 --> 00:18:44
When I learned it, it was
also that stupid trick.
297
00:18:44 --> 00:18:50
So you would like me to put
plus, can I put plus, whatever.
298
00:18:50 --> 00:18:51
What do you want
me to call that?
299
00:18:51 --> 00:18:52
An integrated term?
300
00:18:52 --> 00:18:54
It would be, yeah.
301
00:18:54 --> 00:18:55
I even remember what it is.
302
00:18:55 --> 00:19:03
As you do better than me. u
times w at the, is that good?
303
00:19:03 --> 00:19:06
Yeah, I think.
304
00:19:06 --> 00:19:11
So it's really this part
that I'm now coming to.
305
00:19:11 --> 00:19:14
It's the boundary part
that I'm now coming to.
306
00:19:14 --> 00:19:18
And let me say, so I'm glad you
asked that question because
307
00:19:18 --> 00:19:23
I made it seem unimportant,
where that's not true at all.
308
00:19:23 --> 00:19:27
The boundary condition is part
of the definition of A, and
309
00:19:27 --> 00:19:30
part of the definition
of A transpose.
310
00:19:30 --> 00:19:34
Just the way I'm about to say
free fixed, I had to tell
311
00:19:34 --> 00:19:38
you that for you to
know what a was.
312
00:19:38 --> 00:19:41
Until I tell you the boundary
condition, you don't know
313
00:19:41 --> 00:19:42
what the boundary rows are.
314
00:19:42 --> 00:19:45
You only know the
inside of the matrix.
315
00:19:45 --> 00:19:47
Or one possible inside.
316
00:19:47 --> 00:19:51
So I'm thinking my inside
is going to be minus one,
317
00:19:51 --> 00:19:53
one, minus one, one.
318
00:19:53 --> 00:19:59
So on, as given we
find the differences.
319
00:19:59 --> 00:20:01
Minus one, one.
320
00:20:01 --> 00:20:03
But.
321
00:20:03 --> 00:20:04
Oh no, let's see.
322
00:20:04 --> 00:20:07
So I'm doing free fixed.
323
00:20:07 --> 00:20:08
Is that right?
324
00:20:08 --> 00:20:10
Am I doing free fixed?
325
00:20:10 --> 00:20:16
OK, so am I taking
free at the left end?
326
00:20:16 --> 00:20:17
Yes.
327
00:20:17 --> 00:20:21
Alright, so if I'm free at
the left and fixed at the
328
00:20:21 --> 00:20:23
right end, what's my A?
329
00:20:23 --> 00:20:27
We're getting better
at this, right?
330
00:20:27 --> 00:20:28
Minus one, one.
331
00:20:28 --> 00:20:29
Minus one, one.
332
00:20:29 --> 00:20:30
Minus one, one.
333
00:20:30 --> 00:20:37
Minus one, and the one
here gets chopped off.
334
00:20:37 --> 00:20:41
You could say if you want the
fifth row of A_0, remembering
335
00:20:41 --> 00:20:46
A_0 as the hint on the quiz,
where it had five rows for
336
00:20:46 --> 00:20:48
the full thing, free free.
337
00:20:48 --> 00:20:55
And then when an n got fixed,
the fifth column got removed,
338
00:20:55 --> 00:20:59
and that's my free
fixed matrix.
339
00:20:59 --> 00:21:02
At the left hand end, at
the zero end, it's got
340
00:21:02 --> 00:21:04
the difference in there.
341
00:21:04 --> 00:21:05
Difference, difference,
difference.
342
00:21:05 --> 00:21:11
But here at the right hand end,
it's the fixing, the setting u,
343
00:21:11 --> 00:21:15
whatever it would be. u_5 to
zero, or maybe it's u_4,
344
00:21:15 --> 00:21:17
because it's like
one, two, three.
345
00:21:17 --> 00:21:20
Setting u_4 to zero
knocked that out.
346
00:21:20 --> 00:21:22
OK.
347
00:21:22 --> 00:21:26
All I want to do is
transpose that.
348
00:21:26 --> 00:21:30
And you'll see something that
we maybe didn't notice before.
349
00:21:30 --> 00:21:33
So I transpose it, that's
minus one, one becomes
350
00:21:33 --> 00:21:36
a column, minus one,
one becomes a column.
351
00:21:36 --> 00:21:38
Minus one, one
becomes a column.
352
00:21:38 --> 00:21:41
Minus one, all there is.
353
00:21:41 --> 00:21:50
That row becomes, so this was,
so, have I got it right?
354
00:21:50 --> 00:21:52
Yes.
355
00:21:52 --> 00:21:55
What's happened?
356
00:21:55 --> 00:21:58
A transpose, what are
the boundary conditions
357
00:21:58 --> 00:22:00
going with A transpose?
358
00:22:00 --> 00:22:05
The boundary conditions that
went with a were, let me
359
00:22:05 --> 00:22:08
say first, what were the
boundary conditions with A?
360
00:22:08 --> 00:22:12
Those are going to be
boundary conditions on u.
361
00:22:12 --> 00:22:19
So A has boundary
conditions on u.
362
00:22:19 --> 00:22:26
And A transpose has
boundary conditions on w.
363
00:22:26 --> 00:22:31
Because A transpose acts
on w, and A acts on u.
364
00:22:31 --> 00:22:33
So there was no choice.
365
00:22:33 --> 00:22:36
So now what was the
boundary condition here?
366
00:22:36 --> 00:22:40
The boundary condition
was u_4=0, right?
