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PROFESSOR STRANG: OK.

00:00:22.590 --> 00:00:23.090
All right.

00:00:23.090 --> 00:00:24.630
Good morning.

00:00:24.630 --> 00:00:32.120
So we're doing finite elements.

00:00:32.120 --> 00:00:34.380
The element that we
considered so far

00:00:34.380 --> 00:00:38.500
was the basic linear element.

00:00:38.500 --> 00:00:42.590
Continuous, but of course,
the slopes have jumps.

00:00:42.590 --> 00:00:47.790
The slope was minus one
over delta x for that one.

00:00:47.790 --> 00:00:50.070
This one was a plus
and then a minus.

00:00:50.070 --> 00:00:53.700
So a jump in slope but
no jump in the function.

00:00:53.700 --> 00:00:56.920
So actually, my
abbreviation for that

00:00:56.920 --> 00:01:03.030
would be C zero, saying
that it's continuous

00:01:03.030 --> 00:01:05.890
but no derivative is continuous.

00:01:05.890 --> 00:01:08.280
And now we'll get
to some elements

00:01:08.280 --> 00:01:11.350
where the slope is continuous.

00:01:11.350 --> 00:01:16.970
It's sort of fun to create
these finite elements of higher

00:01:16.970 --> 00:01:18.520
degree.

00:01:18.520 --> 00:01:20.990
It's pretty
straightforward in 1-D.

00:01:20.990 --> 00:01:24.580
And that's where we are now.

00:01:24.580 --> 00:01:27.170
So we'll get second degree
elements and third degree

00:01:27.170 --> 00:01:28.020
elements.

00:01:28.020 --> 00:01:31.530
And that gives us, as
we'll see, higher accuracy.

00:01:31.530 --> 00:01:35.530
So I want to connect the
degree of the polynomials

00:01:35.530 --> 00:01:41.820
to the accuracy of
the approximation.

00:01:41.820 --> 00:01:44.020
Part of that connection
is to recognize

00:01:44.020 --> 00:01:46.950
that these problems have
a strong form, as we know,

00:01:46.950 --> 00:01:49.770
the equation; a
weak form, that's

00:01:49.770 --> 00:01:55.380
the one that has test functions;
and also a minimum form

00:01:55.380 --> 00:01:58.260
that we'll see.

00:01:58.260 --> 00:02:03.830
So how would I get some
quadratics, so second degree

00:02:03.830 --> 00:02:09.650
elements, parabolas,
into the picture?

00:02:09.650 --> 00:02:11.910
You remember Galerkin's idea?

00:02:11.910 --> 00:02:14.270
Choose trial functions.

00:02:14.270 --> 00:02:17.190
And we're taking those to be
the same as the test function.

00:02:17.190 --> 00:02:20.160
So these are the trial
functions we've chosen.

00:02:20.160 --> 00:02:22.700
One, two, three, four of them.

00:02:22.700 --> 00:02:25.590
And they're linear.

00:02:25.590 --> 00:02:28.690
And that limits the
accuracy that you can get,

00:02:28.690 --> 00:02:34.190
because your approximations
then are combinations of those.

00:02:34.190 --> 00:02:37.250
So they're like
broken-line functions,

00:02:37.250 --> 00:02:39.040
linear approximations.

00:02:39.040 --> 00:02:44.650
And the accuracy is not great.

00:02:44.650 --> 00:02:47.220
It's sort of the
lowest level possible.

00:02:47.220 --> 00:02:53.590
So how would you get parabolas?

00:02:53.590 --> 00:02:56.910
So this was first guy.

00:02:56.910 --> 00:03:08.500
The second guy is going to
be continuous and quadratic.

00:03:08.500 --> 00:03:10.880
So it's going to have
new trial functions.

00:03:10.880 --> 00:03:14.300
In addition to these, I'm
going to put in some more.

00:03:14.300 --> 00:03:16.450
Galerkin's happy with that.

00:03:16.450 --> 00:03:18.940
I still proceed as usual.

00:03:18.940 --> 00:03:22.170
My approximation is some
combination of those.

00:03:22.170 --> 00:03:24.050
It's only going
to be continuous.

00:03:24.050 --> 00:03:27.350
So it'll just be a
C zero guy again.

00:03:27.350 --> 00:03:31.090
That means jump in slope.

00:03:31.090 --> 00:03:33.220
The first derivative
isn't there.

00:03:33.220 --> 00:03:35.880
Eventually, I want
to get to a C one

00:03:35.880 --> 00:03:38.080
where the slopes are continuous.

00:03:38.080 --> 00:03:41.970
OK, but how would I
get some quadratics?

00:03:41.970 --> 00:03:47.970
All I want now is my functions,
my space, my combinations,

00:03:47.970 --> 00:03:51.650
should be the
piecewise parabolas

00:03:51.650 --> 00:03:55.590
instead of piecewise linear.

00:03:55.590 --> 00:04:00.450
And the pieces are
broken at the nodes.

00:04:00.450 --> 00:04:04.370
OK, so here is a way to do it.

00:04:04.370 --> 00:04:08.090
Inside each interval,
I'm going to add,

00:04:08.090 --> 00:04:10.250
I'll just call them
bubble functions.

00:04:10.250 --> 00:04:13.200
So these will be
new additional guys.

00:04:13.200 --> 00:04:16.930
So this will be my first phi.

00:04:16.930 --> 00:04:18.720
You remember that half hat?

00:04:18.720 --> 00:04:23.830
Because the problem I was
doing was a free fixed problem.

00:04:23.830 --> 00:04:28.070
That's why I had a
half hat at this end,

00:04:28.070 --> 00:04:32.330
because there was no boundary
condition that my functions had

00:04:32.330 --> 00:04:33.900
to satisfy.

00:04:33.900 --> 00:04:35.560
At this end, there was.

00:04:35.560 --> 00:04:36.370
It was fixed.

00:04:36.370 --> 00:04:40.100
So that's why the hat ended,
and there was no half hat,

00:04:40.100 --> 00:04:42.350
there's no extra function there.

00:04:42.350 --> 00:04:44.770
So I have right now
one, two, three, four.

00:04:44.770 --> 00:04:48.330
I'm going to add four
more bubble functions.

00:04:48.330 --> 00:04:51.060
Each one will be
inside an interval.

00:04:51.060 --> 00:04:53.210
So it'll be a little parabola.

00:04:53.210 --> 00:04:58.250
This is function
number whatever.

00:04:58.250 --> 00:05:02.830
If I number this number one,
let's say, phi_1, now --

00:05:02.830 --> 00:05:04.640
that's probably a
change in numbering --

00:05:04.640 --> 00:05:07.770
phi_2 is going to be my bubble.

00:05:07.770 --> 00:05:10.150
And you see what my
bubble function is?

00:05:10.150 --> 00:05:14.400
It's a function that
goes there and straight.

00:05:14.400 --> 00:05:21.200
So it is continuous, no jumps,
and it is second degree.

00:05:21.200 --> 00:05:24.200
It's a parabola, and
I'll make its height one.

00:05:24.200 --> 00:05:27.320
And then there'll
be another function.

00:05:27.320 --> 00:05:30.090
If I had another
color I could draw it.

00:05:30.090 --> 00:05:33.610
Well, I'll just do it
with broken lines, maybe.

00:05:33.610 --> 00:05:38.720
So there'll be another
bubble function in here,

00:05:38.720 --> 00:05:41.970
a third bubble function
in the third interval,

00:05:41.970 --> 00:05:44.180
and a fourth in the
fourth interval.

00:05:44.180 --> 00:05:50.280
You see that I've now got
my old phi_1, phi_3, phi_5,

00:05:50.280 --> 00:05:54.110
and phi_7 were
the hat functions.

00:05:54.110 --> 00:05:59.910
But now I've got a phi_2,
phi_4, phi_6, and phi_8 that

00:05:59.910 --> 00:06:04.980
are these new trial functions.

00:06:04.980 --> 00:06:07.940
So part of the
message is, we can

00:06:07.940 --> 00:06:11.410
throw in additional functions.

00:06:11.410 --> 00:06:14.280
They don't have
to be polynomials,

00:06:14.280 --> 00:06:17.000
but those are the
simplest choices.

00:06:17.000 --> 00:06:18.410
Why are they simple?

00:06:18.410 --> 00:06:23.310
Because, you remember, that in
the end when I made the choice,

00:06:23.310 --> 00:06:26.320
I have to do various
integrations.

00:06:26.320 --> 00:06:31.940
So you remember that I have to
integrate to find entries K_ij.

00:06:31.940 --> 00:06:34.150
Do you remember what that
integral looked like?

