1
00:00:00 --> 00:00:01
2
00:00:01 --> 00:00:02
The following content is
provided under a Creative
3
00:00:02 --> 00:00:03
Commons license.
4
00:00:03 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to
5
00:00:06 --> 00:00:10
offer high-quality educational
resources for free.
6
00:00:10 --> 00:00:12
To make a donation or to view
additional materials from
7
00:00:12 --> 00:00:16
hundreds of MIT courses, visit
MIT OpenCourseWare
8
00:00:16 --> 00:00:21
at ocw.mit.edu.
9
00:00:21 --> 00:00:22
PROFESSOR STRANG: OK.
10
00:00:22 --> 00:00:23
All right.
11
00:00:23 --> 00:00:24
Good morning.
12
00:00:24 --> 00:00:32
So we're doing finite elements.
13
00:00:32 --> 00:00:36
The element that we considered
so far was the basic linear
14
00:00:36 --> 00:00:40
element, continuous.
15
00:00:40 --> 00:00:42
But of course, the
slopes have jumped.
16
00:00:42 --> 00:00:47
The slope was minus one
over delta x for that one.
17
00:00:47 --> 00:00:50
This one was a plus
and then a minus.
18
00:00:50 --> 00:00:53
So a jump in slope but no
jump in the function.
19
00:00:53 --> 00:01:00
So actually, my abbreviation
for that would be C^0, saying
20
00:01:00 --> 00:01:05
that it's continuous but no
derivative is continuous.
21
00:01:05 --> 00:01:08
And now we'll get to
some elements where the
22
00:01:08 --> 00:01:11
slope is continuous.
23
00:01:11 --> 00:01:16
It's sort of fun to create
these finite elements
24
00:01:16 --> 00:01:18
of higher degree.
25
00:01:18 --> 00:01:20
It's pretty
straightforward in 1-D.
26
00:01:20 --> 00:01:24
And that's where we are now.
27
00:01:24 --> 00:01:26
So we'll get second degree
elements and third
28
00:01:26 --> 00:01:28
degree elements.
29
00:01:28 --> 00:01:31
And that gives us, as we'll
see, higher accuracy.
30
00:01:31 --> 00:01:35
So I want to connect the degree
of the polynomials to the
31
00:01:35 --> 00:01:41
accuracy of the approximation.
32
00:01:41 --> 00:01:45
Part of that connection is to
recognize that these problems
33
00:01:45 --> 00:01:49
have a strong form, as we know,
the equation; a weak form,
34
00:01:49 --> 00:01:52
that's the one that
has test functions.
35
00:01:52 --> 00:01:58
And also a minimum
form that we'll see.
36
00:01:58 --> 00:02:03
So how would I get some
quadratics, so second
37
00:02:03 --> 00:02:09
degree elements, parabolas
into the picture?
38
00:02:09 --> 00:02:11
You remember Gelerkin's idea?
39
00:02:11 --> 00:02:14
Choose trial functions.
40
00:02:14 --> 00:02:17
And we're taking those to be
the same as the test function.
41
00:02:17 --> 00:02:20
So these are the trial
functions we've chosen.
42
00:02:20 --> 00:02:22
One, two, three, four of them.
43
00:02:22 --> 00:02:25
And they're linear.
44
00:02:25 --> 00:02:30
And that limits the accuracy
that you can get, because
45
00:02:30 --> 00:02:34
your approximations then
are combinations of those.
46
00:02:34 --> 00:02:37
So they're like broken
line functions, linear
47
00:02:37 --> 00:02:39
approximations.
48
00:02:39 --> 00:02:44
And the accuracy is not great.
49
00:02:44 --> 00:02:47
It's sort of the lowest
level possible.
50
00:02:47 --> 00:02:53
So how would you get parabolas?
51
00:02:53 --> 00:02:56
So this was first guy.
52
00:02:56 --> 00:03:08
The second guy is going to be
continuous and quadratic.
53
00:03:08 --> 00:03:10
So it's going to have
new trial functions.
54
00:03:10 --> 00:03:14
In addition to these, I'm
going to put in some more.
55
00:03:14 --> 00:03:16
Gelerkin's happy with that.
56
00:03:16 --> 00:03:18
I still proceed as usual.
57
00:03:18 --> 00:03:22
My approximation is some
combination of those.
58
00:03:22 --> 00:03:24
It's only going to
be continuous.
59
00:03:24 --> 00:03:27
So it'll just be
a C^0 guy again.
60
00:03:27 --> 00:03:31
That means jump in slope.
61
00:03:31 --> 00:03:33
The first derivative
isn't there.
62
00:03:33 --> 00:03:36
Eventually, I want to
get to a C^1 where the
63
00:03:36 --> 00:03:38
slopes are continuous.
64
00:03:38 --> 00:03:41
OK, but how would I
get some quadratics?
65
00:03:41 --> 00:03:47
All I want now is my functions,
my space, my combinations
66
00:03:47 --> 00:03:52
should be the piecewise
parabolas instead of
67
00:03:52 --> 00:03:55
piecewise linear.
68
00:03:55 --> 00:04:00
And the pieces are
broken at the nodes.
69
00:04:00 --> 00:04:04
OK, so here is a way to do it.
70
00:04:04 --> 00:04:08
Inside each interval, I'm
going to add, I'll just
71
00:04:08 --> 00:04:10
call them bubble functions.
72
00:04:10 --> 00:04:13
So these will be new
additional guys.
73
00:04:13 --> 00:04:16
So this will be my first phi.
74
00:04:16 --> 00:04:18
You remember that half hat?
75
00:04:18 --> 00:04:23
Because the problem I was doing
was a free fixed problem.
76
00:04:23 --> 00:04:29
That's why I had a half hat at
this end, because there was no
77
00:04:29 --> 00:04:33
boundary condition that my
functions had to satisfy.
78
00:04:33 --> 00:04:35
At this end, there was.
79
00:04:35 --> 00:04:36
It was fixed.
80
00:04:36 --> 00:04:40
So that's why the hat ended,
and there was no half hat,
81
00:04:40 --> 00:04:42
there's no extra
function there.
82
00:04:42 --> 00:04:44
So I have right now
one, two, three, four.
83
00:04:44 --> 00:04:48
I'm going to add four
more bubble functions.
84
00:04:48 --> 00:04:51
Each one will be
inside an interval.
85
00:04:51 --> 00:04:53
So it'll be a little parabola.
86
00:04:53 --> 00:04:58
This is function
number whatever.
87
00:04:58 --> 00:05:03
If I number this number one,
let's say, phi_1 now -- that's
88
00:05:03 --> 00:05:06
probably a change in numbering
-- phi_2 is going
89
00:05:06 --> 00:05:07
to be my bubble.
90
00:05:07 --> 00:05:10
And you see what my
bubble function is?
91
00:05:10 --> 00:05:14
It's a function that goes
there and straight.
92
00:05:14 --> 00:05:21
So it is continuous, no jumps,
and it is second degree.
93
00:05:21 --> 00:05:24
It's a parabola, and I'll
make its height one.
94
00:05:24 --> 00:05:27
And then there'll be
another function.
95
00:05:27 --> 00:05:30
If I had another color
I could draw it.
96
00:05:30 --> 00:05:33
Well, I'll just do it with
broken lines, maybe.
97
00:05:33 --> 00:05:39
So there'll be another bubble
function in here, a third
98
00:05:39 --> 00:05:42
bubble function in the third
interval, and a fourth
99
00:05:42 --> 00:05:44
in the fourth interval.
100
00:05:44 --> 00:05:50
You see that I've now got my
old phi_1, phi_3, phi_5, and
101
00:05:50 --> 00:05:54
phi_7 were the hat functions.
102
00:05:54 --> 00:05:59
But now I've got a phi_2,
phi_4, phi_6, and phi_8 that
103
00:05:59 --> 00:06:04
are these new trial functions.
104
00:06:04 --> 00:06:08
So part of the message
is, we can throw an
105
00:06:08 --> 00:06:11
additional functions.
