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PROFESSOR STRANG: Okay.
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00:00:20 --> 00:00:21
Hi.
11
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So, our goal, certainly, to
reach next week is partial
12
00:00:29 --> 00:00:30
differential equations.
13
00:00:30 --> 00:00:34
Laplace's equation.
14
00:00:34 --> 00:00:38
Additional topics still
to see clearly in 1-D.
15
00:00:39 --> 00:00:44
So one of those topics, these
will both come today, one of
16
00:00:44 --> 00:00:48
those topics is the idea,
still in the finite element
17
00:00:48 --> 00:00:51
world, of element matrices.
18
00:00:51 --> 00:00:58
So you remember, we saw those,
that each bar in the truss
19
00:00:58 --> 00:01:02
could give a piece of A
transpose A that could be
20
00:01:02 --> 00:01:06
stamped in, or assembled, to
use the right word -- I think
21
00:01:06 --> 00:01:09
assembled is maybe used more
than stamped in, but
22
00:01:09 --> 00:01:16
both ok -- into K.
23
00:01:16 --> 00:01:20
For graphs, an edge in
the graph gave us a
24
00:01:20 --> 00:01:23
little [1, -1 ;-1, 1]
25
00:01:23 --> 00:01:25
matrix that could
be stamped in.
26
00:01:25 --> 00:01:29
And now we want to see, how
does that process work
27
00:01:29 --> 00:01:31
for finite elements.
28
00:01:31 --> 00:01:34
Because that's how finite
element matrices are really
29
00:01:34 --> 00:01:37
put together out of
these element matrices.
30
00:01:37 --> 00:01:43
Okay, so that's step one,
that's half today's lecture.
31
00:01:43 --> 00:01:48
Step two, problem two now,
coming from the next section,
32
00:01:48 --> 00:01:51
is fourth order equations.
33
00:01:51 --> 00:01:53
Up to now, all our
differential equations
34
00:01:53 --> 00:01:55
have been second order.
35
00:01:55 --> 00:01:58
Are there fourth order
equations that are important?
36
00:01:58 --> 00:01:59
Yes, there are.
37
00:01:59 --> 00:02:01
For beam bending.
38
00:02:01 --> 00:02:05
So I'll describe that
application, which leads to
39
00:02:05 --> 00:02:07
fourth order equations.
40
00:02:07 --> 00:02:11
Do they fit our A
transpose C A framework?
41
00:02:11 --> 00:02:13
You bet.
42
00:02:13 --> 00:02:16
You know they will.
43
00:02:16 --> 00:02:22
Each additional application in
the framework kind of get us
44
00:02:22 --> 00:02:26
comfortable, familiar with that
framework, what it can do.
45
00:02:26 --> 00:02:28
The A transpose C A.
46
00:02:28 --> 00:02:32
So let me start with
element matrices.
47
00:02:32 --> 00:02:35
I'll get the homework, those
numbers will get posted on
48
00:02:35 --> 00:02:36
the website later today.
49
00:02:36 --> 00:02:41
I just thought I'd put them
down, and I have to figure
50
00:02:41 --> 00:02:43
out what would be a
suitable MATLAB question.
51
00:02:43 --> 00:02:48
So my idea for this homework,
as it really was for the last
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homework, was that you get
some, not a large number of
53
00:02:54 --> 00:02:57
ordinary questions, paper and
pencil questions, and
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00:02:57 --> 00:02:59
one MATLAB question.
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00:02:59 --> 00:03:06
When I wrote MATLAB last,
people interpreted
56
00:03:06 --> 00:03:09
that as last MATLAB.
57
00:03:09 --> 00:03:11
But those words are
not commutative.
58
00:03:11 --> 00:03:13
Right?
59
00:03:13 --> 00:03:16
I just meant -- and it doesn't
really matter, it was a dumb
60
00:03:16 --> 00:03:20
thing to say -- I meant you to
put the MATLAB question
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00:03:20 --> 00:03:23
at the end after the
regular questions.
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00:03:23 --> 00:03:24
Sorry.
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00:03:24 --> 00:03:26
So that MATLAB is due.
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Maybe a bunch of those
turned in on Monday didn't
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00:03:30 --> 00:03:32
include the MATLAB.
66
00:03:32 --> 00:03:34
No penalty.
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00:03:34 --> 00:03:39
I'm talking now about the
MATLAB for the trusses.
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How many have still got
a MATLAB for the truss
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still to turn in?
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00:03:44 --> 00:03:45
A number.
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00:03:45 --> 00:03:46
Oh, not too many.
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00:03:46 --> 00:03:46
Okay.
73
00:03:46 --> 00:03:48
Anyway, just when you can.
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00:03:48 --> 00:03:50
In my envelope is good.
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00:03:50 --> 00:03:50
Okay.
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00:03:50 --> 00:03:55
So it'll be a similar thing for
this week, and because it's
77
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short I said okay, let's get
that cleaned up by Monday.
78
00:04:00 --> 00:04:04
And then we're ready for vector
calculus, partial differential
79
00:04:04 --> 00:04:10
equations, two and three
dimensions, the next big step.
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00:04:10 --> 00:04:15
If I want to illustrate element
matrices, the best place I
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00:04:15 --> 00:04:18
could do it would be to go back
to our piecewise linear
82
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elements in 1-D, and see how an
element now is just one of
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00:04:26 --> 00:04:28
those little intervals.
84
00:04:28 --> 00:04:33
Little pieces of the
whole structure.
85
00:04:33 --> 00:04:38
If it's a bar, or whatever it
is, I've cut it into pieces.
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00:04:38 --> 00:04:42
Here's one piece,
here's another piece.
87
00:04:42 --> 00:04:47
With finite differences, if I
gave you unequally spaced
88
00:04:47 --> 00:04:51
meshes, if my little h is
different from my big H,
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00:04:51 --> 00:04:53
we'd have to think again.
90
00:04:53 --> 00:04:58
I mean there would be some
three point, instead of minus
91
00:04:58 --> 00:05:02
one, two, minus one for
the second difference.
92
00:05:02 --> 00:05:05
It would be a little lopsided,
of course, when the
93
00:05:05 --> 00:05:09
mesh is unequal.
94
00:05:09 --> 00:05:12
We would have to think that
through for finite differences;
95
00:05:12 --> 00:05:15
for finite elements, the
system thinks for us.
96
00:05:15 --> 00:05:18
So I just want to show
how that happens.
97
00:05:18 --> 00:05:19
So I'll take this.
98
00:05:19 --> 00:05:25
What's the element matrix
for our standard equation
99
00:05:25 --> 00:05:27
for that element?
100
00:05:27 --> 00:05:29
Okay.
101
00:05:29 --> 00:05:33
And then it would fit
into the big matrix K.
102
00:05:33 --> 00:05:37
I'm going to come up with a
matrix K that I could do in
103
00:05:37 --> 00:05:40
any other way, but this is
the way it's really done.
104
00:05:40 --> 00:05:46
So I look in that interval.
105
00:05:46 --> 00:05:48
I'm focusing on that interval.
106
00:05:48 --> 00:05:52
So here I've drawn
the basis function.
107
00:05:52 --> 00:05:55
But now I'm not going to
operate so much with
108
00:05:55 --> 00:05:57
basis function.
109
00:05:57 --> 00:05:59
Let me just think about
that interval again.
110
00:05:59 --> 00:06:00
That's the same interval.
111
00:06:00 --> 00:06:03
So those were the basis
functions that went
112
00:06:03 --> 00:06:06
up to height one.
113
00:06:06 --> 00:06:09
One started at height one and
went down, one started at
114
00:06:09 --> 00:06:11
height zero and went up.
115
00:06:11 --> 00:06:14
But it's their
combination, of course.
116
00:06:14 --> 00:06:21
Let me number this node number
zero and this node number one.
117
00:06:21 --> 00:06:27
So we have some
height, U_0, here.