00:06:34.150 --> 00:06:37.790
You certainly remember F_i.

00:06:37.790 --> 00:06:43.790
That was the integral from zero
to one of whatever function

00:06:43.790 --> 00:06:50.850
phi_i we had, times the
f(x)dx, times the load.

00:06:50.850 --> 00:06:54.710
And we computed these.

00:06:54.710 --> 00:06:58.800
You remember we computed these
for the piecewise linear guys.

00:06:58.800 --> 00:07:03.380
But I don't think I wrote
down the expression that we

00:07:03.380 --> 00:07:06.080
were really doing, so
let me just do that.

00:07:06.080 --> 00:07:22.170
It's c(x), there's a du--
No. d phi_j/dx and a dV_i/dx.

00:07:25.890 --> 00:07:29.020
Those were the integrals
that we had to do.

00:07:29.020 --> 00:07:35.320
And we were taking phis
to be the same as V's.

00:07:35.320 --> 00:07:38.210
Maybe I'll just do that here,
because I don't plan to make

00:07:38.210 --> 00:07:40.990
any other choices
at all. d phi_i/dx.

00:07:44.190 --> 00:07:46.770
It's a symmetric matrix now.

00:07:46.770 --> 00:07:52.840
K_ji, because when I'm
choosing phis the same

00:07:52.840 --> 00:07:55.340
as the V's, this is
what it looks like,

00:07:55.340 --> 00:08:00.120
and if I switch j and i, I
don't see any difference.

00:08:00.120 --> 00:08:03.920
So these are the things
that have to be integrated.

00:08:03.920 --> 00:08:06.160
And those are the
ones we did integrate

00:08:06.160 --> 00:08:08.440
when phi was piecewise linear.

00:08:08.440 --> 00:08:11.090
When phi was piecewise
linear, the slope

00:08:11.090 --> 00:08:14.670
was piecewise constant, and
we had really easy integrals.

00:08:14.670 --> 00:08:18.810
Very easy integrals.

00:08:18.810 --> 00:08:22.080
We had to pay
attention to where we

00:08:22.080 --> 00:08:27.500
were was the slope on a minus
interval or on a plus interval,

00:08:27.500 --> 00:08:29.460
but they were easy to compute.

00:08:29.460 --> 00:08:34.190
And they led us back
to the kind of matrix

00:08:34.190 --> 00:08:36.090
that we've seen before.

00:08:36.090 --> 00:08:38.600
The twos and minus ones.

00:08:38.600 --> 00:08:41.840
And our right hand
sides looked familiar.

00:08:41.840 --> 00:08:45.980
Now, we've got new functions.

00:08:45.980 --> 00:08:48.360
We still have the same formulas.

00:08:48.360 --> 00:08:50.600
No change in formulas.

00:08:50.600 --> 00:08:54.140
The system is really
quite successful,

00:08:54.140 --> 00:08:57.530
because these are the things
that we have to compute.

00:08:57.530 --> 00:09:02.710
So now I'll have to
integrate these parabolas,

00:09:02.710 --> 00:09:06.010
these little parabolas,
half of the phis

00:09:06.010 --> 00:09:09.100
will be little parabolas,
and their derivatives

00:09:09.100 --> 00:09:10.610
will be linear.

00:09:10.610 --> 00:09:14.300
So you see, I'll have
more calculations to do.

00:09:14.300 --> 00:09:19.080
Which I don't plan to do,
but more integrations to--

00:09:19.080 --> 00:09:23.650
For example, the
diagonal entry, say

00:09:23.650 --> 00:09:31.520
2, 2, which will come from
that bubble with itself.

00:09:31.520 --> 00:09:36.420
K_22, then, will be
the integral of c(x),

00:09:36.420 --> 00:09:38.690
times the derivative
of that bubble, which

00:09:38.690 --> 00:09:44.050
will be a straight line times
itself, so it would be squared.

00:09:44.050 --> 00:09:48.570
And c is positive,
so this K_22 is

00:09:48.570 --> 00:09:51.360
going to be some
nice positive number.

00:09:51.360 --> 00:09:55.420
But we'll have to
figure out what it is.

00:09:55.420 --> 00:10:01.900
Maybe I'll just say one fact
that we'll come back to.

00:10:01.900 --> 00:10:08.610
That this K is symmetric
positive definite.

00:10:08.610 --> 00:10:11.390
You thought it would be.

00:10:11.390 --> 00:10:15.000
By using the letter K, we
kind of expected it to be.

00:10:15.000 --> 00:10:16.120
And it will be.

00:10:16.120 --> 00:10:21.620
It'll be symmetric because the
phis and the V's are the same.

00:10:21.620 --> 00:10:23.970
And it turns out it's
positive definite.

00:10:23.970 --> 00:10:25.860
So it's just great.

00:10:25.860 --> 00:10:27.850
Just great.

00:10:27.850 --> 00:10:35.400
We have a little more effort,
either to use a formula

00:10:35.400 --> 00:10:41.840
for integrating polynomials,
or using numerical integration.

00:10:41.840 --> 00:10:44.650
One way or another, and I
won't concentrate right now

00:10:44.650 --> 00:10:47.450
on that point, we
get these numbers.

00:10:47.450 --> 00:10:49.220
Okay.

00:10:49.220 --> 00:10:54.200
Here's something
to concentrate on.

00:10:54.200 --> 00:10:58.200
What kind of a
matrix K do we have?

00:10:58.200 --> 00:11:02.450
Where will it be non-zero?

00:11:02.450 --> 00:11:05.780
So it'll be eight
by eight, right?

00:11:05.780 --> 00:11:09.190
I'll follow through
on that choice.

00:11:09.190 --> 00:11:12.610
Just to say, where will
I see non-zeros here?

00:11:12.610 --> 00:11:14.960
Because if you get
that point, you

00:11:14.960 --> 00:11:20.300
see the way things
come together.

00:11:20.300 --> 00:11:22.410
I'll just put a
little x for non-zero.

00:11:22.410 --> 00:11:26.370
So K_11.

00:11:26.370 --> 00:11:29.110
So what's that first row of K?

00:11:29.110 --> 00:11:34.140
It's coming from the
first function, integrated

00:11:34.140 --> 00:11:36.230
against itself.

00:11:36.230 --> 00:11:41.050
K_11, if for 1, 1, we'll
get something there.

00:11:41.050 --> 00:11:46.300
Will we have something
in the 1, 2 position?

00:11:46.300 --> 00:11:50.820
That's my question, do we have
something in the 1, 2 position?

00:11:50.820 --> 00:11:52.370
What's 1, 2?

00:11:52.370 --> 00:11:57.110
That's this function against
the bubble function, yes?

00:11:57.110 --> 00:11:58.100
Right?

00:11:58.100 --> 00:12:00.690
They're non-zero
at the same place.

00:12:00.690 --> 00:12:03.040
We can expect something there.

00:12:03.040 --> 00:12:05.390
What about K_13?

00:12:05.390 --> 00:12:08.410
That's what we've done
before, that's this one

00:12:08.410 --> 00:12:09.680
against this one.

00:12:09.680 --> 00:12:12.130
Yes?

00:12:12.130 --> 00:12:14.640
We expect a non-zero there.

00:12:14.640 --> 00:12:16.070
But then what?

00:12:16.070 --> 00:12:19.600
After that, what will
the rest of that row be?

00:12:19.600 --> 00:12:20.450
Zero.

00:12:20.450 --> 00:12:23.660
Because that first
half hat doesn't

00:12:23.660 --> 00:12:26.490
touch any of the others.

00:12:26.490 --> 00:12:27.710
So let's go on.

00:12:27.710 --> 00:12:31.080
Of course it'll be symmetric.

00:12:31.080 --> 00:12:33.330
I know this much.

00:12:33.330 --> 00:12:43.200
So this is the half hat row, and
this is the first bubble row.

00:12:43.200 --> 00:12:45.720
Because the half hat
was phi_1 and now

00:12:45.720 --> 00:12:47.710
the first bubble is phi_2.

00:12:47.710 --> 00:12:53.020
What non-zeros do we get
in the stiffness matrix?

00:12:53.020 --> 00:13:00.130
Again, we could unconstruct
it entry by entry.

00:13:00.130 --> 00:13:04.030
Another way to construct it
will be element by element,

00:13:04.030 --> 00:13:05.500
stamp them in.

00:13:05.500 --> 00:13:09.710
You're beginning to
see the idea of that.

00:13:09.710 --> 00:13:12.680
So what do I get
for that bubble?

