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PROFESSOR STRANG: Okay.

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Hi.

00:00:21.400 --> 00:00:28.710
So, our goal, certainly,
to reach next week

00:00:28.710 --> 00:00:30.980
is partial
differential equations.

00:00:30.980 --> 00:00:34.990
Laplace's equation.

00:00:34.990 --> 00:00:39.820
Additional topics still
to see clearly in 1-D.

00:00:39.820 --> 00:00:43.700
So one of those topics,
these will both come today,

00:00:43.700 --> 00:00:48.150
one of those topics is the idea,
still in the finite element

00:00:48.150 --> 00:00:51.110
world, of element matrices.

00:00:51.110 --> 00:00:58.590
So you remember, we saw those,
that each bar in the truss

00:00:58.590 --> 00:01:02.080
could give a piece of A
transpose A that could be

00:01:02.080 --> 00:01:05.790
stamped in, or assembled,
to use the right word --

00:01:05.790 --> 00:01:09.810
I think assembled is maybe used
more than stamped in, but both

00:01:09.810 --> 00:01:20.130
okay -- into K. For graphs,
an edge in the graph gave us

00:01:20.130 --> 00:01:25.430
a little [1, -1; -1, 1] matrix
that could be stamped in.

00:01:25.430 --> 00:01:29.640
And now we want to see,
how does that process work

00:01:29.640 --> 00:01:31.110
for finite elements.

00:01:31.110 --> 00:01:34.360
Because that's how finite
element matrices are really

00:01:34.360 --> 00:01:37.710
put together, out of
these element matrices.

00:01:37.710 --> 00:01:43.040
Okay, so that's step one,
that's half today's lecture.

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Step two, problem two now,
coming from the next section,

00:01:48.210 --> 00:01:51.090
is fourth order equations.

00:01:51.090 --> 00:01:53.180
Up to now, all our
differential equations

00:01:53.180 --> 00:01:55.330
have been second order.

00:01:55.330 --> 00:01:58.010
Are there fourth order
equations that are important?

00:01:58.010 --> 00:01:59.490
Yes, there are.

00:01:59.490 --> 00:02:01.230
For beam bending.

00:02:01.230 --> 00:02:05.150
So I'll describe that
application, which leads

00:02:05.150 --> 00:02:07.170
to fourth order equations.

00:02:07.170 --> 00:02:11.580
Do they fit our A
transpose C A framework?

00:02:11.580 --> 00:02:13.000
You bet.

00:02:13.000 --> 00:02:16.760
You know they will.

00:02:16.760 --> 00:02:21.870
Each additional application
in the framework kind of

00:02:21.870 --> 00:02:25.560
gets us comfortable,
familiar with that framework,

00:02:25.560 --> 00:02:26.690
what it can do.

00:02:26.690 --> 00:02:28.490
The A transpose C A.

00:02:28.490 --> 00:02:32.000
So let me start with
element matrices.

00:02:32.000 --> 00:02:34.350
I'll get the homework,
those numbers

00:02:34.350 --> 00:02:36.760
will get posted on the
website later today.

00:02:36.760 --> 00:02:38.910
I just thought
I'd put them down,

00:02:38.910 --> 00:02:43.010
and I have to figure out what
would be a suitable MATLAB

00:02:43.010 --> 00:02:43.880
question.

00:02:43.880 --> 00:02:47.830
So my idea for this
homework, as it really

00:02:47.830 --> 00:02:50.760
was for the last
homework, was that you

00:02:50.760 --> 00:02:55.860
get some, not a large number
of ordinary questions, paper

00:02:55.860 --> 00:02:59.780
and pencil questions,
and one MATLAB question.

00:02:59.780 --> 00:03:06.580
When I wrote "MATLAB last",
people interpreted that

00:03:06.580 --> 00:03:09.570
as "last MATLAB".

00:03:09.570 --> 00:03:11.380
But those words are
not commutative.

00:03:11.380 --> 00:03:13.100
Right?

00:03:13.100 --> 00:03:15.930
I just meant -- and it
doesn't really matter,

00:03:15.930 --> 00:03:19.490
it was a dumb thing to say --
I meant you to put the MATLAB

00:03:19.490 --> 00:03:23.770
question at the end after
the regular questions.

00:03:23.770 --> 00:03:24.450
Sorry.

00:03:24.450 --> 00:03:27.560
So that MATLAB is
due-- Maybe a bunch

00:03:27.560 --> 00:03:32.050
of those turned in on Monday
didn't include the MATLAB.

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No penalty.

00:03:34.380 --> 00:03:39.600
I'm talking now about the
MATLAB for the trusses.

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How many have still got a MATLAB
for the truss still to turn in?

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A number.

00:03:45.070 --> 00:03:45.961
Oh, not too many.

00:03:45.961 --> 00:03:46.460
Okay.

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Anyway, just when you can.

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In my envelope is good.

00:03:50.350 --> 00:03:50.860
Okay.

00:03:50.860 --> 00:03:55.200
So it'll be a similar
thing for this week,

00:03:55.200 --> 00:03:58.260
and because it's
short I said okay,

00:03:58.260 --> 00:04:00.540
let's get that
cleaned up by Monday.

00:04:00.540 --> 00:04:04.990
And then we're ready for vector
calculus, partial differential

00:04:04.990 --> 00:04:10.830
equations, two and three
dimensions, the next big step.

00:04:10.830 --> 00:04:13.870
If I want to illustrate
element matrices,

00:04:13.870 --> 00:04:16.450
the best place I
could do it would

00:04:16.450 --> 00:04:20.660
be to go back to our piecewise
linear elements in 1-D,

00:04:20.660 --> 00:04:26.140
and see how an
element now is just

00:04:26.140 --> 00:04:28.950
one of those little intervals.

00:04:28.950 --> 00:04:33.530
Little pieces of
the whole structure.

00:04:33.530 --> 00:04:38.160
If it's a bar, or whatever it
is, I've cut it into pieces.

00:04:38.160 --> 00:04:42.510
Here's one piece,
here's another piece.

00:04:42.510 --> 00:04:46.470
With finite differences,
if I gave you

00:04:46.470 --> 00:04:49.890
unequally spaced meshes,
if my little h is

00:04:49.890 --> 00:04:53.710
different from my big H,
we'd have to think again.

00:04:53.710 --> 00:04:58.080
I mean there would be
some three-point-- Instead

00:04:58.080 --> 00:05:02.620
of minus one, two, minus one
for the second difference,

00:05:02.620 --> 00:05:05.450
it would be a little
lopsided, of course,

00:05:05.450 --> 00:05:09.560
when the mesh is unequal.

00:05:09.560 --> 00:05:12.690
We would have to think that
through for finite differences;

00:05:12.690 --> 00:05:15.920
for finite elements, the
system thinks for us.

00:05:15.920 --> 00:05:18.270
So I just want to
show how that happens.

00:05:18.270 --> 00:05:19.670
So I'll take this.

00:05:19.670 --> 00:05:25.570
What's the element matrix
for our standard equation

00:05:25.570 --> 00:05:27.320
for that element?

00:05:27.320 --> 00:05:29.580
Okay.

00:05:29.580 --> 00:05:32.210
And then it would fit
into the big matrix

00:05:32.210 --> 00:05:35.960
K. I'm going to come
up with a matrix K

00:05:35.960 --> 00:05:38.250
that I could do
in any other way,

00:05:38.250 --> 00:05:40.580
but this is the way
it's really done.

00:05:40.580 --> 00:05:46.070
So I look in that interval.

00:05:46.070 --> 00:05:48.010
I'm focusing on that interval.

00:05:48.010 --> 00:05:52.700
So here I've drawn
the basis functions.

00:05:52.700 --> 00:05:57.090
But now I'm not going to operate
so much with basis functions.

00:05:57.090 --> 00:05:59.270
Let me just think about
that interval again.

00:05:59.270 --> 00:06:00.810
That's the same interval.

00:06:00.810 --> 00:06:02.670
So those were the
basis functions

00:06:02.670 --> 00:06:06.280
that went up to height one.

00:06:06.280 --> 00:06:08.710
One started at height
one and went down,

00:06:08.710 --> 00:06:11.250
one started at height
zero and went up.

00:06:11.250 --> 00:06:14.740
But it's their
combination, of course.

00:06:14.740 --> 00:06:21.210
Let me number this node number
zero and this node number one.