118
00:06:27 --> 00:06:31
This is coming from
U_0 times that hat.
119
00:06:31 --> 00:06:35
And some other height, U_1,
here, which is coming
120
00:06:35 --> 00:06:38
from U_1 times that hat.
121
00:06:38 --> 00:06:46
But inside this interval, my U
in this interval is U_0 times
122
00:06:46 --> 00:06:51
that first hat function, the
phi_0, the coming down hat.
123
00:06:51 --> 00:06:57
Plus U_1 times the phi_1
function, the going up hat.
124
00:06:57 --> 00:07:01
It's a linear function, so
it's going to be there.
125
00:07:01 --> 00:07:06
That's my graph -- no, yes.
126
00:07:06 --> 00:07:07
This is U(x).
127
00:07:07 --> 00:07:09
128
00:07:09 --> 00:07:10
U(x).
129
00:07:11 --> 00:07:12
Okay.
130
00:07:12 --> 00:07:13
Right?
131
00:07:13 --> 00:07:17
I'm focusing inside
one element.
132
00:07:17 --> 00:07:20
One interval.
133
00:07:20 --> 00:07:23
I've numbered them zero
to one, but of course
134
00:07:23 --> 00:07:25
that distance is H.
135
00:07:25 --> 00:07:28
Okay, so there's my function.
136
00:07:28 --> 00:07:28
Right?
137
00:07:28 --> 00:07:32
Everybody agrees that this
combination, it starts out
138
00:07:32 --> 00:07:38
right, it ends up right, at
height U_1, and it's linear.
139
00:07:38 --> 00:07:40
So that's got to be right.
140
00:07:40 --> 00:07:48
So really, what's the
contribution from that element?
141
00:07:48 --> 00:07:57
I look at the quantity, and I'm
always taking the phi functions
142
00:07:57 --> 00:07:58
to equal the V functions.
143
00:07:58 --> 00:08:06
So I can write this as c(x)
times dU/dx squared, dx.
144
00:08:06 --> 00:08:11
145
00:08:11 --> 00:08:16
Over the whole thing, over the
whole interval, that would
146
00:08:16 --> 00:08:21
be the U transpose K U.
147
00:08:21 --> 00:08:28
This, everybody remembers the
K part, of course, comes from
148
00:08:28 --> 00:08:30
the left side of the problem.
149
00:08:30 --> 00:08:34
It comes from integrations.
150
00:08:34 --> 00:08:40
And this dU is now a
combination of all the -- with
151
00:08:40 --> 00:08:48
weights U_0, U_1, U_2, U_3 --
if I plug in for capital U and
152
00:08:48 --> 00:08:55
do all the integrals,
I'll see what K is.
153
00:08:55 --> 00:08:56
Now what's the point?
154
00:08:56 --> 00:09:00
The point is, when I did those
integrals, the question is, how
155
00:09:00 --> 00:09:03
am I going to do the integrals?
156
00:09:03 --> 00:09:08
Do I do them the way I did
before, was I watched what
157
00:09:08 --> 00:09:15
phi_0, one phi times another
phi and I integrated.
158
00:09:15 --> 00:09:17
That was successful.
159
00:09:17 --> 00:09:23
But the new way is, integrate
it an interval at a time.
160
00:09:23 --> 00:09:26
Do the integrals.
161
00:09:26 --> 00:09:29
So this would be K global.
162
00:09:29 --> 00:09:36
Now I'm going to go
just from zero to H.
163
00:09:36 --> 00:09:41
If I go from just zero to
H, that's going to give
164
00:09:41 --> 00:09:43
me the K element piece.
165
00:09:43 --> 00:09:47
The little piece that comes
from this little interval.
166
00:09:47 --> 00:09:54
And on that little interval,
this is my formula for U.
167
00:09:54 --> 00:09:59
I'm just hoping you'll sort of
see this as a reasonable idea,
168
00:09:59 --> 00:10:03
and then when we do the
integral you'll see it.
169
00:10:03 --> 00:10:04
It clicks.
170
00:10:04 --> 00:10:07
Okay, how does it click?
171
00:10:07 --> 00:10:08
So what's my plan?
172
00:10:08 --> 00:10:11
My plan is, here's my function.
173
00:10:11 --> 00:10:13
There is its picture.
174
00:10:13 --> 00:10:15
I'm just going to
plug it into here.
175
00:10:15 --> 00:10:17
Oh, it's going to be
simple, isn't it?
176
00:10:17 --> 00:10:23
What's the slope of that
function in that element?
177
00:10:23 --> 00:10:24
On that interval?
178
00:10:24 --> 00:10:25
What's the dU/dx?
179
00:10:26 --> 00:10:33
So instead of doing every full
integral for each separate phi,
180
00:10:33 --> 00:10:40
I'm doing every element
integral for both phis.
181
00:10:40 --> 00:10:44
See, these two phis are both
coming in to that element.
182
00:10:44 --> 00:10:46
Okay, so what's dU/dx.
183
00:10:48 --> 00:10:51
I didn't realize how neat
this that's going to be.
184
00:10:51 --> 00:10:55
So what's dU/dx
in this element?
185
00:10:55 --> 00:10:56
I'll use a different
board here.
186
00:10:56 --> 00:11:01
So it's an integral, then,
from zero to H of whatever
187
00:11:01 --> 00:11:04
my c(x) might be.
188
00:11:04 --> 00:11:05
And I'll make a
comment on that.
189
00:11:05 --> 00:11:08
But my focus here
is, what's dU/dx?
190
00:11:09 --> 00:11:12
What's the derivative?
191
00:11:12 --> 00:11:14
And then I'm going
to square it.
192
00:11:14 --> 00:11:18
For this function, for that
picture, the slope is
193
00:11:18 --> 00:11:24
obviously (U_1-U_0)/H Right?
194
00:11:24 --> 00:11:26
The slope is (U_1-U_0)/H.
195
00:11:26 --> 00:11:30
196
00:11:30 --> 00:11:31
And I'm squaring it. dx.
197
00:11:31 --> 00:11:34
198
00:11:34 --> 00:11:35
Okay.
199
00:11:35 --> 00:11:37
Let me take c(x)=1.
200
00:11:37 --> 00:11:39
201
00:11:39 --> 00:11:44
Just to see clearly
what's going on.
202
00:11:44 --> 00:11:47
So c(x) is just
going to be one.
203
00:11:47 --> 00:11:55
Okay, so I'm claiming that
this is my U transpose.
204
00:11:55 --> 00:11:56
My little piece.
205
00:11:56 --> 00:11:58
Why is it only a little piece?
206
00:11:58 --> 00:12:02
Because it only involves
two of the U's.
207
00:12:02 --> 00:12:07
It's going to be a little two
by two element matrix, that
208
00:12:07 --> 00:12:09
comes from this element.
209
00:12:09 --> 00:12:13
And then it's going to to be
put into the big K, the global
210
00:12:13 --> 00:12:16
K, in its proper place.
211
00:12:16 --> 00:12:17
Okay, well.
212
00:12:17 --> 00:12:19
It's trivial, right?
213
00:12:19 --> 00:12:22
This is a constant, this is
a constant, the integral
214
00:12:22 --> 00:12:32
is just H times U_1-U_0,
squared, over H squared.
215
00:12:32 --> 00:12:35
Because that's getting squared,
the interval was length H.
216
00:12:35 --> 00:12:36
I think I just have a 1/H.
217
00:12:38 --> 00:12:45
So this is my U
transpose K element U.
218
00:12:45 --> 00:12:48
And now I want to pick
out, what's that matrix?
219
00:12:48 --> 00:12:51
What's that little two
by two matrix that only
220
00:12:51 --> 00:12:55
touches these two U's?
221
00:12:55 --> 00:12:59
What's the matrix -- well,
since it only touches these
222
00:12:59 --> 00:13:02
two U's, you can tell me.