367
00:22:40 --> 00:22:44
That was what I meant by
that guy getting fixed.
368
00:22:44 --> 00:22:49
Now, and no boundary condition
at u_0, it was free.
369
00:22:49 --> 00:22:51
Now, what are the boundary
conditions that go
370
00:22:51 --> 00:22:52
with A transpose?
371
00:22:52 --> 00:22:55
And remember, A transpose
is multiplying w.
372
00:22:55 --> 00:22:59
I'm going to put w here,
so what are the boundary
373
00:22:59 --> 00:23:02
conditions that go
with A transpose?
374
00:23:02 --> 00:23:05
This thing, nothing
got knocked off.
375
00:23:05 --> 00:23:09
The boundary condition came up
here for A transpose, the
376
00:23:09 --> 00:23:13
battery condition was w_0=0.
377
00:23:13 --> 00:23:21
That got knocked out,
w_0 to be zero.
378
00:23:21 --> 00:23:22
No surprise.
379
00:23:22 --> 00:23:26
Free fixed, this is the free
end at the left, this is the
380
00:23:26 --> 00:23:29
fixed end at the right.
381
00:23:29 --> 00:23:32
Did you ever notice that the
matrix does it for you?
382
00:23:32 --> 00:23:37
I mean, when you transpose that
matrix it just automatically
383
00:23:37 --> 00:23:44
built in the correct boundary
conditions on w by, you started
384
00:23:44 --> 00:23:48
with the conditions on u, you
transpose the matrix and you've
385
00:23:48 --> 00:23:51
discovered what the boundary
conditions on w are.
386
00:23:51 --> 00:23:58
And I'm going to do the same
for the continuous problem.
387
00:23:58 --> 00:24:01
I'm going to do the same for
the continuous problem, so
388
00:24:01 --> 00:24:08
the continuous free fixed.
389
00:24:08 --> 00:24:14
OK, what's the boundary
condition on u?
390
00:24:14 --> 00:24:18
If it's free fixed, I just want
you to repeat this, on the
391
00:24:18 --> 00:24:22
integral zero to one for
functions u(x), w(x)
392
00:24:22 --> 00:24:24
instead of for vectors.
393
00:24:24 --> 00:24:27
What's the boundary condition
on u, if I have a free
394
00:24:27 --> 00:24:30
fixed problem? u(1)=0.
395
00:24:30 --> 00:24:35
396
00:24:35 --> 00:24:38
u over one equals zero.
397
00:24:38 --> 00:24:40
So this is the boundary
conditions that goes with
398
00:24:40 --> 00:24:42
A in the free fixed case.
399
00:24:42 --> 00:24:52
And this is part of A.
400
00:24:52 --> 00:24:57
That is part of A.
401
00:24:57 --> 00:25:01
I don't know what a is until I
know its boundary condition.
402
00:25:01 --> 00:25:04
Just the way I don't know
what this matrix is.
403
00:25:04 --> 00:25:06
It could have been A_0, it
could have lost one column,
404
00:25:06 --> 00:25:08
it could have lost two
columns, whatever.
405
00:25:08 --> 00:25:11
I don't know until I've
told you the boundary
406
00:25:11 --> 00:25:12
condition on u.
407
00:25:12 --> 00:25:16
And then transposing is going
to tell me, automatically,
408
00:25:16 --> 00:25:19
without any further input, the
boundary condition that
409
00:25:19 --> 00:25:21
goes on the adjoint.
410
00:25:21 --> 00:25:26
So what's the boundary
condition on w that goes
411
00:25:26 --> 00:25:28
as part of A transpose?
412
00:25:28 --> 00:25:30
Well, you're going to tell me.
413
00:25:30 --> 00:25:36
Tell me. w(0) should be zero.
414
00:25:36 --> 00:25:39
It came out automatically,
naturally.
415
00:25:39 --> 00:25:43
This is a big distinction
between boundary conditions.
416
00:25:43 --> 00:25:48
I would call that an essential
boundary condition.
417
00:25:48 --> 00:25:51
I had to start with it, I
had to decide on that.
418
00:25:51 --> 00:25:54
And then this, I call a
natural boundary condition.
419
00:25:54 --> 00:25:59
Or there are even two guys'
names, which are associated
420
00:25:59 --> 00:26:01
with these two types.
421
00:26:01 --> 00:26:05
So maybe a first chance to
just mention these names.
422
00:26:05 --> 00:26:14
Because you'll often see,
reading some paper, maybe a
423
00:26:14 --> 00:26:16
little on the mathematical
side, you'll see
424
00:26:16 --> 00:26:18
that word used.
425
00:26:18 --> 00:26:20
The guy's name with this
sort of boundary condition
426
00:26:20 --> 00:26:22
is a French name.
427
00:26:22 --> 00:26:25
Not so easy to say, Dirichlet.
428
00:26:25 --> 00:26:28
I'll say it more
often in the future.
429
00:26:28 --> 00:26:33
Anyway, I would call that a
Dirichlet condition and you
430
00:26:33 --> 00:26:35
would say it's a fixed
boundary condition.
431
00:26:35 --> 00:26:37
And if you were doing heat
flow you would say it's
432
00:26:37 --> 00:26:39
a fixed temperature.
433
00:26:39 --> 00:26:40
Whatever.
434
00:26:40 --> 00:26:45
Fixed, is really the
word to remember there.
435
00:26:45 --> 00:26:51
OK, and then I guess I better
give Germany a shot here, too.
436
00:26:51 --> 00:26:56
So the boundary condition
is associated with
437
00:26:56 --> 00:27:04
the name of Neumann.
