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PROFESSOR STRANG: Okay.

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00:00:20,610 --> 00:00:21,400
Hi.

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00:00:21,400 --> 00:00:28,710
So, our goal, certainly,
to reach next week

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00:00:28,710 --> 00:00:30,980
is partial
differential equations.

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00:00:30,980 --> 00:00:34,990
Laplace's equation.

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00:00:34,990 --> 00:00:39,820
Additional topics still
to see clearly in 1-D.

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00:00:39,820 --> 00:00:43,700
So one of those topics,
these will both come today,

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00:00:43,700 --> 00:00:48,150
one of those topics is the idea,
still in the finite element

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00:00:48,150 --> 00:00:51,110
world, of element matrices.

18
00:00:51,110 --> 00:00:58,590
So you remember, we saw those,
that each bar in the truss

19
00:00:58,590 --> 00:01:02,080
could give a piece of A
transpose A that could be

20
00:01:02,080 --> 00:01:05,790
stamped in, or assembled,
to use the right word --

21
00:01:05,790 --> 00:01:09,810
I think assembled is maybe used
more than stamped in, but both

22
00:01:09,810 --> 00:01:20,130
okay -- into K. For graphs,
an edge in the graph gave us

23
00:01:20,130 --> 00:01:25,430
a little [1, -1; -1, 1] matrix
that could be stamped in.

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00:01:25,430 --> 00:01:29,640
And now we want to see,
how does that process work

25
00:01:29,640 --> 00:01:31,110
for finite elements.

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00:01:31,110 --> 00:01:34,360
Because that's how finite
element matrices are really

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00:01:34,360 --> 00:01:37,710
put together, out of
these element matrices.

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00:01:37,710 --> 00:01:43,040
Okay, so that's step one,
that's half today's lecture.

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00:01:43,040 --> 00:01:48,210
Step two, problem two now,
coming from the next section,

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00:01:48,210 --> 00:01:51,090
is fourth order equations.

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00:01:51,090 --> 00:01:53,180
Up to now, all our
differential equations

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00:01:53,180 --> 00:01:55,330
have been second order.

33
00:01:55,330 --> 00:01:58,010
Are there fourth order
equations that are important?

34
00:01:58,010 --> 00:01:59,490
Yes, there are.

35
00:01:59,490 --> 00:02:01,230
For beam bending.

36
00:02:01,230 --> 00:02:05,150
So I'll describe that
application, which leads

37
00:02:05,150 --> 00:02:07,170
to fourth order equations.

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00:02:07,170 --> 00:02:11,580
Do they fit our A
transpose C A framework?

39
00:02:11,580 --> 00:02:13,000
You bet.

40
00:02:13,000 --> 00:02:16,760
You know they will.

41
00:02:16,760 --> 00:02:21,870
Each additional application
in the framework kind of

42
00:02:21,870 --> 00:02:25,560
gets us comfortable,
familiar with that framework,

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00:02:25,560 --> 00:02:26,690
what it can do.

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00:02:26,690 --> 00:02:28,490
The A transpose C A.

45
00:02:28,490 --> 00:02:32,000
So let me start with
element matrices.

46
00:02:32,000 --> 00:02:34,350
I'll get the homework,
those numbers

47
00:02:34,350 --> 00:02:36,760
will get posted on the
website later today.

48
00:02:36,760 --> 00:02:38,910
I just thought
I'd put them down,

49
00:02:38,910 --> 00:02:43,010
and I have to figure out what
would be a suitable MATLAB

50
00:02:43,010 --> 00:02:43,880
question.

51
00:02:43,880 --> 00:02:47,830
So my idea for this
homework, as it really

52
00:02:47,830 --> 00:02:50,760
was for the last
homework, was that you

53
00:02:50,760 --> 00:02:55,860
get some, not a large number
of ordinary questions, paper

54
00:02:55,860 --> 00:02:59,780
and pencil questions,
and one MATLAB question.

55
00:02:59,780 --> 00:03:06,580
When I wrote "MATLAB last",
people interpreted that

56
00:03:06,580 --> 00:03:09,570
as "last MATLAB".

57
00:03:09,570 --> 00:03:11,380
But those words are
not commutative.

58
00:03:11,380 --> 00:03:13,100
Right?

59
00:03:13,100 --> 00:03:15,930
I just meant -- and it
doesn't really matter,

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00:03:15,930 --> 00:03:19,490
it was a dumb thing to say --
I meant you to put the MATLAB

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00:03:19,490 --> 00:03:23,770
question at the end after
the regular questions.

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00:03:23,770 --> 00:03:24,450
Sorry.

63
00:03:24,450 --> 00:03:27,560
So that MATLAB is
due-- Maybe a bunch

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00:03:27,560 --> 00:03:32,050
of those turned in on Monday
didn't include the MATLAB.

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00:03:32,050 --> 00:03:34,380
No penalty.

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00:03:34,380 --> 00:03:39,600
I'm talking now about the
MATLAB for the trusses.

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00:03:39,600 --> 00:03:44,500
How many have still got a MATLAB
for the truss still to turn in?

