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PROFESSOR STRANG: So, let's
see, you probably guessed on
10
00:00:25 --> 00:00:31
that quiz problem three,
it wasn't what I meant.
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00:00:31 --> 00:00:34
I get a zero for that problem.
12
00:00:34 --> 00:00:37
But you'll get probably good
numbers, so that's an election
13
00:00:37 --> 00:00:41
gift if it comes out that way.
14
00:00:41 --> 00:00:46
So they're all in the hands
of the TAs to be graded.
15
00:00:46 --> 00:00:48
We have a holiday
Monday, I think.
16
00:00:48 --> 00:00:52
We come back to Fourier.
17
00:00:52 --> 00:00:57
Now, so we just have a
concentrated shot at Fourier,
18
00:00:57 --> 00:01:01
just about eight or nine
lectures in November.
19
00:01:01 --> 00:01:05
So stay with it and that'll
be of course the subject
20
00:01:05 --> 00:01:07
of the third quiz.
21
00:01:07 --> 00:01:09
Which will have no mistakes.
22
00:01:09 --> 00:01:14
It'll be solved by the TAs in
advance and we'll spot things.
23
00:01:14 --> 00:01:19
So, and if we have the quizzes
to return to you by Wednesday
24
00:01:19 --> 00:01:20
that will be great.
25
00:01:20 --> 00:01:26
I hope so, but they
have a big job.
26
00:01:26 --> 00:01:31
A little bit, looking far ahead
at the end of Fourier the quiz
27
00:01:31 --> 00:01:35
is December 4th, I think
that's a Thursday.
28
00:01:35 --> 00:01:39
And that's the end
of the course.
29
00:01:39 --> 00:01:41
So December 4th, so we'll
be ending the course
30
00:01:41 --> 00:01:45
a little bit early.
31
00:01:45 --> 00:01:52
Because I'll be in Hong
Kong, to tell the truth.
32
00:01:52 --> 00:01:55
And and we've done a lot, and
with the review sessions
33
00:01:55 --> 00:01:57
we're really doing well.
34
00:01:57 --> 00:02:03
So, that's the future.
35
00:02:03 --> 00:02:09
Fourier, today is an important
day too, finite elements in
36
00:02:09 --> 00:02:15
2-D, that's a major part of
computational science
37
00:02:15 --> 00:02:17
and engineering.
38
00:02:17 --> 00:02:21
The finite element idea, the
idea of using polynomials, you
39
00:02:21 --> 00:02:28
can find in some early papers
by Courant, a mathematician in
40
00:02:28 --> 00:02:36
New York, and by a guy in China
neat guy named Fung Kong But
41
00:02:36 --> 00:02:39
those papers were sort of,
you could do it this
42
00:02:39 --> 00:02:41
way if you wanted.
43
00:02:41 --> 00:02:46
It was really the structural
engineers in Berkeley and
44
00:02:46 --> 00:02:50
elsewhere who made it
happen ten years later.
45
00:02:50 --> 00:02:54
And the whole idea
has just blossomed.
46
00:02:54 --> 00:02:56
Continues to grow.
47
00:02:56 --> 00:03:04
So I had an early book, in the
`70s, actually, about the
48
00:03:04 --> 00:03:07
mathematical underpinnings.
49
00:03:07 --> 00:03:11
The math basis for the
finite element method.
50
00:03:11 --> 00:03:15
And many other finite
element books.
51
00:03:15 --> 00:03:18
Professor Bathe you
know, teaches a full
52
00:03:18 --> 00:03:20
of course on that.
53
00:03:20 --> 00:03:26
But, I think we can get the
idea of finite elements here.
54
00:03:26 --> 00:03:30
We did them in 1-D, and now
there's a MATLAB problem and
55
00:03:30 --> 00:03:33
I'd like to just describe
that particular problem
56
00:03:33 --> 00:03:36
if I can, as an example.
57
00:03:36 --> 00:03:40
And, of course, you would use
the code that's printed in the
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book, and that's available on
the website just to download.
59
00:03:48 --> 00:03:53
But the problem is not
on a square domain.
60
00:03:53 --> 00:03:57
It starts on a circle, so that
the first lines of the code,
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00:03:57 --> 00:04:04
the calling the MATLAB command
square grid, are
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00:04:04 --> 00:04:09
not applicable.
63
00:04:09 --> 00:04:12
So you have to create,
then, a mesh.
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00:04:12 --> 00:04:17
Well, I have a suggested mesh
so I'll draw that, and then
65
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from that you want to make a
list of all the node points.
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A list, P, of - so what the
code needs is two lists.
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00:04:32 --> 00:04:39
Well, let me draw a picture
of, well, it's a circle.
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00:04:39 --> 00:04:43
And I'm going to be solving
Poisson's equation.
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The equation will be
-u_xx-u_yy=4, in the circle.
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00:04:51 --> 00:04:54
So it's Poisson but with a
constant right hand side.
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00:04:54 --> 00:05:01
That will mean that all the
integrals of F times v, all the
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00:05:01 --> 00:05:05
right hand side of our discrete
equation will be, the integrals
73
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are all easy because we
just have a constant there
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times the trial function.
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00:05:11 --> 00:05:14
OK, and then on the boundary
is going to be u=0.
76
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On the boundary.
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So it's a classic problem.
78
00:05:19 --> 00:05:23
And we can say what
the solution is.
79
00:05:23 --> 00:05:25
So it's one with a
known solution.
80
00:05:25 --> 00:05:29
I think it would be x squared.
81
00:05:29 --> 00:05:34
No, I guess one, one minus
x squared minus y squared.
82
00:05:34 --> 00:05:36
This should all be
on the .086 site.
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00:05:36 --> 00:05:38
I just didn't have a chance
to look this morning
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00:05:38 --> 00:05:40
to be sure it got up.
85
00:05:40 --> 00:05:42
So you can watch, here.
86
00:05:42 --> 00:05:44
So that, I hope, does
solve the problem.
87
00:05:44 --> 00:05:48
Two x derivatives give us
a two, two y derivatives
88
00:05:48 --> 00:05:50
another two, so we get four.
89
00:05:50 --> 00:05:54
So we know the answer; the
question is, and I'm interested
90
00:05:54 --> 00:06:02
in this question, for research
reasons too, is what's the
91
00:06:02 --> 00:06:10
error when you go to a polygon?
92
00:06:10 --> 00:06:16
You go to a, these curved
boundaries don't get
93
00:06:16 --> 00:06:18
correctly saved.
94
00:06:18 --> 00:06:22
You approximate them
by straight lines.
95
00:06:22 --> 00:06:24
That would be the first idea.
96
00:06:24 --> 00:06:28
And with this, all this
symmetry, let's keep the
97
00:06:28 --> 00:06:31
problem nice and use
a regular polygon.
98
00:06:31 --> 00:06:34
So maybe I'll try to draw
one with about eight
99
00:06:34 --> 00:06:38
sides, but, OK.
100
00:06:38 --> 00:06:45
So we impose u=0
at these nodes.
101
00:06:45 --> 00:06:50
So u is zero at those nodes,
and then we have a mesh.
102
00:06:50 --> 00:06:53
So we want to create a mesh.
103
00:06:53 --> 00:06:59
OK, so with all the symmetry
here, the natural idea would be
104
00:06:59 --> 00:07:11
to start with eight pieces, or
M pieces if I have, this is a
105
00:07:11 --> 00:07:18
regular M side, let's say, and
I'll take M to be eight
106
00:07:18 --> 00:07:19
in this picture.
107
00:07:19 --> 00:07:24
And I think we can work
on just one triangle.
108
00:07:24 --> 00:07:28
By rotational symmetry,
all those triangles are
109
00:07:28 --> 00:07:29
going to be the same.
110
00:07:29 --> 00:07:35
So I think our domain is really
this one triangle here.
