1
00:00:00 --> 00:00:01
2
00:00:01 --> 00:00:03
The following content is
provided under a Creative
3
00:00:03 --> 00:00:03
Commons license.
4
00:00:03 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to
5
00:00:06 --> 00:00:09
offer high-quality educational
resources for free.
6
00:00:09 --> 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 --> 00:00:20
at ocw.mit.edu.
9
00:00:20 --> 00:00:22
PROFESSOR STRANG: So.
10
00:00:22 --> 00:00:27
I got the preparation
for finite elements.
11
00:00:27 --> 00:00:30
Again we're in one dimension,
because that's where you can
12
00:00:30 --> 00:00:35
see first and most clearly
how the system works.
13
00:00:35 --> 00:00:40
So the system was, really to
begin with the weak form that
14
00:00:40 --> 00:00:43
I introduced last time.
15
00:00:43 --> 00:00:46
The Galerkin idea that I
introduced just at the very end
16
00:00:46 --> 00:00:53
of last time, and that idea is
that instead of the continuous
17
00:00:53 --> 00:00:57
differential equations where
Galerkin's idea is how do you
18
00:00:57 --> 00:01:02
make it discrete, and he'll
choose some trial functions.
19
00:01:02 --> 00:01:03
Those are functions.
20
00:01:03 --> 00:01:05
And some test function.
21
00:01:05 --> 00:01:08
And I didn't get to say,
but I'll say now, very
22
00:01:08 --> 00:01:10
often they're the same.
23
00:01:10 --> 00:01:15
So very often the phis
are the same as the v's.
24
00:01:15 --> 00:01:20
And probably they will be
in all my examples today.
25
00:01:20 --> 00:01:28
And then what's new today is
what choices would you make.
26
00:01:28 --> 00:01:33
You have pretty wide choice,
but there are some natural
27
00:01:33 --> 00:01:35
ones to start with.
28
00:01:35 --> 00:01:38
And so that's where
we are today.
29
00:01:38 --> 00:01:44
And then also the other part of
today is how do we get from all
30
00:01:44 --> 00:01:48
that preparation to the
equations that we
31
00:01:48 --> 00:01:49
actually solve.
32
00:01:49 --> 00:01:52
The KU=F; where does
the K come from, where
33
00:01:52 --> 00:01:54
does the F come from?
34
00:01:54 --> 00:01:57
The F is going to come, of
course, somehow from this right
35
00:01:57 --> 00:02:01
hand side and the K is going
to come from the left side.
36
00:02:01 --> 00:02:06
OK, so that's the overall
direction if you want to know
37
00:02:06 --> 00:02:10
how finite elements, work
that's the key line there.
38
00:02:10 --> 00:02:14
And then here I've recalled
what the step was, here's
39
00:02:14 --> 00:02:16
our differential equation.
40
00:02:16 --> 00:02:17
And there's its weak form.
41
00:02:17 --> 00:02:19
This is the weak form.
42
00:02:19 --> 00:02:22
And let me put this in.
43
00:02:22 --> 00:02:25
Here I've reproduced
what we reached last
44
00:02:25 --> 00:02:30
time, the weak form.
45
00:02:30 --> 00:02:34
That was the strong form with
its boundary conditions; now
46
00:02:34 --> 00:02:37
we get the weak form with
its boundary conditions.
47
00:02:37 --> 00:02:43
And the weak form involves u,
so it's the boundary conditions
48
00:02:43 --> 00:02:51
on u, the fixed ones that
get used in the weak form.
49
00:02:51 --> 00:02:58
The free ones don't appear in
the weak form, but by the magic
50
00:02:58 --> 00:03:04
of integration by parts, the
free ones will sort of come
51
00:03:04 --> 00:03:06
out in a natural way.
52
00:03:06 --> 00:03:10
But we have to build in the
essential Dirichlet boundary
53
00:03:10 --> 00:03:15
conditions like that
fixed right hand n.
54
00:03:15 --> 00:03:22
OK, and because I think of v as
a little movement away from u,
55
00:03:22 --> 00:03:27
but my u's all fixed at the
end, therefore v has
56
00:03:27 --> 00:03:29
to be fixed, too.
57
00:03:29 --> 00:03:36
OK, so this is important.
but we haven't made
58
00:03:36 --> 00:03:38
it down to Earth yet.
59
00:03:38 --> 00:03:46
We were doing a lot of ideas
last time, which are the center
60
00:03:46 --> 00:03:49
of things, but now let's
get down to Earth.
61
00:03:49 --> 00:03:54
And down to Earth really means,
what functions do we choose?
62
00:03:54 --> 00:03:57
And then how do we
get the equation.
63
00:03:57 --> 00:04:01
So that's the job today, the
principle job and this is of
64
00:04:01 --> 00:04:06
course what you would actually
code up to run a finite
65
00:04:06 --> 00:04:08
element simulation.
66
00:04:08 --> 00:04:10
You would make a decision
on these functions.
67
00:04:10 --> 00:04:15
And let's start with
piecewise linear functions.
68
00:04:15 --> 00:04:22
So a typical phi is maybe
centered at a node, so
69
00:04:22 --> 00:04:25
it goes up and down.
70
00:04:25 --> 00:04:33
So if that's node two, let's
number these nodes, one,
71
00:04:33 --> 00:04:40
two, three, four, and five.
72
00:04:40 --> 00:04:44
Well, that's five there but
I'm going to have, let's
73
00:04:44 --> 00:04:46
see; what do I have here?
74
00:04:46 --> 00:04:57
At a typical node two, my
function is zero except in the
75
00:04:57 --> 00:04:59
integrals that touch node two.
76
00:04:59 --> 00:05:00
So here's phi_2.
77
00:05:02 --> 00:05:02
phi_2(x).
78
00:05:03 --> 00:05:04
That's it's graph.
79
00:05:04 --> 00:05:07
It comes along, it's piecewise
linear up, it's peacewise
80
00:05:07 --> 00:05:12
linear back down, and
it's zero again.
81
00:05:12 --> 00:05:12
So that would be phi_2.
82
00:05:13 --> 00:05:15
Then phi_1, there'll
be a phi_1.
83
00:05:15 --> 00:05:18
84
00:05:18 --> 00:05:20
Like so, exactly similar.
85
00:05:20 --> 00:05:24
The whole point is keep
the system simple.
86
00:05:24 --> 00:05:29
That's what the finite
element idea is.
87
00:05:29 --> 00:05:32
Use simple functions
for the phis.
88
00:05:32 --> 00:05:35
And I'm taking them also to
be the test function v.
89
00:05:35 --> 00:05:36
Keep those simple.
90
00:05:36 --> 00:05:40
Now, what about at
the boundaries?
91
00:05:40 --> 00:05:44
OK, well we've said our
functions have to be,
92
00:05:44 --> 00:05:46
I'm doing free-fixed.
93
00:05:46 --> 00:05:53
So fixed comes to zero
there and all my
94
00:05:53 --> 00:05:54
functions will do that.
95
00:05:54 --> 00:05:58
But this end is free,
so watch this.
96
00:05:58 --> 00:06:02
This is another what
I'll call a half-hat.
97
00:06:02 --> 00:06:05
If I call those hat
functions is that OK?
98
00:06:05 --> 00:06:08
It makes a nice short word.
99
00:06:08 --> 00:06:13
So these are hat functions.
100
00:06:13 --> 00:06:16
And then a fuller description
would be piecewise linear,
101
00:06:16 --> 00:06:20
but hat functions is clear.
102
00:06:20 --> 00:06:23
OK, now notice I'm sticking
in here what I maybe
103
00:06:23 --> 00:06:26
could call a half-hat.
104
00:06:26 --> 00:06:32
Because my v and my phis, my
functions are not constrained
105
00:06:32 --> 00:06:33
at that left hand end.