223
00:13:02 --> 00:13:02
I want
224
00:13:02 --> 00:13:03
a U_0, U_1.
225
00:13:03 --> 00:13:06
226
00:13:06 --> 00:13:08
Sorry, let me make a
little more space.
227
00:13:08 --> 00:13:13
And you can tell me what
matrix goes in there.
228
00:13:13 --> 00:13:15
I want this all to match up.
229
00:13:15 --> 00:13:15
U_0, U_1.
230
00:13:16 --> 00:13:20
Now here is the two by two
element matrix, .
231
00:13:20 --> 00:13:25
232
00:13:25 --> 00:13:29
What's the two by two
matrix that will correctly
233
00:13:29 --> 00:13:31
produce this answer?
234
00:13:31 --> 00:13:35
I'm just shooting
for that answer.
235
00:13:35 --> 00:13:37
What do I have here?
236
00:13:37 --> 00:13:44
This is U_1 squared minus
2U_0*U_1, plus a U_0 squared.
237
00:13:44 --> 00:13:45
Right?
238
00:13:45 --> 00:13:48
And I have to remember the
1/H, so it's automatically
239
00:13:48 --> 00:13:51
going to come out right.
240
00:13:51 --> 00:13:57
So there's a 1/H, shall I
remember that first off.
241
00:13:57 --> 00:14:00
1/H is part of my
element matrix.
242
00:14:00 --> 00:14:09
And then, what are the numbers
that go inside that matrix?
243
00:14:09 --> 00:14:11
We had practice with this.
244
00:14:11 --> 00:14:14
You remember when we talked
about positive definite
245
00:14:14 --> 00:14:17
matrices, way back
in chapter one?
246
00:14:17 --> 00:14:23
The point was that we could
look at eigenvalues, or
247
00:14:23 --> 00:14:24
pivots, or something.
248
00:14:24 --> 00:14:27
But the core idea was energy.
249
00:14:27 --> 00:14:31
The core idea was that energy,
that quadratic, and that's
250
00:14:31 --> 00:14:32
what we're looking at again.
251
00:14:32 --> 00:14:34
That's the energy.
252
00:14:34 --> 00:14:39
That's the energy, right there,
and this is the energy, this is
253
00:14:39 --> 00:14:45
the energy in the finite
element subspace.
254
00:14:45 --> 00:14:49
All I'm saying is, what matrix,
what two by two matrix,
255
00:14:49 --> 00:14:54
goes with U_1 squared minus
2U_0*U_1 plus U_0 squared.
256
00:14:54 --> 00:14:56
Just tell me what to
put in that matrix.
257
00:14:56 --> 00:14:58
What do I put in here?
258
00:14:58 --> 00:14:59
One, right.
259
00:14:59 --> 00:15:00
Because it's
multiplying U_0 U_0.
260
00:15:01 --> 00:15:03
What do I put there?
261
00:15:03 --> 00:15:04
Minus one.
262
00:15:04 --> 00:15:05
Good.
263
00:15:05 --> 00:15:10
Because I have a minus two, it
comes in, minus one comes in
264
00:15:10 --> 00:15:12
twice, and a one goes there.
265
00:15:12 --> 00:15:16
So there, with the 1/H
included, is K_e.
266
00:15:17 --> 00:15:20
K element, for that element.
267
00:15:20 --> 00:15:25
For the big H element.
268
00:15:25 --> 00:15:30
You'll say big deal, because
we've seen this thing before.
269
00:15:30 --> 00:15:35
Notice what is nice.
270
00:15:35 --> 00:15:38
First of all, notice
how nice that is.
271
00:15:38 --> 00:15:40
It's particularly nice,
of course, because I
272
00:15:40 --> 00:15:43
took c(x) to be one.
273
00:15:43 --> 00:15:44
So let me make a comment.
274
00:15:44 --> 00:15:47
Suppose c(x) was not one.
275
00:15:47 --> 00:15:50
What would I do?
276
00:15:50 --> 00:15:57
Suppose c(x), suppose I have
some variable stiffness
277
00:15:57 --> 00:15:57
in the material.
278
00:15:57 --> 00:16:05
Suppose the material could
be changing width, so its
279
00:16:05 --> 00:16:09
stiffness would change.
280
00:16:09 --> 00:16:12
So in other words, I'd have
a variable, c(x), that I
281
00:16:12 --> 00:16:15
should do the integral.
282
00:16:15 --> 00:16:19
Probably finite element systems
aren't set up to actually
283
00:16:19 --> 00:16:22
do the exact integral.
284
00:16:22 --> 00:16:23
What would they do?
285
00:16:23 --> 00:16:27
They would take, for that
simple integration, they
286
00:16:27 --> 00:16:32
would probably just take
c(x) at the midpoint.
287
00:16:32 --> 00:16:35
So there's a numerical
integration here in the
288
00:16:35 --> 00:16:39
creation of these
element matrices.
289
00:16:39 --> 00:16:43
And numerical immigration
is just, take a suitable
290
00:16:43 --> 00:16:46
combination of the
values at a few points.
291
00:16:46 --> 00:16:48
You do know Simpson's rule?
292
00:16:48 --> 00:16:53
Simpson's rule, that's a
pretty high level rule.
293
00:16:53 --> 00:16:57
My suggestion there was
just a midpoint rule.
294
00:16:57 --> 00:17:00
Just take c(x) at the
middle of the interval.
295
00:17:00 --> 00:17:02
Then it would factor out.
296
00:17:02 --> 00:17:07
So I should really
put a c here.
297
00:17:07 --> 00:17:10
A c should really be
coming in there.
298
00:17:10 --> 00:17:13
And you expected that, right,
from the A transpose C A.
299
00:17:13 --> 00:17:16
We always saw a C
in the middle.
300
00:17:16 --> 00:17:17
It really should be there.
301
00:17:17 --> 00:17:21
When I took c to be
1, I didn't see it.
302
00:17:21 --> 00:17:23
So what am I doing?
303
00:17:23 --> 00:17:30
I'm approximating c(x)
by c at halfway.
304
00:17:30 --> 00:17:31
Approximately.
305
00:17:31 --> 00:17:37
I would replace this
unpleasant, possibly varying,
306
00:17:37 --> 00:17:42
function by c at a point
and use that value.
307
00:17:42 --> 00:17:46
So numerical integration is one
part of the picture that we
308
00:17:46 --> 00:17:50
won't go into all the
different rules.
309
00:17:50 --> 00:17:52
There's a rectangle rule,
there's a trapezoid
310
00:17:52 --> 00:17:55
rule, very good.
311
00:17:55 --> 00:17:58
There's the Simpson's
rule that's better.
312
00:17:58 --> 00:18:03
As I get higher order elements,
like those cubic elements I
313
00:18:03 --> 00:18:06
spoke about, the numerical
integration has to keep up.
314
00:18:06 --> 00:18:10
If I was integrating cubic
stuff, I wouldn't use
315
00:18:10 --> 00:18:12
such a cheap rule.
316
00:18:12 --> 00:18:15
I would go up to Simpson's
rule, or Gauss's rule,
317
00:18:15 --> 00:18:16
or somebody's rule.
318
00:18:16 --> 00:18:19
Anyway, that's the c part.
319
00:18:19 --> 00:18:23
Here's the part that
stamps into the matrix.
320
00:18:23 --> 00:18:29
Notice, by the way, when I
stamp it in, tell me how it's
321
00:18:29 --> 00:18:30
going to look stamped in.
322
00:18:30 --> 00:18:32
And then I've completed it.
323
00:18:32 --> 00:18:34
So here's my big K.
324
00:18:34 --> 00:18:36
I wish I had a little
more room for it.
325
00:18:36 --> 00:18:40
Okay, here's my big K.
326
00:18:40 --> 00:18:43
So that interval that I drew
there, the H interval, will
327
00:18:43 --> 00:18:46
stamp in here, some two by two.