106
00:06:11 --> 00:06:15
They don't have to be
polynomials, but those are
107
00:06:15 --> 00:06:17
the simplest choices.
108
00:06:17 --> 00:06:18
Why are they simple?
109
00:06:18 --> 00:06:23
Because, you remember, that in
the end when I made the choice,
110
00:06:23 --> 00:06:26
I have to do various
integrations.
111
00:06:26 --> 00:06:30
So you remember that I have to
integrate to find entries K_ij.
112
00:06:31 --> 00:06:34
Do you remember what that
interval looked like?
113
00:06:34 --> 00:06:36
You certainly remember F_i.
114
00:06:37 --> 00:06:43
That was the integral from zero
to one of whatever function
115
00:06:43 --> 00:06:50
phi_i we had, times the
F(x)dx times the load.
116
00:06:50 --> 00:06:54
And we computed these.
117
00:06:54 --> 00:06:58
You remember we computed these
for the piecewise linear guys.
118
00:06:58 --> 00:07:04
But I don't think I wrote down
the expression that were really
119
00:07:04 --> 00:07:06
doing, so let me just do that.
120
00:07:06 --> 00:07:20
It's c(x)du, no. d
phi_j/dx and a dV_i/dx.
121
00:07:20 --> 00:07:25
122
00:07:25 --> 00:07:29
Those were the integrals
that we had to do.
123
00:07:29 --> 00:07:35
And we were taking phis
to be the same as V's.
124
00:07:35 --> 00:07:38
Maybe I'll just do that here,
because I don't plan to make
125
00:07:38 --> 00:07:40
any other choices at
all. d phi_i/dx.
126
00:07:40 --> 00:07:44
127
00:07:44 --> 00:07:46
It's a symmetric matrix now.
128
00:07:46 --> 00:07:54
K_ji, because when I'm choosing
phis the same as the V's, this
129
00:07:54 --> 00:07:58
is what it looks like, and if I
switch j and i, I don't
130
00:07:58 --> 00:08:00
see any difference.
131
00:08:00 --> 00:08:03
So these are the things that
have to be integrated.
132
00:08:03 --> 00:08:07
And those are the ones we
did integrate when phi
133
00:08:07 --> 00:08:08
was piecewise linear.
134
00:08:08 --> 00:08:11
When phi was piecewise linear,
the slope was piecewise
135
00:08:11 --> 00:08:14
constant, and we had
really easy integrals.
136
00:08:14 --> 00:08:18
Very easy integrals.
137
00:08:18 --> 00:08:24
We had to pay attention to
where we were was the slope
138
00:08:24 --> 00:08:28
on a minus interval or on
a plus interval, but they
139
00:08:28 --> 00:08:29
were easy to compute.
140
00:08:29 --> 00:08:34
And they led us back to
the kind of matrix that
141
00:08:34 --> 00:08:36
we've seen before.
142
00:08:36 --> 00:08:38
The twos and minus ones.
143
00:08:38 --> 00:08:41
And our right hand
sides looked familiar.
144
00:08:41 --> 00:08:45
Now, we've got new functions.
145
00:08:45 --> 00:08:48
We still have the
same formulas.
146
00:08:48 --> 00:08:50
No change in formulas.
147
00:08:50 --> 00:08:55
The system is really quite
successful, because these
148
00:08:55 --> 00:08:57
are the things that
we have to compute.
149
00:08:57 --> 00:09:03
So now I'll have to integrate
these parabolas, these little
150
00:09:03 --> 00:09:08
parabolas, half of the phis
will be little parabolas, and
151
00:09:08 --> 00:09:10
their derivatives
will be linear.
152
00:09:10 --> 00:09:14
So you see, I'll have
more calculations to do.
153
00:09:14 --> 00:09:19
Which I don't plan to do,
but more integrations.
154
00:09:19 --> 00:09:25
For example, the diagonal
entry, say, 2, 2, which
155
00:09:25 --> 00:09:31
will come from that
bubble with itself.
156
00:09:31 --> 00:09:37
K_22, then, will be the
integral of c(x), times the
157
00:09:37 --> 00:09:39
derivative of that bubble,
which will be a straight
158
00:09:39 --> 00:09:44
line times itself, so
it would be squared.
159
00:09:44 --> 00:09:49
And c is positive, so this
K_22 is going to be some
160
00:09:49 --> 00:09:51
nice positive number.
161
00:09:51 --> 00:09:55
But we'll have to
figure out what it is.
162
00:09:55 --> 00:10:01
Maybe I'll just say one fact
that we'll come back to.
163
00:10:01 --> 00:10:08
That this K is symmetric
positive definite.
164
00:10:08 --> 00:10:11
You thought it would be.
165
00:10:11 --> 00:10:15
By using the letter K, we
kind of expected it to be.
166
00:10:15 --> 00:10:16
And it will be.
167
00:10:16 --> 00:10:21
It'll be symmetric because the
phis and the V's are the same.
168
00:10:21 --> 00:10:23
And it turns out it's
positive definite.
169
00:10:23 --> 00:10:25
So it's just great.
170
00:10:25 --> 00:10:27
Just great.
171
00:10:27 --> 00:10:35
We have a little more effort,
either to use a formula for
172
00:10:35 --> 00:10:41
integrating polynomials, or
using numerical integration.
173
00:10:41 --> 00:10:45
One way or another, and I won't
concentrate right now on that
174
00:10:45 --> 00:10:47
point, we get these numbers.
175
00:10:47 --> 00:10:49
Okay.
176
00:10:49 --> 00:10:54
Here's something to
concentrate on.
177
00:10:54 --> 00:10:58
What kind of a matrix
K do we have?
178
00:10:58 --> 00:11:02
Where will it be non-zero?
179
00:11:02 --> 00:11:05
So it'll be eight
by eight, right?
180
00:11:05 --> 00:11:09
I'll follow through
on that choice.
181
00:11:09 --> 00:11:12
Just to say, where will
I see non-zeros here?
182
00:11:12 --> 00:11:16
Because if you get that
point, you see the way
183
00:11:16 --> 00:11:20
things come together.
184
00:11:20 --> 00:11:22
I'll just put a little
x for non-zero.
185
00:11:22 --> 00:11:23
So K_11.
186
00:11:26 --> 00:11:29
So what's that first row of K?
187
00:11:29 --> 00:11:34
It's coming from the first
function, integrated
188
00:11:34 --> 00:11:36
against itself.
189
00:11:36 --> 00:11:41
K_11, if for 1, 1, we'll
get something there.
190
00:11:41 --> 00:11:46
Will we have something
in the 1, 2 position?
191
00:11:46 --> 00:11:50
That's my question, do we have
something in the 1, 2 position?
192
00:11:50 --> 00:11:52
What's 1, 2?
193
00:11:52 --> 00:11:57
That's this function against
the bubble function, yes?
194
00:11:57 --> 00:11:58
Right?
195
00:11:58 --> 00:12:00
They're non-zero at
the same place.
196
00:12:00 --> 00:12:03
We can expect something there.
197
00:12:03 --> 00:12:03
What about K_13?
198
00:12:05 --> 00:12:08
That's what we've done
before, that's this
199
00:12:08 --> 00:12:09
one against this one.
200
00:12:09 --> 00:12:12
Yes?
201
00:12:12 --> 00:12:14
We expect a non-zero there.
202
00:12:14 --> 00:12:16
But then what?
203
00:12:16 --> 00:12:19
After that, what will the
rest of that row be?
204
00:12:19 --> 00:12:20
Zero.
205
00:12:20 --> 00:12:25
Because that first half
hat doesn't touch
206
00:12:25 --> 00:12:26
any of the others.
207
00:12:26 --> 00:12:27
So let's go on.
208
00:12:27 --> 00:12:31
Of course it'll be symmetric.
209
00:12:31 --> 00:12:33
I know this much.
210
00:12:33 --> 00:12:39
So this is the half hat
row, and this is the
211
00:12:39 --> 00:12:43
first bubble row.
212
00:12:43 --> 00:12:45
Because the half hat
was phi_1 and now the
213
00:12:45 --> 00:12:47
first bubble is phi_2.