106
00:06:33 --> 00:06:34
That's free.
107
00:06:34 --> 00:06:38
So there's a phi_1,
and there's a phi_0.
108
00:06:39 --> 00:06:42
And a phi_3, and a phi_4.
109
00:06:42 --> 00:06:47
4 So altogether, I'm
going to have five.
110
00:06:47 --> 00:06:49
And I've started the
numbering at zero, I
111
00:06:49 --> 00:06:50
guess it just happens.
112
00:06:50 --> 00:06:52
So let's accept that.
113
00:06:52 --> 00:06:56
So I'm going to have
five functions.
114
00:06:56 --> 00:07:00
And they will also be my
five test functions.
115
00:07:00 --> 00:07:04
So let me first think of them
as the trial functions.
116
00:07:04 --> 00:07:05
Phi.
117
00:07:05 --> 00:07:07
So what was the point
about trial functions?
118
00:07:07 --> 00:07:12
The point about is my
approximate, my finite element
119
00:07:12 --> 00:07:18
solution u(x) is going to be a
combination of those.
120
00:07:18 --> 00:07:26
Some u_0, times the phi_0
function up to u_4 times
121
00:07:26 --> 00:07:29
the phi_4 function.
122
00:07:29 --> 00:07:33
And these are my unknowns.
u_0, u_1, those numbers.
123
00:07:33 --> 00:07:35
I have five unknowns.
124
00:07:35 --> 00:07:38
And I'm using hat functions.
125
00:07:38 --> 00:07:43
So I should, really, why don't
I draw a graph of u(x).
126
00:07:45 --> 00:07:48
So I don't need the
words piecewise linear.
127
00:07:48 --> 00:07:53
You see, I have a kind
of space of function.
128
00:07:53 --> 00:08:02
My functions are all, my
approximations are combinations
129
00:08:02 --> 00:08:05
of these fixed basis functions.
130
00:08:05 --> 00:08:09
You could call these phis,
trial functions, basis
131
00:08:09 --> 00:08:14
functions, you're always
choosing in applied math.
132
00:08:14 --> 00:08:19
A bunch of functions whose
combinations are going
133
00:08:19 --> 00:08:21
to be, is what you're
going to work with.
134
00:08:21 --> 00:08:24
And here I'm making
them hat functions.
135
00:08:24 --> 00:08:29
OK, so what does a combination
of those guys look like?
136
00:08:29 --> 00:08:31
I'll even erase the
word hat functions;
137
00:08:31 --> 00:08:32
we won't forget that.
138
00:08:32 --> 00:08:37
So what would a combination of,
here's the same integral, zero,
139
00:08:37 --> 00:08:44
one, with the same mesh, one,
two, three, four guys inside.
140
00:08:44 --> 00:08:50
What would, it just helps
the visualization to see.
141
00:08:50 --> 00:08:57
If I combine these guys,
so those were despite the
142
00:08:57 --> 00:08:59
way it might look, that's
one separate function.
143
00:08:59 --> 00:09:02
Two, three, four and five.
144
00:09:02 --> 00:09:06
And now suppose I
take a combination.
145
00:09:06 --> 00:09:09
What kind of a
function do I get?
146
00:09:09 --> 00:09:14
How would you describe this,
a combination like this?
147
00:09:14 --> 00:09:15
Of those five guys?
148
00:09:15 --> 00:09:18
What will it look like?
149
00:09:18 --> 00:09:25
Between every node it will
be a straight line, right?
150
00:09:25 --> 00:09:28
Because all these guys are
straight, between those.
151
00:09:28 --> 00:09:31
So a typical function
will start at what?
152
00:09:31 --> 00:09:37
This height will be u_0,
because, well let me
153
00:09:37 --> 00:09:38
draw a few things.
154
00:09:38 --> 00:09:42
OK, so I'll put u_0 a little
higher because I want
155
00:09:42 --> 00:09:43
to end up down at zero.
156
00:09:43 --> 00:09:49
So that might be the height
u_0, u_1 might be pretty close.
157
00:09:49 --> 00:09:51
u_2 might be coming down a bit.
158
00:09:51 --> 00:09:54
Coming down a bit,
coming down a bit.
159
00:09:54 --> 00:09:55
And ending at zero.
160
00:09:55 --> 00:10:01
So those were meant
to be corners.
161
00:10:01 --> 00:10:03
That wasn't a very good bit.
162
00:10:03 --> 00:10:05
There, OK.
163
00:10:05 --> 00:10:10
It wasn't meant to be, yeah.
164
00:10:10 --> 00:10:11
So that height is u_0.
165
00:10:11 --> 00:10:13
Now, notice why?
166
00:10:13 --> 00:10:15
Why is that?
167
00:10:15 --> 00:10:19
Why is the coefficient of
phi_0 exactly the height,
168
00:10:19 --> 00:10:24
the displacement, at zero?
169
00:10:24 --> 00:10:29
Because all the other
phis are zero.
170
00:10:29 --> 00:10:34
All the other phis are
zero at this point.
171
00:10:34 --> 00:10:35
Only this guy's coming in.
172
00:10:35 --> 00:10:43
So we'll only see in this, at
this point see, even phi_1
173
00:10:43 --> 00:10:45
has dropped to zero.
174
00:10:45 --> 00:10:48
So we're only going to
see phi_0 times u_0.
175
00:10:48 --> 00:10:54
So, I should have said, we take
all these heights to be one.
176
00:10:54 --> 00:10:56
Height is one.
177
00:10:56 --> 00:10:59
Course it doesn't matter
because we're multiplying by
178
00:10:59 --> 00:11:00
u's, but so let's settle.
179
00:11:00 --> 00:11:02
They all have heights one.
180
00:11:02 --> 00:11:05
And this is sort of a
key part of the finite
181
00:11:05 --> 00:11:07
that element idea.
182
00:11:07 --> 00:11:12
You see, Galerkin, when he
created just two or three
183
00:11:12 --> 00:11:17
functions phi, he tried to
follow the exact solution.
184
00:11:17 --> 00:11:21
He had an idea in his mind
what the solution to
185
00:11:21 --> 00:11:22
the problem will be.
186
00:11:22 --> 00:11:24
And he wanted to take two or
three functions that would
187
00:11:24 --> 00:11:26
get him close to it.
188
00:11:26 --> 00:11:31
Here we choose the functions,
they're in the finite element
189
00:11:31 --> 00:11:34
library before we even know
what the equation is, or
190
00:11:34 --> 00:11:35
the boundary conditions.
191
00:11:35 --> 00:11:42
These functions are like,
the hat function choice.
192
00:11:42 --> 00:11:48
And they have this beautiful
sort of a connect to nodes.
193
00:11:48 --> 00:11:52
In a way where in fact I got
involved in finite elements
194
00:11:52 --> 00:11:55
in the first place just to
understand what's the
195
00:11:55 --> 00:11:58
difference between finite
elements and finite
196
00:11:58 --> 00:12:00
differences.
197
00:12:00 --> 00:12:04
Because the finite elements, as
you'll see, are associated with
198
00:12:04 --> 00:12:09
nodes. phi_1 is the only guy
that's not zero at node one.
199
00:12:09 --> 00:12:12
And therefore, what
will this height be?
200
00:12:12 --> 00:12:14
What's that height?
201
00:12:14 --> 00:12:16
So that maybe comes
down a little.
202
00:12:16 --> 00:12:19
What's this height? u_1.
203
00:12:19 --> 00:12:24
It's the coefficient of phi_1,
because phi_1 is sitting there.
204
00:12:24 --> 00:12:26
And all the other phis
are zero there, so this
205
00:12:26 --> 00:12:28
height is really u_1.
206
00:12:28 --> 00:12:33
And u_2, and u_3, and
u_4, and u_5 is zero.
207
00:12:33 --> 00:12:41
So that was u_1,
u_2, and so on.