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00:03:44,500 --> 00:03:45,070
A number.

69
00:03:45,070 --> 00:03:45,961
Oh, not too many.

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00:03:45,961 --> 00:03:46,460
Okay.

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00:03:46,460 --> 00:03:48,610
Anyway, just when you can.

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00:03:48,610 --> 00:03:50,350
In my envelope is good.

73
00:03:50,350 --> 00:03:50,860
Okay.

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00:03:50,860 --> 00:03:55,200
So it'll be a similar
thing for this week,

75
00:03:55,200 --> 00:03:58,260
and because it's
short I said okay,

76
00:03:58,260 --> 00:04:00,540
let's get that
cleaned up by Monday.

77
00:04:00,540 --> 00:04:04,990
And then we're ready for vector
calculus, partial differential

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00:04:04,990 --> 00:04:10,830
equations, two and three
dimensions, the next big step.

79
00:04:10,830 --> 00:04:13,870
If I want to illustrate
element matrices,

80
00:04:13,870 --> 00:04:16,450
the best place I
could do it would

81
00:04:16,450 --> 00:04:20,660
be to go back to our piecewise
linear elements in 1-D,

82
00:04:20,660 --> 00:04:26,140
and see how an
element now is just

83
00:04:26,140 --> 00:04:28,950
one of those little intervals.

84
00:04:28,950 --> 00:04:33,530
Little pieces of
the whole structure.

85
00:04:33,530 --> 00:04:38,160
If it's a bar, or whatever it
is, I've cut it into pieces.

86
00:04:38,160 --> 00:04:42,510
Here's one piece,
here's another piece.

87
00:04:42,510 --> 00:04:46,470
With finite differences,
if I gave you

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00:04:46,470 --> 00:04:49,890
unequally spaced meshes,
if my little h is

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00:04:49,890 --> 00:04:53,710
different from my big H,
we'd have to think again.

90
00:04:53,710 --> 00:04:58,080
I mean there would be
some three-point-- Instead

91
00:04:58,080 --> 00:05:02,620
of minus one, two, minus one
for the second difference,

92
00:05:02,620 --> 00:05:05,450
it would be a little
lopsided, of course,

93
00:05:05,450 --> 00:05:09,560
when the mesh is unequal.

94
00:05:09,560 --> 00:05:12,690
We would have to think that
through for finite differences;

95
00:05:12,690 --> 00:05:15,920
for finite elements, the
system thinks for us.

96
00:05:15,920 --> 00:05:18,270
So I just want to
show how that happens.

97
00:05:18,270 --> 00:05:19,670
So I'll take this.

98
00:05:19,670 --> 00:05:25,570
What's the element matrix
for our standard equation

99
00:05:25,570 --> 00:05:27,320
for that element?

100
00:05:27,320 --> 00:05:29,580
Okay.

101
00:05:29,580 --> 00:05:32,210
And then it would fit
into the big matrix

102
00:05:32,210 --> 00:05:35,960
K. I'm going to come
up with a matrix K

103
00:05:35,960 --> 00:05:38,250
that I could do
in any other way,

104
00:05:38,250 --> 00:05:40,580
but this is the way
it's really done.

105
00:05:40,580 --> 00:05:46,070
So I look in that interval.

106
00:05:46,070 --> 00:05:48,010
I'm focusing on that interval.

107
00:05:48,010 --> 00:05:52,700
So here I've drawn
the basis functions.

108
00:05:52,700 --> 00:05:57,090
But now I'm not going to operate
so much with basis functions.

109
00:05:57,090 --> 00:05:59,270
Let me just think about
that interval again.

110
00:05:59,270 --> 00:06:00,810
That's the same interval.

111
00:06:00,810 --> 00:06:02,670
So those were the
basis functions

112
00:06:02,670 --> 00:06:06,280
that went up to height one.

113
00:06:06,280 --> 00:06:08,710
One started at height
one and went down,

114
00:06:08,710 --> 00:06:11,250
one started at height
zero and went up.

115
00:06:11,250 --> 00:06:14,740
But it's their
combination, of course.

116
00:06:14,740 --> 00:06:21,210
Let me number this node number
zero and this node number one.

117
00:06:21,210 --> 00:06:27,380
So we have some
height, U_0, here.

118
00:06:27,380 --> 00:06:31,670
This is coming from
U_0 times that hat.

119
00:06:31,670 --> 00:06:35,000
And some other
height, U_1, here,

120
00:06:35,000 --> 00:06:38,100
which is coming from
U_1 times that hat.

121
00:06:38,100 --> 00:06:46,250
But inside this interval, my U
in this interval is U_0 times

122
00:06:46,250 --> 00:06:51,090
that first hat function, the
phi_0, the coming down hat.

123
00:06:51,090 --> 00:06:57,780
Plus U_1 times the phi_1
function, the going up hat.

124
00:06:57,780 --> 00:07:01,480
It's a linear function,
so it's going to be there.

125
00:07:01,480 --> 00:07:03,970
That's my graph-- No.

126
00:07:03,970 --> 00:07:06,540
Yes.

127
00:07:06,540 --> 00:07:07,140
This is U(x).