00:13:12.680 --> 00:13:17.870
I just look to see which
elements touch that bubble.

00:13:17.870 --> 00:13:21.870
And which ones do?

00:13:21.870 --> 00:13:26.340
One, two and three,
and not four.

00:13:26.340 --> 00:13:28.020
Right?

00:13:28.020 --> 00:13:31.020
In that row, we only get
-- so from that bubble,

00:13:31.020 --> 00:13:33.000
I think we only get that much.

00:13:33.000 --> 00:13:35.530
Now, we're not quite
seeing the picture yet.

00:13:35.530 --> 00:13:38.450
Let me go to the next hat.

00:13:38.450 --> 00:13:45.020
The hat, phi_2, and
then I'll do the bubble.

00:13:45.020 --> 00:13:48.150
Oh, no, sorry, the
hat's numbered phi_3,

00:13:48.150 --> 00:13:53.480
and then the next bubble
is numbered phi_4.

00:13:53.480 --> 00:13:55.680
Where do I get zeros?

00:13:55.680 --> 00:13:58.400
You can tell me,
where do I get zeros?

00:13:58.400 --> 00:14:05.210
From inner products, from
these guys, when i is three.

00:14:05.210 --> 00:14:10.180
So which phis does phi
number three overlap?

00:14:10.180 --> 00:14:12.210
That's all I'm asking.

00:14:12.210 --> 00:14:14.550
Does it overlap number one?

00:14:14.550 --> 00:14:15.460
Yes.

00:14:15.460 --> 00:14:18.530
Does it overlap phi number two?

00:14:18.530 --> 00:14:20.940
You want to highlight,
so we're now

00:14:20.940 --> 00:14:24.100
looking at phi_3, at this hat.

00:14:24.100 --> 00:14:29.790
God, where's it gone?

00:14:29.790 --> 00:14:32.460
That's the one we're doing now?

00:14:32.460 --> 00:14:34.640
So what does it overlap?

00:14:34.640 --> 00:14:39.120
It overlaps the half hat, does
it overlap the first bubble?

00:14:39.120 --> 00:14:39.970
Yes.

00:14:39.970 --> 00:14:41.370
Does it overlap itself?

00:14:41.370 --> 00:14:42.200
Yes.

00:14:42.200 --> 00:14:45.620
Does it overlap
the second bubble?

00:14:45.620 --> 00:14:46.200
Yes.

00:14:46.200 --> 00:14:49.130
Does it overlap the next hat?

00:14:49.130 --> 00:14:50.490
Yes.

00:14:50.490 --> 00:14:53.040
And then all zeros.

00:14:53.040 --> 00:14:55.110
Okay, and now do one more row.

00:14:55.110 --> 00:14:56.570
Bubble four.

00:14:56.570 --> 00:15:01.090
So now I'm looking at this
guy, this next bubble. phi_4.

00:15:01.090 --> 00:15:03.880
What does that overlap?

00:15:03.880 --> 00:15:08.090
Does it overlap
the first half hat?

00:15:08.090 --> 00:15:08.850
Nope.

00:15:08.850 --> 00:15:13.130
Of course, symmetry
told us that.

00:15:13.130 --> 00:15:16.250
Does the second bubble
overlap the first bubble?

00:15:16.250 --> 00:15:17.550
No.

00:15:17.550 --> 00:15:19.480
Big point: zero there.

00:15:19.480 --> 00:15:25.440
Does the second bubble
overlap the hat?

00:15:25.440 --> 00:15:26.860
Yes.

00:15:26.860 --> 00:15:28.920
Does the second
bubble overlap itself?

00:15:28.920 --> 00:15:31.140
Certainly, on the diagonal
we have something.

00:15:31.140 --> 00:15:34.840
Does the second bubble
overlap the next hat, phi_5?

00:15:34.840 --> 00:15:35.900
Yes.

00:15:35.900 --> 00:15:39.140
And that's it.

00:15:39.140 --> 00:15:39.860
I think.

00:15:39.860 --> 00:15:44.680
The second level does not
overlap the following bubble.

00:15:44.680 --> 00:15:51.170
I don't know if you see what
pattern we're getting here.

00:15:51.170 --> 00:15:54.590
Those were special rows, because
that was only a half hat.

00:15:54.590 --> 00:15:56.960
These are typical rows.

00:15:56.960 --> 00:16:03.080
A typical hat function, that row
is showing us five non-zeros,

00:16:03.080 --> 00:16:06.210
because it overlaps itself,
the neighboring hats,

00:16:06.210 --> 00:16:08.080
and the neighboring bubbles.

00:16:08.080 --> 00:16:12.210
But the bubble row
only has three,

00:16:12.210 --> 00:16:15.750
because a bubble overlaps
itself, the neighboring

00:16:15.750 --> 00:16:20.840
hat on each side, but not
the neighboring bubbles.

00:16:20.840 --> 00:16:23.100
So we have only three non-zeros.

00:16:23.100 --> 00:16:28.730
Do you see that the
next row will have five?

00:16:28.730 --> 00:16:30.170
Will I get it right?

00:16:30.170 --> 00:16:31.120
I hope so.

00:16:31.120 --> 00:16:38.340
The next row we'll have,
I think they'd be here.

00:16:38.340 --> 00:16:43.790
And then the next row will have
only three guys, maybe here,

00:16:43.790 --> 00:16:48.150
here, here.

00:16:48.150 --> 00:16:51.580
Well, it's certainly
a band matrix.

00:16:51.580 --> 00:16:54.530
So you could say, okay,
it's a band matrix.

00:16:54.530 --> 00:16:59.510
I wouldn't call it
tri-diagonal anymore.

00:16:59.510 --> 00:17:03.130
If I showed you that matrix and
said, what kind of a matrix,

00:17:03.130 --> 00:17:05.860
you'd say a band matrix.

00:17:05.860 --> 00:17:09.500
If you wanted to tell me
that it had five bands,

00:17:09.500 --> 00:17:12.740
you could maybe say
penta-diagonal, or something.

00:17:12.740 --> 00:17:15.460
But it's easy to
work with, of course.

00:17:15.460 --> 00:17:17.810
That's the point
of finite elements,

00:17:17.810 --> 00:17:20.900
is that all the
functions are local,

00:17:20.900 --> 00:17:28.520
so that we get all zeros when
trial functions don't overlap.

00:17:28.520 --> 00:17:33.130
My additional point
was just a small one

00:17:33.130 --> 00:17:38.570
that's not a big deal, but it's
a little bit worth noticing.

00:17:38.570 --> 00:17:48.330
These rows with only three
entries, three non-zeros.

00:17:48.330 --> 00:17:53.800
I guess what I want to say
is I have to solve eight

00:17:53.800 --> 00:17:56.250
equations and eight unknowns.

00:17:56.250 --> 00:17:59.880
And the normal way to do it
would be just elimination.

00:17:59.880 --> 00:18:01.670
LU, that would work fine.

00:18:01.670 --> 00:18:08.690
Start from the top,
eliminate, and you've got it.

00:18:08.690 --> 00:18:12.240
And of course in one dimension,
nobody would do anything else.

00:18:12.240 --> 00:18:15.130
That would be simple.

00:18:15.130 --> 00:18:18.170
I just want to
say, these bubbles,

00:18:18.170 --> 00:18:22.990
by giving me extra zeros, I
could eliminate the bubbles

00:18:22.990 --> 00:18:24.470
first.

00:18:24.470 --> 00:18:28.170
Can I just make this
point but not labor it?

00:18:28.170 --> 00:18:32.020
I could eliminate
the bubbles first.

00:18:32.020 --> 00:18:39.520
I could use this equation to
express the bubble coefficient

00:18:39.520 --> 00:18:41.190
in terms of its neighbors.

00:18:41.190 --> 00:18:43.770
I could use this one to
express the bubble coefficient

00:18:43.770 --> 00:18:45.220
in terms of it neighbors.

00:18:45.220 --> 00:18:49.500
And I could plug back
into the other equations.

00:18:49.500 --> 00:18:54.310
I could simplify this.

00:18:54.310 --> 00:18:57.250
I could get the bubbles
done first if I wanted.

00:18:57.250 --> 00:19:03.320
I can see that to go
into the gory details

00:19:03.320 --> 00:19:05.290
is probably not wise.

00:19:05.290 --> 00:19:08.340
But bubbles are easy to do.

00:19:08.340 --> 00:19:17.390
However there are
better elements.

00:19:17.390 --> 00:19:20.500
So that's my discussion
of quadratic elements,

00:19:20.500 --> 00:19:22.400
almost complete.

00:19:22.400 --> 00:19:26.750
It's not a big favorite,
because cubics are better.