208
00:12:41 --> 00:12:45
What I'm saying is, and we'll
see it happen, is that this
209
00:12:45 --> 00:12:49
KU=F equation that we finally
get to is going to look
210
00:12:49 --> 00:12:53
very like a finite
difference equation.
211
00:12:53 --> 00:12:56
But it's coming from this
different direction.
212
00:12:56 --> 00:13:06
And this way allows many more
possibilities, gets things
213
00:13:06 --> 00:13:09
sort of more naturally right.
214
00:13:09 --> 00:13:13
It less, with the finite
difference equation, we had
215
00:13:13 --> 00:13:18
to go in there and decide
what it should be.
216
00:13:18 --> 00:13:22
With finite elements, our
decision is just the phis.
217
00:13:22 --> 00:13:25
Once we've decided the
phis, Galerkin tells us
218
00:13:25 --> 00:13:26
what the equation is.
219
00:13:26 --> 00:13:31
And we'll get to the
KU=F, what it is.
220
00:13:31 --> 00:13:37
But here I'm getting you to
see what our approximate
221
00:13:37 --> 00:13:39
solution can look like.
222
00:13:39 --> 00:13:43
And this beautiful fact that
the coefficients in here
223
00:13:43 --> 00:13:45
have a physical meaning.
224
00:13:45 --> 00:13:49
They're actually the
displacements at the nodes
225
00:13:49 --> 00:13:51
for these simple functions.
226
00:13:51 --> 00:13:54
OK.
227
00:13:54 --> 00:13:56
You got a picture of what the
trial functions are, some
228
00:13:56 --> 00:14:02
people would think about the
functions as these guys.
229
00:14:02 --> 00:14:07
Other people might think of
this picture, the combinations.
230
00:14:07 --> 00:14:11
So those are the individual
basis functions.
231
00:14:11 --> 00:14:15
That's a typical combination
of the typical one.
232
00:14:15 --> 00:14:21
OK, so we're looking for an
equation for these u's.
233
00:14:21 --> 00:14:23
Five equations, of course.
234
00:14:23 --> 00:14:27
Because we've got five u's.
235
00:14:27 --> 00:14:30
So that's my final step here.
236
00:14:30 --> 00:14:33
What are the five equations
for the five u's.
237
00:14:33 --> 00:14:35
And those are the equations
that I'm going to call KU=F.
238
00:14:36 --> 00:14:38
OK.
239
00:14:38 --> 00:14:40
So here's now a
critical moment.
240
00:14:40 --> 00:14:43
Where do the
equations come from?
241
00:14:43 --> 00:14:47
Well, the equations come
from the weak form.
242
00:14:47 --> 00:14:50
So I take the weak form.
243
00:14:50 --> 00:14:59
And in for u, for u here,
I guess it's just there.
244
00:14:59 --> 00:15:00
I put this.
245
00:15:00 --> 00:15:02
I put capital U.
246
00:15:02 --> 00:15:10
So so can I just copy, this
is the weak form for before
247
00:15:10 --> 00:15:11
we've made it discrete.
248
00:15:11 --> 00:15:14
Before we've chosen n phis.
249
00:15:14 --> 00:15:16
OK, now let's choose the
n phis, so now this'll
250
00:15:16 --> 00:15:18
be the weak form.
251
00:15:18 --> 00:15:27
Again, the weak form
with Galerkin.
252
00:15:27 --> 00:15:28
After the decision.
253
00:15:28 --> 00:15:31
So it'll be the integral,
from zero to one of this
254
00:15:31 --> 00:15:33
c(x), whatever it is.
255
00:15:33 --> 00:15:39
Times the u(x), times the
dU/dx, right? dU/dx,
256
00:15:39 --> 00:15:42
so what is dU/dx?
257
00:15:43 --> 00:15:46
Oh, you have to pay attention.
258
00:15:46 --> 00:15:47
This was a true solution.
259
00:15:47 --> 00:15:49
Little u.
260
00:15:49 --> 00:15:51
But now this is
where I'm working.
261
00:15:51 --> 00:15:54
I'm working with capital U.
262
00:15:54 --> 00:15:59
So instead of the d little
u/dx, it's d capital U/dx.
263
00:16:00 --> 00:16:02
Maybe I'll put it in
there. d capital U/dx.
264
00:16:03 --> 00:16:06
And then I'll put down
here what it is.
265
00:16:06 --> 00:16:07
What is it?
266
00:16:07 --> 00:16:13
It's u_0*phi_0, can I
use prime just to, or
267
00:16:13 --> 00:16:18
d phi_0/dx, whatever?
268
00:16:18 --> 00:16:20
Can I use prime for derivative?
269
00:16:20 --> 00:16:22
Just save a little space.
270
00:16:22 --> 00:16:25
So this is the
derivative of my guy.
271
00:16:25 --> 00:16:33
u_1*phi_1'(x), up to whatever
it was. u_4*phi_4'(x).
272
00:16:33 --> 00:16:37
273
00:16:37 --> 00:16:40
That's what that term is.
274
00:16:40 --> 00:16:43
And that multiplies dV/dx.
275
00:16:43 --> 00:16:50
Where V is, here V is
any test function.
276
00:16:50 --> 00:16:53
Any test, function only
required to have V(1)=0.
277
00:16:54 --> 00:17:00
But now, I'm going discrete.
278
00:17:00 --> 00:17:04
So instead of any test
function, I'll use
279
00:17:04 --> 00:17:06
these five functions.
280
00:17:06 --> 00:17:08
So I've got five functions.
281
00:17:08 --> 00:17:11
The phis are the same
as the V's, then.
282
00:17:11 --> 00:17:15
V_1, V_2, V_3 and V_4.
283
00:17:15 --> 00:17:17
Same guys.
284
00:17:17 --> 00:17:22
So now I'll put in dV,
can I say dV_i/dx?
285
00:17:22 --> 00:17:27
286
00:17:27 --> 00:17:32
And on the right hand side I
have the integral from zero to
287
00:17:32 --> 00:17:47
one of F(x)*V_i(x)dx, i is
zero, one, two, three, or four.
288
00:17:47 --> 00:17:50
I'm testing against five V's.
289
00:17:50 --> 00:17:55
So I have this equation
for five different V's.
290
00:17:55 --> 00:17:58
So i equals zero, one, two,
three, four, gives me
291
00:17:58 --> 00:18:00
my five equations.
292
00:18:00 --> 00:18:02
Here are my five unknowns.
293
00:18:02 --> 00:18:05
This is my five by five system.
294
00:18:05 --> 00:18:08
Let me just step back a minute
so you see what happened there.
295
00:18:08 --> 00:18:11
So what do you have to do?
296
00:18:11 --> 00:18:19
You chose the basis functions,
the phis and the V's.
297
00:18:19 --> 00:18:23
Then you just plug into the
weak form, you plug in dU/dx,
298
00:18:23 --> 00:18:25
is coming from there.
299
00:18:25 --> 00:18:27
So this is dU/dx.
300
00:18:27 --> 00:18:32
You have to plug dV/dx, you
have to do the integrals.
301
00:18:32 --> 00:18:34
But to do the integrals, that's
something we didn't have
302
00:18:34 --> 00:18:36
in finite differences.
303
00:18:36 --> 00:18:39
Finite elements involves doing
the integrals, left side
304
00:18:39 --> 00:18:42
and right hand side.
305
00:18:42 --> 00:18:48
OK, so and here we have five
different integrals to do.
306
00:18:48 --> 00:18:52
We have F(x) times each V.
307
00:18:52 --> 00:18:56
This will be, this
number will be F_i.
308
00:18:57 --> 00:19:06
So my f vector is going to be
an F_0, F_1, down to F_4.
309
00:19:06 --> 00:19:10
The five guys that I get
from these five integrals.
310
00:19:10 --> 00:19:13
Alright.