128
00:07:07,140 --> 00:07:09,960


129
00:07:09,960 --> 00:07:11,230
U(x).

130
00:07:11,230 --> 00:07:12,550
Okay.

131
00:07:12,550 --> 00:07:13,980
Right?

132
00:07:13,980 --> 00:07:17,030
I'm focusing inside one element.

133
00:07:17,030 --> 00:07:20,740
One interval.

134
00:07:20,740 --> 00:07:23,340
I've numbered them zero
to one, but of course

135
00:07:23,340 --> 00:07:25,070
that distance is H.

136
00:07:25,070 --> 00:07:28,150
Okay, so there's my function.

137
00:07:28,150 --> 00:07:28,740
Right?

138
00:07:28,740 --> 00:07:31,780
Everybody agrees that
this combination,

139
00:07:31,780 --> 00:07:37,690
it starts out right, it ends
up right, at height U_1,

140
00:07:37,690 --> 00:07:38,870
and it's linear.

141
00:07:38,870 --> 00:07:40,840
So that's got to be right.

142
00:07:40,840 --> 00:07:48,940
So really, what's the
contribution from that element?

143
00:07:48,940 --> 00:07:55,070
I look at the quantity,
that-- And I'm always

144
00:07:55,070 --> 00:07:58,750
taking the phi functions
to equal the V functions.

145
00:07:58,750 --> 00:08:06,090
So I can write this as c(x)
times dU/dx squared, dx.

146
00:08:06,090 --> 00:08:11,250


147
00:08:11,250 --> 00:08:16,680
Over the whole thing, over
the whole interval, that

148
00:08:16,680 --> 00:08:23,630
would be the U
transpose K U. This,

149
00:08:23,630 --> 00:08:28,220
everybody remembers
the K part, of course,

150
00:08:28,220 --> 00:08:30,440
comes from the left
side of the problem.

151
00:08:30,440 --> 00:08:34,330
It comes from integrations.

152
00:08:34,330 --> 00:08:40,070
And this d-- U is now a
combination of all the--

153
00:08:40,070 --> 00:08:43,490
With weights U_0, U_1, U_2, U_3.

154
00:08:43,490 --> 00:08:53,220
If I plug in for capital U
and do all the integrals,

155
00:08:53,220 --> 00:08:55,050
I'll see what K is.

156
00:08:55,050 --> 00:08:56,520
Now what's the point?

157
00:08:56,520 --> 00:08:59,840
The point is, when I
did those integrals,

158
00:08:59,840 --> 00:09:03,810
the question is, how am I
going to do the integrals?

159
00:09:03,810 --> 00:09:08,340
Do I do them-- The way I
did before, was I watched

160
00:09:08,340 --> 00:09:15,350
what phi_0, one phi times
another phi and I integrated.

161
00:09:15,350 --> 00:09:17,330
That was successful.

162
00:09:17,330 --> 00:09:23,590
But the new way is, integrate
it an interval at a time.

163
00:09:23,590 --> 00:09:26,320
Do the integrals.

164
00:09:26,320 --> 00:09:29,500
So this would be K_global.

165
00:09:29,500 --> 00:09:36,490
Now I'm going to go
just from zero to H.

166
00:09:36,490 --> 00:09:40,615
If I go from just zero
to H, that's going

167
00:09:40,615 --> 00:09:43,610
to give me the K_element piece.

168
00:09:43,610 --> 00:09:47,250
The little piece that comes
from this little interval.

169
00:09:47,250 --> 00:09:54,500
And on that little interval,
this is my formula for U.

170
00:09:54,500 --> 00:09:59,670
I'm just hoping you'll sort of
see this as a reasonable idea,

171
00:09:59,670 --> 00:10:03,370
and then when we do the
integral you'll see it.

172
00:10:03,370 --> 00:10:04,650
It clicks.

173
00:10:04,650 --> 00:10:07,140
Okay, how does it click?

174
00:10:07,140 --> 00:10:08,720
So what's my plan?

175
00:10:08,720 --> 00:10:11,230
My plan is, here's my function.

176
00:10:11,230 --> 00:10:13,280
There is its picture.

177
00:10:13,280 --> 00:10:15,890
I'm just going to
plug it into here.

178
00:10:15,890 --> 00:10:17,920
Oh, it's going to
be simple, isn't it?

179
00:10:17,920 --> 00:10:23,350
What's the slope of that
function in that element?

180
00:10:23,350 --> 00:10:24,600
On that interval?

181
00:10:24,600 --> 00:10:26,300
What's the dU/dx?

182
00:10:26,300 --> 00:10:33,360
So instead of doing every full
integral for each separate phi,

183
00:10:33,360 --> 00:10:40,380
I'm doing every element
integral for both phis.

184
00:10:40,380 --> 00:10:44,980
See, these two phis are both
coming in to that element.

185
00:10:44,980 --> 00:10:48,780
Okay, so what's dU/dx.

186
00:10:48,780 --> 00:10:51,930
I didn't realize how neat
this that's going to be.

187
00:10:51,930 --> 00:10:55,150
So what's dU/dx in this element?