111
00:07:35 --> 00:07:36
That's where we're working.
112
00:07:36 --> 00:07:39
And in that triangle,
I think we have zero
113
00:07:39 --> 00:07:43
boundary conditions.
114
00:07:43 --> 00:07:50
And across this edge I
think we have natural
115
00:07:50 --> 00:07:51
boundary conditions.
116
00:07:51 --> 00:07:55
Slope zero, if I see
the picture correctly.
117
00:07:55 --> 00:07:58
The rotational symmetry
would mean that things
118
00:07:58 --> 00:08:00
are not changing.
119
00:08:00 --> 00:08:02
That every triangle
is the same.
120
00:08:02 --> 00:08:05
So I think on these
boundaries it's the Neumann
121
00:08:05 --> 00:08:06
condition, dU/dn=0.
122
00:08:06 --> 00:08:10
123
00:08:10 --> 00:08:15
And I'm frankly not sure what
to do at the origin, so I'll
124
00:08:15 --> 00:08:18
maybe just try both
ways and see.
125
00:08:18 --> 00:08:22
OK, so there is a real problem.
126
00:08:22 --> 00:08:25
Of course, it's artificial
in the sense that
127
00:08:25 --> 00:08:27
we know the answer.
128
00:08:27 --> 00:08:34
But it's a real open question
of what does the error look
129
00:08:34 --> 00:08:36
like, from doing that.
130
00:08:36 --> 00:08:41
So that's the goal and let me
just say the problem I'll ask
131
00:08:41 --> 00:08:46
you to do, and it probably is
quite enough to be ready for
132
00:08:46 --> 00:08:51
next Friday, is to use
piecewise linear elements.
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00:08:51 --> 00:08:55
Which is what I'm going to do.
134
00:08:55 --> 00:08:59
What every discussion of
finite elements will begin
135
00:08:59 --> 00:09:00
with, linear elements.
136
00:09:00 --> 00:09:05
Those pyramids that I
spoke about it at the
137
00:09:05 --> 00:09:06
end of last time.
138
00:09:06 --> 00:09:09
So that's what I hope, but
actually I would be highly
139
00:09:09 --> 00:09:14
interested if anybody got
into the problem, to
140
00:09:14 --> 00:09:16
try quadratic elements.
141
00:09:16 --> 00:09:25
So I'll just say here,
second-degree quadratic
142
00:09:25 --> 00:09:34
polynomials would
be more accurate.
143
00:09:34 --> 00:09:36
Would be more accurate.
144
00:09:36 --> 00:09:44
So, in other words, this is a
first type of finite element
145
00:09:44 --> 00:09:48
called P_1, for polynomials
of degree one.
146
00:09:48 --> 00:09:52
These guys, I would call P_2,
for polynomials of degree two.
147
00:09:52 --> 00:09:59
And I've mentioned here the
possibility of using quads.
148
00:09:59 --> 00:10:04
Instead of triangles, if I had
squares for example, the
149
00:10:04 --> 00:10:06
simplest element
would be a Q_1.
150
00:10:08 --> 00:10:13
So these are, if I manage today
to tell you about how to use
151
00:10:13 --> 00:10:19
P_1 and P_2 and Q_1,
you're on your way.
152
00:10:19 --> 00:10:25
And for the requirements of
this course, P_1 is the
153
00:10:25 --> 00:10:27
first point to understand.
154
00:10:27 --> 00:10:28
OK.
155
00:10:28 --> 00:10:32
While I'm speaking about codes
and meshes, let me draw the
156
00:10:32 --> 00:10:38
mesh I proposed in the
homework problem.
157
00:10:38 --> 00:10:42
So I thought OK, we just have
to have a simple mesh here.
158
00:10:42 --> 00:10:47
So let me draw that line in.
159
00:10:47 --> 00:10:48
I know all these points, right?
160
00:10:48 --> 00:10:52
This is 0, 0 here.
161
00:10:52 --> 00:10:57
And that point's on the circle.
162
00:10:57 --> 00:11:01
And so is this, that point
might be, so what's
163
00:11:01 --> 00:11:04
the angle there?
164
00:11:04 --> 00:11:07
That angle is probably pi/8.
165
00:11:09 --> 00:11:13
The whole angle would be 2pi/8,
and eight of them would
166
00:11:13 --> 00:11:14
go all the way around.
167
00:11:14 --> 00:11:16
So I think that angle is pi/8.
168
00:11:17 --> 00:11:20
And so this point
would be cos(pi/8).
169
00:11:22 --> 00:11:23
sin(pi/8).
170
00:11:23 --> 00:11:29
We know where they are.
171
00:11:29 --> 00:11:32
But that's going to be
what we have to list.
172
00:11:32 --> 00:11:34
We have to list where
are the coordinates of
173
00:11:34 --> 00:11:35
all the mesh points.
174
00:11:35 --> 00:11:40
So let me describe the rest of
the mesh points and then see
175
00:11:40 --> 00:11:43
what that list would look like.
176
00:11:43 --> 00:11:51
And once you've created the
list, the code will take over.
177
00:11:51 --> 00:11:55
And then you plot the results
and see what's going on.
178
00:11:55 --> 00:11:59
So let me suggest a mesh.
179
00:11:59 --> 00:12:01
It's pretty straightforward.
180
00:12:01 --> 00:12:12
I just divided this piece into
N, this center thing into N,
181
00:12:12 --> 00:12:15
so I called that distance h.
182
00:12:15 --> 00:12:20
So Nh gets me out to here.
183
00:12:20 --> 00:12:25
Whatever that is, that's
cos(pi/8), I guess, that's the
184
00:12:25 --> 00:12:27
x coordinate of that line.
185
00:12:27 --> 00:12:31
So those are mesh points and
then let me keep drawing these.
186
00:12:31 --> 00:12:34
So these will be mesh
points too, and these
187
00:12:34 --> 00:12:37
will be mesh points too.
188
00:12:37 --> 00:12:44
So at this point, I've got
probably 12 or 13 mesh points.
189
00:12:44 --> 00:12:47
But I've got quads, right?
190
00:12:47 --> 00:12:49
Well, I've got a couple of
triangles here that I'm not
191
00:12:49 --> 00:12:51
going to touch, those are fine.
192
00:12:51 --> 00:12:58
But these are quads and they
could be used with the Q_1
193
00:12:58 --> 00:13:02
element, but I'm thinking
let's stay with triangles.
194
00:13:02 --> 00:13:06
So I just suggested to put
in triangles, put in these
195
00:13:06 --> 00:13:10
diagonals, keeping symmetry.
196
00:13:10 --> 00:13:13
And so there's the mesh.
197
00:13:13 --> 00:13:15
There's the mesh.
198
00:13:15 --> 00:13:19
And then what does
the code ask for?
199
00:13:19 --> 00:13:22
So it's got 13 mesh points.
200
00:13:22 --> 00:13:25
And the code, first of all it
wants a list of the coordinates
201
00:13:25 --> 00:13:27
of all those mesh points.
202
00:13:27 --> 00:13:32
So that the code will, and I
better number them, of course.
203
00:13:32 --> 00:13:36
So let me number them one,
shall I number them the center
204
00:13:36 --> 00:13:42
guys first, one, two, three,
four, five, and then up here
205
00:13:42 --> 00:13:44
six, seven, eight, nine,, now
I don't know if this
206
00:13:44 --> 00:13:45
is a good numbering.
207
00:13:45 --> 00:13:48
Ten, 11, 12, 13.
208
00:13:48 --> 00:13:57
Why don't we, just to have some
consistency within plans,
209
00:13:57 --> 00:14:00
why don't you, don't
have to take N=4.
210
00:14:01 --> 00:14:06
I hope you'll take, what did
I take N as four or five?
211
00:14:06 --> 00:14:08
Yeah, right.
212
00:14:08 --> 00:14:09
N is 4.