214
00:12:47 --> 00:12:53
What non-zeros do we get
in the stiffness matrix?
215
00:12:53 --> 00:13:00
Again, we could unconstruct
it entry by entry.
216
00:13:00 --> 00:13:03
Another way to construct
it will be element by
217
00:13:03 --> 00:13:05
element, stamp them in.
218
00:13:05 --> 00:13:09
You're beginning to
see the idea of that.
219
00:13:09 --> 00:13:12
So what do I get
for that bubble?
220
00:13:12 --> 00:13:17
I just look to see which
elements touch that bubble.
221
00:13:17 --> 00:13:21
And which ones do?
222
00:13:21 --> 00:13:26
One, two and three,
and not four.
223
00:13:26 --> 00:13:28
Right?
224
00:13:28 --> 00:13:31
In that row, we only get --
so from that bubble, I think
225
00:13:31 --> 00:13:33
we only get that much.
226
00:13:33 --> 00:13:35
Now, we're not quite
seeing the picture yet.
227
00:13:35 --> 00:13:38
Let me go to the next hat.
228
00:13:38 --> 00:13:45
The hat, phi_2, and then
I'll do the bubble.
229
00:13:45 --> 00:13:49
Oh, no, sorry, the hat's
numbered phi_3, and then the
230
00:13:49 --> 00:13:51
next bubble is numbered phi_4.
231
00:13:53 --> 00:13:55
Where do I get zeros?
232
00:13:55 --> 00:13:58
You can tell me, where
do I get zeros?
233
00:13:58 --> 00:14:05
From inner products, from
these guys, when i is three.
234
00:14:05 --> 00:14:10
So which phis does phi
number three overlap?
235
00:14:10 --> 00:14:12
That's all I'm asking.
236
00:14:12 --> 00:14:14
Does it overlap number one?
237
00:14:14 --> 00:14:15
Yes.
238
00:14:15 --> 00:14:18
Does it overlap phi number two?
239
00:14:18 --> 00:14:21
You want to highlight,
so we're now looking
240
00:14:21 --> 00:14:24
at phi_3, at this hat.
241
00:14:24 --> 00:14:29
God, where's it gone?
242
00:14:29 --> 00:14:32
That's the one we're doing now?
243
00:14:32 --> 00:14:34
So what does it overlap?
244
00:14:34 --> 00:14:39
It overlaps the half hat, does
it overlap the first bubble?
245
00:14:39 --> 00:14:39
Yes.
246
00:14:39 --> 00:14:41
Does it overlap itself?
247
00:14:41 --> 00:14:42
Yes.
248
00:14:42 --> 00:14:45
Does it overlap the
second bubble?
249
00:14:45 --> 00:14:46
Yes.
250
00:14:46 --> 00:14:49
Does it overlap the next hat?
251
00:14:49 --> 00:14:50
Yes.
252
00:14:50 --> 00:14:53
And then all zeros.
253
00:14:53 --> 00:14:55
Okay, and now do one more row.
254
00:14:55 --> 00:14:56
Bubble four.
255
00:14:56 --> 00:15:01
So now I'm looking at this
guy, this next bubble. phi_4.
256
00:15:01 --> 00:15:03
What does that overlap?
257
00:15:03 --> 00:15:08
Does it overlap the
first half hat?
258
00:15:08 --> 00:15:08
Nope.
259
00:15:08 --> 00:15:13
Of course, symmetry
told us that.
260
00:15:13 --> 00:15:16
Does the second bubble
overlap the first bubble?
261
00:15:16 --> 00:15:17
No.
262
00:15:17 --> 00:15:19
Big point: zero there.
263
00:15:19 --> 00:15:25
Does the second bubble
overlap the hat?
264
00:15:25 --> 00:15:26
Yes.
265
00:15:26 --> 00:15:28
Does the second bubble
overlap itself?
266
00:15:28 --> 00:15:31
Certainly, on the diagonal
we have something.
267
00:15:31 --> 00:15:34
Does the second bubble
overlap the next hat, phi_5?
268
00:15:34 --> 00:15:35
Yes.
269
00:15:35 --> 00:15:39
And that's it.
270
00:15:39 --> 00:15:39
I think.
271
00:15:39 --> 00:15:44
The second level does not
overlap the following bubble.
272
00:15:44 --> 00:15:51
I don't know if you see what
pattern we're getting here.
273
00:15:51 --> 00:15:52
Those were special
rows, because that
274
00:15:52 --> 00:15:54
was only a half hat.
275
00:15:54 --> 00:15:56
These are typical rows.
276
00:15:56 --> 00:16:02
A typical hat function, that
row is showing us five
277
00:16:02 --> 00:16:06
non-zeros, because it overlaps
itself, the neighboring hats,
278
00:16:06 --> 00:16:08
and the neighboring bubbles.
279
00:16:08 --> 00:16:13
But the bubble row only has
three, because a bubble
280
00:16:13 --> 00:16:17
overlaps itself, the
neighboring hat on each
281
00:16:17 --> 00:16:20
side, but not the
neighboring bubbles.
282
00:16:20 --> 00:16:23
So we have only
three non-zeros.
283
00:16:23 --> 00:16:28
Do you see that the next
row will have five?
284
00:16:28 --> 00:16:30
Will I get it right?
285
00:16:30 --> 00:16:31
I hope so.
286
00:16:31 --> 00:16:38
The next row we'll have,
I think they'd be here.
287
00:16:38 --> 00:16:42
And then the next row will
have only three guys,
288
00:16:42 --> 00:16:48
maybe here, here, here.
289
00:16:48 --> 00:16:51
Well, it's certainly
a band matrix.
290
00:16:51 --> 00:16:54
So you could say, okay,
it's a band matrix.
291
00:16:54 --> 00:16:59
I wouldn't call it
tri-diagonal anymore.
292
00:16:59 --> 00:17:03
If I showed you that matrix and
said, what kind of a matrix,
293
00:17:03 --> 00:17:05
you'd say a band matrix.
294
00:17:05 --> 00:17:09
If you wanted to tell me that
it had five bands, you could
295
00:17:09 --> 00:17:12
maybe say penta-diagonal,
or something.
296
00:17:12 --> 00:17:15
But it's easy to work
with, of course.
297
00:17:15 --> 00:17:19
That's the point of finite
elements, is that all the
298
00:17:19 --> 00:17:25
functions are local, so that
we get all zeros when trial
299
00:17:25 --> 00:17:28
functions don't overlap.
300
00:17:28 --> 00:17:34
My additional point was just a
small one that's not a big
301
00:17:34 --> 00:17:38
deal, but it's a little
bit worth noticing.
302
00:17:38 --> 00:17:48
These rows with only three
entries, three non-zeros.
303
00:17:48 --> 00:17:54
I guess what I want to say is I
have to solve eight equations
304
00:17:54 --> 00:17:56
and eight unknowns.
305
00:17:56 --> 00:17:59
And the normal way to do it
would be just elimination.
306
00:17:59 --> 00:18:01
LU, that would work fine.
307
00:18:01 --> 00:18:08
Start from the top, eliminate,
and you've got it.
308
00:18:08 --> 00:18:12
And of course in one dimension,
nobody would do anything else.
309
00:18:12 --> 00:18:15
That would be simple.
310
00:18:15 --> 00:18:19
I just want to say, these
bubbles, by giving me extra
311
00:18:19 --> 00:18:24
zeros, I could eliminate
the bubbles first.
312
00:18:24 --> 00:18:28
Can I just make this
point but not labor it?
313
00:18:28 --> 00:18:32
I could eliminate
the bubbles first.
314
00:18:32 --> 00:18:39
I could use this equation to
express the bubble coefficient
315
00:18:39 --> 00:18:41
in terms of its neighbors.
316
00:18:41 --> 00:18:43
I could use this one to express
the bubble coefficient in
317
00:18:43 --> 00:18:45
terms of it neighbors.
318
00:18:45 --> 00:18:49
And I could plug back into
the other equations.
319
00:18:49 --> 00:18:54
I could simplify this.