188
00:10:55,150 --> 00:10:56,800
I'll use a different board here.

189
00:10:56,800 --> 00:10:59,360
So it's an integral,
then, from zero

190
00:10:59,360 --> 00:11:04,240
to H of whatever
my c(x) might be.

191
00:11:04,240 --> 00:11:05,870
And I'll make a comment on that.

192
00:11:05,870 --> 00:11:09,370
But my focus here
is, what's dU/dx?

193
00:11:09,370 --> 00:11:12,550
What's the derivative?

194
00:11:12,550 --> 00:11:14,470
And then I'm going to square it.

195
00:11:14,470 --> 00:11:18,080
For this function,
for that picture,

196
00:11:18,080 --> 00:11:24,720
the slope is obviously
(U_1-U_0)/H. Right?

197
00:11:24,720 --> 00:11:31,670
The slope is (U_1-U_0)/H.
And I'm squaring it. dx.

198
00:11:31,670 --> 00:11:34,360


199
00:11:34,360 --> 00:11:35,330
Okay.

200
00:11:35,330 --> 00:11:39,910
Let me take c(x)=1.

201
00:11:39,910 --> 00:11:44,750
Just to see clearly
what's going on.

202
00:11:44,750 --> 00:11:47,790
So c(x) is just going to be one.

203
00:11:47,790 --> 00:11:55,290
Okay, so I'm claiming that
this is my U transpose.

204
00:11:55,290 --> 00:11:56,980
My little piece.

205
00:11:56,980 --> 00:11:58,880
Why is it only a little piece?

206
00:11:58,880 --> 00:12:02,530
Because it only
involves two of the U's.

207
00:12:02,530 --> 00:12:07,680
It's going to be a little two
by two element matrix, that

208
00:12:07,680 --> 00:12:09,690
comes from this element.

209
00:12:09,690 --> 00:12:14,420
And then it's going to to be put
into the big K, the global K,

210
00:12:14,420 --> 00:12:16,110
in its proper place.

211
00:12:16,110 --> 00:12:17,740
Okay, well.

212
00:12:17,740 --> 00:12:19,200
It's trivial, right?

213
00:12:19,200 --> 00:12:21,460
This is a constant,
this is a constant,

214
00:12:21,460 --> 00:12:30,130
the integral is just H
times U_1-U_0, squared,

215
00:12:30,130 --> 00:12:32,250
over H squared.

216
00:12:32,250 --> 00:12:34,680
Because that's getting
squared, the interval was

217
00:12:34,680 --> 00:12:38,700
length H. I think
I just have a 1/H.

218
00:12:38,700 --> 00:12:45,080
So this is my U
transpose K element U.

219
00:12:45,080 --> 00:12:48,370
And now I want to pick
out, what's that matrix?

220
00:12:48,370 --> 00:12:52,610
What's that little two by two
matrix that only touches these

221
00:12:52,610 --> 00:12:55,190
two U's?

222
00:12:55,190 --> 00:12:59,160
What's the matrix-- Well,
since it only touches

223
00:12:59,160 --> 00:13:02,110
these two U's, you can tell me.

224
00:13:02,110 --> 00:13:03,050
I want a [U 0, U 1].

225
00:13:03,050 --> 00:13:06,030


226
00:13:06,030 --> 00:13:08,940
Sorry, let me make
a little more space.

227
00:13:08,940 --> 00:13:13,010
And you can tell me what
matrix goes in there.

228
00:13:13,010 --> 00:13:16,790
I want this all to
match up. [U 0, U 1].

229
00:13:16,790 --> 00:13:20,390
Now here is the two by two
element matrix. [U 0, U 1].

230
00:13:20,390 --> 00:13:25,850


231
00:13:25,850 --> 00:13:29,180
What's the two by two
matrix that will correctly

232
00:13:29,180 --> 00:13:31,330
produce this answer?

233
00:13:31,330 --> 00:13:35,840
I'm just shooting
for that answer.

234
00:13:35,840 --> 00:13:37,480
What do I have here?

235
00:13:37,480 --> 00:13:44,100
This is U_1 squared minus
2U_0*U_1, plus a U_0 squared.

236
00:13:44,100 --> 00:13:45,430
Right?

237
00:13:45,430 --> 00:13:48,280
And I have to remember the
1/H, so it's automatically

238
00:13:48,280 --> 00:13:51,780
going to come out right.

239
00:13:51,780 --> 00:13:57,360
So there's a 1/H, shall I
remember that first off.

240
00:13:57,360 --> 00:14:00,720
1/H is part of my
element matrix.

241
00:14:00,720 --> 00:14:09,200
And then, what are the numbers
that go inside that matrix?

242
00:14:09,200 --> 00:14:11,410
We had practice with this.

243
00:14:11,410 --> 00:14:13,360
You remember when
we talked about

244
00:14:13,360 --> 00:14:17,440
positive definite matrices,
way back in chapter one?

245
00:14:17,440 --> 00:14:23,630
The point was that we could
look at eigenvalues, or pivots,

246
00:14:23,630 --> 00:14:24,580
or something.