213
00:14:09 --> 00:14:13
But you'll want to
try different N's.
214
00:14:13 --> 00:14:18
N=4 would be a good crude start
to see what's going on, but
215
00:14:18 --> 00:14:23
then I hope you'll go higher
and get better accuracy.
216
00:14:23 --> 00:14:27
And you can see how the
accuracy improves, how
217
00:14:27 --> 00:14:31
you get closer to that
as n gets bigger.
218
00:14:31 --> 00:14:35
OK, but got the nodes numbered.
219
00:14:35 --> 00:14:38
Oh, I better number
the triangles.
220
00:14:38 --> 00:14:40
OK, how shall we
number the triangles?
221
00:14:40 --> 00:14:43
Shall we do along
the top, or this?
222
00:14:43 --> 00:14:44
I don't know.
223
00:14:44 --> 00:14:48
What do you want to do for the
numbering of the triangles?
224
00:14:48 --> 00:14:51
Maybe run along the top, and
then run along the bottom,
225
00:14:51 --> 00:14:55
because then it'll
practically be a copy.
226
00:14:55 --> 00:14:59
So I didn't leave myself much
space, but one, two, three,
227
00:14:59 --> 00:15:04
four, five, six, seven.
228
00:15:04 --> 00:15:08
Seven triangles along the top
and seven along the bottom, so
229
00:15:08 --> 00:15:14
I have 14 triangles
in this mesh.
230
00:15:14 --> 00:15:23
So it's a mesh with 14
triangles and 13 nodes.
231
00:15:23 --> 00:15:26
And I know the positions
of everyone, right?
232
00:15:26 --> 00:15:29
I know the x,y coordinates
of every one.
233
00:15:29 --> 00:15:34
So what the code will want is
a list of those coordinates.
234
00:15:34 --> 00:15:39
So a list, P, P will be
a list of coordinates.
235
00:15:39 --> 00:15:44
The first guy will be
of, the nodes so 13
236
00:15:44 --> 00:15:50
rows, three columns.
237
00:15:50 --> 00:15:53
So it's a little 13 by three
matrix that tells you
238
00:15:53 --> 00:15:55
where all the nodes are.
239
00:15:55 --> 00:15:59
So the first one on that
list would be (0,0).
240
00:16:01 --> 00:16:03
that's for node one.
241
00:16:03 --> 00:16:06
And the second one would be
whatever the coordinates of
242
00:16:06 --> 00:16:11
that are, something zero.
(h,0), I guess it is.
243
00:16:11 --> 00:16:18
The third one will be (2h,0),
and so on, and then complete
244
00:16:18 --> 00:16:21
the list of 13 positions.
245
00:16:21 --> 00:16:28
So you are then told the code
where all the nodes are.
246
00:16:28 --> 00:16:30
What else do you
have to tell it?
247
00:16:30 --> 00:16:32
Not much.
248
00:16:32 --> 00:16:35
You now have to tell it
about the triangles.
249
00:16:35 --> 00:16:39
So now for every, why do
I say three columns?
250
00:16:39 --> 00:16:44
Maybe only two, is it?
251
00:16:44 --> 00:16:46
You see the point already.
252
00:16:46 --> 00:16:53
I've forgotten, maybe, yeah.
253
00:16:53 --> 00:16:58
I don't see why.
254
00:16:58 --> 00:17:02
Well for triangles, I'm going
to need three, for nodes
255
00:17:02 --> 00:17:03
maybe it's only got two.
256
00:17:03 --> 00:17:07
Maybe it's 13 by two.
257
00:17:07 --> 00:17:10
I don't see why I need three.
258
00:17:10 --> 00:17:11
But, anyway.
259
00:17:11 --> 00:17:15
Then the other list
is triangles.
260
00:17:15 --> 00:17:20
So this will be the list,
t, and so it takes
261
00:17:20 --> 00:17:22
triangle number one.
262
00:17:22 --> 00:17:24
Which is right here.
263
00:17:24 --> 00:17:26
That very first triangle.
264
00:17:26 --> 00:17:31
And what does it have to
tell us about triangle one?
265
00:17:31 --> 00:17:32
The three nodes.
266
00:17:32 --> 00:17:37
If it tells us the three node
numbers, and this list, P, gave
267
00:17:37 --> 00:17:40
their positions, we've got it.
268
00:17:40 --> 00:17:45
So how many triangles
did that I have?
269
00:17:45 --> 00:17:46
14?
270
00:17:46 --> 00:17:56
So t will be 14 by three, and
so the first guy will be just
271
00:17:56 --> 00:18:00
one, node number one,
node number two, and
272
00:18:00 --> 00:18:02
node number six.
273
00:18:02 --> 00:18:06
That will tell us which
is the first triangle.
274
00:18:06 --> 00:18:10
And the second triangle, I
guess I've drawn from two.
275
00:18:10 --> 00:18:14
Two, seven to six, right?
276
00:18:14 --> 00:18:19
That's a very skinny triangle
up there but it's the one that
277
00:18:19 --> 00:18:22
started at node two, went up
to seven and back to six.
278
00:18:22 --> 00:18:26
So a list like that.
279
00:18:26 --> 00:18:32
And then the code
will do the rest.
280
00:18:32 --> 00:18:35
I hope.
281
00:18:35 --> 00:18:38
Almost all the rest.
282
00:18:38 --> 00:18:41
The code will create
the matrix K.
283
00:18:41 --> 00:18:47
It'll create the matrix K
with, it'll be singular.
284
00:18:47 --> 00:18:51
Boundary conditions
won't yet be in there.
285
00:18:51 --> 00:18:56
And then a final step after
that K, or maybe we could call
286
00:18:56 --> 00:19:02
it K_0 is created, a final
step will be to fix u.
287
00:19:02 --> 00:19:04
At least at these three points.
288
00:19:04 --> 00:19:08
So these three will
be boundary nodes.
289
00:19:08 --> 00:19:12
And as I say, I'm not too sure
about that one, I apologize.
290
00:19:12 --> 00:19:14
At those boundary nodes
I'm going to take the
291
00:19:14 --> 00:19:16
values to be zero.
292
00:19:16 --> 00:19:21
So this is going to be zero
along the whole edge, because
293
00:19:21 --> 00:19:24
if it's zero there, zero there,
zero there and zero there, and
294
00:19:24 --> 00:19:27
if it's linear, it's zero.
295
00:19:27 --> 00:19:35
So the final, sort of,
subroutine in the code, the
296
00:19:35 --> 00:19:39
final group of commands you
want to impose, zeroes here,
297
00:19:39 --> 00:19:46
that should then make the
matrix K invertible, and then
298
00:19:46 --> 00:19:49
you've got KU=F to solve.
299
00:19:49 --> 00:19:54
So what the code is doing
is creating K and F.
300
00:19:54 --> 00:19:57
You see the overall picture?
301
00:19:57 --> 00:20:00
I jumped right into this
particular mesh, particular
302
00:20:00 --> 00:20:06
problem, but now I really
should back up to
303
00:20:06 --> 00:20:09
where it starts.
304
00:20:09 --> 00:20:13
This this is going to be
the weak form of Laplace,
305
00:20:13 --> 00:20:15
Poisson, maybe I'll make a
306
00:20:15 --> 00:20:24
little space to put in
Poisson's name too.
307
00:20:24 --> 00:20:29
You have a picture already of
what this weak form is about,
308
00:20:29 --> 00:20:32
so now I'm really backing
up to the start.
309
00:20:32 --> 00:20:36
I take the equation and
I get its weak form.
310
00:20:36 --> 00:20:39
And remember that's in the
continuous case, as it
311
00:20:39 --> 00:20:41
was in the quiz problem.
312
00:20:41 --> 00:20:45
The first step is the
continuous weak form, and then
313
00:20:45 --> 00:20:53
the second step is choose test
functions and, trial - I'm
314
00:20:53 --> 00:20:59
sorry, gosh, I'm in
bad shape here.