311
00:19:13 --> 00:19:17
And the K matrix is
sitting here somewhere.
312
00:19:17 --> 00:19:20
That's the last thing, that's
the final thing is to see
313
00:19:20 --> 00:19:23
what is the K matrix.
314
00:19:23 --> 00:19:24
Which is coming.
315
00:19:24 --> 00:19:28
This is somehow K's times
U's are sitting here.
316
00:19:28 --> 00:19:30
F's are sitting over there.
317
00:19:30 --> 00:19:33
So it may be good to
see the f first.
318
00:19:33 --> 00:19:34
So do you see this now?
319
00:19:34 --> 00:19:38
We made the choices,
then what's our job?
320
00:19:38 --> 00:19:44
Our next job is to do
all the integrations.
321
00:19:44 --> 00:19:47
Integrate my
function against V.
322
00:19:47 --> 00:19:52
Let's make the natural first
example, let F be one.
323
00:19:52 --> 00:19:56
First example, let F be one.
324
00:19:56 --> 00:19:57
All right.
325
00:19:57 --> 00:20:07
So if F is one, I'm going
to find the system KU=F.
326
00:20:08 --> 00:20:14
So if I know F(x) is one, then
I have everything I need
327
00:20:14 --> 00:20:16
to find these numbers.
328
00:20:16 --> 00:20:24
OK, actually we can probably
do those by, you can probably
329
00:20:24 --> 00:20:26
tell me what they are.
330
00:20:26 --> 00:20:30
If F(x) is one, now.
331
00:20:30 --> 00:20:31
So this is example one.
332
00:20:31 --> 00:20:33
F(x) is a constant.
333
00:20:33 --> 00:20:37
One we have solved before.
334
00:20:37 --> 00:20:41
What's the integral of V_0?
335
00:20:41 --> 00:20:43
Right, that's what
I have to do.
336
00:20:43 --> 00:20:47
What's the integral of this
F(x) being one, I'm just
337
00:20:47 --> 00:20:49
asking, I just have V_0(x)?
338
00:20:49 --> 00:20:53
The integral of V_0(x),
and let me again draw
339
00:20:53 --> 00:20:55
V_0, which is phi_0.
340
00:20:56 --> 00:21:01
It's a half hat and then
it goes along at zero.
341
00:21:01 --> 00:21:04
What's the integral
of that function?
342
00:21:04 --> 00:21:06
One, yeah.
343
00:21:06 --> 00:21:08
How do I think about
that integral?
344
00:21:08 --> 00:21:10
It's the area of the triangle.
345
00:21:10 --> 00:21:11
It's the area.
346
00:21:11 --> 00:21:13
That's what an integral
is, it's the area.
347
00:21:13 --> 00:21:20
So the area is, I've
got delta x there.
348
00:21:20 --> 00:21:22
Right, delta x as the base.
349
00:21:22 --> 00:21:28
One as the height, and you see
the formula for the area of
350
00:21:28 --> 00:21:32
a triangle, it's got a half
in there somewhere, right?
351
00:21:32 --> 00:21:33
A half.
352
00:21:33 --> 00:21:36
OK, can I factor
out the delta x?
353
00:21:36 --> 00:21:38
Because the delta x
is going to come in.
354
00:21:38 --> 00:21:41
I think there's a half there.
355
00:21:41 --> 00:21:43
And then what about F_1?
356
00:21:43 --> 00:21:51
What's the integral of this
times V_1(x), the next V?
357
00:21:51 --> 00:21:56
It's the area under
this dashed function.
358
00:21:56 --> 00:22:00
Which is now the basis 2
delta x, so I get a one.
359
00:22:00 --> 00:22:02
Is that right?
360
00:22:02 --> 00:22:06
I get a one, one, one, one.
361
00:22:06 --> 00:22:10
OK, so that was obviously
not too tough, right?
362
00:22:10 --> 00:22:17
That was straightforward
and notice something.
363
00:22:17 --> 00:22:20
Even here; in fact,
we see it here.
364
00:22:20 --> 00:22:24
I don't know if you
remember about that half.
365
00:22:24 --> 00:22:29
Do you remember something about
when we did finite differences
366
00:22:29 --> 00:22:33
and we had a free boundary?
367
00:22:33 --> 00:22:36
And we lost an order
of accuracy if we
368
00:22:36 --> 00:22:37
didn't do it right?
369
00:22:37 --> 00:22:40
Do you remember that?
370
00:22:40 --> 00:22:43
At the fixed boundary we
were fine, but with finite
371
00:22:43 --> 00:22:46
differences that's a free
boundary where I was
372
00:22:46 --> 00:22:48
using the matrix T.
373
00:22:48 --> 00:22:52
With one minus one
at the top row.
374
00:22:52 --> 00:22:55
I lost an order of accuracy.
375
00:22:55 --> 00:22:59
Unless I made some change
on the right hand side.
376
00:22:59 --> 00:23:01
Look what's happening.
377
00:23:01 --> 00:23:04
The finite element method is
making the change for me
378
00:23:04 --> 00:23:06
on the right hand side.
379
00:23:06 --> 00:23:11
So the finite element method is
going to automatically keep
380
00:23:11 --> 00:23:13
the second order accuracy.
381
00:23:13 --> 00:23:15
Keep the second order accuracy.
382
00:23:15 --> 00:23:18
So that's a key point.
383
00:23:18 --> 00:23:23
That these piecewise linear
functions are associated
384
00:23:23 --> 00:23:27
with second order accuracy.
385
00:23:27 --> 00:23:29
Later we'll move
up to parabolas.
386
00:23:29 --> 00:23:32
To cubics that will
move up the order of
387
00:23:32 --> 00:23:35
accuracy in a nice way.
388
00:23:35 --> 00:23:40
Where with finite differences
we would have had to create new
389
00:23:40 --> 00:23:42
finite difference formulas.
390
00:23:42 --> 00:23:47
Our minus one, two, minus one
formula, that was good for
391
00:23:47 --> 00:23:49
second order accuracy.
392
00:23:49 --> 00:23:53
Then we would have to figure
out in the quiz and partly
393
00:23:53 --> 00:23:57
started it, what if there's a
c(x) in there, what do you do?
394
00:23:57 --> 00:24:00
Finite differences, is
more thinking involved.
395
00:24:00 --> 00:24:04
Finite elements is like
just press the button.
396
00:24:04 --> 00:24:07
Well, there's a little more to
it than that of course, because
397
00:24:07 --> 00:24:09
it's taking a whole lecture.
398
00:24:09 --> 00:24:17
But in the end it's more
systematic, you could say.
399
00:24:17 --> 00:24:19
So that's the F.
400
00:24:19 --> 00:24:22
Now are you ready for the K?
401
00:24:22 --> 00:24:24
So this is the key part, OK?
402
00:24:24 --> 00:24:30
So you have to can get
this thing to simplify.
403
00:24:30 --> 00:24:33
So what am I looking for here?
404
00:24:33 --> 00:24:39
This whole left hand side
should be K times U.
405
00:24:39 --> 00:24:45
So I'm looking to see what
multiplies, I'm looking to
406
00:24:45 --> 00:24:46
make sense out of this.
407
00:24:46 --> 00:24:48
What's the first equation?
408
00:24:48 --> 00:24:52
Right, so the first equation or
the zeroth equation, I guess.
409
00:24:52 --> 00:24:57
The zeroth equation, the one
that'll run along and have this
410
00:24:57 --> 00:25:02
right hand side, the zeroth
equation is the equation
411
00:25:02 --> 00:25:05
when i is zero.
412
00:25:05 --> 00:25:10
It's the equation that
comes from testing our
413
00:25:10 --> 00:25:13
weak form for V_0.
414
00:25:13 --> 00:25:17
For that particular form.
415
00:25:17 --> 00:25:21
Maybe I'll just start
over on this board.