328
00:18:46 --> 00:18:47
Right?
329
00:18:47 --> 00:18:51
Now where will the similar
thing coming from this guy
330
00:18:51 --> 00:18:55
-- maybe it has to be
numbered minus one.
331
00:18:55 --> 00:18:58
Sorry about that.
332
00:18:58 --> 00:19:00
That's another little interval.
333
00:19:00 --> 00:19:02
I'll do the same thing
on that interval.
334
00:19:02 --> 00:19:06
I'll get a little element
matrix, two by two guy,
335
00:19:06 --> 00:19:08
for K for that element.
336
00:19:08 --> 00:19:14
And where will it fit
in to the big k?
337
00:19:14 --> 00:19:16
Does it fit in up here,
let me just ask you.
338
00:19:16 --> 00:19:19
Does it fit in up there?
339
00:19:19 --> 00:19:23
Yes, no?
340
00:19:23 --> 00:19:30
I'm assembling, stamping
in the small two by twos
341
00:19:30 --> 00:19:32
into the full n by n.
342
00:19:32 --> 00:19:37
And if I draw the
picture you'll see it.
343
00:19:37 --> 00:19:43
So when I do the two by two for
that big H interval, it goes
344
00:19:43 --> 00:19:46
there, let's say, then I just
want to say where does the
345
00:19:46 --> 00:19:52
two by two go for the
interval to the left?
346
00:19:52 --> 00:19:54
Does it go there?
347
00:19:54 --> 00:19:55
Nope.
348
00:19:55 --> 00:19:57
How does it go?
349
00:19:57 --> 00:19:59
It overlaps.
350
00:19:59 --> 00:19:59
Right?
351
00:19:59 --> 00:20:02
It overlaps.
352
00:20:02 --> 00:20:04
Why does it overlap?
353
00:20:04 --> 00:20:10
Because the phi_0, this guy,
is acting on the right, and
354
00:20:10 --> 00:20:12
also acting on the left.
355
00:20:12 --> 00:20:21
The U_0 is active, is partly
controlling the slope this
356
00:20:21 --> 00:20:23
way, and also that way.
357
00:20:23 --> 00:20:26
The U_0 is in common
the two intervals.
358
00:20:26 --> 00:20:31
Anytime any unknowns, any mesh
points that are in common to
359
00:20:31 --> 00:20:34
two elements, we're going
to have an overlap
360
00:20:34 --> 00:20:36
when we assemble.
361
00:20:36 --> 00:20:39
And so it'll just sit,
it'll sit right there.
362
00:20:39 --> 00:20:44
And so there is
the diagonal guy.
363
00:20:44 --> 00:20:47
And maybe you could tell me
what number will go there.
364
00:20:47 --> 00:20:49
What number would
actually go there?
365
00:20:49 --> 00:20:55
And then you'll see the whole
point of assembly, stamping in.
366
00:20:55 --> 00:21:00
Well, what number goes
there from this?
367
00:21:00 --> 00:21:01
One times the c/H.
368
00:21:03 --> 00:21:11
So in here would be the c, can
I call it c right, or c_H, c on
369
00:21:11 --> 00:21:16
the big H interval,
divided by the H.
370
00:21:16 --> 00:21:20
Because that one, we
had to get it right.
371
00:21:20 --> 00:21:23
And then what will go in
that very same spot?
372
00:21:23 --> 00:21:29
So add it in coming from
the small h interval.
373
00:21:29 --> 00:21:33
The same thing, it'll be
this one, like shifted
374
00:21:33 --> 00:21:37
up, moved over.
375
00:21:37 --> 00:21:41
So it'll be coming from that
one, so it'll be a c on
376
00:21:41 --> 00:21:45
the little h interval,
divided by little h.
377
00:21:45 --> 00:21:50
That would be the
diagonal entry of K.
378
00:21:50 --> 00:21:52
That's what we would
see right there.
379
00:21:52 --> 00:21:57
Over here, should I write
a typical row of K?
380
00:21:57 --> 00:22:00
Typical row of K, when I
do that, is going to have
381
00:22:00 --> 00:22:06
this c_h/h, plus c_H/H.
382
00:22:08 --> 00:22:11
That's like the two, right?
383
00:22:11 --> 00:22:13
That's like the two.
384
00:22:13 --> 00:22:18
And what goes here?
385
00:22:18 --> 00:22:23
What will the entry be that
sits there when I assemble?
386
00:22:23 --> 00:22:25
Just this guy.
387
00:22:25 --> 00:22:26
Right?
388
00:22:26 --> 00:22:28
Just this guy, times that.
389
00:22:28 --> 00:22:31
The entry here will
be the minus c_H/H.
390
00:22:33 --> 00:22:34
That'll be the entry.
391
00:22:34 --> 00:22:38
This is the diagonal one, this
is the one to the right, and
392
00:22:38 --> 00:22:40
what's the one to the left?
393
00:22:40 --> 00:22:45
What's the one here?
394
00:22:45 --> 00:22:47
Well, you know what it is.
395
00:22:47 --> 00:22:53
It's going to come from
the minus and it'll
396
00:22:53 --> 00:22:55
be the minus c_h/h.
397
00:22:55 --> 00:22:58
398
00:22:58 --> 00:22:59
Look at that.
399
00:22:59 --> 00:23:02
That just shows
you how it works.
400
00:23:02 --> 00:23:06
And again, you can look
at that, page 299
401
00:23:06 --> 00:23:09
to 300 in the book.
402
00:23:09 --> 00:23:16
You see that if the c's are the
same, if the h's are the same,
403
00:23:16 --> 00:23:20
then we're looking again at our
minus one, two, minus one.
404
00:23:20 --> 00:23:25
Times whatever c over h, to
keep it dimensionally right.
405
00:23:25 --> 00:23:25
Do you see that?
406
00:23:25 --> 00:23:27
It's just simple.
407
00:23:27 --> 00:23:28
Simple idea.
408
00:23:28 --> 00:23:35
The point is that each interval
can be done separately.
409
00:23:35 --> 00:23:36
It's a simple idea in 1-D.
410
00:23:36 --> 00:23:40
411
00:23:40 --> 00:23:45
It's a key idea in 2-D,
where we have triangles,
412
00:23:45 --> 00:23:47
we have tetrahedra, tets.
413
00:23:47 --> 00:23:50
We'll see this in two
dimensions, later in this
414
00:23:50 --> 00:23:56
chapter, when we're doing
Laplace's equation.
415
00:23:56 --> 00:23:58
It's just fun to see it work.
416
00:23:58 --> 00:24:02
You'll have different
triangles, say
417
00:24:02 --> 00:24:04
column triangles.
418
00:24:04 --> 00:24:13
So that phi -- do you
want me to look ahead?
419
00:24:13 --> 00:24:15
Just ten seconds, to triangles?
420
00:24:15 --> 00:24:19
So imagine we have triangles
here, so we have piecewise,
421
00:24:19 --> 00:24:21
we have little pyramids.
422
00:24:21 --> 00:24:24
Instead of hat functions,
they grow to pyramids.
423
00:24:24 --> 00:24:29
So there's a pyramid guy whose
height is one there, and drops
424
00:24:29 --> 00:24:31
to zero in all these places.
425
00:24:31 --> 00:24:34
And that's our phi.
426
00:24:34 --> 00:24:37
Our trial function,
test function, will be
427
00:24:37 --> 00:24:41
pyramid function, then.
428
00:24:41 --> 00:24:45
And I can do integrals that
way, or I can take the integral
429
00:24:45 --> 00:24:48
over a typical triangle.
430
00:24:48 --> 00:24:53
So a typical triangle is
involved with three, now I've
431
00:24:53 --> 00:24:57
three functions, in the
linear case, controlling
432
00:24:57 --> 00:24:59
inside that triangle.
433
00:24:59 --> 00:25:04
So what will be the size
of the element matrix?