315
00:20:59 --> 00:21:02
Because they're the same.
316
00:21:02 --> 00:21:04
This isn't my worst
error, today.
317
00:21:04 --> 00:21:11
But those are trial functions,
and these are test functions.
318
00:21:11 --> 00:21:14
OK, questions at this point,
because I've, yeah, thank you.
319
00:21:14 --> 00:21:15
Good.
320
00:21:15 --> 00:21:16
Let's look at this picture.
321
00:21:16 --> 00:21:22
AUDIENCE: [INAUDIBLE]
322
00:21:22 --> 00:21:30
PROFESSOR STRANG:
Because, we did.
323
00:21:30 --> 00:21:30
That's right.
324
00:21:30 --> 00:21:40
So because my question is what
is the continuous problem, I
325
00:21:40 --> 00:21:44
would like to solve has
u=0 on the polygon.
326
00:21:44 --> 00:21:50
So in a way you can forget
the circle, where we
327
00:21:50 --> 00:21:52
know the answer now.
328
00:21:52 --> 00:21:57
We really are looking on the
polygon, and I would like to
329
00:21:57 --> 00:22:01
know what's the solution
like on that polygon.
330
00:22:01 --> 00:22:06
And then so there are two
steps, the first step was
331
00:22:06 --> 00:22:09
start with a circle,
we have the answer.
332
00:22:09 --> 00:22:14
Second step is go to a polygon,
continuous problem, Poisson's
333
00:22:14 --> 00:22:16
equation in the polygon.
334
00:22:16 --> 00:22:18
How different is
that from this?
335
00:22:18 --> 00:22:23
Because this will not satisfy
the polygon boundary
336
00:22:23 --> 00:22:24
conditions.
337
00:22:24 --> 00:22:28
So that's the circle answer.
338
00:22:28 --> 00:22:32
Then the question is,
what's the polygon answer.
339
00:22:32 --> 00:22:36
And I don't know that.
340
00:22:36 --> 00:22:39
You may say a regular
polygon, you can't do that.
341
00:22:39 --> 00:22:41
I didn't think you can.
342
00:22:41 --> 00:22:44
It's amazing, but probably
a triangle or a square.
343
00:22:44 --> 00:22:49
So if M is three or four,
probably some formulas
344
00:22:49 --> 00:22:50
would be available.
345
00:22:50 --> 00:22:54
But I think once we get higher,
I don't know the answer to the
346
00:22:54 --> 00:22:59
Dirichlet problem, to Poisson's
equation on a polygon.
347
00:22:59 --> 00:23:02
On a regular polygon and that's
what I would really like
348
00:23:02 --> 00:23:03
to know more about.
349
00:23:03 --> 00:23:06
And how do I find
out more about it?
350
00:23:06 --> 00:23:08
By finite elements.
351
00:23:08 --> 00:23:11
With your help.
352
00:23:11 --> 00:23:16
Taking that polygon, breaking
it into a mesh, looking only
353
00:23:16 --> 00:23:22
at one triangle just for
simplicity, and getting
354
00:23:22 --> 00:23:24
u finite elements.
355
00:23:24 --> 00:23:26
Well, I should say u_p_1.
356
00:23:26 --> 00:23:30
357
00:23:30 --> 00:23:34
That's the finite element
solution using linear.
358
00:23:34 --> 00:23:40
I would really like to know
u_p_2, the finite element
359
00:23:40 --> 00:23:43
solution which will be better.
360
00:23:43 --> 00:23:45
If I use quadratics.
361
00:23:45 --> 00:23:47
So now I get the fun of
describing the linear
362
00:23:47 --> 00:23:51
elements, the quadratic
elements, the quads.
363
00:23:51 --> 00:23:53
But did I answer
that question OK?
364
00:23:53 --> 00:23:54
Yeah.
365
00:23:54 --> 00:23:59
So this is the problem I would
like to know the answer to.
366
00:23:59 --> 00:24:04
If I have this equation, zero
boundary conditions on a
367
00:24:04 --> 00:24:09
regular polygon with M
sides, what's the answer?
368
00:24:09 --> 00:24:13
And it's going to be close to
this, but it won't be the same.
369
00:24:13 --> 00:24:17
Because this does not vanish
on the polygon edges.
370
00:24:17 --> 00:24:22
And I would like to
compare the slopes, too.
371
00:24:22 --> 00:24:25
So the homework problem asked
you not only to compare
372
00:24:25 --> 00:24:32
u circle with u_p_1,
but also the slopes.
373
00:24:32 --> 00:24:37
The slopes here are easy,
slopes here are easy because
374
00:24:37 --> 00:24:39
it's a bunch of flat functions.
375
00:24:39 --> 00:24:43
So the slopes are just
constant in each triangle.
376
00:24:43 --> 00:24:48
OK, I'm guessing that the error
gets smaller as you go in.
377
00:24:48 --> 00:24:52
I think that if you plot the
error, it'll be largest out
378
00:24:52 --> 00:24:53
here and get small there.
379
00:24:53 --> 00:24:56
But remains to be seen.
380
00:24:56 --> 00:24:57
So I hope you enjoy
381
00:24:57 --> 00:24:59
- yeah, good.
382
00:24:59 --> 00:25:05
AUDIENCE: [INAUDIBLE]
383
00:25:05 --> 00:25:07
PROFESSOR STRANG:
Rather than seven.
384
00:25:07 --> 00:25:13
AUDIENCE: [INAUDIBLE]
385
00:25:13 --> 00:25:17
PROFESSOR STRANG:
No, the middle.
386
00:25:17 --> 00:25:21
There's nothing magic about
any particular mesh.
387
00:25:21 --> 00:25:31
I just chose this mesh as
pretty good, and actually,
388
00:25:31 --> 00:25:34
I'm imagining M could
get pretty big.
389
00:25:34 --> 00:25:35
That would be interesting.
390
00:25:35 --> 00:25:40
M=8 would be interesting,
M=16, M=1,024, now then
391
00:25:40 --> 00:25:42
I'd really get interested.
392
00:25:42 --> 00:25:50
OK, but so if M is 1,024, then
this side would be very small.
393
00:25:50 --> 00:25:52
Right?
394
00:25:52 --> 00:25:55
And I just wanted
more triangles.
395
00:25:55 --> 00:25:58
Actually, I would like
more than I've got.
396
00:25:58 --> 00:26:05
I'd like, if M was really
big, then probably N should
397
00:26:05 --> 00:26:07
be at least that big.
398
00:26:07 --> 00:26:09
So I should have a thousand
this way, if this
399
00:26:09 --> 00:26:11
is just a tiny bit.
400
00:26:11 --> 00:26:14
I just want little tiny
h, and then, yeah.
401
00:26:14 --> 00:26:18
Actually, that might
not be too bad.
402
00:26:18 --> 00:26:25
If m anM N were roughly
comparable, then that length
403
00:26:25 --> 00:26:28
would be roughly comparable
to these lengths.
404
00:26:28 --> 00:26:33
And the triangles would
be pretty good shape.
405
00:26:33 --> 00:26:36
And that's what
you're looking for.
406
00:26:36 --> 00:26:43
I think there's a lot of
experiments to be done here.
407
00:26:43 --> 00:26:45
So, I'm thinking
then of M and N.
408
00:26:45 --> 00:26:48
Here I took N to be just four.
409
00:26:48 --> 00:26:50
When M was eight.
410
00:26:50 --> 00:26:51
That's fine.
411
00:26:51 --> 00:26:56
But if you keep M and N roughly
the same size, then you've got
412
00:26:56 --> 00:27:00
triangles that are not
too long and skinny.
413
00:27:00 --> 00:27:04
I'll tell you when
you might want.
414
00:27:04 --> 00:27:07
So generally you want
nice shaped triangles.