438
00:27:04 --> 00:27:09
So if I said a Dirichlet
problem, a total Dirichlet
439
00:27:09 --> 00:27:13
problem, I would be speaking
about fixed fixed.
440
00:27:13 --> 00:27:15
And if I spoke about a
Neumann problem I would be
441
00:27:15 --> 00:27:17
talking about free free.
442
00:27:17 --> 00:27:21
And this problem is Dirichlet
at one end, Neumann
443
00:27:21 --> 00:27:22
at the other.
444
00:27:22 --> 00:27:26
Anyway, so essential
and natural.
445
00:27:26 --> 00:27:31
And now of course I'm hoping
that that's going to make this
446
00:27:31 --> 00:27:38
this boundary term go away.
447
00:27:38 --> 00:27:41
OK, now I'm paying attention
to this thing that you
448
00:27:41 --> 00:27:42
made me write. uw.
449
00:27:43 --> 00:27:45
OK, what happens there?
450
00:27:46 --> 00:27:48
uw?
451
00:27:48 --> 00:27:53
Oh, right this isn't bad
because it shows that there's a
452
00:27:53 --> 00:27:59
boundary condition, I've got
some little deals going.
453
00:27:59 --> 00:28:06
But do you see that
that becomes zero?
454
00:28:06 --> 00:28:10
Why isn't zero at the
top end, at one?
455
00:28:10 --> 00:28:14
When I take u times w at
one, why do I get zero?
456
00:28:14 --> 00:28:17
Because u(1) is zero.
457
00:28:17 --> 00:28:18
Good.
458
00:28:18 --> 00:28:20
And it's the bottom end.
459
00:28:20 --> 00:28:25
When I take uw at the other
boundary, why do I get zero?
460
00:28:25 --> 00:28:27
Because of w.
461
00:28:27 --> 00:28:30
You see that w was needed.
462
00:28:30 --> 00:28:34
That w(0) was needed because
there was no controlling u(0).
463
00:28:35 --> 00:28:38
I had no control of u
at the left hand end,
464
00:28:38 --> 00:28:39
because it was free.
465
00:28:39 --> 00:28:43
So the control has
to come from w.
466
00:28:43 --> 00:28:47
And so w naturally had to
be zero, because I wasn't
467
00:28:47 --> 00:28:51
controlling u at that
left hand, free end.
468
00:28:51 --> 00:28:57
So one way or the other,
the integration by
469
00:28:57 --> 00:28:59
parts is the key.
470
00:28:59 --> 00:29:09
So that said, what I critically
wanted to say about
471
00:29:09 --> 00:29:14
transposing, taking the
adjoint, except I was just
472
00:29:14 --> 00:29:17
going to add a comment about
this square A versus
473
00:29:17 --> 00:29:20
rectangular.
474
00:29:20 --> 00:29:24
And this was a case
of square, right?
475
00:29:24 --> 00:29:27
This was a case, this free
fixed case, this example I
476
00:29:27 --> 00:29:29
happened to pick was squared.
477
00:29:29 --> 00:29:32
A_0, the free-free guy
that was a hint on the
478
00:29:32 --> 00:29:34
quiz, was rectangular.
479
00:29:34 --> 00:29:38
The fixed fixed, which was
also on the quiz, was
480
00:29:38 --> 00:29:40
also rectangular.
481
00:29:40 --> 00:29:44
It was what, four by
three or something.
482
00:29:44 --> 00:29:47
This A is four by four.
483
00:29:47 --> 00:29:57
And what is especially
nice when it's squared?
484
00:29:57 --> 00:30:01
If our problem happens to give
a square matrix, in the trust
485
00:30:01 --> 00:30:05
case if the number of
displacement unknowns happens
486
00:30:05 --> 00:30:09
to equal the number of bars, so
m equals n, I have
487
00:30:09 --> 00:30:11
a square matrix A.
488
00:30:11 --> 00:30:16
And this guy's invertible,
so it's all good.
489
00:30:16 --> 00:30:18
Oh, that may be
the whole point.
490
00:30:18 --> 00:30:21
That if it's rectangular
I wouldn't talk
491
00:30:21 --> 00:30:22
about its inverse.
492
00:30:22 --> 00:30:27
But this is a square matrix,
so A itself has an inverse.
493
00:30:27 --> 00:30:32
Instead of having, as I usually
have, to deal with A transpose
494
00:30:32 --> 00:30:36
C A all at once, let me put
this comment because it's
495
00:30:36 --> 00:30:40
just a small one up here.
496
00:30:40 --> 00:30:41
Right under these words.
497
00:30:41 --> 00:30:43
Square versus rectangular.
498
00:30:43 --> 00:30:50
Square A, and let's say
invertible, otherwise we're in
499
00:30:50 --> 00:30:55
the unstable K, so you know
what I mean, in the network
500
00:30:55 --> 00:31:02
problems the number of nodes
matches the number of edges?
501
00:31:02 --> 00:31:10
In the spring problem we
have free fixed situations.
502
00:31:10 --> 00:31:12
Anyway, A comes out square.
503
00:31:12 --> 00:31:15
Whatever the application.
504
00:31:15 --> 00:31:19
If it comes out square,
what is especially good?
505
00:31:19 --> 00:31:22
It comes out square, what's
especially good is that
506
00:31:22 --> 00:31:23
it has an inverse.
507
00:31:23 --> 00:31:29
So that in this square case,
this K inverse is A transpose
508
00:31:29 --> 00:31:35
C A inverse, can be split.