416
00:25:21 --> 00:25:23
Then I can write a formula,
but I'd rather you
417
00:25:23 --> 00:25:26
see how it comes.
418
00:25:26 --> 00:25:31
So I'm looking at
equation zero.
419
00:25:31 --> 00:25:34
So take i=0.
420
00:25:34 --> 00:25:36
421
00:25:36 --> 00:25:40
So I have my left side is
my integral, of c(x).
422
00:25:41 --> 00:25:44
Times this combination
that I wrote, U_0*phi_0'
423
00:25:44 --> 00:25:44
+...+u_4*phi_4'.
424
00:25:44 --> 00:25:52
425
00:25:52 --> 00:26:02
Times dV_0/dx*dx equal, and on
the right side is where I got
426
00:26:02 --> 00:26:05
the F_0, which I already
figured out to be
427
00:26:05 --> 00:26:07
delta x times a half.
428
00:26:07 --> 00:26:10
It's the left side that
I'm worrying about.
429
00:26:10 --> 00:26:14
OK, you see what's
happening here?
430
00:26:14 --> 00:26:19
This is some matrix.
431
00:26:19 --> 00:26:22
Its zeroth row is
what we're finding.
432
00:26:22 --> 00:26:34
Multiplying U_0, U_1, U_2, U_3,
and U_4 equaling the F vector.
433
00:26:34 --> 00:26:37
I'm supposed to be getting the
first row of the matrix, the
434
00:26:37 --> 00:26:42
top row of the matrix,
from the top V.
435
00:26:42 --> 00:26:44
OK, so let's just do these.
436
00:26:44 --> 00:26:47
We've got integrals
to do again.
437
00:26:47 --> 00:26:50
Alright, what is dV_0/dx?
438
00:26:50 --> 00:26:55
439
00:26:55 --> 00:26:56
Do you see?
440
00:26:56 --> 00:26:57
Let me just see?
441
00:26:57 --> 00:27:02
What number is going in here?
442
00:27:02 --> 00:27:03
What number is going in there?
443
00:27:03 --> 00:27:06
Yeah, if we see
that we're golden.
444
00:27:06 --> 00:27:08
What number is going in there?
445
00:27:08 --> 00:27:15
That's the thing that
multiplies U_0 in the first row
446
00:27:15 --> 00:27:19
that means I should use V_0, so
this is the point,
447
00:27:19 --> 00:27:21
this is K_00.
448
00:27:22 --> 00:27:25
And what's its formula?
449
00:27:25 --> 00:27:28
You realize I'm starting
the count at zero because
450
00:27:28 --> 00:27:29
all these counts.
451
00:27:29 --> 00:27:30
So what is K_00?
452
00:27:33 --> 00:27:35
It's an integral.
453
00:27:35 --> 00:27:40
Of what? c(x), good.
454
00:27:40 --> 00:27:47
Times this guy, because
it's multiplying times
455
00:27:47 --> 00:27:52
this guy, V_0. dx.
456
00:27:52 --> 00:27:54
That's what you have to do.
457
00:27:54 --> 00:28:00
That's what you have to do.
c(x) times phi', it's phi_0',
458
00:28:00 --> 00:28:03
that's what would sit there.
459
00:28:03 --> 00:28:11
And maybe, well, let's
figure that one, shall we?
460
00:28:11 --> 00:28:15
I have to know c(x), right,
that's part of the problem.
461
00:28:15 --> 00:28:17
What would you like me
to choose for c(x)?
462
00:28:18 --> 00:28:18
One.
463
00:28:18 --> 00:28:20
Thank you.
464
00:28:20 --> 00:28:22
I'll choose one.
465
00:28:22 --> 00:28:24
Let this be one.
466
00:28:24 --> 00:28:27
Or I could make it capital C
and you would see a capital
467
00:28:27 --> 00:28:30
C appearing everywhere,
but let's make it one.
468
00:28:30 --> 00:28:32
So what are we doing now?
469
00:28:32 --> 00:28:33
What's our equation?
470
00:28:33 --> 00:28:37
Our right hand side is one,
our c(x) is one, our equation
471
00:28:37 --> 00:28:39
has reduced to -U''=1.
472
00:28:41 --> 00:28:44
The first equation
in the course.
473
00:28:44 --> 00:28:50
So we're back to September
the 3rd or whatever it was.
474
00:28:50 --> 00:28:54
But doing it now by
finite elements.
475
00:28:54 --> 00:28:58
OK, so let c(x) be one and tell
me what this integral is.
476
00:28:58 --> 00:29:03
So c(x) is now, we're taking
in our problem we're
477
00:29:03 --> 00:29:05
supposing it's one.
478
00:29:05 --> 00:29:11
Let me just say suppose
it's function.
479
00:29:11 --> 00:29:20
Then we have lots of integrals
to do involving that function.
480
00:29:20 --> 00:29:24
And we might not do them
exactly, that would be alright.
481
00:29:24 --> 00:29:29
It's certainly totally OK to do
the integrals approximately,
482
00:29:29 --> 00:29:33
because we're doing everything
else approximately.
483
00:29:33 --> 00:29:36
So we just have to be sure that
we do the integrals with
484
00:29:36 --> 00:29:39
sufficient accuracy so that we
don't lose accuracy
485
00:29:39 --> 00:29:41
in the integrals.
486
00:29:41 --> 00:29:44
Of course, with a one
we're going to do the
487
00:29:44 --> 00:29:45
integral exactly.
488
00:29:45 --> 00:29:49
But if c(x) was some variable
function, I wouldn't have to do
489
00:29:49 --> 00:29:52
it exactly, I would just have
to do it with enough accuracy
490
00:29:52 --> 00:29:59
so that I don't lose extra
accuracy beyond what I'm losing
491
00:29:59 --> 00:30:02
in the whole Galerkin
approximation.
492
00:30:02 --> 00:30:05
OK, ready for that number.
493
00:30:05 --> 00:30:09
What number comes out of that?
phi_0', let's graph phi_0'.
494
00:30:11 --> 00:30:14
And of course it's the same
as V_0', so can I put
495
00:30:14 --> 00:30:16
a little graph here?
496
00:30:16 --> 00:30:20
Here is zero to one, and
I'm going to graph phi_0'.
497
00:30:22 --> 00:30:23
So what's phi_0'?
498
00:30:23 --> 00:30:26
499
00:30:26 --> 00:30:30
Oh, it negative, isn't it?
500
00:30:30 --> 00:30:32
My little graph isn't
going to work.
501
00:30:32 --> 00:30:35
I didn't even have room.
502
00:30:35 --> 00:30:39
It's negative.
503
00:30:39 --> 00:30:40
So I'll just write it in words.
504
00:30:40 --> 00:30:44
It's the same as V_0',
and what is it?
505
00:30:44 --> 00:30:50
Tell me what it is, what's the
derivative of that function?
506
00:30:50 --> 00:30:52
It's what?
507
00:30:52 --> 00:30:54
Negative one.
508
00:30:54 --> 00:30:57
Wait a minute.
509
00:30:57 --> 00:31:01
Yeah, it's going to have
a certain value, yeah.
510
00:31:01 --> 00:31:04
You can tell me what it is
beyond that point real fast.
511
00:31:04 --> 00:31:13
So it's something up to delta.
512
00:31:13 --> 00:31:15
So what is the slope?
513
00:31:15 --> 00:31:18
What's that slope
there, of phi_0?
514
00:31:20 --> 00:31:25
It's not negative one, because
remember, what's the base here?
515
00:31:25 --> 00:31:30
That's not the point
one, I'm sorry.
516
00:31:30 --> 00:31:36
All these were delta x's.
517
00:31:36 --> 00:31:38
Those were just
numbering the nodes.
518
00:31:38 --> 00:31:43
But the actual length is
scale is the delta x scale.