434
00:25:04 --> 00:25:07
Can you sort of see how the
system is going to work?
435
00:25:07 --> 00:25:09
And then we'll make
it work in 2-D.
436
00:25:09 --> 00:25:12
437
00:25:12 --> 00:25:16
Every node, every mesh
point, corresponds, has
438
00:25:16 --> 00:25:21
a pyramid function, has
a U that goes with it.
439
00:25:21 --> 00:25:23
Those U's are the unknowns.
440
00:25:23 --> 00:25:27
And how many of those unknowns
are operating inside
441
00:25:27 --> 00:25:29
that triangle?
442
00:25:29 --> 00:25:30
Three.
443
00:25:30 --> 00:25:34
So what will be the size of the
element matrix, the non-zero
444
00:25:34 --> 00:25:36
part of the element matrix?
445
00:25:36 --> 00:25:37
Three by three.
446
00:25:37 --> 00:25:39
What else could it be?
447
00:25:39 --> 00:25:41
So we'll see what
it looks like.
448
00:25:41 --> 00:25:44
And we'll have integrals
over triangles.
449
00:25:44 --> 00:25:48
So that's good.
450
00:25:48 --> 00:25:49
Okay, thanks.
451
00:25:49 --> 00:25:54
Exactly halfway through the
hour is exactly that first
452
00:25:54 --> 00:25:57
topic of element matrices.
453
00:25:57 --> 00:25:58
Done.
454
00:25:58 --> 00:26:02
Okay.
455
00:26:02 --> 00:26:05
Let me take two deep
breaths, and move to
456
00:26:05 --> 00:26:08
fourth order equations.
457
00:26:08 --> 00:26:11
Fourth order equations.
458
00:26:11 --> 00:26:12
For the bending of a beam.
459
00:26:12 --> 00:26:16
So I'd better draw a beam.
460
00:26:16 --> 00:26:19
This is a 1-D problem, still.
461
00:26:19 --> 00:26:24
This is a 1-D problem, still.
462
00:26:24 --> 00:26:28
To keep it 1-D, this
better be a thin beam.
463
00:26:28 --> 00:26:32
So this is a thin beam.
464
00:26:32 --> 00:26:34
And the loads, what's
the difference?
465
00:26:34 --> 00:26:36
What's here?
466
00:26:36 --> 00:26:40
It looks like a bar,
pretty much, right?
467
00:26:40 --> 00:26:47
But the difference is, the
load is acting this way.
468
00:26:47 --> 00:26:49
The load is acting
that way on the beam.
469
00:26:49 --> 00:26:51
Maybe two loads.
470
00:26:51 --> 00:26:53
Maybe a uniform load.
471
00:26:53 --> 00:26:56
Maybe the weight of the beam.
472
00:26:56 --> 00:27:00
But it's transverse.
473
00:27:00 --> 00:27:04
It's in the perpendicular,
it's transverse to the beam.
474
00:27:04 --> 00:27:06
It's this way.
475
00:27:06 --> 00:27:09
So the beam bends.
476
00:27:09 --> 00:27:11
Let me do a fixed free.
477
00:27:11 --> 00:27:15
So this'll be a
fixed free beam.
478
00:27:15 --> 00:27:27
Fixed free, the word for fixed
free would be cantilever beam.
479
00:27:27 --> 00:27:28
Okay.
480
00:27:28 --> 00:27:33
So what happens if I
impose those loads.
481
00:27:33 --> 00:27:37
Well, the beam bends.
482
00:27:37 --> 00:27:43
So the displacement is now
downwards, is now not the
483
00:27:43 --> 00:27:47
direction of the rod, the
displacement I'm interested in
484
00:27:47 --> 00:27:49
is perpendicular to the beam.
485
00:27:49 --> 00:27:50
Downwards.
486
00:27:50 --> 00:27:56
Okay, so we can start
on our framework.
487
00:27:56 --> 00:28:00
So this is displacement. u(x).
488
00:28:02 --> 00:28:07
I'll stay with the
same letters.
489
00:28:07 --> 00:28:11
So you know I'm going to
have an A, that will take
490
00:28:11 --> 00:28:21
me to whatever this is
going to get called.
491
00:28:21 --> 00:28:24
What's the quantity there?
492
00:28:24 --> 00:28:29
It's going to be, let's see.
493
00:28:29 --> 00:28:30
What happens?
494
00:28:30 --> 00:28:33
So this is just geometric now.
495
00:28:33 --> 00:28:38
Let me put in the easy part, C.
496
00:28:38 --> 00:28:40
That'll be sort of the
bending stiffness.
497
00:28:40 --> 00:28:41
Right?
498
00:28:41 --> 00:28:44
This'll be the
bending stiffness.
499
00:28:44 --> 00:28:49
Because the beam
is going to bend.
500
00:28:49 --> 00:28:56
And over here I'll
get a suitable w.
501
00:28:56 --> 00:29:01
So when the beam bends,
it's curvature.
502
00:29:01 --> 00:29:04
Curvature of the beam
is what's produced.
503
00:29:04 --> 00:29:08
It's not stretching of the
beam; it's curving of the beam.
504
00:29:08 --> 00:29:14
So this quantity, e, will be
the curvature that comes
505
00:29:14 --> 00:29:16
from the displacement.
506
00:29:16 --> 00:29:22
If I displace these beams,
suppose I put a node here,
507
00:29:22 --> 00:29:24
it's going to bend that down.
508
00:29:24 --> 00:29:26
The bar will curve.
509
00:29:26 --> 00:29:29
So the curvature, e.
510
00:29:29 --> 00:29:31
Now then, the question
is, what is this A;
511
00:29:31 --> 00:29:33
what is the curvature?
512
00:29:33 --> 00:29:38
Well, do you remember?
513
00:29:38 --> 00:29:40
You're on the ball if you
remember the formula
514
00:29:40 --> 00:29:42
for curvature.
515
00:29:42 --> 00:29:44
It's a horrible
formula, actually.
516
00:29:44 --> 00:29:47
But that's only because we're
going to make it better.
517
00:29:47 --> 00:29:52
You remember the curvature,
it was in calculus.
518
00:29:52 --> 00:29:54
Yes, you all remember this.
519
00:29:54 --> 00:29:58
Suppose I have a graph.
520
00:29:58 --> 00:30:02
I know its slope, that's
become easy now, right?
521
00:30:02 --> 00:30:02
Calculus.
522
00:30:02 --> 00:30:08
But the curvature of it, what
derivative did it involve?
523
00:30:08 --> 00:30:09
Second derivative.
524
00:30:09 --> 00:30:13
And was it the second
derivative exactly?
525
00:30:13 --> 00:30:18
No, unfortunately there's
some term which is horrible.
526
00:30:18 --> 00:30:24
One plus the first derivative
squared, all square root.
527
00:30:24 --> 00:30:33
But I'm just going to
take u double prime.
528
00:30:33 --> 00:30:34
So this is an approximation.
529
00:30:34 --> 00:30:37
So what is A now?
530
00:30:37 --> 00:30:41
What's my matrix, A, that gets
me from u -- or my operator,
531
00:30:41 --> 00:30:45
A, that gets me from u to e?
532
00:30:45 --> 00:30:48
What's the e=Au equation?
533
00:30:48 --> 00:30:52
A is just second derivative.
534
00:30:52 --> 00:30:53
That's something new.
535
00:30:53 --> 00:30:58
A is second derivative.
536
00:30:58 --> 00:31:00
And why do I do that?
537
00:31:00 --> 00:31:05
Because I assume small
curvature, small displacement.
538
00:31:05 --> 00:31:11
I assume that u' squared is
very small compared to one.
539
00:31:11 --> 00:31:16
So it's just slightly bent.
540
00:31:16 --> 00:31:21
A beam that goes way down here,
I'd have to go nonlinear.
541
00:31:21 --> 00:31:28
But if I want to keep things
linear, I approximate this will
542
00:31:28 --> 00:31:36
be much smaller than this, so
one is fine that term goes.