00:06:21.210 --> 00:06:27.380
So we have some
height, U_0, here.

00:06:27.380 --> 00:06:31.670
This is coming from
U_0 times that hat.

00:06:31.670 --> 00:06:35.000
And some other
height, U_1, here,

00:06:35.000 --> 00:06:38.100
which is coming from
U_1 times that hat.

00:06:38.100 --> 00:06:46.250
But inside this interval, my U
in this interval is U_0 times

00:06:46.250 --> 00:06:51.090
that first hat function, the
phi_0, the coming down hat.

00:06:51.090 --> 00:06:57.780
Plus U_1 times the phi_1
function, the going up hat.

00:06:57.780 --> 00:07:01.480
It's a linear function,
so it's going to be there.

00:07:01.480 --> 00:07:03.970
That's my graph-- No.

00:07:03.970 --> 00:07:06.540
Yes.

00:07:06.540 --> 00:07:07.140
This is U(x).

00:07:09.960 --> 00:07:11.230
U(x).

00:07:11.230 --> 00:07:12.550
Okay.

00:07:12.550 --> 00:07:13.980
Right?

00:07:13.980 --> 00:07:17.030
I'm focusing inside one element.

00:07:17.030 --> 00:07:20.740
One interval.

00:07:20.740 --> 00:07:23.340
I've numbered them zero
to one, but of course

00:07:23.340 --> 00:07:25.070
that distance is H.

00:07:25.070 --> 00:07:28.150
Okay, so there's my function.

00:07:28.150 --> 00:07:28.740
Right?

00:07:28.740 --> 00:07:31.780
Everybody agrees that
this combination,

00:07:31.780 --> 00:07:37.690
it starts out right, it ends
up right, at height U_1,

00:07:37.690 --> 00:07:38.870
and it's linear.

00:07:38.870 --> 00:07:40.840
So that's got to be right.

00:07:40.840 --> 00:07:48.940
So really, what's the
contribution from that element?

00:07:48.940 --> 00:07:55.070
I look at the quantity,
that-- And I'm always

00:07:55.070 --> 00:07:58.750
taking the phi functions
to equal the V functions.

00:07:58.750 --> 00:08:06.090
So I can write this as c(x)
times dU/dx squared, dx.

00:08:11.250 --> 00:08:16.680
Over the whole thing, over
the whole interval, that

00:08:16.680 --> 00:08:23.630
would be the U
transpose K U. This,

00:08:23.630 --> 00:08:28.220
everybody remembers
the K part, of course,

00:08:28.220 --> 00:08:30.440
comes from the left
side of the problem.

00:08:30.440 --> 00:08:34.330
It comes from integrations.

00:08:34.330 --> 00:08:40.070
And this d-- U is now a
combination of all the--

00:08:40.070 --> 00:08:43.490
With weights U_0, U_1, U_2, U_3.

00:08:43.490 --> 00:08:53.220
If I plug in for capital U
and do all the integrals,

00:08:53.220 --> 00:08:55.050
I'll see what K is.

00:08:55.050 --> 00:08:56.520
Now what's the point?

00:08:56.520 --> 00:08:59.840
The point is, when I
did those integrals,

00:08:59.840 --> 00:09:03.810
the question is, how am I
going to do the integrals?

00:09:03.810 --> 00:09:08.340
Do I do them-- The way I
did before, was I watched

00:09:08.340 --> 00:09:15.350
what phi_0, one phi times
another phi and I integrated.

00:09:15.350 --> 00:09:17.330
That was successful.

00:09:17.330 --> 00:09:23.590
But the new way is, integrate
it an interval at a time.

00:09:23.590 --> 00:09:26.320
Do the integrals.

00:09:26.320 --> 00:09:29.500
So this would be K_global.

00:09:29.500 --> 00:09:36.490
Now I'm going to go
just from zero to H.

00:09:36.490 --> 00:09:40.615
If I go from just zero
to H, that's going

00:09:40.615 --> 00:09:43.610
to give me the K_element piece.

00:09:43.610 --> 00:09:47.250
The little piece that comes
from this little interval.

00:09:47.250 --> 00:09:54.500
And on that little interval,
this is my formula for U.

00:09:54.500 --> 00:09:59.670
I'm just hoping you'll sort of
see this as a reasonable idea,

00:09:59.670 --> 00:10:03.370
and then when we do the
integral you'll see it.

00:10:03.370 --> 00:10:04.650
It clicks.

00:10:04.650 --> 00:10:07.140
Okay, how does it click?

00:10:07.140 --> 00:10:08.720
So what's my plan?

00:10:08.720 --> 00:10:11.230
My plan is, here's my function.

00:10:11.230 --> 00:10:13.280
There is its picture.

00:10:13.280 --> 00:10:15.890
I'm just going to
plug it into here.

00:10:15.890 --> 00:10:17.920
Oh, it's going to
be simple, isn't it?

00:10:17.920 --> 00:10:23.350
What's the slope of that
function in that element?

00:10:23.350 --> 00:10:24.600
On that interval?

00:10:24.600 --> 00:10:26.300
What's the dU/dx?

00:10:26.300 --> 00:10:33.360
So instead of doing every full
integral for each separate phi,

00:10:33.360 --> 00:10:40.380
I'm doing every element
integral for both phis.

00:10:40.380 --> 00:10:44.980
See, these two phis are both
coming in to that element.

00:10:44.980 --> 00:10:48.780
Okay, so what's dU/dx.

00:10:48.780 --> 00:10:51.930
I didn't realize how neat
this that's going to be.

00:10:51.930 --> 00:10:55.150
So what's dU/dx in this element?

00:10:55.150 --> 00:10:56.800
I'll use a different board here.

00:10:56.800 --> 00:10:59.360
So it's an integral,
then, from zero

00:10:59.360 --> 00:11:04.240
to H of whatever
my c(x) might be.

00:11:04.240 --> 00:11:05.870
And I'll make a comment on that.

00:11:05.870 --> 00:11:09.370
But my focus here
is, what's dU/dx?

00:11:09.370 --> 00:11:12.550
What's the derivative?

00:11:12.550 --> 00:11:14.470
And then I'm going to square it.

00:11:14.470 --> 00:11:18.080
For this function,
for that picture,

00:11:18.080 --> 00:11:24.720
the slope is obviously
(U_1-U_0)/H. Right?

00:11:24.720 --> 00:11:31.670
The slope is (U_1-U_0)/H.
And I'm squaring it. dx.

00:11:34.360 --> 00:11:35.330
Okay.

00:11:35.330 --> 00:11:39.910
Let me take c(x)=1.

00:11:39.910 --> 00:11:44.750
Just to see clearly
what's going on.

00:11:44.750 --> 00:11:47.790
So c(x) is just going to be one.

00:11:47.790 --> 00:11:55.290
Okay, so I'm claiming that
this is my U transpose.

00:11:55.290 --> 00:11:56.980
My little piece.

00:11:56.980 --> 00:11:58.880
Why is it only a little piece?

00:11:58.880 --> 00:12:02.530
Because it only
involves two of the U's.

00:12:02.530 --> 00:12:07.680
It's going to be a little two
by two element matrix, that

00:12:07.680 --> 00:12:09.690
comes from this element.

00:12:09.690 --> 00:12:14.420
And then it's going to to be put
into the big K, the global K,

00:12:14.420 --> 00:12:16.110
in its proper place.

00:12:16.110 --> 00:12:17.740
Okay, well.

00:12:17.740 --> 00:12:19.200
It's trivial, right?

00:12:19.200 --> 00:12:21.460
This is a constant,
this is a constant,

00:12:21.460 --> 00:12:30.130
the integral is just H
times U_1-U_0, squared,

00:12:30.130 --> 00:12:32.250
over H squared.

00:12:32.250 --> 00:12:34.680
Because that's getting
squared, the interval was

00:12:34.680 --> 00:12:38.700
length H. I think
I just have a 1/H.

00:12:38.700 --> 00:12:45.080
So this is my U
transpose K element U.

00:12:45.080 --> 00:12:48.370
And now I want to pick
out, what's that matrix?

00:12:48.370 --> 00:12:52.610
What's that little two by two
matrix that only touches these

00:12:52.610 --> 00:12:55.190
two U's?

00:12:55.190 --> 00:12:59.160
What's the matrix-- Well,
since it only touches

00:12:59.160 --> 00:13:02.110
these two U's, you can tell me.