509
00:31:35 --> 00:31:40
This allows us to separate,
to do what you better
510
00:31:40 --> 00:31:45
not do otherwise.
511
00:31:45 --> 00:31:50
In other words, we are three
steps, which usually mash
512
00:31:50 --> 00:31:53
together and we can't separate
them and we have to deal with
513
00:31:53 --> 00:31:55
the whole matrix at once.
514
00:31:55 --> 00:31:58
In this square case,
they do separate.
515
00:31:58 --> 00:32:02
And so that's worth noticing.
516
00:32:02 --> 00:32:10
It means that we can solve
backwards, we can solve
517
00:32:10 --> 00:32:13
these three one at a time.
518
00:32:13 --> 00:32:15
The inverses can be
done separately.
519
00:32:15 --> 00:32:20
When A and A transpose are
square, then from this
520
00:32:20 --> 00:32:23
equation I can find w.
521
00:32:23 --> 00:32:27
From knowing w I can find
e, just by inverting C.
522
00:32:27 --> 00:32:30
By knowing e I can find
u, just by inverting A.
523
00:32:30 --> 00:32:32
You see the three steps?
524
00:32:32 --> 00:32:34
You could invert that, and then
you can invert the middle
525
00:32:34 --> 00:32:36
step, and you can invert A.
526
00:32:36 --> 00:32:37
And you've got u.
527
00:32:37 --> 00:32:41
So the square case
is worth noticing.
528
00:32:41 --> 00:32:46
It special enough that
in this case we would
529
00:32:46 --> 00:32:48
have an easy problem.
530
00:32:48 --> 00:32:54
And this case is called,
for trusses and mechanics,
531
00:32:54 --> 00:32:55
there's a name for this.
532
00:32:55 --> 00:32:57
And unfortunately
it's a little long.
533
00:32:57 --> 00:33:02
Statically, that's not the
key word, determinant
534
00:33:02 --> 00:33:07
is the key word.
535
00:33:07 --> 00:33:13
Statically determinate,
that goes with squares.
536
00:33:13 --> 00:33:16
And you can guess what the
rectangular matrix a, what
537
00:33:16 --> 00:33:19
word would I use, what's the
opposite of determinate?
538
00:33:19 --> 00:33:23
It's got to be indeterminate.
539
00:33:23 --> 00:33:26
Rectangular a will
be indeterminate.
540
00:33:26 --> 00:33:32
And all that is referring to
is the fact that in the
541
00:33:32 --> 00:33:40
determinate case the forces
determine the stresses.
542
00:33:40 --> 00:33:45
You don't have to take that,
we'd get all three together;
543
00:33:45 --> 00:33:48
mix them, invert, go backwards.
544
00:33:48 --> 00:33:48
All that.
545
00:33:48 --> 00:33:53
You just can do them one at the
time in this determinate case.
546
00:33:53 --> 00:33:58
And now I guess I'd better say,
so here's the matrix case.
547
00:33:58 --> 00:34:00
But now in this chapter
I always have to do
548
00:34:00 --> 00:34:02
the continuous part.
549
00:34:02 --> 00:34:09
So let me just stay with
free fixed, and what is
550
00:34:09 --> 00:34:11
this balance equation?
551
00:34:11 --> 00:34:17
So this is my force balance.
552
00:34:17 --> 00:34:23
I didn't give it its moment
but its moment has come now.
553
00:34:23 --> 00:34:29
So the force balance equation
is -dw/dx, because A transpose
554
00:34:29 --> 00:34:35
is minus a derivative equal
f(x). and my free boundary
555
00:34:35 --> 00:34:43
condition, my free n, gave
me w of what was it?
556
00:34:43 --> 00:34:45
The Neumann guy gave me w(0)=0.
557
00:34:45 --> 00:34:49
558
00:34:49 --> 00:34:52
And what's the point?
559
00:34:52 --> 00:34:54
Do you see what I'm saying?
560
00:34:54 --> 00:34:57
I'm saying that this free
fixed is a beautiful
561
00:34:57 --> 00:34:59
example of determinate.
562
00:34:59 --> 00:35:05
Square matrix A in the matrix
case and the parallel and
563
00:35:05 --> 00:35:08
the continuous cases.
564
00:35:08 --> 00:35:09
I can solve that for w(x).
565
00:35:09 --> 00:35:12
566
00:35:12 --> 00:35:14
I can solve directly for w(x).
567
00:35:14 --> 00:35:18
568
00:35:18 --> 00:35:23
Without involving, you see I
didn't have to know c(x).
569
00:35:24 --> 00:35:26
I hadn't even got that far.
570
00:35:26 --> 00:35:28
I'm just going backwards now.
571
00:35:28 --> 00:35:35
I can solve this, just the way
I can invert that matrix.
572
00:35:35 --> 00:35:37
Inverting the matrix
here is the same as
573
00:35:37 --> 00:35:39
solving the equation.
574
00:35:39 --> 00:35:41
You see I have a first
order first derivative?
575
00:35:41 --> 00:35:44
I mean, it's so trivial, right?
576
00:35:44 --> 00:35:48
It's the equation you solved in
the final problem of the quiz,
577
00:35:48 --> 00:35:50
where an f was a
delta function.
578
00:35:50 --> 00:35:55
It was simple because it was
a square determinate problem
579
00:35:55 --> 00:35:58
with one condition on w.
580
00:35:58 --> 00:36:08
When both conditions are on u,
then it's not square any more.