543
00:31:36 --> 00:31:39
And now the next
step will be easy.
544
00:31:39 --> 00:31:43
What's the bending moment?
545
00:31:43 --> 00:31:47
This will be called
the bending moment.
546
00:31:47 --> 00:31:50
And let me use the
letter w again.
547
00:31:50 --> 00:31:54
I should really capital
M for bending moment.
548
00:31:54 --> 00:32:04
That will be the stiffness
times the curvature.
549
00:32:04 --> 00:32:11
So that's the force, the way
the spring had a restoring
550
00:32:11 --> 00:32:13
force by Hooke's law.
551
00:32:13 --> 00:32:15
This is the equivalent
of Hooke's law.
552
00:32:15 --> 00:32:19
But this restoring force
is not pulling back,
553
00:32:19 --> 00:32:20
it's bending back.
554
00:32:20 --> 00:32:24
It's torquing back.
555
00:32:24 --> 00:32:28
And then, of course you know,
there'll be an equilibrium
556
00:32:28 --> 00:32:30
equation to balance the load.
557
00:32:30 --> 00:32:32
So this load is the f(x).
558
00:32:32 --> 00:32:34
559
00:32:34 --> 00:32:36
And you know what that'll be.
560
00:32:36 --> 00:32:38
Because you know it'll
involve A transpose.
561
00:32:38 --> 00:32:46
And the transpose of that, do
you want to just make a guess?
562
00:32:46 --> 00:32:49
I shouldn't use transpose, of
course, but I'll use it again.
563
00:32:49 --> 00:32:53
It'll be the same; it'll
be second derivative.
564
00:32:53 --> 00:32:57
I mentioned we had a minus sign
with the first derivative, but
565
00:32:57 --> 00:33:00
now we're going to have two
minus signs, so it's going to
566
00:33:00 --> 00:33:04
come out symmetric, second
derivative, and the equation
567
00:33:04 --> 00:33:06
here will be w''=f(x).
568
00:33:06 --> 00:33:13
569
00:33:13 --> 00:33:16
Our framework is working.
570
00:33:16 --> 00:33:25
We have plus boundary
conditions on u.
571
00:33:25 --> 00:33:32
And here we have plus
boundary conditions on w.
572
00:33:32 --> 00:33:40
So those parts we have
not yet mentioned.
573
00:33:40 --> 00:33:43
And of course, that
depends on my picture.
574
00:33:43 --> 00:33:46
While we're at it, why don't we
figure out, what do you think
575
00:33:46 --> 00:33:49
is the boundary condition here?
576
00:33:49 --> 00:33:51
And how many boundary
conditions am I going to
577
00:33:51 --> 00:33:53
look for all together?
578
00:33:53 --> 00:33:55
Four all together.
579
00:33:55 --> 00:33:58
Because I have a fourth
order equation.
580
00:33:58 --> 00:34:01
There will be four arbitrary
constants until I plug
581
00:34:01 --> 00:34:02
in boundary conditions.
582
00:34:02 --> 00:34:05
So I'm looking for four
boundary conditions, two at
583
00:34:05 --> 00:34:07
this end, two at that end now.
584
00:34:07 --> 00:34:10
And what will be the
two at the fixed end?
585
00:34:10 --> 00:34:15
At the fixed end,
obviously, it's built in.
586
00:34:15 --> 00:34:18
Built in.
587
00:34:18 --> 00:34:22
Slightly different words
sometimes for the beam problem.
588
00:34:22 --> 00:34:25
Here I'll have u=0 and u'=0.
589
00:34:25 --> 00:34:28
590
00:34:28 --> 00:34:36
Those apply with A;
those go with A.
591
00:34:36 --> 00:34:39
Those are the essential
conditions, the Dirichlet
592
00:34:39 --> 00:34:43
conditions, the ones I
must impose all the way.
593
00:34:43 --> 00:34:47
And now at this free
end, what do you think?
594
00:34:47 --> 00:34:50
Well, it's great.
595
00:34:50 --> 00:34:52
It's w=0, and w'=0.
596
00:34:52 --> 00:34:55
597
00:34:55 --> 00:34:58
It's just beautiful,
the way it all works.
598
00:34:58 --> 00:35:01
So that's a completely
fixed free.
599
00:35:01 --> 00:35:05
Then why don't I draw in, just
while we're talking about
600
00:35:05 --> 00:35:08
boundary conditions,
an alternative.
601
00:35:08 --> 00:35:10
So here's my beam.
602
00:35:10 --> 00:35:23
And now you see, it's under a
load, f(x), transverse load.
603
00:35:23 --> 00:35:25
Now that would be different
boundary conditions.
604
00:35:25 --> 00:35:32
Anybody know the name of a
beam that's set up like that?
605
00:35:32 --> 00:35:35
Simply supported.
606
00:35:35 --> 00:35:39
You don't need beam theory, and
I don't know beam theory, to
607
00:35:39 --> 00:35:43
tell the truth, to
do these problems.
608
00:35:43 --> 00:35:48
So that's a simply
supported beam.
609
00:35:48 --> 00:35:51
And what are the boundary
conditions that go with that?
610
00:35:51 --> 00:35:53
Well this is u=0.
611
00:35:53 --> 00:35:57
612
00:35:57 --> 00:36:03
No displacement, it sits
there on that support.
613
00:36:03 --> 00:36:07
And what else is happening
at that support?
614
00:36:07 --> 00:36:10
There's no bending moment.
615
00:36:10 --> 00:36:11
Nobody's here.
616
00:36:11 --> 00:36:13
Right?
617
00:36:13 --> 00:36:14
So it's w=0.
618
00:36:14 --> 00:36:16
619
00:36:16 --> 00:36:18
And at this end, too.
620
00:36:18 --> 00:36:22
Also at this end, u(1)
is sitting there,
621
00:36:22 --> 00:36:25
and w(1) is zero.
622
00:36:25 --> 00:36:26
Yeah.
623
00:36:26 --> 00:36:31
That would be the boundary
conditions, four of them, for
624
00:36:31 --> 00:36:34
a fourth order equation that
we'll just write down in a
625
00:36:34 --> 00:36:36
minute, for simply supported.
626
00:36:36 --> 00:36:38
And we could have a mix.
627
00:36:38 --> 00:36:41
This could be simply
supported here, free here.
628
00:36:41 --> 00:36:42
I think.
629
00:36:42 --> 00:36:48
Or maybe, could it be, or
maybe that's too risky.
630
00:36:48 --> 00:36:52
Would that be a singular
case, simply supported?
631
00:36:52 --> 00:36:54
Huh.
632
00:36:54 --> 00:37:00
So as always with boundary
conditions, some are unstable.
633
00:37:00 --> 00:37:04
Some are not going to
determine all four constants.
634
00:37:04 --> 00:37:06
Just the way free free
didn't work, right?
635
00:37:06 --> 00:37:11
Free free for a rod didn't
determine anything, it left
636
00:37:11 --> 00:37:15
a whole rigid motion.
637
00:37:15 --> 00:37:22
Maybe u=0, w=0 at one end,
and free at the other end;
638
00:37:22 --> 00:37:24
it sounds risky to me.
639
00:37:24 --> 00:37:27
But we can see.
640
00:37:27 --> 00:37:27
Okay.
641
00:37:27 --> 00:37:31
So, do you get the general
picture of the beam?
642
00:37:31 --> 00:37:32
So what's the equation?
643
00:37:32 --> 00:37:36
What's A transpose C A, when
I put it all together?
644
00:37:36 --> 00:37:42
I'll use that space to
put in A transpose C A.
645
00:37:42 --> 00:37:45
Continuous, we're talking here.
646
00:37:45 --> 00:37:48
Right now we've got
differential equations.
647
00:37:48 --> 00:37:51
So what's the differential
equation, A transpose C A?