320
00:18:54 --> 00:18:57
I could get the bubbles
done first if I wanted.
321
00:18:57 --> 00:19:03
I can see that to go into
the gory details is
322
00:19:03 --> 00:19:05
probably not wise.
323
00:19:05 --> 00:19:08
But bubbles are easy to do.
324
00:19:08 --> 00:19:17
However there are
better elements.
325
00:19:17 --> 00:19:20
So that's my discussion
of quadratic elements,
326
00:19:20 --> 00:19:22
almost complete.
327
00:19:22 --> 00:19:26
It's not a big favorite,
because cubics are better.
328
00:19:26 --> 00:19:28
So why are cubics better?
329
00:19:28 --> 00:19:31
Why are cubics better?
330
00:19:31 --> 00:19:34
So you're going to say,
okay, upgrade to cubics.
331
00:19:34 --> 00:19:38
How shall I do that?
332
00:19:38 --> 00:19:43
And I want to say a word
about the error here.
333
00:19:43 --> 00:19:45
Of course, the reason
quadratics are better
334
00:19:45 --> 00:19:48
than cubics...
335
00:19:48 --> 00:19:50
Sorry, the reason why
quadratics are better than
336
00:19:50 --> 00:19:54
linear, and cubics will be
better than quadratics is
337
00:19:54 --> 00:20:00
I'm getting more accuracy.
338
00:20:00 --> 00:20:06
Suppose my true solution may
be some curve like that.
339
00:20:06 --> 00:20:07
Okay.
340
00:20:07 --> 00:20:12
My piecewise linear elements,
suppose the piecewise linear
341
00:20:12 --> 00:20:16
elements happen to be, as they
would in a special model
342
00:20:16 --> 00:20:21
problem, right on the
money, at the nodes.
343
00:20:21 --> 00:20:23
Usually they won't be.
344
00:20:23 --> 00:20:27
But what would be the
error in that one?
345
00:20:27 --> 00:20:30
Well, no error at all at the
nodes as I've drawn it.
346
00:20:30 --> 00:20:32
But that's not what
I'm interested in.
347
00:20:32 --> 00:20:35
I'm interested in,
how big is that?
348
00:20:35 --> 00:20:42
How far off is the
displacement?
349
00:20:42 --> 00:20:47
What's the maximum error
in the displacement.
350
00:20:47 --> 00:20:48
Do you have any idea?
351
00:20:48 --> 00:20:52
If this is size h delta x.
352
00:20:52 --> 00:21:01
Shall I call it delta x or h.
353
00:21:01 --> 00:21:08
How far does a curving
function escape from the...
354
00:21:08 --> 00:21:10
I need to blow
that up, don't I?
355
00:21:10 --> 00:21:15
So I have a curving function
and a linear function, and I
356
00:21:15 --> 00:21:19
want to know how far apart
they are over a distance
357
00:21:19 --> 00:21:21
of length delta x.
358
00:21:21 --> 00:21:24
What's this scale?
359
00:21:24 --> 00:21:28
That's the question.
360
00:21:28 --> 00:21:32
It's just good, it'll
have a simple answer and
361
00:21:32 --> 00:21:35
it's great to know it.
362
00:21:35 --> 00:21:38
Anybody want to make a guess?
363
00:21:38 --> 00:21:41
Is that scale of size delta x?
364
00:21:41 --> 00:21:44
Is it of size delta x
squared, size delta x cubed?
365
00:21:44 --> 00:21:48
It's that exponent of delta
x that is telling me
366
00:21:48 --> 00:21:50
how big is the error?
367
00:21:50 --> 00:21:55
And it's easy to find once
you get the hang of it.
368
00:21:55 --> 00:21:57
Anybody want to make a guess?
369
00:21:57 --> 00:21:59
Delta x?
370
00:21:59 --> 00:21:59
Squared.
371
00:21:59 --> 00:22:02
Squared would be
the right guess.
372
00:22:02 --> 00:22:07
Squared would be
the right guess.
373
00:22:07 --> 00:22:12
I could just turn that
picture if we wanted,
374
00:22:12 --> 00:22:17
to this is delta x.
375
00:22:17 --> 00:22:22
Now it would look like
that, pretty much.
376
00:22:22 --> 00:22:25
Doesn't have to be symmetric,
of course, because this could
377
00:22:25 --> 00:22:26
be a complicated function.
378
00:22:26 --> 00:22:32
But when I focus on a little
delta x interval, every
379
00:22:32 --> 00:22:38
function looks like a
little polynomial.
380
00:22:38 --> 00:22:41
The error there, let's see.
381
00:22:41 --> 00:22:55
What would that function be?
382
00:22:55 --> 00:22:56
I could go forever on this.
383
00:22:56 --> 00:23:04
But look, if the slope is
something, whatever, let
384
00:23:04 --> 00:23:05
me change numbers here.
385
00:23:05 --> 00:23:14
Let me call it from zero to y,
what would be a little parabola
386
00:23:14 --> 00:23:20
that has a slope of one,
let's say, at both ends.
387
00:23:20 --> 00:23:22
What would that parabola be?
388
00:23:22 --> 00:23:24
We probably have
seen that before.
389
00:23:24 --> 00:23:34
If I wanted a slope of one at
both ends, the polynomial
390
00:23:34 --> 00:23:39
would be something
like... what would it be?
391
00:23:39 --> 00:23:41
Sorry, tell me that
little polynomial.
392
00:23:41 --> 00:23:44
It's a polynomial in x,
it's just a quadratic.
393
00:23:44 --> 00:23:52
Its slope is one, so it
maybe starts with an x.
394
00:23:52 --> 00:23:55
I've got to bring it down here.
395
00:23:55 --> 00:24:01
It's x times one
minus x over....
396
00:24:01 --> 00:24:07
I didn't like y ever
in the first place.
397
00:24:07 --> 00:24:08
What do I want to put there?
398
00:24:08 --> 00:24:11
I don't want to put a one.
399
00:24:11 --> 00:24:18
That would make it
look big. y is there.
400
00:24:18 --> 00:24:24
Okay, I think that
quadratic is zero at zero,
401
00:24:24 --> 00:24:26
because of that term.
402
00:24:26 --> 00:24:30
It's zero at x=y,
because of that term.
403
00:24:30 --> 00:24:31
It's second degree.
404
00:24:31 --> 00:24:35
And I think its height is
a maximum right there.
405
00:24:35 --> 00:24:37
And what is that height?
406
00:24:37 --> 00:24:40
At y/2, this is
y/2, this is y/2.
407
00:24:41 --> 00:24:44
That height is y
squared over four.
408
00:24:44 --> 00:24:46
That's what I was shooting for.
409
00:24:46 --> 00:24:48
The square.
410
00:24:48 --> 00:24:52
That in a little interval of
length y, for length delta x,
411
00:24:52 --> 00:24:58
if I draw a little parabola and
I'm matching at the ends, then
412
00:24:58 --> 00:25:02
the height it reaches
is like y squared.
413
00:25:02 --> 00:25:04
That's the scale.
414
00:25:04 --> 00:25:09
So my conclusion is that if I
use these basic hat function
415
00:25:09 --> 00:25:17
elements, the error I get is --
so can I list the errors? --
416
00:25:17 --> 00:25:22
the error is delta x squared.
417
00:25:22 --> 00:25:24
That's the displacement error.
418
00:25:24 --> 00:25:28
The error in U.
419
00:25:28 --> 00:25:31
I'm not proving anything.
420
00:25:31 --> 00:25:36
The careful discussion of
the accuracy is a later
421
00:25:36 --> 00:25:38
section in the book.
422
00:25:38 --> 00:25:43
But I'm trying to make the main
point, is that if we're fitting
423
00:25:43 --> 00:25:47
functions by straight lines,
then we have an error
424
00:25:47 --> 00:25:48
of delta x squared.
425
00:25:48 --> 00:25:51
And what's the slope error?
426
00:25:51 --> 00:25:56
What do you think is
the slope error?
427
00:25:56 --> 00:25:58
Because for us that
slope is important.
428
00:25:58 --> 00:26:02
That's the error in the
stretching and the strain.