581
00:36:08 --> 00:36:10
OK for that point?
582
00:36:10 --> 00:36:13
Determinate versus
indeterminate.
583
00:36:13 --> 00:36:14
OK.
584
00:36:14 --> 00:36:19
So that's sort of, and
I could do examples.
585
00:36:19 --> 00:36:21
Maybe I've asked you on
the homework to take
586
00:36:21 --> 00:36:22
a particular f(x).
587
00:36:23 --> 00:36:26
I hope it was a free fixed
problem, if I was feeling good
588
00:36:26 --> 00:36:29
that day, because free fixed
you'll be able to get
589
00:36:29 --> 00:36:31
w(x) right away.
590
00:36:31 --> 00:36:34
If it was fixed-fixed then
I apologize it's going to
591
00:36:34 --> 00:36:39
take you a little bit
longer to get to w.
592
00:36:39 --> 00:36:41
To get to u.
593
00:36:41 --> 00:36:41
OK.
594
00:36:41 --> 00:36:47
But this, of course-
I just integrate.
595
00:36:47 --> 00:36:55
Inverting a difference matrix
is just integrating a function.
596
00:36:55 --> 00:36:56
Good.
597
00:36:56 --> 00:37:07
OK, so this lecture so far was
the transition from vectors
598
00:37:07 --> 00:37:11
and matrices to functions
and continuous problems.
599
00:37:11 --> 00:37:15
And then, of course we're going
to get deep into that because
600
00:37:15 --> 00:37:18
we got partial differential
equations ahead.
601
00:37:18 --> 00:37:22
But today let's stay in one
dimension and introduce
602
00:37:22 --> 00:37:23
finite elements.
603
00:37:23 --> 00:37:26
OK.
604
00:37:26 --> 00:37:34
Ready for finite elements,
so that's now a major step.
605
00:37:34 --> 00:37:38
Finite differences, maybe I'll
mention this, probably in this
606
00:37:38 --> 00:37:41
afternoon's review session,
where I'll just be open
607
00:37:41 --> 00:37:43
to homework problems.
608
00:37:43 --> 00:37:47
I'll say something more about
truss examples, and I might
609
00:37:47 --> 00:37:50
say something about finite
differences for this.
610
00:37:50 --> 00:37:53
But really, it's finite
elements that get
611
00:37:53 --> 00:37:55
introduced right now.
612
00:37:55 --> 00:37:57
So let me do that.
613
00:37:57 --> 00:38:02
Finite elements and
introducing them.
614
00:38:02 --> 00:38:06
OK.
615
00:38:06 --> 00:38:10
So the prep, the getting ready
for finite elements is to get
616
00:38:10 --> 00:38:23
hold of something called the
weak form of the equation.
617
00:38:23 --> 00:38:28
So that's going to be a
statement of, the finite
618
00:38:28 --> 00:38:30
elements aren't appearing yet.
619
00:38:30 --> 00:38:32
Matrices are not appearing yet.
620
00:38:32 --> 00:38:38
I'm talking about the
differential equation.
621
00:38:38 --> 00:38:41
But what do I mean
by this weak form?
622
00:38:41 --> 00:38:44
OK, let me just go
for it directly.
623
00:38:44 --> 00:38:46
You see the equation up there?
624
00:38:46 --> 00:38:47
Let me copy it.
625
00:38:47 --> 00:38:52
So here's the strong form.
626
00:38:52 --> 00:38:55
The strong form is, you would
say, the ordinary equation.
627
00:38:55 --> 00:39:01
Strong form is what our
equation is, minus d/dx
628
00:39:01 --> 00:39:03
of c(x), du/dx=f(x).
629
00:39:03 --> 00:39:11
630
00:39:11 --> 00:39:14
OK, that's the strong form.
631
00:39:14 --> 00:39:15
That's the equation.
632
00:39:15 --> 00:39:19
Now, how do I get
to the weak form?
633
00:39:19 --> 00:39:22
Let me just go to it directly
and then over the next
634
00:39:22 --> 00:39:27
days we'll see why it's so
natural and important.
635
00:39:27 --> 00:39:31
If I go for it directly,
what I do is this.
636
00:39:31 --> 00:39:35
I multiply both sides of the
equation by something I'll
637
00:39:35 --> 00:39:37
call a test function.
638
00:39:37 --> 00:39:41
And I'll try to systematically
use the letter v for the test
639
00:39:41 --> 00:39:48
function. u will be the
solution. v isn't the solution,
640
00:39:48 --> 00:39:54
v is like any function that I
test this equation this way.
641
00:39:54 --> 00:39:57
I'm just multiplying both sides
by the same thing, some v(x).
642
00:39:58 --> 00:39:58
Any v(x).
643
00:40:00 --> 00:40:03
We'll see if there are
any limitations, OK?
644
00:40:03 --> 00:40:14
And I integrate.
645
00:40:14 --> 00:40:15
OK.
646
00:40:15 --> 00:40:18
So you're going to say
nope, no problem.
647
00:40:18 --> 00:40:22
I integrate from zero to one.
648
00:40:22 --> 00:40:30
Alright, this would
be true for f.
649
00:40:30 --> 00:40:34
So now I'll erase the word
strong form, because the
650
00:40:34 --> 00:40:37
strong form isn't on
the board anymore.
651
00:40:37 --> 00:40:40
It's the weak form now
that we're looking at.
652
00:40:40 --> 00:40:44
And this is for any, and I'll
put "any" in quotes just
653
00:40:44 --> 00:40:51
because eventually I'll say
a little more about this.