519
00:31:43 --> 00:31:46
So now tell me what it is.
520
00:31:46 --> 00:31:52
The derivative is negative
one over delta x, right?
521
00:31:52 --> 00:31:56
It dropped by one when it
went across by delta x.
522
00:31:56 --> 00:32:02
And this is only
up to node one.
523
00:32:02 --> 00:32:06
Up to delta x and then
zero afterwards.
524
00:32:06 --> 00:32:09
This is a key point.
525
00:32:09 --> 00:32:14
That all our
functions are local.
526
00:32:14 --> 00:32:16
Our functions are local.
527
00:32:16 --> 00:32:18
What does that mean?
528
00:32:18 --> 00:32:24
You can tell me what am I going
to get when I integrate, for
529
00:32:24 --> 00:32:29
example, when later on I might
be integrating phi_1'
530
00:32:29 --> 00:32:31
against V_4'.
531
00:32:31 --> 00:32:34
532
00:32:34 --> 00:32:37
What's the answer?
533
00:32:37 --> 00:32:39
This is the key point.
534
00:32:39 --> 00:32:43
Later, when I'm looking for the
one, four entry, when I'm
535
00:32:43 --> 00:32:47
looking there, I'm going to
do an integral of phi_1'.
536
00:32:49 --> 00:32:54
I'll erase for a moment
and do this in my head.
537
00:32:54 --> 00:32:59
When I integrate
phi_1' against V_4'.
538
00:33:01 --> 00:33:06
Maybe it's the fourth row.
539
00:33:06 --> 00:33:09
And the first guy over, maybe
it's this guy I'm doing.
540
00:33:09 --> 00:33:11
Doesn't matter a whole lot.
541
00:33:11 --> 00:33:18
Because the answer is, when I
integrate phi_1' against V_4'
542
00:33:18 --> 00:33:22
just, it's nice to
get the easy ones.
543
00:33:22 --> 00:33:23
It's zero.
544
00:33:23 --> 00:33:26
Why is it zero?
545
00:33:26 --> 00:33:31
Why is the integral of
phi_1' against V_4' zero?
546
00:33:31 --> 00:33:37
Because these phis are local.
phi_1' is only non-zero here.
547
00:33:37 --> 00:33:41
V_4' is only
non-zero over here.
548
00:33:41 --> 00:33:45
The two don't overlap.
549
00:33:45 --> 00:33:48
Anywhere the one is not
zero, the other is zero.
550
00:33:48 --> 00:33:52
So that's a zero there.
551
00:33:52 --> 00:33:59
In fact, our overlaps, I'm just
sort of looking ahead here.
552
00:33:59 --> 00:34:05
Our overlaps, a phi overlaps
itself, of course.
553
00:34:05 --> 00:34:08
And its right hand neighbor
and its left hand neighbor.
554
00:34:08 --> 00:34:13
But nobody two or
three or more away.
555
00:34:13 --> 00:34:17
I think our K, all our
integrals are going to be zero
556
00:34:17 --> 00:34:21
outside, we'll have another
tri-diagonal matrix.
557
00:34:21 --> 00:34:24
We're going to have
zeroes all here.
558
00:34:24 --> 00:34:37
And we'll only have entries
where phi against V when
559
00:34:37 --> 00:34:40
they're either the same
or just differ by one.
560
00:34:40 --> 00:34:45
So we'll only have
three diagonals.
561
00:34:45 --> 00:34:48
OK, we were about to find
out what that number is.
562
00:34:48 --> 00:34:54
So the slope of this is minus
one over delta x, and that's -
563
00:34:54 --> 00:34:59
I'm sorry, let me go
back to zero, zero.
564
00:34:59 --> 00:35:02
OK.
565
00:35:02 --> 00:35:06
This is what we're keeping
our fingers crossed for.
566
00:35:06 --> 00:35:09
What's that number?
567
00:35:09 --> 00:35:15
So I have this thing,
actually is it just squared?
568
00:35:15 --> 00:35:17
And that's the slope.
569
00:35:17 --> 00:35:21
And then the phi and the V
I'm choosing the same, so
570
00:35:21 --> 00:35:23
that's the slope again.
571
00:35:23 --> 00:35:28
I think I'm just getting one
over delta x squared for that
572
00:35:28 --> 00:35:30
times that times the one.
573
00:35:30 --> 00:35:31
So what's K_00?
574
00:35:33 --> 00:35:35
One over delta x.
575
00:35:35 --> 00:35:37
Where'd the delta x come from?
576
00:35:37 --> 00:35:43
Because we're only integrating
over, it looks like zero to
577
00:35:43 --> 00:35:46
one but they're all zero.
578
00:35:46 --> 00:35:50
We're really only integrating,
the only reality was
579
00:35:50 --> 00:35:51
out to node one.
580
00:35:51 --> 00:35:53
Out to delta x.
581
00:35:53 --> 00:35:59
You see that the number there,
the number here on the
582
00:35:59 --> 00:36:08
diagonal is one over delta x.
583
00:36:08 --> 00:36:14
OK, how about doing
K_11 for me?
584
00:36:14 --> 00:36:17
So again, now these guys
will be the same guys.
585
00:36:17 --> 00:36:19
It's a square.
586
00:36:19 --> 00:36:24
No it's the integral phi_1'
against V_1', they're the same.
587
00:36:24 --> 00:36:31
And what is phi_1', which
is the same as V_1'?
588
00:36:32 --> 00:36:35
What's the derivative now?
589
00:36:35 --> 00:36:39
It's, ah.
590
00:36:39 --> 00:36:42
What's the slope
of this function?
591
00:36:42 --> 00:36:46
It goes up and goes
back down, right?
592
00:36:46 --> 00:36:51
I have a plus part, so the
slope going up is the
593
00:36:51 --> 00:36:53
one over delta x.
594
00:36:53 --> 00:36:57
And then the slope coming down
is minus one over delta x.
595
00:36:57 --> 00:37:05
So this was up to delta
x and then to two delta
596
00:37:05 --> 00:37:08
x, and then zero.
597
00:37:08 --> 00:37:12
That's a much more typical
thing, this smoke goes the
598
00:37:12 --> 00:37:16
function, the hat function goes
up to the top of the hat.
599
00:37:16 --> 00:37:18
Back down.
600
00:37:18 --> 00:37:21
The slope up and the
slope down are easy.
601
00:37:21 --> 00:37:23
And now the integral's easy.
602
00:37:23 --> 00:37:28
So I'm just squaring, well,
when I square it, this
603
00:37:28 --> 00:37:30
squared is the one
over delta x squared.
604
00:37:30 --> 00:37:34
This is the same, because
the minus will get squared.
605
00:37:34 --> 00:37:35
So what's K_11?
606
00:37:37 --> 00:37:41
What's K_11 now?
607
00:37:41 --> 00:37:46
Have you got K_11 in your head?
608
00:37:46 --> 00:37:50
This is one over
delta x squared.
609
00:37:50 --> 00:37:54
And now what is the integral?
610
00:37:54 --> 00:37:57
Two over delta x.
611
00:37:57 --> 00:38:04
Because now we're integrating
from zero to two delta x,
612
00:38:04 --> 00:38:07
because that's where my
functions are going out
613
00:38:07 --> 00:38:09
from zero to node two.
614
00:38:09 --> 00:38:12
If the function's numbered one.
615
00:38:12 --> 00:38:17
So it's two delta x times this;
I think we get a two over
616
00:38:17 --> 00:38:22
delta x on that diagonal.
617
00:38:22 --> 00:38:28
Would you care to guess
the rest of the diagonal?
618
00:38:28 --> 00:38:34
Yes, you tell me what's
K_22 and K_33 and K_44?
619
00:38:36 --> 00:38:37
They're all the same.
620
00:38:37 --> 00:38:39
We're just shifting over.