00:19:26.750 --> 00:19:28.810
So why are cubics better?

00:19:28.810 --> 00:19:31.050
Why are cubics better?

00:19:31.050 --> 00:19:34.470
So you're going to say,
okay, upgrade to cubics.

00:19:34.470 --> 00:19:38.970
How shall I do that?

00:19:38.970 --> 00:19:43.590
And I want to say a word
about the error here.

00:19:43.590 --> 00:19:45.700
Of course, the
reason quadratics are

00:19:45.700 --> 00:19:49.800
better than cubics-- Sorry,
the reason why quadratics

00:19:49.800 --> 00:19:52.050
are better than
linear, and cubics

00:19:52.050 --> 00:20:00.850
will be better than quadratics
is I'm getting more accuracy.

00:20:00.850 --> 00:20:06.360
Suppose my true solution
may be some curve like that.

00:20:06.360 --> 00:20:07.190
Okay.

00:20:07.190 --> 00:20:11.510
My piecewise linear
elements, suppose

00:20:11.510 --> 00:20:14.350
the piecewise linear
elements happen to be,

00:20:14.350 --> 00:20:19.690
as they would in a special model
problem, right on the money,

00:20:19.690 --> 00:20:21.930
at the nodes.

00:20:21.930 --> 00:20:23.410
Usually they won't be.

00:20:23.410 --> 00:20:27.410
But what would be the
error in that one?

00:20:27.410 --> 00:20:30.370
Well, no error at all at
the nodes as I've drawn it.

00:20:30.370 --> 00:20:32.840
But that's not what
I'm interested in.

00:20:32.840 --> 00:20:35.810
I'm interested in,
how big is that?

00:20:35.810 --> 00:20:42.740
How far off is the displacement?

00:20:42.740 --> 00:20:47.260
What's the maximum error
in the displacement?

00:20:47.260 --> 00:20:48.460
Do you have any idea?

00:20:48.460 --> 00:20:51.440
If this is size h.

00:20:51.440 --> 00:21:01.950
Delta x, shall I call
it delta x, or h?

00:21:01.950 --> 00:21:08.430
How far does a curving function
escape from the-- I need

00:21:08.430 --> 00:21:10.420
to blow that up, don't I?

00:21:10.420 --> 00:21:15.320
So I have a curving function
and a linear function,

00:21:15.320 --> 00:21:17.270
and I want to know
how far apart they

00:21:17.270 --> 00:21:21.800
are over a distance
of length delta x.

00:21:21.800 --> 00:21:24.920
What's this scale?

00:21:24.920 --> 00:21:28.180
That's the question.

00:21:28.180 --> 00:21:31.990
It's just good, it'll
have a simple answer

00:21:31.990 --> 00:21:35.010
and it's great to know it.

00:21:35.010 --> 00:21:38.530
Anybody want to make a guess?

00:21:38.530 --> 00:21:41.360
Is that scale of size delta x?

00:21:41.360 --> 00:21:44.980
Is it of size delta x
squared, size delta x cubed?

00:21:44.980 --> 00:21:48.300
It's that exponent of
delta x that is telling me

00:21:48.300 --> 00:21:50.730
how big is the error?

00:21:50.730 --> 00:21:55.860
And it's easy to find once
you get the hang of it.

00:21:55.860 --> 00:21:57.630
Anybody want to make a guess?

00:21:57.630 --> 00:21:59.350
Delta x?

00:21:59.350 --> 00:21:59.990
Squared.

00:21:59.990 --> 00:22:02.020
Squared would be
the right guess.

00:22:02.020 --> 00:22:07.580
Squared would be
the right guess.

00:22:07.580 --> 00:22:13.550
I could just turn that picture,
if we wanted, to-- Again, this

00:22:13.550 --> 00:22:17.190
is delta x.

00:22:17.190 --> 00:22:22.100
Now it would look like
that, pretty much.

00:22:22.100 --> 00:22:23.910
Doesn't have to be
symmetric, of course,

00:22:23.910 --> 00:22:26.920
because this could be
a complicated function.

00:22:26.920 --> 00:22:31.750
But when I focus on a
little delta x interval,

00:22:31.750 --> 00:22:38.460
every function looks
like a little polynomial.

00:22:38.460 --> 00:22:41.280
The error there, let's see.

00:22:41.280 --> 00:22:55.000
What would that function be?

00:22:55.000 --> 00:22:56.910
I could go forever on this.

00:22:56.910 --> 00:22:59.520
But look, if the
slope is something,

00:22:59.520 --> 00:23:05.990
whatever, let me
change numbers here.

00:23:05.990 --> 00:23:11.500
Let me call it from
zero to y, what

00:23:11.500 --> 00:23:18.020
would be a little parabola
that has a slope of one,

00:23:18.020 --> 00:23:20.440
let's say, at both ends.

00:23:20.440 --> 00:23:22.360
What would that parabola be?

00:23:22.360 --> 00:23:24.190
We probably have
seen that before.

00:23:24.190 --> 00:23:33.990
If I wanted a slope
of one at both ends,

00:23:33.990 --> 00:23:35.580
the polynomial
would be something

00:23:35.580 --> 00:23:39.540
like-- What would it be?

00:23:39.540 --> 00:23:41.590
Sorry, tell me that
little polynomial.

00:23:41.590 --> 00:23:44.940
It's a polynomial in x,
it's just a quadratic.

00:23:44.940 --> 00:23:52.190
Its slope is one, so it
maybe starts with an x.

00:23:52.190 --> 00:23:55.400
I've got to bring it down here.

00:23:55.400 --> 00:24:02.080
Is it x times one
minus x over-- I

00:24:02.080 --> 00:24:07.774
didn't like y ever
in the first place.

00:24:07.774 --> 00:24:08.940
What do I want to put there?

00:24:08.940 --> 00:24:11.520
I don't want to put a one.

00:24:11.520 --> 00:24:18.240
That would make it
look big. y is there.

00:24:18.240 --> 00:24:24.930
Okay, I think that
quadratic is zero at zero,

00:24:24.930 --> 00:24:26.520
because of that term.

00:24:26.520 --> 00:24:30.160
It's zero at x=y,
because of that term.

00:24:30.160 --> 00:24:31.710
It's second degree.

00:24:31.710 --> 00:24:35.270
And I think its height
is a maximum right there.

00:24:35.270 --> 00:24:37.680
And what is that height?

00:24:37.680 --> 00:24:41.120
At y/2, this is
y/2, this is y/2.

00:24:41.120 --> 00:24:44.470
That height is y
squared over four.

00:24:44.470 --> 00:24:46.810
That's what I was shooting for.

00:24:46.810 --> 00:24:48.210
The square.

00:24:48.210 --> 00:24:52.690
That, in a little interval of
length y, for length delta x,

00:24:52.690 --> 00:24:56.770
if I draw a little parabola
and I'm matching at the ends,

00:24:56.770 --> 00:25:02.040
then the height it
reaches is like y squared.

00:25:02.040 --> 00:25:04.180
That's the scale.

00:25:04.180 --> 00:25:09.760
So my conclusion is that if I
use these basic hat function

00:25:09.760 --> 00:25:17.330
elements, the error I get
is-- So can I list the errors?

00:25:17.330 --> 00:25:22.190
The error is delta x squared.

00:25:22.190 --> 00:25:24.700
That's the displacement error.

00:25:24.700 --> 00:25:28.610
The error in u.

00:25:28.610 --> 00:25:31.330
I'm not proving anything.

00:25:31.330 --> 00:25:35.390
The careful discussion
of the accuracy

00:25:35.390 --> 00:25:38.590
is a later section in the book.

00:25:38.590 --> 00:25:40.900
But I'm trying to
make the main point,

00:25:40.900 --> 00:25:45.370
is that if we're fitting
functions by straight lines,

00:25:45.370 --> 00:25:48.540
then we have an error
of delta x squared.

00:25:48.540 --> 00:25:51.350
And what's the slope error?

00:25:51.350 --> 00:25:56.390
What do you think
is the slope error?

00:25:56.390 --> 00:25:58.980
Because for us that
slope is important.

00:25:58.980 --> 00:26:02.170
That's the error in the
stretching and the strain.

00:26:02.170 --> 00:26:05.130
So the error in the
function is delta x squared.

00:26:05.130 --> 00:26:10.600
The error in the slope will be
one order less, just delta x.

00:26:10.600 --> 00:26:13.200
Okay, I'll come
back to all this.

00:26:13.200 --> 00:26:16.920
Now, make a guess.