415
00:27:07 --> 00:27:13
You don't want angles very
small or very large, usually.
416
00:27:13 --> 00:27:20
But there would be, anybody in
Course 16 can imagine that if I
417
00:27:20 --> 00:27:29
am computing the flow field
past a wing, that long, thin
418
00:27:29 --> 00:27:33
triangles in the direction
of the wing are natural.
419
00:27:33 --> 00:27:41
I mean, somehow a problem
like true aerodynamics is
420
00:27:41 --> 00:27:43
by no means isotropic.
421
00:27:43 --> 00:27:46
I mean, the direction of the
wing is kind of critical to
422
00:27:46 --> 00:27:49
whether the plane flies, right?
423
00:27:49 --> 00:27:54
So don't make the
wing vertical.
424
00:27:54 --> 00:27:57
And if you want accuracy, then
you have long, thin triangles
425
00:27:57 --> 00:27:59
in the direction of the flow.
426
00:27:59 --> 00:28:02
But here we're not
doing a flow problem.
427
00:28:02 --> 00:28:06
We haven't got shocks, or
trailing edges, and other
428
00:28:06 --> 00:28:10
horrible stuff that
makes planes fly.
429
00:28:10 --> 00:28:12
We just got Poisson's equation.
430
00:28:12 --> 00:28:13
OK.
431
00:28:13 --> 00:28:15
Thanks for those good
questions, another one.
432
00:28:15 --> 00:28:19
AUDIENCE: [INAUDIBLE]
433
00:28:19 --> 00:28:20
PROFESSOR STRANG: What would
the dimension look like?
434
00:28:20 --> 00:28:24
Ah, would you like me to show
you something about quadratics?
435
00:28:24 --> 00:28:27
Yeah.
436
00:28:27 --> 00:28:30
Shall I jump into quadratics,
it's kind of fun.
437
00:28:30 --> 00:28:37
Quadratics, so let me just do,
so I'll come back to the weak
438
00:28:37 --> 00:28:41
form, right it's totally,
oh I'll do it now.
439
00:28:41 --> 00:28:44
It's so simple I don't
want to forget it.
440
00:28:44 --> 00:28:48
The weak form, so I write the
equation down, -u_xx-u_yy=f.
441
00:28:48 --> 00:28:51
442
00:28:51 --> 00:28:57
This is the continuous weak
form equal f(x,y), OK?
443
00:28:57 --> 00:29:00
So that's the strong form.
444
00:29:00 --> 00:29:04
And I've made it the Laplace
in here to keep it simple,
445
00:29:04 --> 00:29:06
on any right hand side.
446
00:29:06 --> 00:29:09
OK, how do I get
to the weak form?
447
00:29:09 --> 00:29:14
Just remind me, I multiply
both sides by any
448
00:29:14 --> 00:29:15
test function v(x,y).
449
00:29:17 --> 00:29:23
Multiply by v(x,y), and
then what do I do?
450
00:29:23 --> 00:29:28
I integrate over
the whole region.
451
00:29:28 --> 00:29:35
So that's the weak form,
dxdy, this is for all v,
452
00:29:35 --> 00:29:40
all v(x,y), all, I'll say
all admissible v(x,y).
453
00:29:42 --> 00:29:45
So that's the weak form.
454
00:29:45 --> 00:29:51
If this holds for all this
great family of v's, the idea
455
00:29:51 --> 00:29:55
behind it is, that if this
holds for all these trial
456
00:29:55 --> 00:29:59
functions, test functions,
v(x,y), the only way that can
457
00:29:59 --> 00:30:03
happen is for this to
actually equal that.
458
00:30:03 --> 00:30:12
That's a fundamental lemma in
this part of math, and of
459
00:30:12 --> 00:30:17
course it has to be spelled out
more than I'm doing in words.
460
00:30:17 --> 00:30:22
But the idea is that if these
hold for such a large class of
461
00:30:22 --> 00:30:25
v(x,y), then the only way that
can happen is for the
462
00:30:25 --> 00:30:26
strong form to hold.
463
00:30:26 --> 00:30:29
For this to actually
match this.
464
00:30:29 --> 00:30:31
OK, so that's the start.
465
00:30:31 --> 00:30:35
But then what's the next
step in the weak form?
466
00:30:35 --> 00:30:38
I I like the right hand
side but I'm not so crazy
467
00:30:38 --> 00:30:39
about the left hand side.
468
00:30:39 --> 00:30:43
I'm not crazy about it because
this says second derivatives
469
00:30:43 --> 00:30:49
of u, and my little roof
functions, pyramid functions,
470
00:30:49 --> 00:30:51
haven't got second derivatives.
471
00:30:51 --> 00:30:55
So I would be dead in the water
without doing the natural step
472
00:30:55 --> 00:30:58
that makes everything
beautiful, which is?
473
00:30:58 --> 00:31:00
Integration by parts.
474
00:31:00 --> 00:31:01
Integrate by parts.
475
00:31:01 --> 00:31:05
Move derivatives
of of u, on to v.
476
00:31:05 --> 00:31:10
One derivative onto v, off
of u, so then u and v
477
00:31:10 --> 00:31:12
each have one derivative.
478
00:31:12 --> 00:31:17
I can use my piecewise linear,
piecewise quadratics, all my
479
00:31:17 --> 00:31:20
finite elements are
going to go fine.
480
00:31:20 --> 00:31:22
So I integrate by parts.
481
00:31:22 --> 00:31:28
So integrate by parts, and
what is that mean in 2-D?
482
00:31:28 --> 00:31:30
Of course I have a
double integral here.
483
00:31:30 --> 00:31:37
So integrate by parts, that
mean you the Green's formula.
484
00:31:37 --> 00:31:43
That was the key point
of this Green, or
485
00:31:43 --> 00:31:47
Gauss-Green's, formula.
486
00:31:47 --> 00:32:01
Can I do it first in, this is
-div(grad u), times vdxdy, we
487
00:32:01 --> 00:32:03
can write out all the terms.
488
00:32:03 --> 00:32:05
We can use vector notation.
489
00:32:05 --> 00:32:09
I could use that nabla,
that upside down triangle
490
00:32:09 --> 00:32:10
notation, or whatever.
491
00:32:10 --> 00:32:13
But maybe good to see it
a few different ways.
492
00:32:13 --> 00:32:15
So what's the point?
493
00:32:15 --> 00:32:20
When I integrate by parts, that
minus disappears to a plus, I
494
00:32:20 --> 00:32:26
have a double integral then,
and these derivatives move off
495
00:32:26 --> 00:32:32
of, I'm taking one derivative
off of here, the divergence
496
00:32:32 --> 00:32:35
moves over there, but when
the divergence moves
497
00:32:35 --> 00:32:38
onto v it becomes?
498
00:32:38 --> 00:32:39
The transpose.
499
00:32:39 --> 00:32:40
It becomes gradient.
500
00:32:40 --> 00:32:52
And so this is gradient
view, gradient of v. dxdy,
501
00:32:52 --> 00:32:54
plus boundary terms.
502
00:32:54 --> 00:32:58
The integral of,
what is, let's see.
503
00:32:58 --> 00:33:02
504
00:33:02 --> 00:33:11
What do I have in this
integral, I have grad u dot n,
505
00:33:11 --> 00:33:14
times v around the boundary.
506
00:33:14 --> 00:33:18
And that's with my boundary
conditions that's going
507
00:33:18 --> 00:33:20
to be gone, so I can
come back to that.
508
00:33:20 --> 00:33:26
Now, you all looked a little
uncertain when I wrote
509
00:33:26 --> 00:33:29
Green's formula this way.
510
00:33:29 --> 00:33:33
For this problem I can
write it more easily.
511
00:33:33 --> 00:33:36
This is my left side.
512
00:33:36 --> 00:33:40
I want to write the answer, I
just want to write this weak
513
00:33:40 --> 00:33:44
form in a much simpler form.