429
00:26:02 --> 00:26:05
So the error in the function
is delta x squared.
430
00:26:05 --> 00:26:10
The error in the slope will be
one order less, just delta x.
431
00:26:10 --> 00:26:13
Okay, I'll come
back to all this.
432
00:26:13 --> 00:26:16
Now, make a guess.
433
00:26:16 --> 00:26:23
Suppose I include these
bubble functions.
434
00:26:23 --> 00:26:31
With delta x as my length scale
horizontally, what will be
435
00:26:31 --> 00:26:33
the scale of the error?
436
00:26:33 --> 00:26:38
What do you guess is the
expected error in displacement
437
00:26:38 --> 00:26:44
for a general problem, for
a general c(x) and F(x).
438
00:26:44 --> 00:26:47
439
00:26:47 --> 00:26:49
Which I won't get
exactly right, but how
440
00:26:49 --> 00:26:51
close will I come?
441
00:26:51 --> 00:26:56
I'll come within delta
x to what power?
442
00:26:56 --> 00:26:58
Make a guess, please.
443
00:26:58 --> 00:27:01
Four is an optimist.
444
00:27:01 --> 00:27:02
I won't get up to four.
445
00:27:02 --> 00:27:03
Cubed.
446
00:27:03 --> 00:27:04
I'd only get cubed.
447
00:27:04 --> 00:27:10
I'll get one by increasing the
degree of the polynomial by
448
00:27:10 --> 00:27:15
one, I'll get one
degree better.
449
00:27:15 --> 00:27:20
So it you could look
at it this way.
450
00:27:20 --> 00:27:23
Suppose I have any function.
451
00:27:23 --> 00:27:26
This is a another way to
think about the accuracy.
452
00:27:26 --> 00:27:28
Suppose I have any
function F(x).
453
00:27:28 --> 00:27:32
454
00:27:32 --> 00:27:39
The whole point of calculus is
that I could start, if I start
455
00:27:39 --> 00:27:47
where it is at zero, then I add
in F'(0), the slope times x.
456
00:27:47 --> 00:27:54
Then I add in 1/2 F''(0)
times x squared, and so on.
457
00:27:54 --> 00:27:56
Right?
458
00:27:56 --> 00:28:00
It's called the Taylor series.
459
00:28:00 --> 00:28:03
And we're not paying any
attention to convergence,
460
00:28:03 --> 00:28:06
or high order.
461
00:28:06 --> 00:28:10
It's the early terms
that I'm interested in.
462
00:28:10 --> 00:28:14
And the point is that if my
functions include linear
463
00:28:14 --> 00:28:21
functions, which the hats did,
they will be able to get these
464
00:28:21 --> 00:28:25
terms right, and this will be
the error that I missed.
465
00:28:25 --> 00:28:28
I'm just looking to see what's
the first term in the Taylor
466
00:28:28 --> 00:28:31
series that I will not get.
467
00:28:31 --> 00:28:33
And if I only have hat
functions, I can't
468
00:28:33 --> 00:28:34
get an x squared.
469
00:28:34 --> 00:28:36
I can't get a parabola.
470
00:28:36 --> 00:28:41
But when I go here and include
the x squareds, I can
471
00:28:41 --> 00:28:43
get that term right.
472
00:28:43 --> 00:28:47
So then it'll be the
1/6 f triple prime x
473
00:28:47 --> 00:28:49
cubed that I miss.
474
00:28:49 --> 00:28:55
So the error will be
the next missing term.
475
00:28:55 --> 00:28:59
Okay, so that's thoughts
about the error.
476
00:28:59 --> 00:29:04
And of course that's why those
elements are better than these.
477
00:29:04 --> 00:29:07
They take more work,
but they are worth it.
478
00:29:07 --> 00:29:10
But now I want to tell you
about the next elements.
479
00:29:10 --> 00:29:13
Cubics.
480
00:29:13 --> 00:29:18
Where you're going to expect
to get delta x to the fourth.
481
00:29:18 --> 00:29:22
So now we're getting
serious accuracy.
482
00:29:22 --> 00:29:24
Now we're getting
good accuracy.
483
00:29:24 --> 00:29:28
Of course our problem is not
the most difficult problem.
484
00:29:28 --> 00:29:28
It's in 1-D.
485
00:29:29 --> 00:29:30
But this is good.
486
00:29:30 --> 00:29:34
Okay.
487
00:29:34 --> 00:29:38
This was now the fun in the
golden age of finite elements.
488
00:29:38 --> 00:29:41
To construct cubics.
489
00:29:41 --> 00:29:45
What shall I use as basis
functions for cubics?
490
00:29:45 --> 00:29:49
So I want to have a
cubic in each piece.
491
00:29:49 --> 00:29:54
First of all suppose I just
want no more than that.
492
00:29:54 --> 00:30:02
Suppose I'm happy with just
continuous functions and
493
00:30:02 --> 00:30:06
I let the slope jump.
494
00:30:06 --> 00:30:09
What new trial function
shall I put in?
495
00:30:09 --> 00:30:12
So I'm going to put in
new trial functions.
496
00:30:12 --> 00:30:14
What will they look like?
497
00:30:14 --> 00:30:15
Little cubics?
498
00:30:15 --> 00:30:18
Little third degree bit pieces.
499
00:30:18 --> 00:30:20
Instead of parabolas,
they'll be little
500
00:30:20 --> 00:30:23
pieces of third degree.
501
00:30:23 --> 00:30:27
And I could put in
four more bubbles.
502
00:30:27 --> 00:30:29
Four cubic bubbles.
503
00:30:29 --> 00:30:37
So I would be up to twelve
degrees, twelve by twelve
504
00:30:37 --> 00:30:40
matrices, twelve functions.
505
00:30:40 --> 00:30:43
And for that size delta
x, that would give me
506
00:30:43 --> 00:30:44
delta x to the fourth.
507
00:30:44 --> 00:30:47
So that would be okay.
508
00:30:47 --> 00:30:49
There's a better idea.
509
00:30:49 --> 00:30:52
You can see that I left space.
510
00:30:52 --> 00:30:59
I'm going to make the
slope also continuous.
511
00:30:59 --> 00:31:03
I'm not going to allow
jumps in slope.
512
00:31:03 --> 00:31:06
Think how will I do that?
513
00:31:06 --> 00:31:10
So I'm going to call those C
-- what will I call that when
514
00:31:10 --> 00:31:11
the slope is continuous?
515
00:31:11 --> 00:31:15
The first derivative, I'll
call that C^1, continuous
516
00:31:15 --> 00:31:17
first derivative.
517
00:31:17 --> 00:31:19
Okay.
518
00:31:19 --> 00:31:23
Now I'm actually in section
3.2, where these better
519
00:31:23 --> 00:31:28
elements, these really nifty
elements are constructed.
520
00:31:28 --> 00:31:31
C^1 continuous slope cubics.
521
00:31:31 --> 00:31:32
Okay.
522
00:31:32 --> 00:31:33
Ready for those?
523
00:31:33 --> 00:31:37
What shall be my trial function
for continuous slope cubics?
524
00:31:37 --> 00:31:39
So I have to start again.
525
00:31:39 --> 00:31:44
I have to start again because
the hat functions are out now.
526
00:31:44 --> 00:31:47
Those hat functions
have a jump in slope.
527
00:31:47 --> 00:31:53
The bubble functions
have a jump in slope.
528
00:31:53 --> 00:31:59
I'm rethinking here to
create a better element.
529
00:31:59 --> 00:32:00
Okay.
530
00:32:00 --> 00:32:05
So let's just think, if we've
got a chance at it, how
531
00:32:05 --> 00:32:07
could these elements work?
532
00:32:07 --> 00:32:09
Okay, so here is
the idea, then.
533
00:32:09 --> 00:32:15
Here is my interval.
534
00:32:15 --> 00:32:18
Zero to one, and here's
a typical interval.
535
00:32:18 --> 00:32:24
And now at a typical node, like
node one, I plan to have as
536
00:32:24 --> 00:32:28
unknowns the height of the
function, as before,
537
00:32:28 --> 00:32:29
and also the slope.