00:26:16.920 --> 00:26:23.810
Suppose I include
these bubble functions.

00:26:23.810 --> 00:26:30.780
With delta x as my length
scale horizontally,

00:26:30.780 --> 00:26:33.220
what will be the
scale of the error?

00:26:33.220 --> 00:26:38.930
What do you guess is the
expected error in displacement

00:26:38.930 --> 00:26:44.610
for a general problem, for
a general c(x) and f(x).

00:26:47.560 --> 00:26:51.160
Which I won't get exactly right,
but how close will I come?

00:26:51.160 --> 00:26:56.040
I'll come within
delta x to what power?

00:26:56.040 --> 00:26:58.410
Make a guess, please.

00:26:58.410 --> 00:27:01.020
Four is an optimist.

00:27:01.020 --> 00:27:02.861
I won't get up to four.

00:27:02.861 --> 00:27:03.360
Cubed.

00:27:03.360 --> 00:27:04.490
I'd only get cubed.

00:27:04.490 --> 00:27:10.330
I'll get one by increasing
the degree of the polynomial

00:27:10.330 --> 00:27:15.160
by one, I'll get
one degree better.

00:27:15.160 --> 00:27:20.380
So you could look
at it this way.

00:27:20.380 --> 00:27:23.820
Suppose I have any function.

00:27:23.820 --> 00:27:26.500
This is a another way to
think about the accuracy.

00:27:26.500 --> 00:27:28.210
Suppose I have
any function F(x).

00:27:32.340 --> 00:27:37.920
The whole point of calculus
is that I could start,

00:27:37.920 --> 00:27:46.600
if I start where it is at zero,
then I add in F'(0), the slope,

00:27:46.600 --> 00:27:47.990
times x.

00:27:47.990 --> 00:27:54.480
Then I add in 1/2 F''(0)
times x squared, and so on.

00:27:54.480 --> 00:27:56.120
Right?

00:27:56.120 --> 00:28:00.000
It's called the Taylor series.

00:28:00.000 --> 00:28:03.990
And we're not paying any
attention to convergence,

00:28:03.990 --> 00:28:06.080
or high order.

00:28:06.080 --> 00:28:10.380
It's the early terms
that I'm interested in.

00:28:10.380 --> 00:28:14.440
And the point is that
if my functions include

00:28:14.440 --> 00:28:19.380
linear functions,
which the hats did,

00:28:19.380 --> 00:28:22.010
they will be able to
get these terms right,

00:28:22.010 --> 00:28:25.950
and this will be the
error that I missed.

00:28:25.950 --> 00:28:28.840
I'm just looking to see what's
the first term in the Taylor

00:28:28.840 --> 00:28:31.240
series that I will not get.

00:28:31.240 --> 00:28:32.890
And if I only have
hat functions,

00:28:32.890 --> 00:28:34.520
I can't get an x squared.

00:28:34.520 --> 00:28:36.050
I can't get a parabola.

00:28:36.050 --> 00:28:41.390
But when I go here and
include the x squareds,

00:28:41.390 --> 00:28:43.060
I can get that term right.

00:28:43.060 --> 00:28:48.040
So then it'll be the 1/6
f triple prime x cubed

00:28:48.040 --> 00:28:49.220
that I miss.

00:28:49.220 --> 00:28:55.240
So the error will be
the next missing term.

00:28:55.240 --> 00:28:59.550
Okay, so that's thoughts
about the error.

00:28:59.550 --> 00:29:04.010
And of course that's why those
elements are better than these.

00:29:04.010 --> 00:29:07.050
They take more work,
but they are worth it.

00:29:07.050 --> 00:29:10.870
But now I want to tell you
about the next elements.

00:29:10.870 --> 00:29:13.680
Cubics.

00:29:13.680 --> 00:29:18.600
Where you're going to expect
to get delta x to the fourth.

00:29:18.600 --> 00:29:22.250
So now we're getting
serious accuracy.

00:29:22.250 --> 00:29:24.740
Now we're getting good accuracy.

00:29:24.740 --> 00:29:28.230
Of course our problem is not
the most difficult problem.

00:29:28.230 --> 00:29:30.740
It's in 1-D. But this is good.

00:29:30.740 --> 00:29:34.540
Okay.

00:29:34.540 --> 00:29:38.620
This was now the fun in the
golden age of finite elements.

00:29:38.620 --> 00:29:41.900
To construct cubics.

00:29:41.900 --> 00:29:45.860
What shall I use as basis
functions for cubics?

00:29:45.860 --> 00:29:49.240
So I want to have a
cubic in each piece.

00:29:49.240 --> 00:29:54.870
First of all suppose I just
want no more than that.

00:29:54.870 --> 00:30:02.150
Suppose I'm happy with
just continuous functions

00:30:02.150 --> 00:30:06.270
and I let the slope jump.

00:30:06.270 --> 00:30:09.830
What new trial function
shall I put in?

00:30:09.830 --> 00:30:12.160
So I'm going to put in
new trial functions.

00:30:12.160 --> 00:30:14.570
What will they look like?

00:30:14.570 --> 00:30:15.910
Little cubics?

00:30:15.910 --> 00:30:18.490
Little third degree pieces.

00:30:18.490 --> 00:30:23.430
Instead of parabolas, they'll be
little pieces of third degree.

00:30:23.430 --> 00:30:27.850
And I could put in
four more bubbles.

00:30:27.850 --> 00:30:29.020
Four cubic bubbles.

00:30:29.020 --> 00:30:37.450
So I would be up to
twelve degrees, twelve

00:30:37.450 --> 00:30:40.180
by twelve matrices,
twelve functions.

00:30:40.180 --> 00:30:43.850
And for that size delta x,
that would give me delta

00:30:43.850 --> 00:30:44.830
x to the fourth.

00:30:44.830 --> 00:30:47.780
So that would be okay.

00:30:47.780 --> 00:30:49.360
There's a better idea.

00:30:49.360 --> 00:30:52.530
You can see that I left space.

00:30:52.530 --> 00:30:59.470
I'm going to make the
slope also continuous.

00:30:59.470 --> 00:31:03.850
I'm not going to
allow jumps in slope.

00:31:03.850 --> 00:31:06.400
Think, how will I do that?

00:31:06.400 --> 00:31:10.280
So I'm going to call those
C-- what will I call that when

00:31:10.280 --> 00:31:11.600
the slope is continuous?

00:31:11.600 --> 00:31:14.490
The first derivative,
I'll call that C one,

00:31:14.490 --> 00:31:17.280
continuous first derivative.

00:31:17.280 --> 00:31:19.070
Okay.

00:31:19.070 --> 00:31:22.040
Now I'm actually in
section 3.2, where

00:31:22.040 --> 00:31:26.050
these better elements,
these really nifty elements

00:31:26.050 --> 00:31:28.870
are constructed.

00:31:28.870 --> 00:31:31.740
C^1, continuous slope, cubics.

00:31:31.740 --> 00:31:32.330
Okay.

00:31:32.330 --> 00:31:33.540
Ready for those?

00:31:33.540 --> 00:31:37.940
What shall be my trial function
for continuous slope cubics?

00:31:37.940 --> 00:31:39.750
So I have to start again.

00:31:39.750 --> 00:31:44.090
I have to start again because
the hat functions are out now.

00:31:44.090 --> 00:31:47.460
Those hat functions
have a jump in slope.

00:31:47.460 --> 00:31:53.520
The bubble functions
have a jump in slope.

00:31:53.520 --> 00:31:59.640
I'm rethinking here to
create a better element.

00:31:59.640 --> 00:32:00.140
Okay.

00:32:00.140 --> 00:32:05.610
So let's just think, if
we've got a chance at it,

00:32:05.610 --> 00:32:07.200
how could these elements work?

00:32:07.200 --> 00:32:09.590
Okay, so here is the idea, then.

00:32:09.590 --> 00:32:15.490
Here is my interval.

00:32:15.490 --> 00:32:18.370
Zero to one, and here's
a typical interval.

00:32:18.370 --> 00:32:22.190
And now at a typical
node, like node one,

00:32:22.190 --> 00:32:27.040
I plan to have as unknowns
the height of the function, as

00:32:27.040 --> 00:32:29.730
before, and also the slope.

00:32:29.730 --> 00:32:33.270
So I want the function,
my trial function

00:32:33.270 --> 00:32:38.850
is going to have some
height and some slope.

00:32:38.850 --> 00:32:41.540
And at node two,
it's going to have

00:32:41.540 --> 00:32:44.380
some height and some slope.

00:32:44.380 --> 00:32:47.310
And here's the question.