654
00:40:51 --> 00:40:56
I'll write the
equation this way.
655
00:40:56 --> 00:41:02
And you might think, OK, if
this has to hold for every
656
00:41:02 --> 00:41:09
v(x), I could let v(x) be
concentrated in a little area.
657
00:41:09 --> 00:41:11
And this would have to hold,
then I could try another
658
00:41:11 --> 00:41:16
v(x), concentrated
around other points.
659
00:41:16 --> 00:41:23
You can maybe feel that if this
holds for every v(x), then
660
00:41:23 --> 00:41:25
I can get back to
the strong form.
661
00:41:25 --> 00:41:29
If this holds for every
v(x), then somehow that had
662
00:41:29 --> 00:41:31
better be the same as that.
663
00:41:31 --> 00:41:39
Because if this was f(x+1) and
this is f(x), then I wouldn't
664
00:41:39 --> 00:41:41
have the equality any more.
665
00:41:41 --> 00:41:43
Should I just say that again?
666
00:41:43 --> 00:41:46
It's just like, at this point
it's just a feeling, that
667
00:41:46 --> 00:41:52
if this is true for every
v(x), then that part had
668
00:41:52 --> 00:41:55
better equal that part.
669
00:41:55 --> 00:41:57
That'll be my way back
to the strong form.
670
00:41:57 --> 00:41:59
It's a little bit like
climbing a hill.
671
00:41:59 --> 00:42:03
Going downhill was easy, I just
multiplied by v and integrated.
672
00:42:03 --> 00:42:05
Nobody objected to that.
673
00:42:05 --> 00:42:10
I'm saying I'll be able to
get back to the strong form
674
00:42:10 --> 00:42:11
with a little patience.
675
00:42:11 --> 00:42:14
But I like the week form.
676
00:42:14 --> 00:42:15
That's the whole point.
677
00:42:15 --> 00:42:19
You've got to begin to
like the week form.
678
00:42:19 --> 00:42:22
If you begin to take
it in and think OK.
679
00:42:22 --> 00:42:25
Now, why do I like it?
680
00:42:25 --> 00:42:28
What am I going to do
to that left side?
681
00:42:28 --> 00:42:30
The right side's cool, right?
682
00:42:30 --> 00:42:31
It looks good.
683
00:42:31 --> 00:42:35
Left side does not
look good to me.
684
00:42:35 --> 00:42:38
When you see something like
that, what do you think?
685
00:42:38 --> 00:42:42
Today's lecture has already
said what to think.
686
00:42:42 --> 00:42:46
What should I do to
make that look better?
687
00:42:46 --> 00:42:48
I should, yep.
688
00:42:48 --> 00:42:50
Integrate by parts.
689
00:42:50 --> 00:42:54
If I integrate by parts, you
see what I don't like about it
690
00:42:54 --> 00:42:58
as it is, is two derivatives
are hitting u, and
691
00:42:58 --> 00:43:01
v is by itself.
692
00:43:01 --> 00:43:05
And I want it to be symmetric.
693
00:43:05 --> 00:43:08
I'm going to integrate this
by parts, this is minus the
694
00:43:08 --> 00:43:10
derivative of something.
695
00:43:10 --> 00:43:12
Times v.
696
00:43:12 --> 00:43:15
And when I integrate by
parts, I'm going to have,
697
00:43:15 --> 00:43:17
it'll be an integral.
698
00:43:17 --> 00:43:19
And can you integrate
by parts now?
699
00:43:19 --> 00:43:24
I mean, you probably haven't
thought about integration
700
00:43:24 --> 00:43:26
by parts for a while.
701
00:43:26 --> 00:43:29
Just think of it as taking the
derivative off of this, so
702
00:43:29 --> 00:43:32
it leaves that by itself.
703
00:43:32 --> 00:43:41
And putting it onto v, so it's
dv/dx, and remembering the
704
00:43:41 --> 00:43:43
minus sign, but we have a minus
sign so now it's
705
00:43:43 --> 00:43:45
coming up plus.
706
00:43:45 --> 00:43:46
That's the weak form.
707
00:43:46 --> 00:43:50
Can I put a circle
around the weak form?
708
00:43:50 --> 00:43:53
Well, that wasn't
exactly a circle.
709
00:43:53 --> 00:43:54
OK.
710
00:43:54 --> 00:43:57
But that's the weak form.
711
00:43:57 --> 00:44:01
For every v, this is, I could
give you a physical
712
00:44:01 --> 00:44:10
interpretation but I won't do
it just this minute.
713
00:44:10 --> 00:44:14
This is going to
hold for any v.
714
00:44:14 --> 00:44:16
That's the weak form.
715
00:44:16 --> 00:44:17
OK.
716
00:44:17 --> 00:44:18
Good.
717
00:44:18 --> 00:44:25
Now, why did I want to do that?
718
00:44:25 --> 00:44:29
The person who reminds me
about boundary conditions
719
00:44:29 --> 00:44:31
should remind me again.
720
00:44:31 --> 00:44:33
That when I did this
integration by parts, there
721
00:44:33 --> 00:44:36
should have been also.
722
00:44:36 --> 00:44:39
What's the integrated
part now, that has to be
723
00:44:39 --> 00:44:43
evaluated at zero and one?
724
00:44:43 --> 00:44:47
This c, so it's that
times that, right?
725
00:44:47 --> 00:44:53
It's that c(x)du/dv times v(x).
726
00:44:53 --> 00:44:56
727
00:44:56 --> 00:44:57
Maybe minus.