648
00:37:51 --> 00:37:54
So it's the second derivative.
649
00:37:54 --> 00:37:59
I'm just going backwards around
the framework, as always.
650
00:37:59 --> 00:38:05
The second derivative of
this, and this is c(x) times
651
00:38:05 --> 00:38:14
e(x), and e(x) is second
derivative of u=f(x).
652
00:38:14 --> 00:38:19
653
00:38:19 --> 00:38:21
Good.
654
00:38:21 --> 00:38:27
In ten minutes, we've written
down the framework, some
655
00:38:27 --> 00:38:32
possible boundary conditions,
and the combined A
656
00:38:32 --> 00:38:34
transpose C A equation.
657
00:38:34 --> 00:38:37
I mean, we're ready to go.
658
00:38:37 --> 00:38:43
We've got the pattern
to think about this.
659
00:38:43 --> 00:38:47
So let's see, what
should we do first?
660
00:38:47 --> 00:38:53
I would say the first thing
to do is, let c be one
661
00:38:53 --> 00:38:55
and solve some problems.
662
00:38:55 --> 00:39:00
Let c be one, and consider
-- so if c is one, it's a
663
00:39:00 --> 00:39:05
fourth derivative equation.
664
00:39:05 --> 00:39:08
Should we take uniform load?
665
00:39:08 --> 00:39:10
Yeah.
666
00:39:10 --> 00:39:14
How does a beam bend
under its own weight?
667
00:39:14 --> 00:39:18
So it's just one, or
whatever constant.
668
00:39:18 --> 00:39:21
So it's constant load, it's
just its own weight, it's going
669
00:39:21 --> 00:39:24
to sag a little in the middle.
670
00:39:24 --> 00:39:26
What's the solution
to that equation?
671
00:39:26 --> 00:39:32
And what shall I take as
boundary conditions?
672
00:39:32 --> 00:39:35
Let me do the simply
supported one.
673
00:39:35 --> 00:39:37
Because that would
be kind of nice.
674
00:39:37 --> 00:39:41
So it's simply supported, it's
sagging under its own weight,
675
00:39:41 --> 00:39:49
with u(0)=0, u''(0)=0,
because that's the w.
676
00:39:49 --> 00:39:53
And u(1)=0, u''(1)=0.
677
00:39:53 --> 00:39:55
678
00:39:55 --> 00:39:59
Whatever.
679
00:39:59 --> 00:40:03
I don't know that I'll have the
patience to go through and plug
680
00:40:03 --> 00:40:05
in all four boundary conditions
to determine all
681
00:40:05 --> 00:40:07
four constants.
682
00:40:07 --> 00:40:09
Just get me to that point.
683
00:40:09 --> 00:40:13
Get me to a solution u(x), the
general solution here that's
684
00:40:13 --> 00:40:20
got four constants
in it is what?
685
00:40:20 --> 00:40:21
Okay.
686
00:40:21 --> 00:40:23
Okay, think again.
687
00:40:23 --> 00:40:24
What are we looking at?
688
00:40:24 --> 00:40:28
We're looking at a linear
differential equation.
689
00:40:28 --> 00:40:31
Linear problem.
690
00:40:31 --> 00:40:35
I'm asking for the general
solution to a linear equation.
691
00:40:35 --> 00:40:37
What's the general set up?
692
00:40:37 --> 00:40:41
General set up is,
particular solution plus
693
00:40:41 --> 00:40:42
nullspace solution.
694
00:40:42 --> 00:40:43
Right?
695
00:40:43 --> 00:40:46
You see an equation like that,
looking for the general
696
00:40:46 --> 00:40:50
solution, tell me one
particular solution and then
697
00:40:50 --> 00:40:54
tell me all the solutions when
it has zero on the right,
698
00:40:54 --> 00:40:56
and we've got everybody.
699
00:40:56 --> 00:41:00
So that was true for matrices,
it was true for Ax=b,
700
00:41:00 --> 00:41:03
it's just as true for
differential equations.
701
00:41:03 --> 00:41:06
So what's one
particular solution?
702
00:41:06 --> 00:41:13
What's one function whose
fourth derivative is one?
703
00:41:13 --> 00:41:16
Yes?
704
00:41:16 --> 00:41:18
What am I looking for here?
705
00:41:18 --> 00:41:22
1/4 of x to the -- no, what?
706
00:41:22 --> 00:41:23
1/24, is it?
707
00:41:23 --> 00:41:31
1/24 of x to the fourth?
x to the fourth over 24.
708
00:41:31 --> 00:41:33
Because four derivatives --
so we're thinking we're in
709
00:41:33 --> 00:41:35
the polynomial world here.
710
00:41:35 --> 00:41:36
Just as we were with u''.
711
00:41:39 --> 00:41:42
With the bar it was x squared
over two, the particular
712
00:41:42 --> 00:41:46
solution, now we're up
to x fourth over 24.
713
00:41:46 --> 00:41:50
So that's the fourth
derivative is one, good.
714
00:41:50 --> 00:41:55
So I'm seeing a fourth
degree bending there.
715
00:41:55 --> 00:41:58
And now what about the null
space solutions, the
716
00:41:58 --> 00:42:02
homogeneous solutions.
717
00:42:02 --> 00:42:06
This accounts for the one, now
what are the possibilities
718
00:42:06 --> 00:42:11
if it was a zero?
719
00:42:11 --> 00:42:13
You're going to tell me
the whole bunch, right?
720
00:42:13 --> 00:42:23
A plus Bx plus Cx
squared plus Dx cubed.
721
00:42:23 --> 00:42:28
Because all of those have
fourth derivatives equal zero.
722
00:42:28 --> 00:42:31
So that's the general solution.
723
00:42:31 --> 00:42:33
Okay.
724
00:42:33 --> 00:42:36
So whatever the boundary
conditions are, they
725
00:42:36 --> 00:42:42
determine A, B, C, D.
726
00:42:42 --> 00:42:45
We're not that far away
from Monday's lecture
727
00:42:45 --> 00:42:47
on fitting cubics.
728
00:42:47 --> 00:42:50
Actually we're
really close to it.
729
00:42:50 --> 00:42:53
When we use finite elements,
we're going to use
730
00:42:53 --> 00:42:57
exactly those cubics.
731
00:42:57 --> 00:43:01
I'll get to that point.
732
00:43:01 --> 00:43:05
Let me take the other model
problem, that everybody
733
00:43:05 --> 00:43:07
knows what's coming.
734
00:43:07 --> 00:43:12
What's the other right hand
side that this course lives
735
00:43:12 --> 00:43:14
and dies on -- lives on.
736
00:43:14 --> 00:43:15
Delta function.
737
00:43:15 --> 00:43:16
Right.
738
00:43:16 --> 00:43:18
Delta at some point.
739
00:43:18 --> 00:43:21
So that's a point load then.
740
00:43:21 --> 00:43:29
I can make it whatever the
boundary conditions are.
741
00:43:29 --> 00:43:30
Right.
742
00:43:30 --> 00:43:32
Good point.
743
00:43:32 --> 00:43:35
This boring stuff will
just repeat, right?
744
00:43:35 --> 00:43:37
That's the null space solution.
745
00:43:37 --> 00:43:41
But now, what is a
particular solution?
746
00:43:41 --> 00:43:44
The particular solution
has become interesting.
747
00:43:44 --> 00:43:48
The particular solution here
was straightforward, simple,
748
00:43:48 --> 00:43:50
a good one to do first.
749
00:43:50 --> 00:43:55
What's a particular
solution to a point load?
750
00:43:55 --> 00:44:01
So instead of having
distributed load here, I'm
751
00:44:01 --> 00:44:05
putting a heavy weight
here at this point, a.
752
00:44:05 --> 00:44:12
And it's heavy weight, I'm
multiplying the delta function
753
00:44:12 --> 00:44:16
by one, I could multiply by
some l for load or something,
754
00:44:16 --> 00:44:18
but let's just keep it simple.