00:32:47.310 --> 00:32:49.080
Here's the good point.

00:32:49.080 --> 00:32:55.390
That those four numbers, the
two heights and the two slopes,

00:32:55.390 --> 00:33:00.200
that gives me four
things, four quantities.

00:33:00.200 --> 00:33:03.720
How many quantities do I
need to determine a cubic?

00:33:03.720 --> 00:33:07.390
So by a cubic, of course,
I mean by a cubic something

00:33:07.390 --> 00:33:14.400
like a_0 plus a_1 x plus a_2
x squared and a_3 x cubed.

00:33:14.400 --> 00:33:18.050
It's called a cubic
because it's x cubed.

00:33:18.050 --> 00:33:20.281
So how many numbers here?

00:33:20.281 --> 00:33:20.780
Four.

00:33:20.780 --> 00:33:22.200
Perfect match.

00:33:22.200 --> 00:33:25.160
There's exactly
one cubic that has

00:33:25.160 --> 00:33:29.440
a specified height and a
specified slope at these two

00:33:29.440 --> 00:33:31.490
ends.

00:33:31.490 --> 00:33:33.910
There's one cubic
that'll do that.

00:33:33.910 --> 00:33:37.220
And then whatever
the height here is

00:33:37.220 --> 00:33:39.700
and whatever the slope
there is, there'll

00:33:39.700 --> 00:33:42.180
be one cubic with that
height and that slope

00:33:42.180 --> 00:33:44.650
that comes into this one.

00:33:44.650 --> 00:33:48.360
And you see that they will
have continuous slope.

00:33:48.360 --> 00:33:51.820
Because of course the slope
is continuous in between;

00:33:51.820 --> 00:33:53.470
it's a polynomial.

00:33:53.470 --> 00:33:56.990
The question is
always at the nodes.

00:33:56.990 --> 00:33:59.660
But I use the same number
coming from the left

00:33:59.660 --> 00:34:00.930
and from the right.

00:34:00.930 --> 00:34:05.250
The slope has become
an extra unknown.

00:34:05.250 --> 00:34:08.420
The slope has become
an extra unknown.

00:34:08.420 --> 00:34:11.110
So I have height,
slope at every point.

00:34:11.110 --> 00:34:17.400
So that's one way to describe
these trial functions now.

00:34:17.400 --> 00:34:27.840
The trial functions have height
and also slope at each node.

00:34:27.840 --> 00:34:29.890
So what does that mean?

00:34:29.890 --> 00:34:32.820
That means that I'm going
to have two unknowns.

00:34:32.820 --> 00:34:35.630
Two functions, two
trial functions,

00:34:35.630 --> 00:34:40.210
each with its own
coefficient at each node.

00:34:40.210 --> 00:34:45.170
So if I take a typical node
there, I want two functions.

00:34:45.170 --> 00:34:49.230
Okay, this is interesting.

00:34:49.230 --> 00:34:52.650
But you see what I'm creating.

00:34:52.650 --> 00:34:55.780
I think I'm going to
get two functions there,

00:34:55.780 --> 00:34:58.610
two functions there,
two functions there,

00:34:58.610 --> 00:35:00.150
two functions there, right?

00:35:00.150 --> 00:35:02.110
Because nobody's
constraining that.

00:35:02.110 --> 00:35:03.320
So I'm up to eight.

00:35:03.320 --> 00:35:05.160
And how many
functions do you think

00:35:05.160 --> 00:35:09.400
I'm going to have
associated with that node?

00:35:09.400 --> 00:35:10.140
Only one.

00:35:10.140 --> 00:35:12.100
Why?

00:35:12.100 --> 00:35:14.950
Because the height is fixed.

00:35:14.950 --> 00:35:20.270
So I think I've got nine
trial functions here.

00:35:20.270 --> 00:35:23.330
And if we can see
what those are,

00:35:23.330 --> 00:35:25.030
then the system will take over.

00:35:25.030 --> 00:35:29.700
They're my phi_1
to phi_9, whatever,

00:35:29.700 --> 00:35:32.740
they plug in here, they
plug in the right-hand side,

00:35:32.740 --> 00:35:37.680
I'll have a nine by
nine stiffness matrix.

00:35:37.680 --> 00:35:40.920
It'll be local again.

00:35:40.920 --> 00:35:45.100
Well, let's see if we can
figure out these functions.

00:35:45.100 --> 00:35:47.030
Okay, so you have the idea?

00:35:47.030 --> 00:35:50.190
I'm expecting two
trial functions.

00:35:50.190 --> 00:35:53.240
One is sort of a round hat.

00:35:53.240 --> 00:35:54.490
All right, let me draw that.

00:35:54.490 --> 00:36:02.720
The round hat function
will be the function--

00:36:02.720 --> 00:36:10.740
These will be the round hats,
and they'll be associated with,

00:36:10.740 --> 00:36:14.190
they give me heights.

00:36:14.190 --> 00:36:19.040
And then I'll also
have an additional one,

00:36:19.040 --> 00:36:21.000
except at the last node.

00:36:21.000 --> 00:36:23.620
And these will be-- I don't
know what to call them yet.

00:36:23.620 --> 00:36:25.340
You'll have to give me a name.

00:36:25.340 --> 00:36:27.950
These will give me the slopes.

00:36:27.950 --> 00:36:28.630
Okay.

00:36:28.630 --> 00:36:30.520
So what does a
round hat look like?

00:36:30.520 --> 00:36:35.110
Now these have to
be, follow my rules,

00:36:35.110 --> 00:36:37.290
they have to be
continuous, their slope

00:36:37.290 --> 00:36:38.850
has to be continuous.

00:36:38.850 --> 00:36:42.430
And I want to take the one
that has height one and zero

00:36:42.430 --> 00:36:43.750
slope there.

00:36:43.750 --> 00:36:47.190
And it should have height
zero and zero slope, here.

00:36:47.190 --> 00:36:52.460
Height zero, zero slope.

00:36:52.460 --> 00:36:54.330
You see what it's going to be?

00:36:54.330 --> 00:36:57.760
This phi, whatever
number it is, it'll

00:36:57.760 --> 00:37:03.070
be the phi whose
coefficient tells me

00:37:03.070 --> 00:37:05.130
the height at node one.

00:37:05.130 --> 00:37:08.670
So here's node one.

00:37:08.670 --> 00:37:10.700
What will it look like?

00:37:10.700 --> 00:37:12.270
What will this function do?

00:37:12.270 --> 00:37:16.270
Well, there is
exactly one cubic that

00:37:16.270 --> 00:37:19.260
starts from zero with
slope zero and ends there,

00:37:19.260 --> 00:37:21.470
ends at one with slope zero.

00:37:21.470 --> 00:37:21.970
Right?

00:37:21.970 --> 00:37:26.590
That's what we said; four
numbers determine that cubic

00:37:26.590 --> 00:37:28.300
in that interval.

00:37:28.300 --> 00:37:34.160
Then there's another cubic that,
with those two numbers again,

00:37:34.160 --> 00:37:37.532
that keeps the continuous
slope, and these two numbers

00:37:37.532 --> 00:37:38.240
in this interval.

00:37:38.240 --> 00:37:41.880
And of course it'll
just be symmetric.

00:37:41.880 --> 00:37:44.020
You see the round hat?

00:37:44.020 --> 00:37:47.710
So that's the basis
function, the trial

00:37:47.710 --> 00:37:52.980
function that has continuous
slopes and heights, of course,

00:37:52.980 --> 00:37:55.370
and it has height
one at that point.

00:37:55.370 --> 00:38:01.100
And now let me draw the one that
has height zero, slope zero;

00:38:01.100 --> 00:38:04.090
height zero, slope zero.

00:38:04.090 --> 00:38:10.610
And what do I want
it to do there?

00:38:10.610 --> 00:38:16.370
What should this
function be like?

00:38:16.370 --> 00:38:19.510
It should be the one that
tells-- Its coefficient will

00:38:19.510 --> 00:38:21.230
tell me the slope.

00:38:21.230 --> 00:38:28.390
So I want it to have a slope
of one and a height of zero.

00:38:28.390 --> 00:38:31.180
Do you see these
functions, shall

00:38:31.180 --> 00:38:35.560
I call these, the height
functions, phi h 1?

00:38:35.560 --> 00:38:38.660
That's the phi, that's the
trial function that tells me

00:38:38.660 --> 00:38:41.780
the height at node one.

00:38:41.780 --> 00:38:43.740
When I take
combinations, it gets

00:38:43.740 --> 00:38:51.270
multiplied by U h 1, which is
exactly the height at node one.