728
00:44:57 --> 00:44:58
Yeah, you're right.
729
00:44:58 --> 00:44:59
Minus.
730
00:44:59 --> 00:45:01
Good.
731
00:45:01 --> 00:45:03
What do I want
this to come out?
732
00:45:03 --> 00:45:05
Zero, of course.
733
00:45:05 --> 00:45:08
I don't want to think
that the same.
734
00:45:08 --> 00:45:11
Alright, so now I'm doing this
free fixed problem still.
735
00:45:11 --> 00:45:16
So what's the deal on
the free fixed problem?
736
00:45:16 --> 00:45:22
Well, let's see.
737
00:45:22 --> 00:45:26
OK, I got the two ends and
I want them to be zero.
738
00:45:26 --> 00:45:38
OK, now at the free end,
I'm not controlling v.
739
00:45:38 --> 00:45:40
I wasn't controlling u
and I'm not going to be
740
00:45:40 --> 00:45:43
controlling its friend v.
741
00:45:43 --> 00:45:45
So this had to be zero.
742
00:45:45 --> 00:45:52
So this part will be
zero at the free end.
743
00:45:52 --> 00:45:56
That boundary condition has
just appeared again naturally.
744
00:45:56 --> 00:45:59
I had to have it because
I had no control over b.
745
00:45:59 --> 00:46:01
And what about at
the fixed end.
746
00:46:01 --> 00:46:07
At the fixed end, which is
that, at the free end.
747
00:46:07 --> 00:46:11
Now, what's up at
the fixed end?
748
00:46:11 --> 00:46:13
What was the fixed end?
749
00:46:13 --> 00:46:17
That's where u was zero.
750
00:46:17 --> 00:46:20
I'm going to make v also zero.
751
00:46:20 --> 00:46:27
So there's, when I said any
v(x), I better put in with v=0
752
00:46:27 --> 00:46:37
at the Dirichlet point, at
fixed point, at fixed end.
753
00:46:37 --> 00:46:40
I need that.
754
00:46:40 --> 00:46:44
I need to know that v
is zero at that end.
755
00:46:44 --> 00:46:45
I had u=0.
756
00:46:46 --> 00:46:50
Here's why I'm fine.
757
00:46:50 --> 00:46:55
So I'm saying that any time I
have a Dirichlet condition, a
758
00:46:55 --> 00:46:59
fixed condition that tells me
u, I think of v and you'll
759
00:46:59 --> 00:47:02
begin to think of v as a
little movement away from
760
00:47:02 --> 00:47:13
u. u is the solution.
761
00:47:13 --> 00:47:16
Now, remind me, this
was free fixed.
762
00:47:16 --> 00:47:21
So the u might have been
something like this.
763
00:47:21 --> 00:47:22
I just draw that.
764
00:47:22 --> 00:47:24
That's my u.
765
00:47:24 --> 00:47:27
This guy was fixed, right?
766
00:47:27 --> 00:47:29
By u.
767
00:47:29 --> 00:47:35
Now, I'm thinking of v's as,
the letter v is very fortunate
768
00:47:35 --> 00:47:38
because it stands for
virtual displacement.
769
00:47:38 --> 00:47:41
A virtual displacement is a
little displacement away from
770
00:47:41 --> 00:47:46
u, but it has to satisfy the
zero, the fixed condition
771
00:47:46 --> 00:47:48
that u satisfied.
772
00:47:48 --> 00:47:51
In other words, the little
virtual v can't move
773
00:47:51 --> 00:47:53
away from zero.
774
00:47:53 --> 00:48:02
So I get this term is
zero at the fixed end.
775
00:48:02 --> 00:48:11
OK. that's the little five
minute time out state to check
776
00:48:11 --> 00:48:13
the boundary condition part.
777
00:48:13 --> 00:48:19
The net result is that that
term's gone and I've got the
778
00:48:19 --> 00:48:21
weak format I've wanted.
779
00:48:21 --> 00:48:26
OK, three minutes to
start to tell you how
780
00:48:26 --> 00:48:30
to use the weak form.
781
00:48:30 --> 00:48:39
So this is called
Galerkin's method.
782
00:48:39 --> 00:48:50
And it starts with
the weak form.
783
00:48:50 --> 00:48:51
So he's Russian.
784
00:48:51 --> 00:48:54
Russia gets into
the picture now.
785
00:48:54 --> 00:48:56
We had France and Germany with
the boundary conditions, now
786
00:48:56 --> 00:49:01
we've got Russia with this
fundamental principle of how to
787
00:49:01 --> 00:49:06
turn a continuous problem
into a discrete problem.
788
00:49:06 --> 00:49:09
That's what Galerkin's
idea does.
789
00:49:09 --> 00:49:13
Instead of a function unknown
I want to have n unknowns.
790
00:49:13 --> 00:49:17
I want to get a discrete
equation which will
791
00:49:17 --> 00:49:19
eventually be kKU=F.
792
00:49:20 --> 00:49:27
So I'm going to get to an
equation KU=F, but not by
793
00:49:27 --> 00:49:29
finite difference, right?
794
00:49:29 --> 00:49:31
I could but, I'm not.
795
00:49:31 --> 00:49:36
I'm doing it this weak
Galerkin finite element way.
796
00:49:36 --> 00:49:42
OK, so if I tell you the
Galerkin idea then next time we
797
00:49:42 --> 00:49:46
bring in, we have libraries
of finite elements.
798
00:49:46 --> 00:49:48
But you have to get the
principle straight.