247
00:14:24,580 --> 00:14:27,880
But the core idea was energy.

248
00:14:27,880 --> 00:14:30,680
The core idea was that
energy, that quadratic,

249
00:14:30,680 --> 00:14:32,510
and that's what we're
looking at again.

250
00:14:32,510 --> 00:14:34,750
That's the energy.

251
00:14:34,750 --> 00:14:39,130
That's the energy, right
there, and this is the energy,

252
00:14:39,130 --> 00:14:45,430
this is the energy in the
finite element subspace.

253
00:14:45,430 --> 00:14:49,020
All I'm saying is, what
matrix, what two by two matrix,

254
00:14:49,020 --> 00:14:54,210
goes with U_1 squared minus
2U_0*U_1 plus U_0 squared.

255
00:14:54,210 --> 00:14:56,220
Just tell me what to
put in that matrix.

256
00:14:56,220 --> 00:14:58,270
What do I put in here?

257
00:14:58,270 --> 00:14:59,220
One, right.

258
00:14:59,220 --> 00:15:01,570
Because it's
multiplying U_0*U_0.

259
00:15:01,570 --> 00:15:03,340
What do I put there?

260
00:15:03,340 --> 00:15:04,310
Minus one.

261
00:15:04,310 --> 00:15:05,150
Good.

262
00:15:05,150 --> 00:15:09,810
Because I have a minus
two, it comes in, minus one

263
00:15:09,810 --> 00:15:12,240
comes in twice, and
a one goes there.

264
00:15:12,240 --> 00:15:17,400
So there, with the
1/H included, is K_e.

265
00:15:17,400 --> 00:15:20,760
K_element, for that element.

266
00:15:20,760 --> 00:15:25,560
For the big H element.

267
00:15:25,560 --> 00:15:30,220
You'll say big deal, because
we've seen this thing before.

268
00:15:30,220 --> 00:15:35,470
Notice what is nice.

269
00:15:35,470 --> 00:15:38,240
First of all, notice
how nice that is.

270
00:15:38,240 --> 00:15:40,110
It's particularly
nice, of course,

271
00:15:40,110 --> 00:15:43,250
because I took c(x) to be one.

272
00:15:43,250 --> 00:15:44,810
So let me make a comment.

273
00:15:44,810 --> 00:15:47,890
Suppose c(x) was not one.

274
00:15:47,890 --> 00:15:50,040
What would I do?

275
00:15:50,040 --> 00:15:57,090
Suppose c(x), suppose I
have some variable stiffness

276
00:15:57,090 --> 00:15:57,980
in the material.

277
00:15:57,980 --> 00:16:03,150
Suppose the material
could be changing width,

278
00:16:03,150 --> 00:16:09,430
so its stiffness would change.

279
00:16:09,430 --> 00:16:12,020
So in other words, I'd
have a variable, c(x),

280
00:16:12,020 --> 00:16:15,350
that I should do the integral.

281
00:16:15,350 --> 00:16:18,615
Probably finite element
systems aren't set up

282
00:16:18,615 --> 00:16:22,030
to actually do the
exact integral.

283
00:16:22,030 --> 00:16:23,800
What would they do?

284
00:16:23,800 --> 00:16:27,430
They would take, for
that simple integration,

285
00:16:27,430 --> 00:16:32,360
they would probably just
take c(x) at the midpoint.

286
00:16:32,360 --> 00:16:34,310
So there's a
numerical integration

287
00:16:34,310 --> 00:16:39,780
here in the creation of
these element matrices.

288
00:16:39,780 --> 00:16:42,310
And numerical
immigration is just,

289
00:16:42,310 --> 00:16:46,560
take a suitable combination
of the values at a few points.

290
00:16:46,560 --> 00:16:48,970
You do know Simpson's rule?

291
00:16:48,970 --> 00:16:53,210
Simpson's rule, that's a
pretty high level rule.

292
00:16:53,210 --> 00:16:57,670
My suggestion there was
just a midpoint rule.

293
00:16:57,670 --> 00:17:00,890
Just take c(x) at the
middle of the interval.

294
00:17:00,890 --> 00:17:02,140
Then it would factor out.

295
00:17:02,140 --> 00:17:07,580
So I should really put a c here.

296
00:17:07,580 --> 00:17:10,080
A c should really
be coming in there.

297
00:17:10,080 --> 00:17:13,560
And you expected that, right,
from the A transpose C A.

298
00:17:13,560 --> 00:17:16,340
We always saw a C in the middle.

299
00:17:16,340 --> 00:17:17,720
It really should be there.

300
00:17:17,720 --> 00:17:21,410
When I took c to be
1, I didn't see it.

301
00:17:21,410 --> 00:17:23,750
So what am I doing?

302
00:17:23,750 --> 00:17:30,950
I'm approximating
c(x) by c at halfway.

303
00:17:30,950 --> 00:17:31,980
Approximately.

304
00:17:31,980 --> 00:17:39,390
I would replace this unpleasant,
possibly varying function by c

305
00:17:39,390 --> 00:17:42,830
at a point and use that value.