538
00:32:29 --> 00:32:35
So I want the function, my
trial function is going to have
539
00:32:35 --> 00:32:38
some height and some slope.
540
00:32:38 --> 00:32:41
And at node two, it's
going to have some
541
00:32:41 --> 00:32:44
height and some slope.
542
00:32:44 --> 00:32:47
And here's the question.
543
00:32:47 --> 00:32:49
Here here's the good point.
544
00:32:49 --> 00:32:53
That those four numbers, the
two heights and the two
545
00:32:53 --> 00:33:00
slopes, that gives me four
things, four quantities.
546
00:33:00 --> 00:33:03
How many quantities do I
need to determine a cubic?
547
00:33:03 --> 00:33:07
So by a cubic, of course, I
mean by a cubic something like
548
00:33:07 --> 00:33:14
a zero plus a one x plus a two
x squared and a three x cubed.
549
00:33:14 --> 00:33:18
It's called a cubic
because it's x cubed.
550
00:33:18 --> 00:33:20
So how many numbers here?
551
00:33:20 --> 00:33:20
Four.
552
00:33:20 --> 00:33:22
Perfect match.
553
00:33:22 --> 00:33:27
There's exactly one cubic that
has a specified height and a
554
00:33:27 --> 00:33:31
specified slope at
these two ends.
555
00:33:31 --> 00:33:33
There's one cubic
that'll do that.
556
00:33:33 --> 00:33:38
And then whatever the height
here is and whatever the slope
557
00:33:38 --> 00:33:42
there is, there'll be one cubic
with that height and that slope
558
00:33:42 --> 00:33:44
that comes into this one.
559
00:33:44 --> 00:33:48
And you see that they will
have continuous slope.
560
00:33:48 --> 00:33:51
Because of course the slope
is continuous in between;
561
00:33:51 --> 00:33:53
it's a polynomial.
562
00:33:53 --> 00:33:56
The question is
always at the nodes.
563
00:33:56 --> 00:33:59
But I use the same number
coming from the left
564
00:33:59 --> 00:34:00
and from the right.
565
00:34:00 --> 00:34:05
The slope has become
an extra unknown.
566
00:34:05 --> 00:34:08
The slope has become
an extra unknown.
567
00:34:08 --> 00:34:11
So I have height slope
at every point.
568
00:34:11 --> 00:34:17
So that's one way to describe
these trial functions now.
569
00:34:17 --> 00:34:27
The trial functions have height
and also slope at each node.
570
00:34:27 --> 00:34:29
So what does that mean?
571
00:34:29 --> 00:34:32
That means that I'm going
to have two unknowns.
572
00:34:32 --> 00:34:36
Two functions, two trial
functions, each with its own
573
00:34:36 --> 00:34:40
coefficient at each node.
574
00:34:40 --> 00:34:45
So if I take a typical node
there, I want two functions.
575
00:34:45 --> 00:34:49
Okay, this is interesting.
576
00:34:49 --> 00:34:52
But you see what I'm creating.
577
00:34:52 --> 00:34:56
I think I'm going to get two
functions there, two functions
578
00:34:56 --> 00:35:00
there, two functions there,
two functions there, right?
579
00:35:00 --> 00:35:02
Because nobody's
constraining that.
580
00:35:02 --> 00:35:03
So I'm up to eight.
581
00:35:03 --> 00:35:06
And how many functions do you
think I'm going to have
582
00:35:06 --> 00:35:09
associated with that node.
583
00:35:09 --> 00:35:10
Only one.
584
00:35:10 --> 00:35:12
Why?
585
00:35:12 --> 00:35:14
Because the height is fixed.
586
00:35:14 --> 00:35:20
So I think I've got nine
trial functions here.
587
00:35:20 --> 00:35:23
And if we can see what
those are, then the
588
00:35:23 --> 00:35:25
system will take over.
589
00:35:25 --> 00:35:31
They're my phi_1 to phi_9,
whatever they plug in here,
590
00:35:31 --> 00:35:33
they plug in the right hand
side, I'll have a nine by
591
00:35:33 --> 00:35:37
nine stiffness matrix.
592
00:35:37 --> 00:35:40
It'll be local again.
593
00:35:40 --> 00:35:45
Well, let's see if we can
figure out these functions.
594
00:35:45 --> 00:35:47
Okay, so you have the idea?
595
00:35:47 --> 00:35:50
I'm expecting two
trial functions.
596
00:35:50 --> 00:35:53
One is sort of a round hat.
597
00:35:53 --> 00:35:54
All right, let me draw that.
598
00:35:54 --> 00:36:03
The round hat function will be
the function -- These will be
599
00:36:03 --> 00:36:10
the round hats, and they'll be
associated with, they
600
00:36:10 --> 00:36:14
give me heights.
601
00:36:14 --> 00:36:19
And then I'll also have an
additional one, except
602
00:36:19 --> 00:36:21
at the last node.
603
00:36:21 --> 00:36:23
And these will be -- I don't
know what to call them yet.
604
00:36:23 --> 00:36:25
You'll have to give me a name.
605
00:36:25 --> 00:36:27
These will give me the slopes.
606
00:36:27 --> 00:36:28
Okay.
607
00:36:28 --> 00:36:30
So what does a round
hat look like?
608
00:36:30 --> 00:36:35
Now these have to be, follow
my rules, they have to be
609
00:36:35 --> 00:36:38
continuous, their slope
has to be continuous.
610
00:36:38 --> 00:36:41
And I want to take the
one that has height one
611
00:36:41 --> 00:36:43
and zero slope there.
612
00:36:43 --> 00:36:47
And it should have height
zero and zero slope, here.
613
00:36:47 --> 00:36:52
Height zero, zero slope.
614
00:36:52 --> 00:36:54
You see what it's going to be?
615
00:36:54 --> 00:37:01
This phi, whatever number it
is, it'll be the phi whose
616
00:37:01 --> 00:37:05
coefficient tells me the
height at node one.
617
00:37:05 --> 00:37:08
So here's node one.
618
00:37:08 --> 00:37:10
What will it look like?
619
00:37:10 --> 00:37:12
What will this function do?
620
00:37:12 --> 00:37:17
Well, there is exactly one
cubic, that starts from zero
621
00:37:17 --> 00:37:21
with slope zero and ends there,
ends at one with slope zero.
622
00:37:21 --> 00:37:21
Right?
623
00:37:21 --> 00:37:24
That's what we said; four
numbers determine that
624
00:37:24 --> 00:37:28
cubic in that interval.
625
00:37:28 --> 00:37:33
Then there's another cubic
that, with those two numbers
626
00:37:33 --> 00:37:36
again, that keeps the
continuous slope, and these
627
00:37:36 --> 00:37:38
two numbers in this interval.
628
00:37:38 --> 00:37:41
And of course it'll
just be symmetric.
629
00:37:41 --> 00:37:44
You see the round hat?
630
00:37:44 --> 00:37:49
So that's the basis function,
the trial function that has
631
00:37:49 --> 00:37:53
continuous slopes and heights,
of course, and it has
632
00:37:53 --> 00:37:55
height one at that point.
633
00:37:55 --> 00:38:00
And now let me draw the one
that has height zero slope
634
00:38:00 --> 00:38:04
zero, height zero slope zero.
635
00:38:04 --> 00:38:10
And what do I want
it to do there?
636
00:38:10 --> 00:38:16
What should this
function be like?
637
00:38:16 --> 00:38:19
It should be the one that
it's coefficient will
638
00:38:19 --> 00:38:21
tell me the slope.
639
00:38:21 --> 00:38:28
So I want it to have a slope
of one and a height of zero.
640
00:38:28 --> 00:38:32
Do you see these functions,
shall I call these the
641
00:38:32 --> 00:38:35
height functions, phi h 1?
642
00:38:35 --> 00:38:38
That's the phi, that's the
trial function that tells
643
00:38:38 --> 00:38:41
me the height at node one.