514
00:33:44 --> 00:33:49
So let say, what
have I got here.
515
00:33:49 --> 00:33:54
Well, all I've got is one
derivative is moving
516
00:33:54 --> 00:33:56
off of u and on to v.
517
00:33:56 --> 00:33:58
And the minus sign is
disappearing, so I have
518
00:33:58 --> 00:34:02
du/dx times dv/dx.
519
00:34:02 --> 00:34:03
Right?
520
00:34:03 --> 00:34:06
One off of u, onto v.
521
00:34:06 --> 00:34:10
The other term, one y
derivative, moving off
522
00:34:10 --> 00:34:12
of this and onto v.
523
00:34:12 --> 00:34:15
Minus sign again going to
a plus. du/dy, dv/dy.
524
00:34:15 --> 00:34:19
525
00:34:19 --> 00:34:20
That's the integral.
526
00:34:20 --> 00:34:22
That's it, that's cool.
527
00:34:22 --> 00:34:24
Easy to do.
528
00:34:24 --> 00:34:29
And on the right hand side
of course I have no change.
529
00:34:29 --> 00:34:31
The integral of
f(x,y)*v(x,y)*dy.
530
00:34:31 --> 00:34:34
531
00:34:34 --> 00:34:36
Now, that's the
weak form, dx/dy.
532
00:34:36 --> 00:34:44
533
00:34:44 --> 00:34:52
Here it is, weak form.
534
00:34:52 --> 00:34:55
That's pretty nice.
535
00:34:55 --> 00:34:58
Beautifully symmetric, though
the matrix that comes up when
536
00:34:58 --> 00:35:04
we plug in finite specific
trial functions and test
537
00:35:04 --> 00:35:07
functions is going to be
a symmetric matrix K.
538
00:35:07 --> 00:35:13
And the integrals of first
derivatives, so as long as our
539
00:35:13 --> 00:35:19
functions, our trial functions
and test functions are
540
00:35:19 --> 00:35:24
continuous, that is,
they shouldn't jump.
541
00:35:24 --> 00:35:28
If the trial functions or test
functions jump, then if I have
542
00:35:28 --> 00:35:31
a jump, then the derivative
would be a delta.
543
00:35:31 --> 00:35:34
I'd have another delta here,
I'd have an integral delta, a
544
00:35:34 --> 00:35:37
delta times delta, and
I don't want that.
545
00:35:37 --> 00:35:39
That's infinite.
546
00:35:39 --> 00:35:45
Those discontinuous elements
would not be conforming, and
547
00:35:45 --> 00:35:48
that's a whole new world of
discontinuous Galerkin.
548
00:35:48 --> 00:35:52
I'd have to impose penalty
stuff, and Professor
549
00:35:52 --> 00:35:54
Peraire I mentioned.
550
00:35:54 --> 00:36:00
And others, Professor Darmofal
in aero are experts on this.
551
00:36:00 --> 00:36:02
We're doing continuous form.
552
00:36:02 --> 00:36:03
CG.
553
00:36:04 --> 00:36:07
Our piecewise linear,
piecewise quadratic,
554
00:36:07 --> 00:36:08
they'll be continuous.
555
00:36:08 --> 00:36:11
All I have to do is
these derivatives.
556
00:36:11 --> 00:36:14
Integrate those things and
that's what the code will do.
557
00:36:14 --> 00:36:19
OK, I've got to the weak form.
558
00:36:19 --> 00:36:22
That's the weak form.
559
00:36:22 --> 00:36:25
Now comes the finite
element idea.
560
00:36:25 --> 00:36:28
So there is our weak
form, now ready for the
561
00:36:28 --> 00:36:31
finite element idea.
562
00:36:31 --> 00:36:34
OK, so what was that idea?
563
00:36:34 --> 00:36:36
That's the continuous problem.
564
00:36:36 --> 00:36:47
Now, the finite element idea
is, plug in U as a combination.
565
00:36:47 --> 00:36:52
Let me write out the terms.
566
00:36:52 --> 00:36:55
You know what's coming here.
567
00:36:55 --> 00:36:58
If I'm using finite elements,
I'm going to choose nice
568
00:36:58 --> 00:37:07
polynomials, phi,
say, N of them.
569
00:37:07 --> 00:37:10
That would be like, one for
every node, so I would
570
00:37:10 --> 00:37:14
have 13 functions here.
571
00:37:14 --> 00:37:18
I'm going to choose the v's
to be the same as the phis.
572
00:37:18 --> 00:37:23
And then, I'm working then
in 13 dimensions instead
573
00:37:23 --> 00:37:26
of infinite dimensions.
574
00:37:26 --> 00:37:29
So what do I do?
575
00:37:29 --> 00:37:33
For this limited subspace, this
finite element subspace, this
576
00:37:33 --> 00:37:38
piecewise polynomial, piecewise
linear subspace, I plug that
577
00:37:38 --> 00:37:44
into the weak form and I test
it against 13 V's,
578
00:37:44 --> 00:37:45
which are phis.
579
00:37:45 --> 00:37:53
So I plug that in,
so now what is K?
580
00:37:53 --> 00:38:00
Now let me just say, so I now
have the integral of, yeah I
581
00:38:00 --> 00:38:03
guess I'd better plug it in.
582
00:38:03 --> 00:38:07
K_ij would then
be the integral.
583
00:38:07 --> 00:38:09
I'm just copying
the weak form in.
584
00:38:09 --> 00:38:22
Of dU/dx, no, sorry I'd better
just plug it in first.
585
00:38:22 --> 00:38:32
dU/dx*dV/dx plus dU/dy*dV/dy,
those are the integrals
586
00:38:32 --> 00:38:33
I have to do.
587
00:38:33 --> 00:38:37
And on the right hand side
I have to do the fV.
588
00:38:37 --> 00:38:40
589
00:38:40 --> 00:38:44
OK, plug that in.
590
00:38:44 --> 00:38:48
That's the integral
over the whole domain.
591
00:38:48 --> 00:38:54
When I plug it in this U
is a combination of known
592
00:38:54 --> 00:39:05
functions and the V's
will be the same guys.
593
00:39:05 --> 00:39:08
So what am I going to get here?
594
00:39:08 --> 00:39:10
It's just as in 1-D.
595
00:39:10 --> 00:39:14
So no new ideas entering here.
596
00:39:14 --> 00:39:20
The new idea's going to enter
when I construct these phis.
597
00:39:20 --> 00:39:23
Let me just say,
though, one thing.
598
00:39:23 --> 00:39:29
In 1-D, we've pretty much
had a choice of, when
599
00:39:29 --> 00:39:31
it was one dimension.
600
00:39:31 --> 00:39:34
Just remember that.
601
00:39:34 --> 00:39:40
In one dimension, when I had
these hat functions, when I had
602
00:39:40 --> 00:39:44
these guys, integrated against
these guys, I pretty much had a
603
00:39:44 --> 00:39:48
choice of did I want to think
about integrating that hat
604
00:39:48 --> 00:39:50
function against that one.
605
00:39:50 --> 00:39:53
Or actually it was
their derivatives.
606
00:39:53 --> 00:39:58
It was the integral of U, yeah.
607
00:39:58 --> 00:40:02
Of phi, what I needed
was all the integrals
608
00:40:02 --> 00:40:04
of phi_i', phi_j'.
609
00:40:06 --> 00:40:12
Those are what I needed,
these go into K.
610
00:40:12 --> 00:40:13
Into the matrix K.
611
00:40:13 --> 00:40:16
In fact, that's what
equals K_ij, the
612
00:40:16 --> 00:40:18
integral of phi prime.
613
00:40:18 --> 00:40:20
In 1-D.
614
00:40:20 --> 00:40:21
OK.
615
00:40:21 --> 00:40:26
Now, what I was going
to say, I could do it
616
00:40:26 --> 00:40:28
this way if I wanted.