00:13:02.110 --> 00:13:03.050
I want a [U 0, U 1].

00:13:06.030 --> 00:13:08.940
Sorry, let me make
a little more space.

00:13:08.940 --> 00:13:13.010
And you can tell me what
matrix goes in there.

00:13:13.010 --> 00:13:16.790
I want this all to
match up. [U 0, U 1].

00:13:16.790 --> 00:13:20.390
Now here is the two by two
element matrix. [U 0, U 1].

00:13:25.850 --> 00:13:29.180
What's the two by two
matrix that will correctly

00:13:29.180 --> 00:13:31.330
produce this answer?

00:13:31.330 --> 00:13:35.840
I'm just shooting
for that answer.

00:13:35.840 --> 00:13:37.480
What do I have here?

00:13:37.480 --> 00:13:44.100
This is U_1 squared minus
2U_0*U_1, plus a U_0 squared.

00:13:44.100 --> 00:13:45.430
Right?

00:13:45.430 --> 00:13:48.280
And I have to remember the
1/H, so it's automatically

00:13:48.280 --> 00:13:51.780
going to come out right.

00:13:51.780 --> 00:13:57.360
So there's a 1/H, shall I
remember that first off.

00:13:57.360 --> 00:14:00.720
1/H is part of my
element matrix.

00:14:00.720 --> 00:14:09.200
And then, what are the numbers
that go inside that matrix?

00:14:09.200 --> 00:14:11.410
We had practice with this.

00:14:11.410 --> 00:14:13.360
You remember when
we talked about

00:14:13.360 --> 00:14:17.440
positive definite matrices,
way back in chapter one?

00:14:17.440 --> 00:14:23.630
The point was that we could
look at eigenvalues, or pivots,

00:14:23.630 --> 00:14:24.580
or something.

00:14:24.580 --> 00:14:27.880
But the core idea was energy.

00:14:27.880 --> 00:14:30.680
The core idea was that
energy, that quadratic,

00:14:30.680 --> 00:14:32.510
and that's what we're
looking at again.

00:14:32.510 --> 00:14:34.750
That's the energy.

00:14:34.750 --> 00:14:39.130
That's the energy, right
there, and this is the energy,

00:14:39.130 --> 00:14:45.430
this is the energy in the
finite element subspace.

00:14:45.430 --> 00:14:49.020
All I'm saying is, what
matrix, what two by two matrix,

00:14:49.020 --> 00:14:54.210
goes with U_1 squared minus
2U_0*U_1 plus U_0 squared.

00:14:54.210 --> 00:14:56.220
Just tell me what to
put in that matrix.

00:14:56.220 --> 00:14:58.270
What do I put in here?

00:14:58.270 --> 00:14:59.220
One, right.

00:14:59.220 --> 00:15:01.570
Because it's
multiplying U_0*U_0.

00:15:01.570 --> 00:15:03.340
What do I put there?

00:15:03.340 --> 00:15:04.310
Minus one.

00:15:04.310 --> 00:15:05.150
Good.

00:15:05.150 --> 00:15:09.810
Because I have a minus
two, it comes in, minus one

00:15:09.810 --> 00:15:12.240
comes in twice, and
a one goes there.

00:15:12.240 --> 00:15:17.400
So there, with the
1/H included, is K_e.

00:15:17.400 --> 00:15:20.760
K_element, for that element.

00:15:20.760 --> 00:15:25.560
For the big H element.

00:15:25.560 --> 00:15:30.220
You'll say big deal, because
we've seen this thing before.

00:15:30.220 --> 00:15:35.470
Notice what is nice.

00:15:35.470 --> 00:15:38.240
First of all, notice
how nice that is.

00:15:38.240 --> 00:15:40.110
It's particularly
nice, of course,

00:15:40.110 --> 00:15:43.250
because I took c(x) to be one.

00:15:43.250 --> 00:15:44.810
So let me make a comment.

00:15:44.810 --> 00:15:47.890
Suppose c(x) was not one.

00:15:47.890 --> 00:15:50.040
What would I do?

00:15:50.040 --> 00:15:57.090
Suppose c(x), suppose I
have some variable stiffness

00:15:57.090 --> 00:15:57.980
in the material.

00:15:57.980 --> 00:16:03.150
Suppose the material
could be changing width,

00:16:03.150 --> 00:16:09.430
so its stiffness would change.

00:16:09.430 --> 00:16:12.020
So in other words, I'd
have a variable, c(x),

00:16:12.020 --> 00:16:15.350
that I should do the integral.

00:16:15.350 --> 00:16:18.615
Probably finite element
systems aren't set up

00:16:18.615 --> 00:16:22.030
to actually do the
exact integral.

00:16:22.030 --> 00:16:23.800
What would they do?

00:16:23.800 --> 00:16:27.430
They would take, for
that simple integration,

00:16:27.430 --> 00:16:32.360
they would probably just
take c(x) at the midpoint.

00:16:32.360 --> 00:16:34.310
So there's a
numerical integration

00:16:34.310 --> 00:16:39.780
here in the creation of
these element matrices.

00:16:39.780 --> 00:16:42.310
And numerical
immigration is just,

00:16:42.310 --> 00:16:46.560
take a suitable combination
of the values at a few points.

00:16:46.560 --> 00:16:48.970
You do know Simpson's rule?

00:16:48.970 --> 00:16:53.210
Simpson's rule, that's a
pretty high level rule.

00:16:53.210 --> 00:16:57.670
My suggestion there was
just a midpoint rule.

00:16:57.670 --> 00:17:00.890
Just take c(x) at the
middle of the interval.

00:17:00.890 --> 00:17:02.140
Then it would factor out.

00:17:02.140 --> 00:17:07.580
So I should really put a c here.

00:17:07.580 --> 00:17:10.080
A c should really
be coming in there.

00:17:10.080 --> 00:17:13.560
And you expected that, right,
from the A transpose C A.

00:17:13.560 --> 00:17:16.340
We always saw a C in the middle.

00:17:16.340 --> 00:17:17.720
It really should be there.

00:17:17.720 --> 00:17:21.410
When I took c to be
1, I didn't see it.

00:17:21.410 --> 00:17:23.750
So what am I doing?

00:17:23.750 --> 00:17:30.950
I'm approximating
c(x) by c at halfway.

00:17:30.950 --> 00:17:31.980
Approximately.

00:17:31.980 --> 00:17:39.390
I would replace this unpleasant,
possibly varying function by c

00:17:39.390 --> 00:17:42.830
at a point and use that value.

00:17:42.830 --> 00:17:46.030
So numerical integration
is one part of the picture

00:17:46.030 --> 00:17:50.320
that-- We won't go into
all the different rules.

00:17:50.320 --> 00:17:52.520
There's a rectangle
rule, there's

00:17:52.520 --> 00:17:55.390
a trapezoid rule, very good.

00:17:55.390 --> 00:17:58.590
There's the Simpson's
rule that's better.

00:17:58.590 --> 00:18:02.890
As I get higher order elements,
like those cubic elements

00:18:02.890 --> 00:18:06.660
I spoke about, the numerical
integration has to keep up.

00:18:06.660 --> 00:18:10.110
If I was integrating
cubic stuff,

00:18:10.110 --> 00:18:12.040
I wouldn't use
such a cheap rule.

00:18:12.040 --> 00:18:15.030
I would go up to Simpson's
rule, or Gauss's rule,

00:18:15.030 --> 00:18:16.440
or somebody's rule.

00:18:16.440 --> 00:18:19.770
Anyway, that's the c part.

00:18:19.770 --> 00:18:23.860
Here's the part that
stamps into the matrix.

00:18:23.860 --> 00:18:28.380
Notice, by the way,
when I stamp it in,

00:18:28.380 --> 00:18:30.540
tell me how it's going
to look stamped in.

00:18:30.540 --> 00:18:32.040
And then I've completed it.

00:18:32.040 --> 00:18:36.930
So here's my big K. I wish I
had a little more room for it.

00:18:36.930 --> 00:18:41.520
Okay, here's my big
K. So that interval

00:18:41.520 --> 00:18:43.500
that I drew there,
the H interval,

00:18:43.500 --> 00:18:46.430
will stamp in here,
some two by two.

00:18:46.430 --> 00:18:47.760
Right?