621
00:38:39 --> 00:38:44
So two over delta x
goes down there.
622
00:38:44 --> 00:38:47
Alright, one more to do.
623
00:38:47 --> 00:38:50
One more integral to do.
624
00:38:50 --> 00:38:53
This next guy.
625
00:38:53 --> 00:38:56
So can you tell me what
do I get now for K_01?
626
00:38:58 --> 00:39:07
K_01, so now this is the case
where I'm in row zero, so this
627
00:39:07 --> 00:39:12
should be V_0, because that
tells me the row I'm in.
628
00:39:12 --> 00:39:13
But phi_1.
629
00:39:13 --> 00:39:17
630
00:39:17 --> 00:39:22
What happens when I integrate,
just see the picture here.
631
00:39:22 --> 00:39:28
Let me just draw it small.
phi_1', so let me draw phi_1.
632
00:39:30 --> 00:39:33
And V_0.
633
00:39:33 --> 00:39:35
OK.
634
00:39:35 --> 00:39:37
But it's the derivatives
that I want.
635
00:39:37 --> 00:39:39
It's the slopes
that I want, OK?
636
00:39:39 --> 00:39:45
So what do I get
from here on out?
637
00:39:45 --> 00:39:48
Zero, because this guy
only got to there.
638
00:39:48 --> 00:39:51
That's the half hat, the first
guy stopped at delta x.
639
00:39:51 --> 00:39:53
So whatever is happening
here is going to be
640
00:39:53 --> 00:39:55
multiplied by zero.
641
00:39:55 --> 00:39:57
So it's just here.
642
00:39:57 --> 00:40:04
One delta x integral
for this one.
643
00:40:04 --> 00:40:07
They just overlap in one
interval, of course.
644
00:40:07 --> 00:40:11
This guy and its neighbor only
overlap in one interval.
645
00:40:11 --> 00:40:15
And what's the deal
about the two slopes?
646
00:40:15 --> 00:40:17
They're opposite.
647
00:40:17 --> 00:40:20
One's coming down,
one's going up.
648
00:40:20 --> 00:40:23
But the slopes are one
over delta x and minus
649
00:40:23 --> 00:40:24
one over delta x.
650
00:40:24 --> 00:40:26
Do you see what's
happening here?
651
00:40:26 --> 00:40:26
I'm integrating.
652
00:40:26 --> 00:40:31
Here I have a slowpoke of one
over delta x, and here I have a
653
00:40:31 --> 00:40:35
slope of minus one over delta
x, so I should multiply those.
654
00:40:35 --> 00:40:41
Minus one over delta x squared
integrate, what goes in K_01?
655
00:40:41 --> 00:40:47
What's that number?
656
00:40:47 --> 00:40:53
It's that times that
integrated, but now the
657
00:40:53 --> 00:40:58
integral is only going really
out to delta x because
658
00:40:58 --> 00:41:01
basically I'm just
stopping there.
659
00:41:01 --> 00:41:03
So but there's a minus now.
660
00:41:03 --> 00:41:05
Because it's not the square.
661
00:41:05 --> 00:41:07
It's this times its neighbor.
662
00:41:07 --> 00:41:10
One's going up, and
one's going down.
663
00:41:10 --> 00:41:13
So it's delta x squared,
and then the length of
664
00:41:13 --> 00:41:23
the integral, this is a
minus one over delta x.
665
00:41:23 --> 00:41:28
Would you care to guess
the rest of this matrix?
666
00:41:28 --> 00:41:32
What's the rest of
that diagonal, above
667
00:41:32 --> 00:41:34
the main diagonal?
668
00:41:34 --> 00:41:36
It's all the same.
669
00:41:36 --> 00:41:44
That stays the same, because
when I do phi_2*V_1, my picture
670
00:41:44 --> 00:41:47
is just like shifted over.
671
00:41:47 --> 00:41:50
But I still have one coming
down, and one going up.
672
00:41:50 --> 00:41:53
When I do phi_3*V_2,
same thing.
673
00:41:53 --> 00:42:00
And if I do phi_1*V_2,
it'll be the same.
674
00:42:00 --> 00:42:04
I'm going to get this minus
one over delta x all the
675
00:42:04 --> 00:42:07
way on that diagonal, also.
676
00:42:07 --> 00:42:08
Symmetry.
677
00:42:08 --> 00:42:10
It's going to come
out symmetric.
678
00:42:10 --> 00:42:14
Actually, since the course
started by speaking about
679
00:42:14 --> 00:42:19
properties of matrices, let me
just say K is going to turn
680
00:42:19 --> 00:42:21
out to be symmetric
positive definite.
681
00:42:21 --> 00:42:24
And what's more, for this
example we recognize
682
00:42:24 --> 00:42:27
K completely.
683
00:42:27 --> 00:42:33
You will say why did you take
so long to get to this result.
684
00:42:33 --> 00:42:40
K is the one over delta x part
times, what's the matrix?
685
00:42:40 --> 00:42:42
It's T.
686
00:42:42 --> 00:42:43
It's T.
687
00:42:43 --> 00:42:49
So it's one, minus one, minus
one, two, minus one, minus one,
688
00:42:49 --> 00:42:53
two, minus one, minus one, two,
minus one, and I guess
689
00:42:53 --> 00:42:57
we had five of them.
690
00:42:57 --> 00:43:01
Oh, but nobody there, right?
691
00:43:01 --> 00:43:02
That's not there.
692
00:43:02 --> 00:43:06
That fixed, why do we
not have a minus one?
693
00:43:06 --> 00:43:09
Because we've got no,
there isn't a six.
694
00:43:09 --> 00:43:10
That would be column.
695
00:43:10 --> 00:43:13
We've got one, two, three,
four, five columns; there's
696
00:43:13 --> 00:43:20
no U_5, there's no V_5,
we've got them all.
697
00:43:20 --> 00:43:27
And the F, so that, so KU, this
thing multiplies U_0 to U_4, to
698
00:43:27 --> 00:43:35
U_4, and it produces F, which
is one over delta,
699
00:43:35 --> 00:43:39
which is what?
700
00:43:39 --> 00:43:44
Oh, delta x is in the
numerator, right.
701
00:43:44 --> 00:43:53
Times a half, one,
one, one and one.
702
00:43:53 --> 00:43:56
That is the finite
element system KU=F.
703
00:43:56 --> 00:44:00
704
00:44:00 --> 00:44:04
For this simple problem.
705
00:44:04 --> 00:44:06
It's exactly what finite
differences did.
706
00:44:06 --> 00:44:11
So you can see why my first
introduction to finite elements
707
00:44:11 --> 00:44:15
was with the question
what's the difference.
708
00:44:15 --> 00:44:21
The finite element community
at that point, this was like
709
00:44:21 --> 00:44:24
the golden age of finite
elements, all this was just
710
00:44:24 --> 00:44:26
beginning to be created.
711
00:44:26 --> 00:44:33
These elements were being used.
712
00:44:33 --> 00:44:36
Especially in civil and
structural engineering, that's
713
00:44:36 --> 00:44:40
where a lot of the earliest
papers came out of.
714
00:44:40 --> 00:44:45
And then, in a model problem
it didn't look anything new.
715
00:44:45 --> 00:44:50
It looked like our original
finite difference matrix.
716
00:44:50 --> 00:44:55
But there were some new things.
717
00:44:55 --> 00:44:59
First, there was this new 1/2,
that we hadn't particularly
718
00:44:59 --> 00:45:02
noticed with finite
differences.
719
00:45:02 --> 00:45:04
We we could catch onto that.
720
00:45:04 --> 00:45:06
Here's a minor difference.
721
00:45:06 --> 00:45:10
You notice that the delta x is
strictly speaking the delta x
722
00:45:10 --> 00:45:15
is up here then, but when I
divide by delta x then I'm back
723
00:45:15 --> 00:45:16
to the finite difference.