617
00:40:28 --> 00:40:30
But you remember the
other way to do it?
618
00:40:30 --> 00:40:33
Was elements at a time.
619
00:40:33 --> 00:40:36
So this was one method here.
620
00:40:36 --> 00:40:43
That found the entries of
K separately, one by one.
621
00:40:43 --> 00:40:47
The other way was take the
elements, one by one.
622
00:40:47 --> 00:40:52
So the other way was take an
element like this element.
623
00:40:52 --> 00:40:56
It's got two functions,
two trial functions
624
00:40:56 --> 00:40:57
are involved there.
625
00:40:57 --> 00:41:02
There's a little two by
two, so this is four.
626
00:41:02 --> 00:41:06
Two by two element matrices.
627
00:41:06 --> 00:41:08
K equals.
628
00:41:08 --> 00:41:12
And the quiz
recalled that part.
629
00:41:12 --> 00:41:13
That approach.
630
00:41:13 --> 00:41:16
So what I want to say is
that's the right way to
631
00:41:16 --> 00:41:18
do it in two dimensions.
632
00:41:18 --> 00:41:20
A triangle at a time.
633
00:41:20 --> 00:41:22
That's the way the
code will do it.
634
00:41:22 --> 00:41:26
It creates these little element
matrices, and then it stamps
635
00:41:26 --> 00:41:30
them into the big matrix K.
636
00:41:30 --> 00:41:32
Alright.
637
00:41:32 --> 00:41:42
So I want to do this integral
one triangle at a time.
638
00:41:42 --> 00:41:45
Is the good way.
639
00:41:45 --> 00:41:47
OK, and that's what
the code will do.
640
00:41:47 --> 00:41:52
Actually, I think that the
best way to learn these
641
00:41:52 --> 00:41:55
steps is just to read
the lines of the code.
642
00:41:55 --> 00:41:59
You can read them in the
book, Page 303 or something.
643
00:41:59 --> 00:42:04
And you'll see it just
doing all the steps
644
00:42:04 --> 00:42:05
that need to be done.
645
00:42:05 --> 00:42:07
One triangle at a time.
646
00:42:07 --> 00:42:11
So, now.
647
00:42:11 --> 00:42:12
Now comes the fun.
648
00:42:12 --> 00:42:15
I get to answer what do
these piecewise linear
649
00:42:15 --> 00:42:16
elements look like.
650
00:42:16 --> 00:42:18
What do the quadratic
elements look like.
651
00:42:18 --> 00:42:24
What do the Q_1 quad
elements look like?
652
00:42:24 --> 00:42:28
This was the golden age of
finite elements, when people
653
00:42:28 --> 00:42:35
invented these ways to create
piecewise polynomials.
654
00:42:35 --> 00:42:37
And it continues.
655
00:42:37 --> 00:42:41
People are still inventing,
I had a email this week,
656
00:42:41 --> 00:42:44
somebody says I've got
spectral elements.
657
00:42:44 --> 00:42:47
People are going higher
and higher degrees.
658
00:42:47 --> 00:42:51
You, know sixth degree,
eighth degree.
659
00:42:51 --> 00:42:53
In order to get more accuracy.
660
00:42:53 --> 00:42:55
OK, let's start with P_1.
661
00:42:57 --> 00:43:02
How do I describe a P_1
element inside a triangle?
662
00:43:02 --> 00:43:11
So in a triangle, the unknowns
will be the value, this has a
663
00:43:11 --> 00:43:14
height U_1, this has a
height U_2, and a height
664
00:43:14 --> 00:43:17
U_3 at those nodes.
665
00:43:17 --> 00:43:24
Inside the triangle, the
function U is linear. a+bx+cy.
666
00:43:24 --> 00:43:31
667
00:43:31 --> 00:43:37
Then, you see that if I know
these three values, then I
668
00:43:37 --> 00:43:39
know these three numbers.
669
00:43:39 --> 00:43:40
And vice versa.
670
00:43:40 --> 00:43:43
There's a three by
three matrix, right?
671
00:43:43 --> 00:43:44
There has to be a three by
672
00:43:44 --> 00:43:47
- any time you see pictures
like this, this is like the
673
00:43:47 --> 00:43:51
good part of 18.085 is to
realize that if I have three
674
00:43:51 --> 00:43:55
numbers here, three values and
I've got three coefficients,
675
00:43:55 --> 00:43:58
that there's some three by
three matrix that connects them
676
00:43:58 --> 00:43:59
that you're going to need.
677
00:43:59 --> 00:44:04
That's like a meta-message
of this course.
678
00:44:04 --> 00:44:13
Is, you've got to translate
between the node values
679
00:44:13 --> 00:44:15
and the coefficients.
680
00:44:15 --> 00:44:19
Because the node values
are the unknowns, right?
681
00:44:19 --> 00:44:22
These are the guys that are
multiplying the pyramid
682
00:44:22 --> 00:44:25
function, this is multiplying
a pyramid function
683
00:44:25 --> 00:44:26
with height one.
684
00:44:26 --> 00:44:31
At that point, going down to
zero, so this one will be a
685
00:44:31 --> 00:44:36
pyramid function of height U_2
times one, going down to zero.
686
00:44:36 --> 00:44:37
And U_3.
687
00:44:38 --> 00:44:42
So we've got a flat
function in here.
688
00:44:42 --> 00:44:44
And it looks exactly like that.
689
00:44:44 --> 00:44:46
OK?
690
00:44:46 --> 00:44:50
So what do I want to say?
691
00:44:50 --> 00:44:54
When we know the positions
of these three nodes
692
00:44:54 --> 00:44:57
from our list, P, right?
693
00:44:57 --> 00:45:00
These were the crucial
things we did.
694
00:45:00 --> 00:45:05
The positions of all the nodes,
we know where they are.
695
00:45:05 --> 00:45:10
Then there has to be a three by
three matrix that will now
696
00:45:10 --> 00:45:13
connect to the coefficients.
697
00:45:13 --> 00:45:15
Why do we want the
coefficients?
698
00:45:15 --> 00:45:18
Because those are what we
do when we integrate.
699
00:45:18 --> 00:45:21
The coefficients are what we
need, we need to integrate
700
00:45:21 --> 00:45:21
dU/dx, dU/dy, dU/dz.
701
00:45:23 --> 00:45:27
Sorry, dU/dx, dU/dy.
702
00:45:27 --> 00:45:31
Are you visualizing
this overall solution
703
00:45:31 --> 00:45:36
capital U, yeah.
704
00:45:36 --> 00:45:40
So what the overall solution
capital U, you should visualize
705
00:45:40 --> 00:45:44
with a combination of all the
little u's, is zero here and
706
00:45:44 --> 00:45:48
then it's going to go up and
these triangles and bend around
707
00:45:48 --> 00:45:51
and, I don't know,
maybe down again.
708
00:45:51 --> 00:45:53
Or maybe, no, maybe
it keeps going up.
709
00:45:53 --> 00:45:58
This is probably the largest
value, because it's the largest
710
00:45:58 --> 00:46:00
value and the correct solution.
711
00:46:00 --> 00:46:11
So is this is probably going to
be the highest point of this.
712
00:46:11 --> 00:46:14
What's the Forbidden
City, right?
713
00:46:14 --> 00:46:20
In China, is in Beijing
is like, or a single,
714
00:46:20 --> 00:46:23
do pagodas have flat?
715
00:46:23 --> 00:46:24
No.
716
00:46:24 --> 00:46:29
We we will meet
pagoda functions.
717
00:46:29 --> 00:46:34
But this would be just an
ordinary western roof, I guess.
718
00:46:34 --> 00:46:35
Just flat pieces.
719
00:46:35 --> 00:46:36
Yeah.
720
00:46:36 --> 00:46:40
OK, see, you've got to see
the whole thing and then
721
00:46:40 --> 00:46:42
you look at each piece.