306
00:17:42,830 --> 00:17:46,030
So numerical integration
is one part of the picture

307
00:17:46,030 --> 00:17:50,320
that-- We won't go into
all the different rules.

308
00:17:50,320 --> 00:17:52,520
There's a rectangle
rule, there's

309
00:17:52,520 --> 00:17:55,390
a trapezoid rule, very good.

310
00:17:55,390 --> 00:17:58,590
There's the Simpson's
rule that's better.

311
00:17:58,590 --> 00:18:02,890
As I get higher order elements,
like those cubic elements

312
00:18:02,890 --> 00:18:06,660
I spoke about, the numerical
integration has to keep up.

313
00:18:06,660 --> 00:18:10,110
If I was integrating
cubic stuff,

314
00:18:10,110 --> 00:18:12,040
I wouldn't use
such a cheap rule.

315
00:18:12,040 --> 00:18:15,030
I would go up to Simpson's
rule, or Gauss's rule,

316
00:18:15,030 --> 00:18:16,440
or somebody's rule.

317
00:18:16,440 --> 00:18:19,770
Anyway, that's the c part.

318
00:18:19,770 --> 00:18:23,860
Here's the part that
stamps into the matrix.

319
00:18:23,860 --> 00:18:28,380
Notice, by the way,
when I stamp it in,

320
00:18:28,380 --> 00:18:30,540
tell me how it's going
to look stamped in.

321
00:18:30,540 --> 00:18:32,040
And then I've completed it.

322
00:18:32,040 --> 00:18:36,930
So here's my big K. I wish I
had a little more room for it.

323
00:18:36,930 --> 00:18:41,520
Okay, here's my big
K. So that interval

324
00:18:41,520 --> 00:18:43,500
that I drew there,
the H interval,

325
00:18:43,500 --> 00:18:46,430
will stamp in here,
some two by two.

326
00:18:46,430 --> 00:18:47,760
Right?

327
00:18:47,760 --> 00:18:51,860
Now where will the similar
thing coming from this guy.

328
00:18:51,860 --> 00:18:55,590
So, maybe it has to
be numbered minus one.

329
00:18:55,590 --> 00:18:58,670
Sorry about that.

330
00:18:58,670 --> 00:19:00,590
That's another little interval.

331
00:19:00,590 --> 00:19:02,990
I'll do the same thing
on that interval.

332
00:19:02,990 --> 00:19:06,400
I'll get a little element
matrix, two by two guy,

333
00:19:06,400 --> 00:19:08,650
for K for that element.

334
00:19:08,650 --> 00:19:14,190
And where will it
fit in to the big K?

335
00:19:14,190 --> 00:19:16,890
Does it fit in up here,
let me just ask you.

336
00:19:16,890 --> 00:19:19,360
Does it fit in up there?

337
00:19:19,360 --> 00:19:23,720
Yes, no?

338
00:19:23,720 --> 00:19:28,870
I'm assembling, stamping
in the small two

339
00:19:28,870 --> 00:19:32,870
by twos into the full n by n.

340
00:19:32,870 --> 00:19:37,290
And if I draw the
picture you'll see it.

341
00:19:37,290 --> 00:19:42,920
So when I do the two by two
for that big H interval,

342
00:19:42,920 --> 00:19:46,530
it goes there, let's say,
then I just want to say where

343
00:19:46,530 --> 00:19:52,720
does the two by two go for
the interval to the left?

344
00:19:52,720 --> 00:19:54,810
Does it go there?

345
00:19:54,810 --> 00:19:55,800
Nope.

346
00:19:55,800 --> 00:19:57,680
How does it go?

347
00:19:57,680 --> 00:19:59,230
It overlaps.

348
00:19:59,230 --> 00:19:59,730
Right?

349
00:19:59,730 --> 00:20:02,880
It overlaps.

350
00:20:02,880 --> 00:20:04,100
Why does it overlap?

351
00:20:04,100 --> 00:20:10,030
Because the phi_0, this
guy, is acting on the right,

352
00:20:10,030 --> 00:20:12,490
and also acting on the left.

353
00:20:12,490 --> 00:20:21,150
The U_0 is active, is
partly controlling the slope

354
00:20:21,150 --> 00:20:23,190
this way, and also that way.

355
00:20:23,190 --> 00:20:26,680
The U_0 is in common
the two intervals.

356
00:20:26,680 --> 00:20:30,490
Anytime any unknowns, any
mesh points that are in

357
00:20:30,490 --> 00:20:32,795
common to two
elements, we're going

358
00:20:32,795 --> 00:20:36,110
to have an overlap
when we assemble.

359
00:20:36,110 --> 00:20:39,710
And so it'll just sit,
it'll sit right there.

360
00:20:39,710 --> 00:20:44,390
And so there is
the diagonal guy.

361
00:20:44,390 --> 00:20:47,700
And maybe you could tell me
what number will go there.

362
00:20:47,700 --> 00:20:49,640
What number would
actually go there?

363
00:20:49,640 --> 00:20:55,980
And then you'll see the whole
point of assembly, stamping in.