00:38:51.270 --> 00:38:53.090
Now what about this guy?

00:38:53.090 --> 00:38:58.640
This guy is going to start
with zero slope at zero.

00:38:58.640 --> 00:39:00.810
It's going to be a cubic,
and there's exactly

00:39:00.810 --> 00:39:02.180
one cubic that'll do it.

00:39:02.180 --> 00:39:04.800
It'll look a little like this.

00:39:04.800 --> 00:39:06.860
Then there'll be
exactly one cubic

00:39:06.860 --> 00:39:10.360
that does that and
gets back to zero.

00:39:10.360 --> 00:39:12.260
You see that that's possible?

00:39:12.260 --> 00:39:15.480
In each interval,
I've got four numbers:

00:39:15.480 --> 00:39:17.210
two heights, two slopes.

00:39:17.210 --> 00:39:23.540
So this would be a picture
of the phi slope at node one

00:39:23.540 --> 00:39:25.290
function.

00:39:25.290 --> 00:39:30.280
So that's a standard function,
it's a cubic, piecewise cubic.

00:39:30.280 --> 00:39:34.200
Local again, because in all
these intervals it's zero.

00:39:34.200 --> 00:39:38.110
And it will be, when I
go to take combinations

00:39:38.110 --> 00:39:40.390
of all these guys,
it'll be multiplied

00:39:40.390 --> 00:39:43.910
by its coefficient, U slope one.

00:39:43.910 --> 00:39:48.420
And then I'll have
nine altogether.

00:39:48.420 --> 00:39:50.950
But those two are
the typical ones.

00:39:50.950 --> 00:39:57.650
Do you do see how that's going?

00:39:57.650 --> 00:40:03.460
It's more subtle
than hat functions.

00:40:03.460 --> 00:40:09.950
Suppose whoever's writing
the finite element code

00:40:09.950 --> 00:40:16.510
gets a formula for those
phis and plugs them

00:40:16.510 --> 00:40:23.600
into the integrals, comes
out with a stiffness matrix.

00:40:23.600 --> 00:40:27.750
Actually, we could even look
at that stiffness matrix.

00:40:27.750 --> 00:40:31.810
This is a good way to
understand the picture.

00:40:31.810 --> 00:40:33.900
Now it'll be nine by nine.

00:40:33.900 --> 00:40:36.130
Right?

00:40:36.130 --> 00:40:41.080
So here we'll have a typical,
this'll be our phi height 1

00:40:41.080 --> 00:40:45.630
row, and this'll be
our phi slope 1 row,

00:40:45.630 --> 00:40:49.970
and this'll be our phi
height 2 row, and so on.

00:40:49.970 --> 00:40:53.590
Of course, I didn't
leave room for all--

00:40:53.590 --> 00:40:59.710
What will a typical row of this
stiffness matrix have in it?

00:40:59.710 --> 00:41:03.750
I'm just asking about the
overlaps. phi_1 height

00:41:03.750 --> 00:41:06.970
certainly overlaps itself.

00:41:06.970 --> 00:41:11.730
Does phi_1 height
overlap phi_1 slope?

00:41:11.730 --> 00:41:13.690
Yes or no?

00:41:13.690 --> 00:41:14.450
Sure.

00:41:14.450 --> 00:41:16.350
Sure.

00:41:16.350 --> 00:41:21.580
Does phi_1 height
overlap phi_2 height?

00:41:21.580 --> 00:41:23.030
Yes.

00:41:23.030 --> 00:41:23.650
Yes.

00:41:23.650 --> 00:41:28.910
Because the phi_2 height
will go up like that.

00:41:28.910 --> 00:41:29.540
You see?

00:41:29.540 --> 00:41:30.840
And the phi_2 slope.

00:41:30.840 --> 00:41:38.920
So actually we'll have,
I think we'll have

00:41:38.920 --> 00:41:43.600
six non-zeros on a typical row.

00:41:43.600 --> 00:41:45.560
Is that right?

00:41:45.560 --> 00:41:46.880
Six non-zeros?

00:41:46.880 --> 00:41:53.650
Because a typical h-- this
is maybe not so typical,

00:41:53.650 --> 00:41:56.040
because to the left of
it there's only one,

00:41:56.040 --> 00:41:59.140
there's no-- No, there are two.

00:41:59.140 --> 00:42:00.050
Right?

00:42:00.050 --> 00:42:03.520
There's a phi_0, phi
h 0 and a phi s 0.

00:42:03.520 --> 00:42:07.210
Sure, there are two
here, the two guys

00:42:07.210 --> 00:42:11.350
here, there's one
height guy, and there's

00:42:11.350 --> 00:42:17.350
one-- what's cooking in that?

00:42:17.350 --> 00:42:21.970
Oh, it's got a slope of one
and it gets back to zero.

00:42:21.970 --> 00:42:25.040
What I'm drawing now
in little dashed lines

00:42:25.040 --> 00:42:29.370
was the phi slope 0.

00:42:29.370 --> 00:42:32.830
The one that gives
me a slope at node

00:42:32.830 --> 00:42:35.000
zero, and this is the one
that gives me a height.

00:42:35.000 --> 00:42:36.050
Yes.

00:42:36.050 --> 00:42:40.910
Do you see it?

00:42:40.910 --> 00:42:46.040
So above this was a phi
slope 0, and stuck in there

00:42:46.040 --> 00:42:54.680
was a phi height 0.

00:42:54.680 --> 00:42:57.190
Six-diagonal matrix.

00:42:57.190 --> 00:43:02.310
I think it helps to draw
that little thing with x's

00:43:02.310 --> 00:43:05.630
and zeros, because then you sort
of see how things are fitting

00:43:05.630 --> 00:43:06.700
together.

00:43:06.700 --> 00:43:09.020
Okay.

00:43:09.020 --> 00:43:18.360
So these functions now, I've
gone into section 3.2 for that.

00:43:18.360 --> 00:43:22.420
I want to go to a
slightly different topic,

00:43:22.420 --> 00:43:28.130
and then I'll come back in
section 3.2 to these cubics.

00:43:28.130 --> 00:43:32.350
So these are C^1 cubics,
continuous slope cubics.

00:43:32.350 --> 00:43:35.460
Very interesting construction.

00:43:35.460 --> 00:43:39.110
Are you seeing how it could
go in more dimensions?

00:43:39.110 --> 00:43:45.040
I mean, that's what we'll
see for Laplace's equation,

00:43:45.040 --> 00:43:49.310
how can you construct
quadratics, cubics in a plane.

00:43:49.310 --> 00:43:52.210
It gets interesting.

00:43:52.210 --> 00:43:56.340
But you'll get the
knack of these guys.

00:43:56.340 --> 00:44:02.920
These are pretty
direct, and very useful.

00:44:02.920 --> 00:44:04.280
So what's the effect?

00:44:04.280 --> 00:44:09.990
The effect is that
we get a matrix.

00:44:09.990 --> 00:44:13.710
It looks quite like
a difference matrix.

00:44:13.710 --> 00:44:16.740
Well, actually, the height
rows and-- the numbers

00:44:16.740 --> 00:44:20.180
in the height rows and the
slope rows look different.

00:44:20.180 --> 00:44:23.880
We're getting
something new here.

00:44:23.880 --> 00:44:29.800
We're getting a matrix, a KU=F,
that's going to give us fourth

00:44:29.800 --> 00:44:31.860
order accuracy.

00:44:31.860 --> 00:44:37.500
So the accuracy has moved up.

00:44:37.500 --> 00:44:39.720
So we've got up to
fourth order accuracy,

00:44:39.720 --> 00:44:44.830
which we could get
by finite differences

00:44:44.830 --> 00:44:47.190
by a lot of patience.

00:44:47.190 --> 00:44:50.360
We get them from finite
elements in a straight way.

00:44:50.360 --> 00:44:53.010
Okay, any question
or discussion?

00:44:53.010 --> 00:45:00.040
I'm talking real fast to get
this new idea of constructing

00:45:00.040 --> 00:45:03.000
finite elements here.

00:45:03.000 --> 00:45:13.190
I do want to say
something about that line.

00:45:13.190 --> 00:45:17.070
Because that's a
part of this business

00:45:17.070 --> 00:45:21.040
of estimating the accuracy.

00:45:21.040 --> 00:45:28.670
It's a key idea in the
background of the Galerkin

00:45:28.670 --> 00:45:29.380
method.

00:45:29.380 --> 00:45:34.490
And the minimum form would
be associated with names

00:45:34.490 --> 00:45:37.060
like Rayleigh and Ritz.

00:45:37.060 --> 00:45:37.930
All right.