644
00:38:41 --> 00:38:46
When I take combinations, it
gets multiplied by U h 1,
645
00:38:46 --> 00:38:51
which is exactly the
height at node one.
646
00:38:51 --> 00:38:53
Now what about this guy?
647
00:38:53 --> 00:38:58
This guy is going to start
with zero slope at zero.
648
00:38:58 --> 00:39:01
It's going to be a cubic,
and there's exactly one
649
00:39:01 --> 00:39:02
cubic that'll do it.
650
00:39:02 --> 00:39:04
It'll look a little like.
651
00:39:04 --> 00:39:07
Then there'll be exactly
one cubic that does that
652
00:39:07 --> 00:39:10
and gets back to zero.
653
00:39:10 --> 00:39:12
You see that that's possible?
654
00:39:12 --> 00:39:15
In each interval, I've
got four numbers: two
655
00:39:15 --> 00:39:17
heights, two slopes.
656
00:39:17 --> 00:39:22
So this would be a picture
of the phi slope at
657
00:39:22 --> 00:39:25
node one function.
658
00:39:25 --> 00:39:30
So that's a standard function,
it's a cubic, piecewise cubic.
659
00:39:30 --> 00:39:34
Local again, because in all
these intervals it's zero.
660
00:39:34 --> 00:39:38
And it will be, when I go to
take combinations of all these
661
00:39:38 --> 00:39:43
guys, it'll be multiplied by
its coefficient, U slope one.
662
00:39:43 --> 00:39:48
And then I'll have
nine all together.
663
00:39:48 --> 00:39:50
But those two are
the typical ones.
664
00:39:50 --> 00:39:57
Do you do see how that's going?
665
00:39:57 --> 00:40:03
It's more subtle
than hat functions.
666
00:40:03 --> 00:40:10
Suppose whoever's writing the
finite element code gets a
667
00:40:10 --> 00:40:17
formula for those phis and
plugs them into the
668
00:40:17 --> 00:40:23
integrals, comes out with
a stiffness matrix.
669
00:40:23 --> 00:40:27
Actually, we could even look
at that stiffness matrix.
670
00:40:27 --> 00:40:31
This is a good way to
understand the picture.
671
00:40:31 --> 00:40:33
Now it'll be nine by nine.
672
00:40:33 --> 00:40:36
Right?
673
00:40:36 --> 00:40:41
So here we'll have a typical,
this'll be our phi height 1
674
00:40:41 --> 00:40:46
row, and this'll be our phi
slope 1 row, and this'll be our
675
00:40:46 --> 00:40:49
phi height 2 row, and so on.
676
00:40:49 --> 00:40:53
Of course, I didn't
leave room for all.
677
00:40:53 --> 00:40:59
What will a typical row of this
stiffness matrix have in it?
678
00:40:59 --> 00:41:03
I'm just asking about the
overlaps. phi 1 height
679
00:41:03 --> 00:41:06
certainly overlaps itself.
680
00:41:06 --> 00:41:11
Does phi 1 height
overlap phi_1 slope?
681
00:41:11 --> 00:41:13
Yes or no?
682
00:41:13 --> 00:41:14
Sure.
683
00:41:14 --> 00:41:16
Sure.
684
00:41:16 --> 00:41:21
Does phi_1 height
overlap phi_2 height?
685
00:41:21 --> 00:41:23
Yes.
686
00:41:23 --> 00:41:23
Yes.
687
00:41:23 --> 00:41:28
Because the phi_2 height
will go up like that.
688
00:41:28 --> 00:41:29
You see?
689
00:41:29 --> 00:41:30
And the phi_2 slope.
690
00:41:30 --> 00:41:41
So actually we'll have, I think
we'll have six non-zeros
691
00:41:41 --> 00:41:43
on a typical row.
692
00:41:43 --> 00:41:45
Is that right?
693
00:41:45 --> 00:41:46
Six non-zeros?
694
00:41:46 --> 00:41:54
Because a typical h -- this is
maybe not so typical, because
695
00:41:54 --> 00:41:59
to the left of it there's only
one -- No, there are two?
696
00:41:59 --> 00:42:00
Right?
697
00:42:00 --> 00:42:03
There's a phi_0, phi
h 0 and a phi s 0.
698
00:42:03 --> 00:42:08
Sure, there are two here, the
two guys here, there's one
699
00:42:08 --> 00:42:17
height guy, and there's one
-- what's cooking in that?
700
00:42:17 --> 00:42:21
Oh, it's got a slope of one
and it gets back to zero.
701
00:42:21 --> 00:42:25
What I'm drawing now in
little dashed lines
702
00:42:25 --> 00:42:29
was the phi slope 0.
703
00:42:29 --> 00:42:33
The one that gives me a slope
at node zero, and this is the
704
00:42:33 --> 00:42:35
one that gives me a height.
705
00:42:35 --> 00:42:36
Yes.
706
00:42:36 --> 00:42:40
Do you see it?
707
00:42:40 --> 00:42:45
So above this was a phi
slope 0, and stuck in
708
00:42:45 --> 00:42:54
there was a phi height 0.
709
00:42:54 --> 00:42:57
Six diagonal matrix.
710
00:42:57 --> 00:43:02
I think it helps to draw that
little thing with x's and
711
00:43:02 --> 00:43:05
zeroes, because then you sort
of see how things are
712
00:43:05 --> 00:43:06
fitting together.
713
00:43:06 --> 00:43:09
Okay.
714
00:43:09 --> 00:43:18
So these functions now, I've
gone into section 3.2 for that.
715
00:43:18 --> 00:43:22
I want to go to a slightly
different topic, and then
716
00:43:22 --> 00:43:28
I'll come back in section
3.2 to these cubics.
717
00:43:28 --> 00:43:32
So these are C^1 cubics,
continuous slope cubics.
718
00:43:32 --> 00:43:35
Very interesting construction.
719
00:43:35 --> 00:43:39
Are you seeing how it could
go in more dimensions?
720
00:43:39 --> 00:43:45
I mean, that's what we'll see
for Laplace's Equation, how
721
00:43:45 --> 00:43:49
can you construct quadratics,
cubics in a plane.
722
00:43:49 --> 00:43:52
It gets interesting.
723
00:43:52 --> 00:43:56
But you'll get the
knack of these guys.
724
00:43:56 --> 00:44:02
These are pretty direct,
and very useful.
725
00:44:02 --> 00:44:04
So what's the effect?
726
00:44:04 --> 00:44:09
The effect is that
we get a matrix.
727
00:44:09 --> 00:44:13
It looks quite like a
difference matrix.
728
00:44:13 --> 00:44:16
Well, actually, the height
rows and the numbers in the
729
00:44:16 --> 00:44:20
height rows and the slopes
rows look different.
730
00:44:20 --> 00:44:23
We're getting
something new here.
731
00:44:23 --> 00:44:29
We're getting matrix, a KU=F,
that's going to give us
732
00:44:29 --> 00:44:31
fourth order accuracy.
733
00:44:31 --> 00:44:37
So the accuracy has moved up.
734
00:44:37 --> 00:44:42
So we've got up to fourth order
accuracy, which we could get by
735
00:44:42 --> 00:44:47
finite differences by
a lot of patience.
736
00:44:47 --> 00:44:50
We get them from finite
elements in a straight way.
737
00:44:50 --> 00:44:53
Okay, any question
or discussion?
738
00:44:53 --> 00:45:00
I'm talking real fast to get
this new idea of constructing
739
00:45:00 --> 00:45:03
finite elements here.
740
00:45:03 --> 00:45:13
I do want to say something
about that line.
741
00:45:13 --> 00:45:17
Because that's a part
of this business of
742
00:45:17 --> 00:45:21
estimating the accuracy.
743
00:45:21 --> 00:45:28
It's a key idea in the
background of the Galerkin
744
00:45:28 --> 00:45:33
method, and the minimum form
would be associated with
745
00:45:33 --> 00:45:37
names like Raleigh and Ritz.
746
00:45:37 --> 00:45:37
All right.
747
00:45:37 --> 00:45:42
I'll just go directly
to that, if I may.