724
00:45:16 --> 00:45:19
I have the one over delta
x squared, it looks like
725
00:45:19 --> 00:45:22
finite differences again.
726
00:45:22 --> 00:45:24
So everything looks the same.
727
00:45:24 --> 00:45:30
But, of course, if c(x)
isn't one or if F(x)
728
00:45:30 --> 00:45:31
isn't one, oh yeah.
729
00:45:31 --> 00:45:37
If c(x) isn't one then
I've got integrals to do.
730
00:45:37 --> 00:45:38
I would approximate those.
731
00:45:38 --> 00:45:41
And I could then come out with
something that would look
732
00:45:41 --> 00:45:43
like a finite differences.
733
00:45:43 --> 00:45:46
Let me take our other
favorite model problem.
734
00:45:46 --> 00:45:51
What would be the F if,
yeah, here's a question.
735
00:45:51 --> 00:45:56
What would be the right side if
my vector, instead of being
736
00:45:56 --> 00:46:00
one, what's my other
favorite choice?
737
00:46:00 --> 00:46:03
Delta.
738
00:46:03 --> 00:46:08
So I take delta at x minus,
let me take delta at
739
00:46:08 --> 00:46:10
x minus 1/4, first.
740
00:46:10 --> 00:46:12
Suppose that's my F.
741
00:46:12 --> 00:46:18
Then I've got to change
all these guys.
742
00:46:18 --> 00:46:20
And what would they be?
743
00:46:20 --> 00:46:25
What would be the new
right hand side when I
744
00:46:25 --> 00:46:33
have this point low?
745
00:46:33 --> 00:46:35
I have to go back to
the integrals, right?
746
00:46:35 --> 00:46:38
I have to go back
to these guys.
747
00:46:38 --> 00:46:43
These integrals, with that new
F, this is now delta of x
748
00:46:43 --> 00:46:48
minus a 1/4, times each V.
749
00:46:48 --> 00:46:52
Times dx, I have to integrate
delta of x minus 1/4
750
00:46:52 --> 00:46:54
against every hat function.
751
00:46:54 --> 00:46:56
And see what it equals?
752
00:46:56 --> 00:46:59
And what will I get?
753
00:46:59 --> 00:47:00
You're going to tell
me right away.
754
00:47:00 --> 00:47:04
What are those integrals?
755
00:47:04 --> 00:47:08
That's a point
load at node one.
756
00:47:08 --> 00:47:14
Times the V integrated
over the whole thing.
757
00:47:14 --> 00:47:18
What do I get?
758
00:47:18 --> 00:47:21
I get a one, yeah.
759
00:47:21 --> 00:47:25
That integral is going to pick
out the value at a quarter.
760
00:47:25 --> 00:47:27
Right, that's what the
delta function does, the
761
00:47:27 --> 00:47:29
spike is at a quarter.
762
00:47:29 --> 00:47:33
Has area one, so it picks
out the V_i at a quarter.
763
00:47:33 --> 00:47:37
V_i at a quarter will be?
764
00:47:37 --> 00:47:46
One for the, I think we
get a .
765
00:47:46 --> 00:47:56
Again a little bit what our
finite differences suggested
766
00:47:56 --> 00:47:59
that we should do.
767
00:47:59 --> 00:48:02
Alright, here's one
final one for today.
768
00:48:02 --> 00:48:05
Suppose the delta function
is not at a node.
769
00:48:05 --> 00:48:10
Suppose it's at 3/8.
770
00:48:10 --> 00:48:15
Or it could be at any point
a, but let me just take a
771
00:48:15 --> 00:48:18
typical, a special one
where I can do it.
772
00:48:18 --> 00:48:21
Suppose the load is at 3/8.
773
00:48:22 --> 00:48:27
What do I get for
the integrals now?
774
00:48:27 --> 00:48:31
So now, it's delta
of x minus 3/8.
775
00:48:31 --> 00:48:40
The spike is at
this point here.
776
00:48:40 --> 00:48:44
That's where delta is
now, spiking at 3/8.
777
00:48:44 --> 00:48:47
Is that right?
778
00:48:47 --> 00:48:50
We had 1/4, tell me what
- oh, I should have
779
00:48:50 --> 00:48:53
had 1/5 before, sorry.
780
00:48:53 --> 00:48:55
Change that on the videotape.
781
00:48:55 --> 00:48:59
All those 1/4s where
that 1/4 was 1/5, and
782
00:48:59 --> 00:49:01
now what do I want?
783
00:49:01 --> 00:49:03
Three?
784
00:49:03 --> 00:49:07
I wanted to take a nice
one that was halfway.
785
00:49:07 --> 00:49:09
I just forgot what halfway was.
786
00:49:09 --> 00:49:11
Where is halfway there?
787
00:49:11 --> 00:49:16
3/10 now for delta.
788
00:49:16 --> 00:49:19
So that was before I
had it for when delta.
789
00:49:19 --> 00:49:28
So previously was delta at x
minus 1/5 and now delta at x
790
00:49:28 --> 00:49:34
minus 3/10, what's the F now?
791
00:49:34 --> 00:49:35
What's the F now?
792
00:49:35 --> 00:49:39
So the spike is right in the
middle between one and two.
793
00:49:39 --> 00:49:43
What's do those integrals
come out to be?
794
00:49:43 --> 00:49:48
If I integrate delta function
times the different
795
00:49:48 --> 00:49:51
hats, what do I get?
796
00:49:51 --> 00:49:52
What do I get, yeah.
797
00:49:52 --> 00:49:56
I get a zero for this first
guy because it didn't
798
00:49:56 --> 00:49:57
touch the half hat.
799
00:49:57 --> 00:50:00
And then what do I get there?
800
00:50:00 --> 00:50:01
Half.
801
00:50:01 --> 00:50:03
And what do I get
at the next one?
802
00:50:03 --> 00:50:04
Half again.
803
00:50:04 --> 00:50:07
And then the other guys
it doesn't touch.
804
00:50:07 --> 00:50:09
You see, it
automatically does it.
805
00:50:09 --> 00:50:11
Does those smart things.
806
00:50:11 --> 00:50:14
It automatically makes
the smart choice.
807
00:50:14 --> 00:50:22
And if the spike was at a, at
any point a, then at that
808
00:50:22 --> 00:50:26
typical point a wherever it
is, like there, spike
809
00:50:26 --> 00:50:27
could be there.
810
00:50:27 --> 00:50:31
Then I would have, what would I
have if the spike was there?
811
00:50:31 --> 00:50:35
I'd have a little bit of phi_3,
and a big bit of phi_4.
812
00:50:36 --> 00:50:38
And the two parts
would add to one.
813
00:50:38 --> 00:50:40
It would take the
right proportion.
814
00:50:40 --> 00:50:47
It would be the proportion, by
however much this spike was
815
00:50:47 --> 00:50:50
near there, it would give
that extra weight to phi_4.
816
00:50:51 --> 00:50:54
OK.
817
00:50:54 --> 00:50:57
So there is the finite
element method.
818
00:50:57 --> 00:51:02
It produced something that you
might say, oh, we knew that.
819
00:51:02 --> 00:51:08
But you've got to see that it
deals automatically with c(x),
820
00:51:08 --> 00:51:12
it deals automatically with
F(x), it deals automatically
821
00:51:12 --> 00:51:13
with the free boundary.
822
00:51:13 --> 00:51:20
You see, the solution there is
going to take sort of a
823
00:51:20 --> 00:51:24
balance, a pretty close
balance, of this half hat with
824
00:51:24 --> 00:51:29
this one, and the solution will
actually be a pretty
825
00:51:29 --> 00:51:30
close to free.
826
00:51:30 --> 00:51:34
It'll be pretty close to having
the right zero slope there.
827
00:51:34 --> 00:51:36
OK, good.