00:45:37.930 --> 00:45:42.610
I'll just go directly
to that, if I may.

00:45:42.610 --> 00:45:49.940
So what I want to do is tell
you, for our model problem,

00:45:49.940 --> 00:45:58.970
I want to tell you the strong
form-- Let me do it this way.

00:45:58.970 --> 00:46:03.720
I'll put the strong
form, the weak form,

00:46:03.720 --> 00:46:09.610
and then I want to add
in the minimum form.

00:46:09.610 --> 00:46:11.340
Okay.

00:46:11.340 --> 00:46:15.720
So the strong form of
our equation was minus

00:46:15.720 --> 00:46:20.190
the derivative of
c*du/dx equal f.

00:46:24.670 --> 00:46:28.180
Okay.

00:46:28.180 --> 00:46:33.460
What was the weak form?

00:46:33.460 --> 00:46:36.070
This is an f(x).

00:46:36.070 --> 00:46:39.880
The weak form, how do
you get to the weak form?

00:46:39.880 --> 00:46:44.110
You multiply both sides by a
test function, you integrate,

00:46:44.110 --> 00:46:47.980
you integrate by parts, and you
get this beautifully symmetric

00:46:47.980 --> 00:46:57.250
form that we have up
there, du/dx*dv/dx*dx,

00:46:57.250 --> 00:47:00.490
equals the integral
of f(x)*v(x)*dx.

00:47:03.740 --> 00:47:08.060
I write that again, just so
you see the nice symmetry

00:47:08.060 --> 00:47:10.280
of that weak form.

00:47:10.280 --> 00:47:21.470
And it's for all test
functions v. Okay.

00:47:21.470 --> 00:47:24.500
I'm shooting for a
third description.

00:47:24.500 --> 00:47:27.760
A third description
of the same problem.

00:47:27.760 --> 00:47:31.600
And it's really neat to
see that you have that.

00:47:31.600 --> 00:47:35.870
Let me just see it first
in the discrete case.

00:47:35.870 --> 00:47:40.680
The discrete case, the strong
form would be A transpose C

00:47:40.680 --> 00:47:42.940
Au=f.

00:47:42.940 --> 00:47:46.430
That's the strong form.

00:47:46.430 --> 00:47:48.030
Right?

00:47:48.030 --> 00:47:50.430
I always like to see
the discrete one first,

00:47:50.430 --> 00:47:52.100
and then the continuous.

00:47:52.100 --> 00:47:55.840
Okay, what would be the weak
form in the discrete case?

00:47:55.840 --> 00:47:59.560
I would multiply by
a vector v, and I

00:47:59.560 --> 00:48:04.180
would take inner products, A
transpose C Au inner product

00:48:04.180 --> 00:48:11.745
with v, equals f inner product
with v. You can use dot,

00:48:11.745 --> 00:48:18.350
or-- So that would
be the weak form.

00:48:18.350 --> 00:48:21.070
I've just taken the dot
product of both sides

00:48:21.070 --> 00:48:27.931
with v. Now you'll see the weak
form better if, what should I

00:48:27.931 --> 00:48:28.430
do?

00:48:28.430 --> 00:48:32.640
What would make that look nice?

00:48:32.640 --> 00:48:38.450
So that's the dot product
of A transpose C Au with v.

00:48:38.450 --> 00:48:41.890
And what do I do to
make that look nice?

00:48:41.890 --> 00:48:44.420
Do you get the idea yet?

00:48:44.420 --> 00:48:47.640
It doesn't look pretty to me.

00:48:47.640 --> 00:48:49.820
It's all lopsided.

00:48:49.820 --> 00:48:50.450
Right?

00:48:50.450 --> 00:48:53.200
So what can I do
with A transpose?

00:48:53.200 --> 00:48:55.700
What's the rule
about A transpose?

00:48:55.700 --> 00:48:57.890
That if I have A
transpose times something,

00:48:57.890 --> 00:49:02.610
dotted with something,
what can I do?

00:49:02.610 --> 00:49:08.640
I can move the A transpose
over to the other guy.

00:49:08.640 --> 00:49:11.230
And what will it
be when I do that?

00:49:11.230 --> 00:49:19.280
So I take it away from here,
and what do I put there?

00:49:19.280 --> 00:49:24.970
A. That's the whole
point of transposes.

00:49:24.970 --> 00:49:28.480
Transposes, you put them on the
other side of the dot product,

00:49:28.480 --> 00:49:32.020
you take the transpose, so
it would be literally, maybe

00:49:32.020 --> 00:49:34.750
A transpose
transpose, which is A.

00:49:34.750 --> 00:49:37.620
What I just did there
is integration by parts.

00:49:37.620 --> 00:49:39.800
Well, summation by
parts, because I'm

00:49:39.800 --> 00:49:41.170
in the discrete case.

00:49:41.170 --> 00:49:44.240
The whole idea of
integration by parts

00:49:44.240 --> 00:49:49.110
amounted to taking A transpose
off of u, off of this,

00:49:49.110 --> 00:49:51.820
and putting A over there.

00:49:51.820 --> 00:49:53.810
Isn't that neat?

00:49:53.810 --> 00:49:58.420
And you see that this CAu,
Av is just what I have here.

00:49:58.420 --> 00:50:03.450
C, A is derivative,
so this is CAu, Av.

00:50:03.450 --> 00:50:07.540
Inner product.

00:50:07.540 --> 00:50:10.730
That's cool.

00:50:10.730 --> 00:50:13.710
That's just like
how it should be.

00:50:13.710 --> 00:50:17.270
I just followed that rule,
that A transpose times

00:50:17.270 --> 00:50:22.540
something, shall I call it
w, inner product with u,

00:50:22.540 --> 00:50:26.630
is the same as (w, Au).

00:50:26.630 --> 00:50:28.650
That if I bring
A transpose over,

00:50:28.650 --> 00:50:31.300
it becomes an A. If
I bring an A over,

00:50:31.300 --> 00:50:33.140
it would become an A transpose.

00:50:33.140 --> 00:50:34.880
All right, what about
the minimum form?

00:50:34.880 --> 00:50:37.320
Have I got one minute
to do the minimum form?

00:50:37.320 --> 00:50:40.130
Yes.

00:50:40.130 --> 00:50:44.220
So what's the minimization
that's hiding behind this?

00:50:44.220 --> 00:50:49.570
The minimization in the
discrete case, do you remember?

00:50:49.570 --> 00:50:53.050
We're looking at Ku=f.

00:50:53.050 --> 00:50:57.860
And some quadratic quantity from
least squares has its minimum

00:50:57.860 --> 00:51:00.230
when Ku=f.

00:51:00.230 --> 00:51:07.170
And it's 1/2 u transpose
Ku minus u transpose f.

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Where K is A transpose C A.
This is the minimum statement

00:51:11.620 --> 00:51:13.620
of the problem.

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That if I look for the u that
minimizes that quadratic,

00:51:17.810 --> 00:51:21.190
it leads me to
the equation Ku=f.

00:51:21.190 --> 00:51:22.860
So that's the minimum statement.

00:51:22.860 --> 00:51:27.330
And if we want it to really
look perfectly like the others,

00:51:27.330 --> 00:51:39.510
I would put in A
transpose C A. Okay.

00:51:39.510 --> 00:51:42.720
Can I write down next time,
because our time is really up.

00:51:42.720 --> 00:51:46.120
It's not fair to-- Al I'm
going to do is write down

00:51:46.120 --> 00:51:47.680
the same thing here.

00:51:47.680 --> 00:51:51.280
I'm minimizing 1/2-- Oh,
I'm going to do it anyway.

00:51:51.280 --> 00:51:58.790
c(x) du/dx squared, minus
the integral of f(x)u(x).

00:52:02.470 --> 00:52:05.130
So that's the minimum problem.

00:52:05.130 --> 00:52:12.400
Minimize over all
u, this quadratic.

00:52:12.400 --> 00:52:16.080
This is the right way
to see these problems.

00:52:16.080 --> 00:52:18.450
You see a differential
equation, which

00:52:18.450 --> 00:52:20.820
we use for finite
differences; you

00:52:20.820 --> 00:52:24.090
see a weak form, which we
use for finite elements;

00:52:24.090 --> 00:52:26.340
and now you see a minimum form.

00:52:26.340 --> 00:52:28.540
Okay, that gives you
something to think about.

00:52:28.540 --> 00:52:31.180
And there'll be a homework
on finite elements

00:52:31.180 --> 00:52:33.370
that'll give you a
chance to use them.

00:52:33.370 --> 00:52:35.300
Okay, thank you.