00:18:47.760 --> 00:18:51.860
Now where will the similar
thing coming from this guy.

00:18:51.860 --> 00:18:55.590
So, maybe it has to
be numbered minus one.

00:18:55.590 --> 00:18:58.670
Sorry about that.

00:18:58.670 --> 00:19:00.590
That's another little interval.

00:19:00.590 --> 00:19:02.990
I'll do the same thing
on that interval.

00:19:02.990 --> 00:19:06.400
I'll get a little element
matrix, two by two guy,

00:19:06.400 --> 00:19:08.650
for K for that element.

00:19:08.650 --> 00:19:14.190
And where will it
fit in to the big K?

00:19:14.190 --> 00:19:16.890
Does it fit in up here,
let me just ask you.

00:19:16.890 --> 00:19:19.360
Does it fit in up there?

00:19:19.360 --> 00:19:23.720
Yes, no?

00:19:23.720 --> 00:19:28.870
I'm assembling, stamping
in the small two

00:19:28.870 --> 00:19:32.870
by twos into the full n by n.

00:19:32.870 --> 00:19:37.290
And if I draw the
picture you'll see it.

00:19:37.290 --> 00:19:42.920
So when I do the two by two
for that big H interval,

00:19:42.920 --> 00:19:46.530
it goes there, let's say,
then I just want to say where

00:19:46.530 --> 00:19:52.720
does the two by two go for
the interval to the left?

00:19:52.720 --> 00:19:54.810
Does it go there?

00:19:54.810 --> 00:19:55.800
Nope.

00:19:55.800 --> 00:19:57.680
How does it go?

00:19:57.680 --> 00:19:59.230
It overlaps.

00:19:59.230 --> 00:19:59.730
Right?

00:19:59.730 --> 00:20:02.880
It overlaps.

00:20:02.880 --> 00:20:04.100
Why does it overlap?

00:20:04.100 --> 00:20:10.030
Because the phi_0, this
guy, is acting on the right,

00:20:10.030 --> 00:20:12.490
and also acting on the left.

00:20:12.490 --> 00:20:21.150
The U_0 is active, is
partly controlling the slope

00:20:21.150 --> 00:20:23.190
this way, and also that way.

00:20:23.190 --> 00:20:26.680
The U_0 is in common
the two intervals.

00:20:26.680 --> 00:20:30.490
Anytime any unknowns, any
mesh points that are in

00:20:30.490 --> 00:20:32.795
common to two
elements, we're going

00:20:32.795 --> 00:20:36.110
to have an overlap
when we assemble.

00:20:36.110 --> 00:20:39.710
And so it'll just sit,
it'll sit right there.

00:20:39.710 --> 00:20:44.390
And so there is
the diagonal guy.

00:20:44.390 --> 00:20:47.700
And maybe you could tell me
what number will go there.

00:20:47.700 --> 00:20:49.640
What number would
actually go there?

00:20:49.640 --> 00:20:55.980
And then you'll see the whole
point of assembly, stamping in.

00:20:55.980 --> 00:21:00.010
Well, what number
goes there from this?

00:21:00.010 --> 00:21:06.490
One times the c/H. So in
here would be the c --

00:21:06.490 --> 00:21:15.670
can I call it c_right, or c_H,
c on the big H interval --

00:21:15.670 --> 00:21:20.850
divided by the H. Because that
one, we had to get it right.

00:21:20.850 --> 00:21:23.910
And then what will go
in that very same spot?

00:21:23.910 --> 00:21:29.540
So add it in coming from
the small h interval.

00:21:29.540 --> 00:21:36.550
The same thing, it'll be
this one, like shifted up,

00:21:36.550 --> 00:21:37.740
moved over.

00:21:37.740 --> 00:21:39.490
So it'll be coming
from that one,

00:21:39.490 --> 00:21:45.120
so it'll be a c on the little h
interval, divided by little h.

00:21:45.120 --> 00:21:50.020
That would be the
diagonal entry of K.

00:21:50.020 --> 00:21:52.290
That's what we would
see right there.

00:21:52.290 --> 00:21:57.050
Over here, should I
write a typical row of K?

00:21:57.050 --> 00:21:59.660
Typical row of K,
when I do that,

00:21:59.660 --> 00:22:10.250
is going to have this c_h/h
plus c_H/H. That's like the two,

00:22:10.250 --> 00:22:11.810
right?

00:22:11.810 --> 00:22:13.150
That's like the two.

00:22:13.150 --> 00:22:18.700
And what goes here?

00:22:18.700 --> 00:22:23.330
What will the entry be that
sits there when I assemble?

00:22:23.330 --> 00:22:25.740
Just this guy.

00:22:25.740 --> 00:22:26.510
Right?

00:22:26.510 --> 00:22:28.340
Just this guy, times that.

00:22:28.340 --> 00:22:34.590
The entry here will be the minus
c_H/H. That'll be the entry.

00:22:34.590 --> 00:22:37.870
This is the diagonal one,
this is the one to the right,

00:22:37.870 --> 00:22:40.810
and what's the one to the left?

00:22:40.810 --> 00:22:45.590
What's the one here?

00:22:45.590 --> 00:22:47.170
Well, you know what it is.

00:22:47.170 --> 00:22:55.350
It's going to come from the
minus and it'll be the minus c

00:22:55.350 --> 00:22:58.420
little h over little h.

00:22:58.420 --> 00:22:59.330
Look at that.

00:22:59.330 --> 00:23:02.360
That just shows
you how it works.

00:23:02.360 --> 00:23:09.190
And again, you can look at that,
page 299 to 300 in the book.

00:23:09.190 --> 00:23:16.100
You see that if the c's are the
same, if the h's are the same,

00:23:16.100 --> 00:23:20.790
then we're looking again at
our minus one, two, minus one.

00:23:20.790 --> 00:23:25.190
Times whatever c over h, to
keep it dimensionally right.

00:23:25.190 --> 00:23:25.930
Do you see that?

00:23:25.930 --> 00:23:27.440
It's just simple.

00:23:27.440 --> 00:23:28.890
Simple idea.

00:23:28.890 --> 00:23:35.150
The point is that each interval
can be done separately.

00:23:35.150 --> 00:23:42.840
It's a simple idea in 1-D.
It's a key idea in 2-D,

00:23:42.840 --> 00:23:47.210
where we have triangles,
we have tetrahedra, tets.

00:23:47.210 --> 00:23:49.710
We'll see this in
two dimensions,

00:23:49.710 --> 00:23:56.030
later in this chapter, when
we're doing Laplace's equation.

00:23:56.030 --> 00:23:58.880
It's just fun to see it work.

00:23:58.880 --> 00:24:04.640
You'll have different triangles,
say call them triangles.

00:24:04.640 --> 00:24:13.300
So that phi-- Do you
want me to look ahead?

00:24:13.300 --> 00:24:15.480
Just ten seconds, to triangles?

00:24:15.480 --> 00:24:19.450
So imagine we have triangles
here, so we have piecewise,

00:24:19.450 --> 00:24:21.350
we have little pyramids.

00:24:21.350 --> 00:24:24.880
Instead of hat functions,
they grow to pyramids.

00:24:24.880 --> 00:24:28.600
So there's a pyramid guy
whose height is one there,

00:24:28.600 --> 00:24:31.860
and drops to zero
in all these places.

00:24:31.860 --> 00:24:34.050
And that's our phi.

00:24:34.050 --> 00:24:36.860
Our trial function,
test function,

00:24:36.860 --> 00:24:41.000
will be a pyramid
function, then.

00:24:41.000 --> 00:24:45.450
And I can do integrals that
way, or I can take the integral

00:24:45.450 --> 00:24:48.960
over a typical triangle.

00:24:48.960 --> 00:24:53.090
So a typical triangle
is involved with three,

00:24:53.090 --> 00:24:57.240
now I've three functions, in
the linear case, controlling

00:24:57.240 --> 00:24:59.390
inside that triangle.

00:24:59.390 --> 00:25:04.640
So what will be the size
of the element matrix?

00:25:04.640 --> 00:25:07.640
Can you sort of see how the
system is going to work?

00:25:07.640 --> 00:25:12.970
And then we'll make
it work in 2-D.

00:25:12.970 --> 00:25:16.530
Every node, every mesh
point, corresponds,

00:25:16.530 --> 00:25:21.370
has a pyramid function,
has a U that goes with it.