799
00:49:48 --> 00:49:51
So it's Galerkin's idea.
800
00:49:51 --> 00:50:01
Galerkin's idea was was
choose trial functions.
801
00:50:01 --> 00:50:11
Let me call them call them t?
802
00:50:11 --> 00:50:16
Have to get the names right.
803
00:50:16 --> 00:50:21
Phi.
804
00:50:21 --> 00:50:24
OK, the Greeks get a shot ok.
805
00:50:24 --> 00:50:29
Trial functions,
phi_1(x) to phi_n(x).
806
00:50:31 --> 00:50:35
OK, so that's a
choice you make.
807
00:50:35 --> 00:50:37
And we have a free choice.
808
00:50:37 --> 00:50:40
And it's a fundamental choice
for all of applied math here.
809
00:50:40 --> 00:50:43
You choose some functions, and
if you choose them well you get
810
00:50:43 --> 00:50:45
a great method, if you choose
them badly you got
811
00:50:45 --> 00:50:47
a lousy method.
812
00:50:47 --> 00:50:49
OK, so you choose trial
functions, and now what's
813
00:50:49 --> 00:50:51
the idea going to be?
814
00:50:51 --> 00:50:59
Your approximate U, approximate
solution will be some
815
00:50:59 --> 00:51:03
combination of this.
816
00:51:03 --> 00:51:09
So combinations of those, let
me call the coefficients U's,
817
00:51:09 --> 00:51:10
because those are the unknowns.
818
00:51:10 --> 00:51:12
Plus U_n*phi_n.
819
00:51:12 --> 00:51:15
820
00:51:15 --> 00:51:16
So those are the unknowns.
821
00:51:16 --> 00:51:23
The n unknowns.
822
00:51:23 --> 00:51:28
I'll even remove that
for the moment.
823
00:51:28 --> 00:51:31
You see, these are
functions of x.
824
00:51:31 --> 00:51:34
And these are numbers.
825
00:51:34 --> 00:51:39
So our unknown, our n unknown
numbers are the coefficients
826
00:51:39 --> 00:51:43
to be decided of the
functions we chose.
827
00:51:43 --> 00:51:47
OK., now I need n equations.
828
00:51:47 --> 00:51:49
I've got n unknowns now,
they're the unknown
829
00:51:49 --> 00:51:52
coefficients of
these functions.
830
00:51:52 --> 00:51:55
I need an equation so
I get n equations by
831
00:51:55 --> 00:52:01
choose test functions.
832
00:52:01 --> 00:52:08
V_1, V_2, up to V(x).
833
00:52:08 --> 00:52:11
Each V will give
me an equation.
834
00:52:11 --> 00:52:14
So I'll have n equations at
the end, I have n unknowns,
835
00:52:14 --> 00:52:16
I'll have a square matrix.
836
00:52:16 --> 00:52:20
And that'll be a linear system.
837
00:52:20 --> 00:52:21
I'll get to KU=F.
838
00:52:22 --> 00:52:24
But do you see how
I'm getting there?
839
00:52:24 --> 00:52:30
I'm getting there by using the
weak form, By using Galerkin's
840
00:52:30 --> 00:52:33
idea of picking some trial
functions, and some test
841
00:52:33 --> 00:52:37
functions, and putting
them into the weak form.
842
00:52:37 --> 00:52:41
So Galerkin's idea is,
take these functions
843
00:52:41 --> 00:52:43
and these functions.
844
00:52:43 --> 00:52:47
And apply the weak form
just to those guys.
845
00:52:47 --> 00:52:50
Not to, the real weak form,
the continuous weak form,
846
00:52:50 --> 00:52:53
was for a whole lot of V.
847
00:52:53 --> 00:52:57
We'll get n equations by
picking n V's, and we'll get n
848
00:52:57 --> 00:53:01
unknowns by picking n phis.
849
00:53:01 --> 00:53:04
So this method, this idea,
was a hundred years older
850
00:53:04 --> 00:53:06
than finite elements.
851
00:53:06 --> 00:53:11
The finite element idea was a
particular choice of these
852
00:53:11 --> 00:53:18
guys, a particular choice of
the phis and the V's as
853
00:53:18 --> 00:53:21
simple polynomials.
854
00:53:21 --> 00:53:25
And you might think well, why
didn't Galerkin try those
855
00:53:25 --> 00:53:26
first, maybe he did.
856
00:53:26 --> 00:53:35
But key is that now with the
computing power we now have
857
00:53:35 --> 00:53:40
compared to Galerkin, we can
choose thousands of functions.
858
00:53:40 --> 00:53:41
If we keep them simple.
859
00:53:41 --> 00:53:46
So that's really what the
finite element brought,
860
00:53:46 --> 00:53:48
finite element brought is.
861
00:53:48 --> 00:53:51
Keep the functions as
simple polynomials and
862
00:53:51 --> 00:53:53
take many of them.
863
00:53:53 --> 00:53:59
Where Galerkin, who didn't have
MATLAB, he probably didn't even
864
00:53:59 --> 00:54:02
have a desk computer, he used
pencil and paper, he
865
00:54:02 --> 00:54:04
took one function.
866
00:54:04 --> 00:54:05
Or maybe two.
867
00:54:05 --> 00:54:08
I mean, that took him a day.
868
00:54:08 --> 00:54:12
But we take thousands of
functions, simple functions,
869
00:54:12 --> 00:54:17
and we'll see on Friday the
steps that get us to KU=F.
870
00:54:17 --> 00:54:21
So this is the prep
for finite elements.