722
00:46:42 --> 00:46:47
Each piece looks like that,
and the integrals are doable.
723
00:46:47 --> 00:46:54
OK, so while I'm going here,
I want to do quadratics.
724
00:46:54 --> 00:46:55
You'll get the idea right away.
725
00:46:55 --> 00:46:59
So, same triangle, now I'm
going to have quadratics.
726
00:46:59 --> 00:47:02
So I'm now going to have, so
this won't be the arrow, this
727
00:47:02 --> 00:47:04
arrow will now go this way.
728
00:47:04 --> 00:47:10
I'm going to have dx squared,
exy, and f y squared.
729
00:47:10 --> 00:47:12
So now how many
coefficients have I got
730
00:47:12 --> 00:47:15
to determine a quadratic?
731
00:47:15 --> 00:47:18
Six, right? a, b, c, d, e, f.
732
00:47:18 --> 00:47:20
How many nodes do I need?
733
00:47:20 --> 00:47:21
Six.
734
00:47:21 --> 00:47:22
Where are they?
735
00:47:22 --> 00:47:29
Well, the natural positions are
those guys in the mid-point.
736
00:47:29 --> 00:47:33
So now, those are
all nodes now.
737
00:47:33 --> 00:47:38
Some nodes are at vertices
of triangles, some
738
00:47:38 --> 00:47:39
nodes are at midpoint.
739
00:47:39 --> 00:47:44
But remember, we've got other
triangles hooking on here, many
740
00:47:44 --> 00:47:48
other triangles, all with
their own six nodes.
741
00:47:48 --> 00:47:51
Well, not their own,
because they share.
742
00:47:51 --> 00:47:53
That's a big point.
743
00:47:53 --> 00:47:59
So there's a grid of triangles,
with nodes for quadratic.
744
00:47:59 --> 00:48:02
And we've got one, two, three,
four, five, six, seven, eight,
745
00:48:02 --> 00:48:09
nine, ten, 11, 12, 13, 14,
15, 16 nodes, I think.
746
00:48:09 --> 00:48:13
And within each triangle,
this is what we've got.
747
00:48:13 --> 00:48:17
So there's a six by six
matrix for each triangle.
748
00:48:17 --> 00:48:21
A six by six matrix which will
connect the values U_1, U_2,
749
00:48:21 --> 00:48:29
U_3, U_4, U_5, U_6
for this triangle.
750
00:48:29 --> 00:48:33
Connect those six heights
with these six numbers.
751
00:48:33 --> 00:48:42
And what will the roof look
like within that triangle?
752
00:48:42 --> 00:48:44
Well, sort of curved.
753
00:48:44 --> 00:48:45
A parabola, right?
754
00:48:45 --> 00:48:49
A parabola somehow in 2-D,
it'll look like this, yeah.
755
00:48:49 --> 00:48:49
Yeah.
756
00:48:49 --> 00:48:52
And here's the key question.
757
00:48:52 --> 00:48:59
Will that roof, that curvy
roof, fit the one over there?
758
00:48:59 --> 00:49:01
Because if it didn't
fit, we're in trouble.
759
00:49:01 --> 00:49:04
This derivative would have a
delta function, and we've got
760
00:49:04 --> 00:49:07
delta functions, and integral
squaring them would
761
00:49:07 --> 00:49:09
give infinite.
762
00:49:09 --> 00:49:10
So here's the question.
763
00:49:10 --> 00:49:16
Why does this roof, using these
six points, fit on to the roof
764
00:49:16 --> 00:49:23
that uses U_7, U_8, U_9,
and U_3 U_4 and U_6?
765
00:49:23 --> 00:49:26
Why do those two
roofs fit together?
766
00:49:26 --> 00:49:30
This one piecewise polynomials?
767
00:49:30 --> 00:49:33
Of course, the
slope will change.
768
00:49:33 --> 00:49:35
But the roof won't have a gap.
769
00:49:35 --> 00:49:37
Water won't go through it.
770
00:49:37 --> 00:49:38
Why's that?
771
00:49:38 --> 00:49:41
Do you see why?
772
00:49:41 --> 00:49:43
Because what do they
share, what do those
773
00:49:43 --> 00:49:46
two curvy roofs share?
774
00:49:46 --> 00:49:48
They share a side.
775
00:49:48 --> 00:49:53
They share the same
values along the side.
776
00:49:53 --> 00:49:57
And are those three values that
are shared along the side
777
00:49:57 --> 00:50:03
sufficient to make it
match all along the side?
778
00:50:03 --> 00:50:03
Yes.
779
00:50:03 --> 00:50:05
That's the important question.
780
00:50:05 --> 00:50:08
Finite elements lives or
dies on that question.
781
00:50:08 --> 00:50:14
The answer is yes, because
along that side, if I just
782
00:50:14 --> 00:50:20
focus on that side, where these
three values are shared on both
783
00:50:20 --> 00:50:23
sides, by the triangle
on both sides.
784
00:50:23 --> 00:50:28
Along that edge, what kind
of a function have I got?
785
00:50:28 --> 00:50:31
It's second degree.
786
00:50:31 --> 00:50:34
This is whatever, when I
restrict this to just run along
787
00:50:34 --> 00:50:37
a line, it's a parabola.
788
00:50:37 --> 00:50:40
And the parabola is determined
by those three values.
789
00:50:40 --> 00:50:42
So having it right at three
points means I have it
790
00:50:42 --> 00:50:44
right the whole way.
791
00:50:44 --> 00:50:44
Yeah.
792
00:50:44 --> 00:50:49
So there you see what quadratic
elements would look like, and
793
00:50:49 --> 00:50:55
you could extend the code in
the book and on the CSE site to
794
00:50:55 --> 00:50:57
work for quadratic elements.
795
00:50:57 --> 00:51:01
And you want to just guess what
cubic elements could look like?
796
00:51:01 --> 00:51:03
I'm sorry, we've run five
minutes over, but maybe
797
00:51:03 --> 00:51:06
finite elements is worth it.
798
00:51:06 --> 00:51:13
So if I had cubic elements,
any idea how many?
799
00:51:13 --> 00:51:18
So I'm now going up to, I'm
adding g x cubed, h, i,
800
00:51:18 --> 00:51:25
j, any idea how many
coefficients I now have?
801
00:51:25 --> 00:51:28
Four new ones plus
these six is ten.
802
00:51:28 --> 00:51:29
I need ten nodes.
803
00:51:29 --> 00:51:33
Where I am I going to put
ten nodes in this triangle?
804
00:51:33 --> 00:51:36
I want to put them, I'd like
to have some on the edges.
805
00:51:36 --> 00:51:39
Because the edges help me make
triangles match each other.
806
00:51:39 --> 00:51:42
They'll just be like
bowling balls.
807
00:51:42 --> 00:51:52
So here's six, oops, that
wouldn't be believable.
808
00:51:52 --> 00:51:54
Is that right?
809
00:51:54 --> 00:51:55
Four, three, two, and one.
810
00:51:55 --> 00:51:56
Yeah.
811
00:51:56 --> 00:51:57
Yeah.
812
00:51:57 --> 00:51:58
OK.
813
00:51:58 --> 00:52:06
So, now I've got a bubble node
inside and I've got four nodes
814
00:52:06 --> 00:52:13
of vertices and two points, at
two 1/3 points, and that
815
00:52:13 --> 00:52:16
will then match the
triangle next to it.
816
00:52:16 --> 00:52:19
Because four points
determine a cubic.
817
00:52:19 --> 00:52:23
There you go, I hope you
have fun, I hope you
818
00:52:23 --> 00:52:24
have a great holiday.
819
00:52:24 --> 00:52:30
I'll see you Wednesday for
Fourier and always open for
820
00:52:30 --> 00:52:32
questions on the MATLAB.