00:25:21.370 --> 00:25:23.860
Those U's are the unknowns.

00:25:23.860 --> 00:25:26.120
And how many of
those unknowns are

00:25:26.120 --> 00:25:29.010
operating inside that triangle?

00:25:29.010 --> 00:25:30.040
Three.

00:25:30.040 --> 00:25:32.930
So what will be the
size of the element

00:25:32.930 --> 00:25:36.690
matrix, the non-zero part
of the element matrix?

00:25:36.690 --> 00:25:37.490
Three by three.

00:25:37.490 --> 00:25:39.040
What else could it be?

00:25:39.040 --> 00:25:41.350
So we'll see what it looks like.

00:25:41.350 --> 00:25:44.660
And we'll have integrals
over triangles.

00:25:44.660 --> 00:25:48.110
So that's good.

00:25:48.110 --> 00:25:49.700
Okay, thanks.

00:25:49.700 --> 00:25:54.840
Exactly halfway through the
hour is exactly that first topic

00:25:54.840 --> 00:25:57.180
of element matrices.

00:25:57.180 --> 00:25:58.520
Done.

00:25:58.520 --> 00:26:02.000
Okay.

00:26:02.000 --> 00:26:05.190
Let me take two deep
breaths, and move

00:26:05.190 --> 00:26:08.660
to fourth order equations.

00:26:08.660 --> 00:26:11.370
Fourth order equations.

00:26:11.370 --> 00:26:12.720
For the bending of a beam.

00:26:12.720 --> 00:26:16.200
So I'd better draw a beam.

00:26:16.200 --> 00:26:19.240
This is a 1-D problem, still.

00:26:19.240 --> 00:26:24.040
This is a 1-D problem, still.

00:26:24.040 --> 00:26:28.190
To keep it 1-D, this
better be a thin beam.

00:26:28.190 --> 00:26:32.220
So this is a thin beam.

00:26:32.220 --> 00:26:34.730
And the loads, what's
the difference?

00:26:34.730 --> 00:26:36.830
What's here?

00:26:36.830 --> 00:26:40.900
It looks like a bar,
pretty much, right?

00:26:40.900 --> 00:26:47.050
But the difference is, the
load is acting this way.

00:26:47.050 --> 00:26:49.890
The load is acting
that way on the beam.

00:26:49.890 --> 00:26:51.710
Maybe two loads.

00:26:51.710 --> 00:26:53.500
Maybe a uniform load.

00:26:53.500 --> 00:26:56.170
Maybe the weight of the beam.

00:26:56.170 --> 00:27:00.150
But it's transverse.

00:27:00.150 --> 00:27:04.760
It's in the perpendicular,
it's transverse to the beam.

00:27:04.760 --> 00:27:06.250
It's this way.

00:27:06.250 --> 00:27:09.860
So the beam bends.

00:27:09.860 --> 00:27:11.470
Let me do a fixed-free.

00:27:11.470 --> 00:27:15.330
So this'll be a fixed-free beam.

00:27:15.330 --> 00:27:19.920
Fixed-free, the
word for fixed-free

00:27:19.920 --> 00:27:27.230
would be cantilever beam.

00:27:27.230 --> 00:27:28.040
Okay.

00:27:28.040 --> 00:27:33.750
So what happens if I
impose those loads.

00:27:33.750 --> 00:27:37.660
Well, the beam bends.

00:27:37.660 --> 00:27:42.800
So the displacement is
now downwards, is now

00:27:42.800 --> 00:27:45.530
not the direction of the rod.

00:27:45.530 --> 00:27:47.280
The displacement
I'm interested in

00:27:47.280 --> 00:27:49.570
is perpendicular to the beam.

00:27:49.570 --> 00:27:50.620
Downwards.

00:27:50.620 --> 00:27:56.800
Okay, so we can start
on our framework.

00:27:56.800 --> 00:28:02.290
So this is displacement. u(x).

00:28:02.290 --> 00:28:07.340
I'll stay with the same letters.

00:28:07.340 --> 00:28:11.340
So you know I'm going
to have an A, that

00:28:11.340 --> 00:28:21.780
will take me to whatever
this is going to get called.

00:28:21.780 --> 00:28:24.820
What's the quantity there?

00:28:24.820 --> 00:28:29.250
It's going to be, let's see.

00:28:29.250 --> 00:28:30.210
What happens?

00:28:30.210 --> 00:28:33.760
So this is just geometric now.

00:28:33.760 --> 00:28:38.400
Let me put in the
easy part, C. That'll

00:28:38.400 --> 00:28:40.800
be sort of the
bending stiffness.

00:28:40.800 --> 00:28:41.300
Right?

00:28:41.300 --> 00:28:44.480
This'll be the
bending stiffness.

00:28:44.480 --> 00:28:49.670
Because the beam
is going to bend.

00:28:49.670 --> 00:28:54.283
And over here I'll
get a suitable w.

00:28:54.283 --> 00:28:59.820
What I get here-- So
when the beam bends...

00:28:59.820 --> 00:29:01.780
Its curvature.

00:29:01.780 --> 00:29:04.360
Curvature of the beam
is what's produced.

00:29:04.360 --> 00:29:08.090
It's not stretching of the
beam; it's curving of the beam.

00:29:08.090 --> 00:29:12.580
So this quantity, e, will
be the curvature that

00:29:12.580 --> 00:29:16.700
comes from the displacement.

00:29:16.700 --> 00:29:22.660
If I displace these beams,
suppose I put a load here,

00:29:22.660 --> 00:29:24.900
it's going to bend that down.

00:29:24.900 --> 00:29:26.580
The bar will curve.

00:29:26.580 --> 00:29:29.080
So the curvature e.

00:29:29.080 --> 00:29:30.960
Now then, the
question is, what is

00:29:30.960 --> 00:29:33.600
this A; what is the curvature?

00:29:33.600 --> 00:29:38.060
Well, do you remember?

00:29:38.060 --> 00:29:39.590
You're on the ball
if you remember

00:29:39.590 --> 00:29:42.180
the formula for curvature.

00:29:42.180 --> 00:29:44.830
It's a horrible
formula, actually.

00:29:44.830 --> 00:29:47.430
But that's only-- We're
going to make it better.

00:29:47.430 --> 00:29:52.390
Do you remember the
curvature, it was in calculus.

00:29:52.390 --> 00:29:54.860
Yes, you all remember this.

00:29:54.860 --> 00:29:58.590
Suppose I have a graph.

00:29:58.590 --> 00:30:02.260
I know its slope, that's
become easy now, right?

00:30:02.260 --> 00:30:02.770
Calculus.

00:30:02.770 --> 00:30:08.180
But the curvature of it, what
derivative did it involve?

00:30:08.180 --> 00:30:09.590
Second derivative.

00:30:09.590 --> 00:30:13.210
And was it the second
derivative exactly?

00:30:13.210 --> 00:30:18.070
No, unfortunately there's
some term which is horrible.

00:30:18.070 --> 00:30:24.980
One plus the first derivative
squared, all square root.

00:30:24.980 --> 00:30:33.010
But I'm just going to
take u double prime.

00:30:33.010 --> 00:30:34.790
So this is an approximation.

00:30:34.790 --> 00:30:37.740
So what is A now?

00:30:37.740 --> 00:30:41.900
What's my matrix A, that gets
me from u-- or my operator

00:30:41.900 --> 00:30:45.270
A, that gets me from u to e?

00:30:45.270 --> 00:30:48.450
What's the e=Au equation?

00:30:48.450 --> 00:30:52.210
A is just second derivative.

00:30:52.210 --> 00:30:53.620
That's something new.

00:30:53.620 --> 00:30:58.070
A is second derivative.

00:30:58.070 --> 00:31:00.610
And why do I do that?

00:31:00.610 --> 00:31:05.730
Because I assume small
curvature, small displacements.

00:31:05.730 --> 00:31:11.650
I assume that u' squared is
very small compared to one.

00:31:11.650 --> 00:31:16.850
So it's just slightly bent.

00:31:16.850 --> 00:31:21.560
A beam that goes way down
here, I'd have to go nonlinear.

00:31:21.560 --> 00:31:26.780
But if I want to keep things
linear, I approximate.

00:31:26.780 --> 00:31:29.380
This will be much
smaller than this,

00:31:29.380 --> 00:31:36.460
so one is fine, that term goes.