755
00:44:18 --> 00:44:23
What's a solution to that?
756
00:44:23 --> 00:44:27
Fourth derivative equals delta.
757
00:44:27 --> 00:44:31
So that means I've now got to
integrate, one way to get the
758
00:44:31 --> 00:44:34
answer here would be to
integrate four times.
759
00:44:34 --> 00:44:35
Right?
760
00:44:35 --> 00:44:40
If I integrate delta four
times, then I've got something
761
00:44:40 --> 00:44:42
whose fourth derivative
will match delta.
762
00:44:42 --> 00:44:46
So do you remember the
integrals of delta?
763
00:44:46 --> 00:44:48
Okay, so I integrate.
764
00:44:48 --> 00:44:51
First integral is, step.
765
00:44:51 --> 00:44:54
Second integral is, ramp.
766
00:44:54 --> 00:44:59
Third integral is,
quadratic, right?
767
00:44:59 --> 00:45:02
This was linear, boop
boop, linear pieces.
768
00:45:02 --> 00:45:09
The next integral is going to
bring me up to quadratic ramp.
769
00:45:09 --> 00:45:11
And the next, fourth one
is going to bring me
770
00:45:11 --> 00:45:14
up to cubic ramp.
771
00:45:14 --> 00:45:15
Cubic.
772
00:45:15 --> 00:45:22
So it's going to be cubic
ramp, is what I get there.
773
00:45:22 --> 00:45:26
So one particular solution
would be a function that's
774
00:45:26 --> 00:45:33
zero, and then at the point a,
it suddenly goes up cubically.
775
00:45:33 --> 00:45:39
So it's zero here, and it's x
cubed over six there, I think.
776
00:45:39 --> 00:45:45
If I do integrate three times,
I'll be up to x cubed over six.
777
00:45:45 --> 00:45:46
Is that right?
778
00:45:46 --> 00:45:49
Yes, cubic.
779
00:45:49 --> 00:45:53
So that's an
interesting function.
780
00:45:53 --> 00:46:00
Of course, these parts
will tilt the function,
781
00:46:00 --> 00:46:05
will change it.
782
00:46:05 --> 00:46:10
So our solution won't look like
this, because I've only got one
783
00:46:10 --> 00:46:14
particular function, and I'll
need these to satisfy the
784
00:46:14 --> 00:46:16
boundary conditions.
785
00:46:16 --> 00:46:18
So there is one
particular solution.
786
00:46:18 --> 00:46:22
The general solution -- yes,
good for us to think out
787
00:46:22 --> 00:46:24
the general solution.
788
00:46:24 --> 00:46:32
What does that picture look
like when I add in this stuff?
789
00:46:32 --> 00:46:34
Very, very important.
790
00:46:34 --> 00:46:36
I'm sorry I don't have more
space for this highly
791
00:46:36 --> 00:46:37
important picture.
792
00:46:37 --> 00:46:42
Okay, so here's my point a.
793
00:46:42 --> 00:46:44
Keep your eye on that point a.
794
00:46:44 --> 00:46:48
Okay, so to the left of it,
I've got some curve, whatever,
795
00:46:48 --> 00:46:51
dut dut dut dut dut, the beam.
796
00:46:51 --> 00:46:55
And to the right of it, I've
got some other curve, whatever
797
00:46:55 --> 00:46:57
it is, dut dut dut dut dut.
798
00:46:57 --> 00:47:01
And what's cooking at point a?
799
00:47:01 --> 00:47:03
What's the jump
condition at point a?
800
00:47:03 --> 00:47:05
That's the critical question.
801
00:47:05 --> 00:47:08
What changes at point a?
802
00:47:08 --> 00:47:13
You remember, this is the
corresponding thing,
803
00:47:13 --> 00:47:16
the analog of our ramp.
804
00:47:16 --> 00:47:18
So what changed for
the ramp at point a?
805
00:47:18 --> 00:47:20
What jumped?
806
00:47:20 --> 00:47:21
The slope.
807
00:47:21 --> 00:47:25
Okay, now the question is
what's going to jump here?
808
00:47:25 --> 00:47:27
What jumped there?
809
00:47:27 --> 00:47:31
Here, did the function jump?
810
00:47:31 --> 00:47:31
Certainly not.
811
00:47:31 --> 00:47:32
Did the slope jump?
812
00:47:32 --> 00:47:33
Certainly not.
813
00:47:33 --> 00:47:35
Did the second derivative jump?
814
00:47:35 --> 00:47:36
No, no.
815
00:47:36 --> 00:47:39
The second derivative
was zero and then zero.
816
00:47:39 --> 00:47:40
What jumped?
817
00:47:40 --> 00:47:42
Third derivative.
818
00:47:42 --> 00:47:45
The third derivative
is allowed to jump.
819
00:47:45 --> 00:47:46
And of course.
820
00:47:46 --> 00:47:51
A jump in the third derivative
produces a delta in the fourth.
821
00:47:51 --> 00:47:52
Right?
822
00:47:52 --> 00:47:55
It just works.
823
00:47:55 --> 00:48:01
So this is a cubic of some
sort, coming from that jump.
824
00:48:01 --> 00:48:06
This is another, a different
cubic, coming from this sort,
825
00:48:06 --> 00:48:09
from this junk, and this.
826
00:48:09 --> 00:48:11
So it's cubic in each piece.
827
00:48:11 --> 00:48:15
Why is it cubic in each piece?
828
00:48:15 --> 00:48:20
Because, what's the equation
in the middle of that piece?
829
00:48:20 --> 00:48:25
What's our differential
equation if I look here?
830
00:48:25 --> 00:48:26
Here's my equation.
831
00:48:26 --> 00:48:29
What is it in the middle
of that piece? u
832
00:48:29 --> 00:48:33
fourth equal zero.
833
00:48:33 --> 00:48:36
The delta function
is zero there.
834
00:48:36 --> 00:48:37
And u fourth is zero here.
835
00:48:37 --> 00:48:41
So of course this
is a cubic spline.
836
00:48:41 --> 00:48:43
We're meeting that neat
word, cubic spline.
837
00:48:43 --> 00:48:46
Those turn out to be very,
very handy functions
838
00:48:46 --> 00:48:48
for other things, too.
839
00:48:48 --> 00:48:52
So we see them here as a
solution to fourth order
840
00:48:52 --> 00:48:55
equations with point
loads are cubic splines.
841
00:48:55 --> 00:49:03
Because the big key point is
that there's a jump here in
842
00:49:03 --> 00:49:05
u''', the third derivative.
843
00:49:05 --> 00:49:10
A jump in the third derivative,
that's what we saw here.
844
00:49:10 --> 00:49:17
And we'll see it if we have
any cubic meeting any cubic.
845
00:49:17 --> 00:49:22
Let me just say, a jump in the
third derivative, your eye
846
00:49:22 --> 00:49:24
probably won't notice it.
847
00:49:24 --> 00:49:28
I mean, it's a
discontinuity, somehow.
848
00:49:28 --> 00:49:30
We don't have the same
polynomial from here to here.
849
00:49:30 --> 00:49:37
But that discontinuity in u''',
it's pretty darn smooth still.
850
00:49:37 --> 00:49:41
The slope is continuous, so
your eye doesn't see a ramp.
851
00:49:41 --> 00:49:45
And even more, the curving is
continuous, the curvature is
852
00:49:45 --> 00:49:50
continuous. e and w are good.
853
00:49:50 --> 00:49:55
It's just a jump in
the third derivative.
854
00:49:55 --> 00:50:01
Okay, so I want to speak about
splines, and more about this,
855
00:50:01 --> 00:50:07
and about finite elements for
beam problems on Friday.
856
00:50:07 --> 00:50:11
And then that will take care of
1-D and we'll move into 2-D.
857
00:50:11 --> 00:50:13
Okay.