364
00:20:55,980 --> 00:21:00,010
Well, what number
goes there from this?

365
00:21:00,010 --> 00:21:06,490
One times the c/H. So in
here would be the c --

366
00:21:06,490 --> 00:21:15,670
can I call it c_right, or c_H,
c on the big H interval --

367
00:21:15,670 --> 00:21:20,850
divided by the H. Because that
one, we had to get it right.

368
00:21:20,850 --> 00:21:23,910
And then what will go
in that very same spot?

369
00:21:23,910 --> 00:21:29,540
So add it in coming from
the small h interval.

370
00:21:29,540 --> 00:21:36,550
The same thing, it'll be
this one, like shifted up,

371
00:21:36,550 --> 00:21:37,740
moved over.

372
00:21:37,740 --> 00:21:39,490
So it'll be coming
from that one,

373
00:21:39,490 --> 00:21:45,120
so it'll be a c on the little h
interval, divided by little h.

374
00:21:45,120 --> 00:21:50,020
That would be the
diagonal entry of K.

375
00:21:50,020 --> 00:21:52,290
That's what we would
see right there.

376
00:21:52,290 --> 00:21:57,050
Over here, should I
write a typical row of K?

377
00:21:57,050 --> 00:21:59,660
Typical row of K,
when I do that,

378
00:21:59,660 --> 00:22:10,250
is going to have this c_h/h
plus c_H/H. That's like the two,

379
00:22:10,250 --> 00:22:11,810
right?

380
00:22:11,810 --> 00:22:13,150
That's like the two.

381
00:22:13,150 --> 00:22:18,700
And what goes here?

382
00:22:18,700 --> 00:22:23,330
What will the entry be that
sits there when I assemble?

383
00:22:23,330 --> 00:22:25,740
Just this guy.

384
00:22:25,740 --> 00:22:26,510
Right?

385
00:22:26,510 --> 00:22:28,340
Just this guy, times that.

386
00:22:28,340 --> 00:22:34,590
The entry here will be the minus
c_H/H. That'll be the entry.

387
00:22:34,590 --> 00:22:37,870
This is the diagonal one,
this is the one to the right,

388
00:22:37,870 --> 00:22:40,810
and what's the one to the left?

389
00:22:40,810 --> 00:22:45,590
What's the one here?

390
00:22:45,590 --> 00:22:47,170
Well, you know what it is.

391
00:22:47,170 --> 00:22:55,350
It's going to come from the
minus and it'll be the minus c

392
00:22:55,350 --> 00:22:58,420
little h over little h.

393
00:22:58,420 --> 00:22:59,330
Look at that.

394
00:22:59,330 --> 00:23:02,360
That just shows
you how it works.

395
00:23:02,360 --> 00:23:09,190
And again, you can look at that,
page 299 to 300 in the book.

396
00:23:09,190 --> 00:23:16,100
You see that if the c's are the
same, if the h's are the same,

397
00:23:16,100 --> 00:23:20,790
then we're looking again at
our minus one, two, minus one.

398
00:23:20,790 --> 00:23:25,190
Times whatever c over h, to
keep it dimensionally right.

399
00:23:25,190 --> 00:23:25,930
Do you see that?

400
00:23:25,930 --> 00:23:27,440
It's just simple.

401
00:23:27,440 --> 00:23:28,890
Simple idea.

402
00:23:28,890 --> 00:23:35,150
The point is that each interval
can be done separately.

403
00:23:35,150 --> 00:23:42,840
It's a simple idea in 1-D.
It's a key idea in 2-D,

404
00:23:42,840 --> 00:23:47,210
where we have triangles,
we have tetrahedra, tets.

405
00:23:47,210 --> 00:23:49,710
We'll see this in
two dimensions,

406
00:23:49,710 --> 00:23:56,030
later in this chapter, when
we're doing Laplace's equation.

407
00:23:56,030 --> 00:23:58,880
It's just fun to see it work.

408
00:23:58,880 --> 00:24:04,640
You'll have different triangles,
say call them triangles.

409
00:24:04,640 --> 00:24:13,300
So that phi-- Do you
want me to look ahead?

410
00:24:13,300 --> 00:24:15,480
Just ten seconds, to triangles?

411
00:24:15,480 --> 00:24:19,450
So imagine we have triangles
here, so we have piecewise,

412
00:24:19,450 --> 00:24:21,350
we have little pyramids.

413
00:24:21,350 --> 00:24:24,880
Instead of hat functions,
they grow to pyramids.

414
00:24:24,880 --> 00:24:28,600
So there's a pyramid guy
whose height is one there,

415
00:24:28,600 --> 00:24:31,860
and drops to zero
in all these places.

416
00:24:31,860 --> 00:24:34,050
And that's our phi.

417
00:24:34,050 --> 00:24:36,860
Our trial function,
test function,

418
00:24:36,860 --> 00:24:41,000
will be a pyramid
function, then.

419
00:24:41,000 --> 00:24:45,450
And I can do integrals that
way, or I can take the integral

420
00:24:45,450 --> 00:24:48,960
over a typical triangle.