00:31:36.460 --> 00:31:39.870
And now the next
step will be easy.

00:31:39.870 --> 00:31:43.110
What's the bending moment?

00:31:43.110 --> 00:31:47.320
This will be called
the bending moment.

00:31:47.320 --> 00:31:50.430
And let me use the
letter w again.

00:31:50.430 --> 00:31:54.340
I should really use capital
M for bending moment.

00:31:54.340 --> 00:32:04.640
That will be the stiffness
times the curvature.

00:32:04.640 --> 00:32:10.700
So that's the force,
the way the spring

00:32:10.700 --> 00:32:13.610
had a restoring
force by Hooke's law.

00:32:13.610 --> 00:32:15.710
This is the equivalent
of Hooke's law.

00:32:15.710 --> 00:32:19.260
But this restoring force
is not pulling back,

00:32:19.260 --> 00:32:20.570
it's bending back.

00:32:20.570 --> 00:32:24.220
It's torquing back.

00:32:24.220 --> 00:32:26.690
And then, of course
you know, there'll

00:32:26.690 --> 00:32:30.660
be an equilibrium equation
to balance the load.

00:32:30.660 --> 00:32:32.030
So this load is the f(x).

00:32:34.760 --> 00:32:36.570
And you know what that'll be.

00:32:36.570 --> 00:32:38.550
Because you know it'll
involve A transpose.

00:32:38.550 --> 00:32:46.080
And the transpose of that, do
you want to just make a guess?

00:32:46.080 --> 00:32:49.800
I shouldn't use transpose, of
course, but I'll use it again.

00:32:49.800 --> 00:32:53.300
It'll be the same; it'll
be second derivative.

00:32:53.300 --> 00:32:57.730
I mentioned we had a minus
sign with the first derivative,

00:32:57.730 --> 00:32:59.580
but now we're going to
have two minus signs,

00:32:59.580 --> 00:33:02.100
so it's going to come out
symmetric, second derivative,

00:33:02.100 --> 00:33:07.740
and the equation here
will be w''=f(x).

00:33:13.380 --> 00:33:16.520
Our framework is working.

00:33:16.520 --> 00:33:25.510
We have plus boundary
conditions on u.

00:33:25.510 --> 00:33:32.930
And here we have plus
boundary conditions on w.

00:33:32.930 --> 00:33:40.220
So those parts we have
not yet mentioned.

00:33:40.220 --> 00:33:43.650
And of course, that
depends on my picture.

00:33:43.650 --> 00:33:46.700
While we're at it, why don't we
figure out, what do you think

00:33:46.700 --> 00:33:49.180
is the boundary condition here?

00:33:49.180 --> 00:33:51.010
And how many boundary
conditions am I

00:33:51.010 --> 00:33:53.960
going to look for altogether?

00:33:53.960 --> 00:33:55.850
Four altogether.

00:33:55.850 --> 00:33:58.120
Because I have a
fourth order equation.

00:33:58.120 --> 00:34:00.260
There will be four
arbitrary constants

00:34:00.260 --> 00:34:02.740
until I plug in
boundary conditions.

00:34:02.740 --> 00:34:04.780
So I'm looking for four
boundary conditions,

00:34:04.780 --> 00:34:07.340
two at this end,
two at that end now.

00:34:07.340 --> 00:34:10.750
And what will be the
two at the fixed end?

00:34:10.750 --> 00:34:15.490
At the fixed end,
obviously, it's built in.

00:34:15.490 --> 00:34:18.180
Built in.

00:34:18.180 --> 00:34:22.110
Slightly different words
sometimes for the beam problem.

00:34:22.110 --> 00:34:25.910
Here I'll have u=0 and u'=0.

00:34:28.920 --> 00:34:36.290
Those apply with
A; those go with A.

00:34:36.290 --> 00:34:38.300
Those are the
essential conditions,

00:34:38.300 --> 00:34:43.270
the Dirichlet conditions, the
ones I must impose all the way.

00:34:43.270 --> 00:34:47.220
And now at this free
end, what do you think?

00:34:47.220 --> 00:34:50.200
Well, it's great.

00:34:50.200 --> 00:34:52.520
It's w=0, and w'=0.

00:34:55.070 --> 00:34:58.880
It's just beautiful,
the way it all works.

00:34:58.880 --> 00:35:01.980
So that's a
completely fixed-free.

00:35:01.980 --> 00:35:04.580
Then why don't I draw
in, just while we're

00:35:04.580 --> 00:35:08.410
talking about boundary
conditions, an alternative.

00:35:08.410 --> 00:35:10.960
So here's my beam.

00:35:10.960 --> 00:35:23.000
And now you see, it's under a
load, f(x), transverse load.

00:35:23.000 --> 00:35:25.530
Now that would be different
boundary conditions.

00:35:25.530 --> 00:35:32.270
Anybody know the name of a
beam that's set up like that?

00:35:32.270 --> 00:35:35.580
Simply supported.

00:35:35.580 --> 00:35:37.660
You don't need
beam theory, and I

00:35:37.660 --> 00:35:42.190
don't know beam theory,
to tell the truth,

00:35:42.190 --> 00:35:43.980
to do these problems.

00:35:43.980 --> 00:35:48.610
So that's a simply
supported beam.

00:35:48.610 --> 00:35:51.120
And what are the boundary
conditions that go with that?

00:35:51.120 --> 00:35:53.400
Well this is u=0.

00:35:57.610 --> 00:36:03.870
No displacement, it sits
there on that support.

00:36:03.870 --> 00:36:07.660
And what else is
happening at that support?

00:36:07.660 --> 00:36:10.500
There's no bending moment.

00:36:10.500 --> 00:36:11.830
Nobody's here.

00:36:11.830 --> 00:36:13.420
Right?

00:36:13.420 --> 00:36:14.470
So it's w=0.

00:36:16.980 --> 00:36:18.890
And at this end, too.

00:36:18.890 --> 00:36:22.860
Also at this end, u at
one is sitting there,

00:36:22.860 --> 00:36:25.560
and w(1) is zero.

00:36:25.560 --> 00:36:26.290
Yeah.

00:36:26.290 --> 00:36:30.900
That would be the boundary
conditions, four of them,

00:36:30.900 --> 00:36:33.380
for a fourth order
equation that we'll just

00:36:33.380 --> 00:36:36.360
write down in a minute,
for simply supported.

00:36:36.360 --> 00:36:38.050
And we could have a mix.

00:36:38.050 --> 00:36:41.890
This could be simply
supported here, free here.

00:36:41.890 --> 00:36:42.540
I think.

00:36:42.540 --> 00:36:48.010
Or maybe, could it be, or
maybe that's too risky.

00:36:48.010 --> 00:36:52.820
Would that be a singular
case, simply supported?

00:36:52.820 --> 00:36:54.190
Huh.

00:36:54.190 --> 00:37:00.910
So as always with boundary
conditions, some are unstable.

00:37:00.910 --> 00:37:04.350
Some are not going to
determine all four constants.

00:37:04.350 --> 00:37:06.740
Just the way free-free
didn't work, right?

00:37:06.740 --> 00:37:11.460
Free-free for a rod
didn't determine anything,

00:37:11.460 --> 00:37:15.720
it left a whole rigid motion.

00:37:15.720 --> 00:37:20.940
Maybe u=0, w=0 at one end,
and free at the other end,

00:37:20.940 --> 00:37:22.300
is that--?

00:37:22.300 --> 00:37:24.590
It sounds risky to me.

00:37:24.590 --> 00:37:27.010
But we can see.

00:37:27.010 --> 00:37:27.910
Okay.

00:37:27.910 --> 00:37:31.540
So, do you get the general
picture of the beam?

00:37:31.540 --> 00:37:32.970
So what's the equation?

00:37:32.970 --> 00:37:36.680
What's A transpose C A,
when I put it all together?

00:37:36.680 --> 00:37:43.370
I'll use that space to put in
A transpose C A. Continuous,

00:37:43.370 --> 00:37:45.490
we're talking here.

00:37:45.490 --> 00:37:48.670
Right now we've got
differential equations.

00:37:48.670 --> 00:37:51.610
So what's the differential
equation, A transpose C A?

00:37:51.610 --> 00:37:54.580
So it's the second derivative.

00:37:54.580 --> 00:37:59.280
I'm just going backwards around
the framework, as always.