748
00:45:42 --> 00:45:50
So what I want to do is tell
you, for our model problem, I
749
00:45:50 --> 00:45:58
want to tell you the strong
form-- let me do it this way.
750
00:45:58 --> 00:46:04
I'll put the strong form, the
weak form, and then I want
751
00:46:04 --> 00:46:09
to add in the minimum form.
752
00:46:09 --> 00:46:11
Okay.
753
00:46:11 --> 00:46:15
So the strong form of our
equation was minus the
754
00:46:15 --> 00:46:18
derivative of c*du/dx=f.
755
00:46:18 --> 00:46:24
756
00:46:24 --> 00:46:28
Okay.
757
00:46:28 --> 00:46:33
What was the weak form?
758
00:46:33 --> 00:46:34
This is an f(x).
759
00:46:36 --> 00:46:39
The weak form, how do you
get to the weak form?
760
00:46:39 --> 00:46:44
You multiply both sides by a
test function, you integrate,
761
00:46:44 --> 00:46:47
you integrate by parts, and you
get this beautifully symmetric
762
00:46:47 --> 00:46:57
form that we have up there,
du/dx*dv/dx*dx, equals the
763
00:46:57 --> 00:46:58
integral of f(x)*v(x)*dx.
764
00:46:58 --> 00:47:03
765
00:47:03 --> 00:47:08
I write that again, just so
you see the nice symmetry
766
00:47:08 --> 00:47:10
of that weak form.
767
00:47:10 --> 00:47:21
And it's for all test
functions v Okay.
768
00:47:21 --> 00:47:24
I'm shooting for a
third description.
769
00:47:24 --> 00:47:27
A third description
of the same problem.
770
00:47:27 --> 00:47:31
And it's really neat to
see that you have that.
771
00:47:31 --> 00:47:35
Let me just see it first
in the discrete case.
772
00:47:35 --> 00:47:39
The discrete case, the
strong form would be
773
00:47:39 --> 00:47:41
A transpose C Au=f.
774
00:47:42 --> 00:47:46
That's the strong form.
775
00:47:46 --> 00:47:48
Right?
776
00:47:48 --> 00:47:50
I always like to see the
discrete one first, and
777
00:47:50 --> 00:47:52
then the continuous.
778
00:47:52 --> 00:47:55
Okay, what would be the weak
form in the discrete case?
779
00:47:55 --> 00:48:00
I would multiply by a vector v,
and I would take inner products
780
00:48:00 --> 00:48:06
A transpose C Au inner product
with v, equals f inner
781
00:48:06 --> 00:48:09
product with v.
782
00:48:09 --> 00:48:12
You can use dot.
783
00:48:12 --> 00:48:18
So that would be the weak form.
784
00:48:18 --> 00:48:24
I've just taking the dot
product of both sides with v.
785
00:48:24 --> 00:48:28
Now you'll see the weak form
better if, what should I do?
786
00:48:28 --> 00:48:32
What would make that look nice?
787
00:48:32 --> 00:48:38
So that's the dot product of
A transpose C Au with v.
788
00:48:38 --> 00:48:41
And what do I do to
make that look nice?
789
00:48:41 --> 00:48:44
Do you get the idea yet?
790
00:48:44 --> 00:48:47
It doesn't look pretty to me.
791
00:48:47 --> 00:48:49
It's all lopsided.
792
00:48:49 --> 00:48:50
Right?
793
00:48:50 --> 00:48:53
So what can I do
with A transpose?
794
00:48:53 --> 00:48:55
What's the rule
about A transpose?
795
00:48:55 --> 00:48:58
That if I have A transpose
times something, dotted with
796
00:48:58 --> 00:49:02
something, what can I do?
797
00:49:02 --> 00:49:08
I can move the A transpose
over to the other guy.
798
00:49:08 --> 00:49:11
And what will it be
when I do that?
799
00:49:11 --> 00:49:19
So I take it away from here,
and what do I put there?
800
00:49:19 --> 00:49:21
A.
801
00:49:21 --> 00:49:24
That's the whole
point of transposes.
802
00:49:24 --> 00:49:28
Transposes, you put them on the
other side of the dot product,
803
00:49:28 --> 00:49:32
you take the transpose, so it
would be literally, maybe A
804
00:49:32 --> 00:49:34
transpose transpose,
which is A.
805
00:49:34 --> 00:49:37
What I just did there is
integration by parts.
806
00:49:37 --> 00:49:40
Well, summation by parts,
because I'm in the
807
00:49:40 --> 00:49:41
discrete case.
808
00:49:41 --> 00:49:46
The whole idea of integration
by parts amounted to taking A
809
00:49:46 --> 00:49:51
transpose off of u, off of
this, and putting a over there.
810
00:49:51 --> 00:49:53
Isn't that neat?
811
00:49:53 --> 00:49:58
And you see that this CAuAv
is just what I have here.
812
00:49:58 --> 00:50:01
C, a is derivative,
so this is CAuAv.
813
00:50:03 --> 00:50:07
Inner product.
814
00:50:07 --> 00:50:10
That's cool.
815
00:50:10 --> 00:50:13
That's just like
how it should be.
816
00:50:13 --> 00:50:18
I just followed that rule, that
A transpose times something,
817
00:50:18 --> 00:50:22
shall I call it w, inner
product with u, is
818
00:50:22 --> 00:50:24
the same as wAu.
819
00:50:26 --> 00:50:30
That if I bring A transpose
over, it becomes an A.
820
00:50:30 --> 00:50:33
If I bring an A over, it
would become an A transpose.
821
00:50:33 --> 00:50:34
All right, what about
the minimum form?
822
00:50:34 --> 00:50:37
Have I got one minute to
do the minimum form?
823
00:50:37 --> 00:50:40
Yes.
824
00:50:40 --> 00:50:44
So what's the minimization
that's hiding behind this?
825
00:50:44 --> 00:50:49
The minimization in the
discrete case, do you remember?
826
00:50:49 --> 00:50:50
We're looking at Ku=f.
827
00:50:53 --> 00:50:57
And some quadratic quantity
from least squares has
828
00:50:57 --> 00:50:58
its minimum when Ku=f.
829
00:51:00 --> 00:51:07
And it's 1/2 u transpose
Ku minus u transpose f.
830
00:51:07 --> 00:51:09
Where K is A transpose C A.
831
00:51:09 --> 00:51:13
This is the minimum
statement of the problem.
832
00:51:13 --> 00:51:18
That if I look for the u that
minimizes that quadratic, it
833
00:51:18 --> 00:51:20
leads me to the equation Ku=f.
834
00:51:21 --> 00:51:22
So that's the
minimum statement.
835
00:51:22 --> 00:51:27
And if we want it to really
look perfectly like the others,
836
00:51:27 --> 00:51:35
I would put in A transpose C A.
837
00:51:35 --> 00:51:39
Okay.
838
00:51:39 --> 00:51:42
Can I write down next time,
because our time is really up.
839
00:51:42 --> 00:51:46
It's not fair to -- all I'm
going to do is write down
840
00:51:46 --> 00:51:47
the same thing here.
841
00:51:47 --> 00:51:51
I'm minimizing 1/2 -- oh,
I'm going to do it anyway.
842
00:51:51 --> 00:51:58
c(x)*du/dx squared, minus
the integral of f(x)u(x).
843
00:51:58 --> 00:52:02
844
00:52:02 --> 00:52:05
So that's the minimum problem.
845
00:52:05 --> 00:52:12
Minimize over all
u, this quadratic.
846
00:52:12 --> 00:52:16
This is the right way
to see these problems.
847
00:52:16 --> 00:52:19
You see a differential
equation, which we use for
848
00:52:19 --> 00:52:22
finite differences; you see a
weak form, which we use for
849
00:52:22 --> 00:52:26
finite elements; and now
you see a minimum form.
850
00:52:26 --> 00:52:28
Okay, that gives you
something to think about.
851
00:52:28 --> 00:52:31
And there'll be a homework on
finite elements that'll give
852
00:52:31 --> 00:52:33
you a chance to use them.
853
00:52:33 --> 00:52:35
Okay, thank you.