00:37:59.280 --> 00:38:05.970
The second derivative of this,
and this is c(x) times e(x),

00:38:05.970 --> 00:38:19.670
and e(x) is second
derivative of u, equal f(x).

00:38:19.670 --> 00:38:21.850
Good.

00:38:21.850 --> 00:38:23.910
In ten minutes,
we've written down

00:38:23.910 --> 00:38:29.920
the framework, some possible
boundary conditions,

00:38:29.920 --> 00:38:34.500
and the combined A
transpose C A equation.

00:38:34.500 --> 00:38:37.340
I mean, we're ready to go.

00:38:37.340 --> 00:38:43.760
We've got the pattern
to think about this.

00:38:43.760 --> 00:38:47.340
So let's see, what
should we do first?

00:38:47.340 --> 00:38:53.300
I would say the first thing
to do is, let c be one

00:38:53.300 --> 00:38:55.350
and solve some problems.

00:38:55.350 --> 00:39:00.620
Let c be one, and
consider-- So if c is one,

00:39:00.620 --> 00:39:05.620
it's a fourth
derivative equation.

00:39:05.620 --> 00:39:08.420
Should we take uniform load?

00:39:08.420 --> 00:39:10.330
Yeah.

00:39:10.330 --> 00:39:14.660
How does a beam bend
under its own weight?

00:39:14.660 --> 00:39:18.350
So it's just one, or
whatever constant.

00:39:18.350 --> 00:39:21.200
So it's constant load,
it's just its own weight,

00:39:21.200 --> 00:39:24.360
it's going to sag a
little in the middle.

00:39:24.360 --> 00:39:26.860
What's the solution
to that equation?

00:39:26.860 --> 00:39:32.790
And what shall I take
as boundary conditions?

00:39:32.790 --> 00:39:35.480
Let me do the simply
supported one.

00:39:35.480 --> 00:39:37.210
Because that would
be kind of nice.

00:39:37.210 --> 00:39:41.060
So it's simply supported, it's
sagging under its own weight,

00:39:41.060 --> 00:39:49.550
with u(0)=0, u''(0)=0,
because that's the w.

00:39:49.550 --> 00:39:55.840
And u(1)=0, u''(1)=0.

00:39:55.840 --> 00:39:59.340
Whatever.

00:39:59.340 --> 00:40:01.860
I don't know that
I'll have the patience

00:40:01.860 --> 00:40:04.760
to go through and plug in
all four boundary conditions

00:40:04.760 --> 00:40:07.090
to determine all four constants.

00:40:07.090 --> 00:40:09.250
Just get me to that point.

00:40:09.250 --> 00:40:12.590
Get me to a solution
u(x), the general solution

00:40:12.590 --> 00:40:20.210
here that's got four
constants in it is what?

00:40:20.210 --> 00:40:21.900
Okay.

00:40:21.900 --> 00:40:23.792
Okay, think again.

00:40:23.792 --> 00:40:24.750
What are we looking at?

00:40:24.750 --> 00:40:28.590
We're looking at a linear
differential equation.

00:40:28.590 --> 00:40:31.710
Linear problem.

00:40:31.710 --> 00:40:35.490
I'm asking for the general
solution to a linear equation.

00:40:35.490 --> 00:40:37.590
What's the general set up?

00:40:37.590 --> 00:40:40.780
General set up is,
particular solution

00:40:40.780 --> 00:40:42.550
plus nullspace solution.

00:40:42.550 --> 00:40:43.330
Right?

00:40:43.330 --> 00:40:46.035
You see an equation
like that, you're

00:40:46.035 --> 00:40:47.940
looking for the
general solution,

00:40:47.940 --> 00:40:51.250
tell me one particular
solution and then tell me

00:40:51.250 --> 00:40:54.470
all the solutions when
it has zero on the right,

00:40:54.470 --> 00:40:56.320
and we've got everybody.

00:40:56.320 --> 00:41:00.520
So that was true for matrices,
it was true for Ax=b,

00:41:00.520 --> 00:41:03.870
it's just as true for
differential equations.

00:41:03.870 --> 00:41:06.660
So what's one
particular solution?

00:41:06.660 --> 00:41:13.680
What's one function whose
fourth derivative is one?

00:41:13.680 --> 00:41:16.350
Yes?

00:41:16.350 --> 00:41:18.730
What am I looking for here?

00:41:18.730 --> 00:41:22.640
1/4 of x to the-- No, what?

00:41:22.640 --> 00:41:23.790
1/24, is it?

00:41:23.790 --> 00:41:31.032
1/24 of x to the fourth?
x to the fourth over 24.

00:41:31.032 --> 00:41:32.490
Because four
derivatives-- So we're

00:41:32.490 --> 00:41:35.320
thinking we're in the
polynomial world here.

00:41:35.320 --> 00:41:39.090
Just as we were with u''.

00:41:39.090 --> 00:41:42.060
With the bar it was
x squared over two,

00:41:42.060 --> 00:41:46.280
the particular solution, now
we're up to x fourth over 24.

00:41:46.280 --> 00:41:50.410
So that's the fourth
derivative is one, good.

00:41:50.410 --> 00:41:55.250
So I'm seeing a fourth
degree bending there.

00:41:55.250 --> 00:41:58.480
And now what about the
null space solutions,

00:41:58.480 --> 00:42:02.140
the homogeneous solutions.

00:42:02.140 --> 00:42:06.590
This accounts for the one,
now what are the possibilities

00:42:06.590 --> 00:42:11.190
if it was a zero?

00:42:11.190 --> 00:42:13.390
You're going to tell me
the whole bunch, right?

00:42:13.390 --> 00:42:23.910
A plus Bx plus Cx
squared plus Dx cubed.

00:42:23.910 --> 00:42:28.230
Because all of those have
fourth derivatives equal zero.

00:42:28.230 --> 00:42:31.570
So that's the general solution.

00:42:31.570 --> 00:42:33.080
Okay.

00:42:33.080 --> 00:42:36.730
So whatever the
boundary conditions are,

00:42:36.730 --> 00:42:43.360
they determine A,
B, C, D. We're not

00:42:43.360 --> 00:42:47.970
that far away from Monday's
lecture on fitting cubics.

00:42:47.970 --> 00:42:50.760
Actually we're
really close to it.

00:42:50.760 --> 00:42:52.640
When we use finite
elements, we're

00:42:52.640 --> 00:42:57.140
going to use exactly
those cubics.

00:42:57.140 --> 00:43:01.510
I'll get to that point.

00:43:01.510 --> 00:43:04.290
Let me take the
other model problem,

00:43:04.290 --> 00:43:07.420
that everybody
knows what's coming.

00:43:07.420 --> 00:43:12.890
What's the other right-hand side
that this course lives and dies

00:43:12.890 --> 00:43:14.760
on-- lives on.

00:43:14.760 --> 00:43:15.590
Delta function.

00:43:15.590 --> 00:43:16.170
Right.

00:43:16.170 --> 00:43:18.970
Delta at some point.

00:43:18.970 --> 00:43:21.890
So that's a point load then.

00:43:21.890 --> 00:43:29.690
I can make it-- Whatever
the boundary conditions are.

00:43:29.690 --> 00:43:30.960
Right.

00:43:30.960 --> 00:43:32.550
Good point.

00:43:32.550 --> 00:43:35.520
This boring stuff will
just repeat, right?

00:43:35.520 --> 00:43:37.810
That's the null space solution.

00:43:37.810 --> 00:43:41.430
But now, what is a
particular solution?

00:43:41.430 --> 00:43:44.370
The particular solution
has become interesting.

00:43:44.370 --> 00:43:48.870
The particular solution here
was straightforward, simple,

00:43:48.870 --> 00:43:50.400
a good one to do first.

00:43:50.400 --> 00:43:55.110
What's a particular
solution to a point load?

00:43:55.110 --> 00:44:01.420
So instead of having
distributed load here,

00:44:01.420 --> 00:44:05.680
I'm putting a heavy weight
here at this point, a.

00:44:05.680 --> 00:44:12.110
And it's heavy weight, I'm
multiplying the delta function

00:44:12.110 --> 00:44:16.190
by one, I could multiply by
some L for load or something,

00:44:16.190 --> 00:44:18.590
but let's just keep it simple.

00:44:18.590 --> 00:44:23.920
What's a solution to that?

00:44:23.920 --> 00:44:27.370
Fourth derivative equals delta.

00:44:27.370 --> 00:44:30.500
So that means I've
now got to integrate,

00:44:30.500 --> 00:44:32.820
one way to get the
answer here would

00:44:32.820 --> 00:44:34.650
be to integrate four times.

00:44:34.650 --> 00:44:35.660
Right?

00:44:35.660 --> 00:44:39.570
If I integrate delta
four times, then

00:44:39.570 --> 00:44:42.940
I've got something whose fourth
derivative will match delta.

00:44:42.940 --> 00:44:46.720
So do you remember the
integrals of delta?

00:44:46.720 --> 00:44:48.050
Okay, so I integrate.

00:44:48.050 --> 00:44:51.520
First integral is: step.

00:44:51.520 --> 00:44:54.860
Second integral is: ramp.

00:44:54.860 --> 00:44:59.290
Third integral is:
quadratic, right?

00:44:59.290 --> 00:45:02.710
This was linear, boop
boop, linear pieces.

00:45:02.710 --> 00:45:09.070
The next integral is going to
bring me up to quadratic ramp.

00:45:09.070 --> 00:45:11.220
And the next,
fourth one, is going

00:45:11.220 --> 00:45:14.661
to bring me up to cubic ramp.

00:45:14.661 --> 00:45:15.160
Cubic.

00:45:15.160 --> 00:45:22.330
So it's going to be cubic
ramp, is what I get there.

00:45:22.330 --> 00:45:27.990
So one particular solution
would be a function that's zero,

00:45:27.990 --> 00:45:33.540
and then at the point a, it
suddenly goes up cubically.

00:45:33.540 --> 00:45:39.840
So it's zero here, and it's x
cubed over six there, I think.

00:45:39.840 --> 00:45:45.340
If I integrate three times,
I'll be up to x cubed over six.

00:45:45.340 --> 00:45:46.910
Is that right?

00:45:46.910 --> 00:45:49.370
Yes, cubic.

00:45:49.370 --> 00:45:53.540
So that's an
interesting function.

00:45:53.540 --> 00:46:05.590
Of course, these parts will tilt
the function, will change it.

00:46:05.590 --> 00:46:08.860
So our solution
won't look like this,

00:46:08.860 --> 00:46:12.320
because I've only got
one particular function,

00:46:12.320 --> 00:46:16.050
and I'll need these to satisfy
the boundary conditions.

00:46:16.050 --> 00:46:18.470
So there is one
particular solution.

00:46:18.470 --> 00:46:22.460
The general solution-- Yes,
good for us to think out

00:46:22.460 --> 00:46:24.450
the general solution.

00:46:24.450 --> 00:46:32.960
What does that picture look
like when I add in this stuff?

00:46:32.960 --> 00:46:34.260
Very, very important.

00:46:34.260 --> 00:46:36.600
I'm sorry I don't have
more space for this highly

00:46:36.600 --> 00:46:37.960
important picture.

00:46:37.960 --> 00:46:42.950
Okay, so here's my point a.

00:46:42.950 --> 00:46:44.930
Keep your eye on that point a.

00:46:44.930 --> 00:46:48.970
Okay, so to the left of it,
I've got some curve, whatever,

00:46:48.970 --> 00:46:51.990
dut dut dut dut dut, the beam.

00:46:51.990 --> 00:46:55.120
And to the right of it, I've
got some other curve, whatever

00:46:55.120 --> 00:46:57.530
it is, dut dut dut dut dut.

00:46:57.530 --> 00:47:01.280
And what's cooking at point a?

00:47:01.280 --> 00:47:03.590
What's the jump
condition at point a?

00:47:03.590 --> 00:47:05.630
That's the critical question.

00:47:05.630 --> 00:47:08.070
What changes at point a?

00:47:08.070 --> 00:47:13.790
You remember, this is
the corresponding thing,

00:47:13.790 --> 00:47:16.130
the analog of our ramp.

00:47:16.130 --> 00:47:18.560
So what changed for
the ramp at point a?

00:47:18.560 --> 00:47:20.420
What jumped?

00:47:20.420 --> 00:47:21.780
The slope.

00:47:21.780 --> 00:47:25.650
Okay, now the question is:
what's going to jump here?

00:47:25.650 --> 00:47:27.580
What jumped there?

00:47:27.580 --> 00:47:31.010
Here, did the function jump?

00:47:31.010 --> 00:47:31.840
Certainly not.

00:47:31.840 --> 00:47:32.830
Did the slope jump?

00:47:32.830 --> 00:47:33.560
Certainly not.

00:47:33.560 --> 00:47:35.040
Did the second derivative jump?

00:47:35.040 --> 00:47:36.810
No, no.

00:47:36.810 --> 00:47:39.020
The second derivative
was zero and then zero.

00:47:39.020 --> 00:47:40.870
What jumped?

00:47:40.870 --> 00:47:42.880
Third derivative.

00:47:42.880 --> 00:47:45.090
The third derivative
is allowed to jump.

00:47:45.090 --> 00:47:46.410
And of course.

00:47:46.410 --> 00:47:51.770
A jump in the third derivative
produces a delta in the fourth.

00:47:51.770 --> 00:47:52.340
Right?

00:47:52.340 --> 00:47:55.070
It just works.

00:47:55.070 --> 00:48:01.400
So this is a cubic of some
sort, coming from that jump.

00:48:01.400 --> 00:48:06.290
This is another, a different
cubic, coming from this sort,

00:48:06.290 --> 00:48:09.140
from this junk, and this.

00:48:09.140 --> 00:48:11.630
So it's cubic in each piece.

00:48:11.630 --> 00:48:15.380
Why is it cubic in each piece?

00:48:15.380 --> 00:48:20.960
Because, what's the equation
in the middle of that piece?

00:48:20.960 --> 00:48:25.630
What's our differential
equation if I look here?

00:48:25.630 --> 00:48:26.650
Here's my equation.

00:48:26.650 --> 00:48:33.960
What is it in the middle of
that piece? u fourth equal zero.

00:48:33.960 --> 00:48:36.310
The delta function
is zero there.

00:48:36.310 --> 00:48:37.620
And u fourth is zero here.

00:48:37.620 --> 00:48:41.140
So of course this
is a cubic spline.

00:48:41.140 --> 00:48:43.710
We're meeting that neat
word, cubic spline.

00:48:43.710 --> 00:48:46.490
Those turn out to be
very, very handy functions

00:48:46.490 --> 00:48:48.140
for other things, too.

00:48:48.140 --> 00:48:51.070
So we see them
here as a solution

00:48:51.070 --> 00:48:55.640
to fourth order equations with
point loads are cubic splines.

00:48:55.640 --> 00:49:04.140
Because the big key point is
that there's a jump here in u

00:49:04.140 --> 00:49:05.980
triple prime, the
third derivative.

00:49:05.980 --> 00:49:10.670
A jump in the third derivative,
that's what we saw here.

00:49:10.670 --> 00:49:17.360
And we'll see it if we have
any cubic meeting any cubic.

00:49:17.360 --> 00:49:21.210
Let me just say, a jump
in the third derivative,

00:49:21.210 --> 00:49:24.350
your eye probably
won't notice it.

00:49:24.350 --> 00:49:28.000
I mean, it's a
discontinuity, somehow.

00:49:28.000 --> 00:49:30.640
We don't have the same
polynomial from here to here.

00:49:30.640 --> 00:49:34.190
But that discontinuity
in u triple prime,

00:49:34.190 --> 00:49:37.000
it's pretty darn smooth still.

00:49:37.000 --> 00:49:41.100
The slope is continuous, so
your eye doesn't see a ramp.

00:49:41.100 --> 00:49:44.940
And even more, the
curving is continuous,

00:49:44.940 --> 00:49:50.450
the curvature is continuous.
e and w are good.

00:49:50.450 --> 00:49:55.290
It's just a jump in
the third derivative.

00:49:55.290 --> 00:50:01.180
Okay, so I want to speak
about splines, and more

00:50:01.180 --> 00:50:05.880
about this, and about finite
elements for beam problems

00:50:05.880 --> 00:50:07.010
on Friday.

00:50:07.010 --> 00:50:11.900
And then that will take care
of 1-D and we'll move into 2-D.

00:50:11.900 --> 00:50:13.150
Okay.