421
00:24:48,960 --> 00:24:53,090
So a typical triangle
is involved with three,

422
00:24:53,090 --> 00:24:57,240
now I've three functions, in
the linear case, controlling

423
00:24:57,240 --> 00:24:59,390
inside that triangle.

424
00:24:59,390 --> 00:25:04,640
So what will be the size
of the element matrix?

425
00:25:04,640 --> 00:25:07,640
Can you sort of see how the
system is going to work?

426
00:25:07,640 --> 00:25:12,970
And then we'll make
it work in 2-D.

427
00:25:12,970 --> 00:25:16,530
Every node, every mesh
point, corresponds,

428
00:25:16,530 --> 00:25:21,370
has a pyramid function,
has a U that goes with it.

429
00:25:21,370 --> 00:25:23,860
Those U's are the unknowns.

430
00:25:23,860 --> 00:25:26,120
And how many of
those unknowns are

431
00:25:26,120 --> 00:25:29,010
operating inside that triangle?

432
00:25:29,010 --> 00:25:30,040
Three.

433
00:25:30,040 --> 00:25:32,930
So what will be the
size of the element

434
00:25:32,930 --> 00:25:36,690
matrix, the non-zero part
of the element matrix?

435
00:25:36,690 --> 00:25:37,490
Three by three.

436
00:25:37,490 --> 00:25:39,040
What else could it be?

437
00:25:39,040 --> 00:25:41,350
So we'll see what it looks like.

438
00:25:41,350 --> 00:25:44,660
And we'll have integrals
over triangles.

439
00:25:44,660 --> 00:25:48,110
So that's good.

440
00:25:48,110 --> 00:25:49,700
Okay, thanks.

441
00:25:49,700 --> 00:25:54,840
Exactly halfway through the
hour is exactly that first topic

442
00:25:54,840 --> 00:25:57,180
of element matrices.

443
00:25:57,180 --> 00:25:58,520
Done.

444
00:25:58,520 --> 00:26:02,000
Okay.

445
00:26:02,000 --> 00:26:05,190
Let me take two deep
breaths, and move

446
00:26:05,190 --> 00:26:08,660
to fourth order equations.

447
00:26:08,660 --> 00:26:11,370
Fourth order equations.

448
00:26:11,370 --> 00:26:12,720
For the bending of a beam.

449
00:26:12,720 --> 00:26:16,200
So I'd better draw a beam.

450
00:26:16,200 --> 00:26:19,240
This is a 1-D problem, still.

451
00:26:19,240 --> 00:26:24,040
This is a 1-D problem, still.

452
00:26:24,040 --> 00:26:28,190
To keep it 1-D, this
better be a thin beam.

453
00:26:28,190 --> 00:26:32,220
So this is a thin beam.

454
00:26:32,220 --> 00:26:34,730
And the loads, what's
the difference?

455
00:26:34,730 --> 00:26:36,830
What's here?

456
00:26:36,830 --> 00:26:40,900
It looks like a bar,
pretty much, right?

457
00:26:40,900 --> 00:26:47,050
But the difference is, the
load is acting this way.

458
00:26:47,050 --> 00:26:49,890
The load is acting
that way on the beam.

459
00:26:49,890 --> 00:26:51,710
Maybe two loads.

460
00:26:51,710 --> 00:26:53,500
Maybe a uniform load.

461
00:26:53,500 --> 00:26:56,170
Maybe the weight of the beam.

462
00:26:56,170 --> 00:27:00,150
But it's transverse.

463
00:27:00,150 --> 00:27:04,760
It's in the perpendicular,
it's transverse to the beam.

464
00:27:04,760 --> 00:27:06,250
It's this way.

465
00:27:06,250 --> 00:27:09,860
So the beam bends.

466
00:27:09,860 --> 00:27:11,470
Let me do a fixed-free.

467
00:27:11,470 --> 00:27:15,330
So this'll be a fixed-free beam.

468
00:27:15,330 --> 00:27:19,920
Fixed-free, the
word for fixed-free

469
00:27:19,920 --> 00:27:27,230
would be cantilever beam.

470
00:27:27,230 --> 00:27:28,040
Okay.

471
00:27:28,040 --> 00:27:33,750
So what happens if I
impose those loads.

472
00:27:33,750 --> 00:27:37,660
Well, the beam bends.

473
00:27:37,660 --> 00:27:42,800
So the displacement is
now downwards, is now

474
00:27:42,800 --> 00:27:45,530
not the direction of the rod.

475
00:27:45,530 --> 00:27:47,280
The displacement
I'm interested in

476
00:27:47,280 --> 00:27:49,570
is perpendicular to the beam.

477
00:27:49,570 --> 00:27:50,620
Downwards.

478
00:27:50,620 --> 00:27:56,800
Okay, so we can start
on our framework.

479
00:27:56,800 --> 00:28:02,290
So this is displacement. u(x).

480
00:28:02,290 --> 00:28:07,340
I'll stay with the same letters.

481
00:28:07,340 --> 00:28:11,340
So you know I'm going
to have an A, that

482
00:28:11,340 --> 00:28:21,780
will take me to whatever
this is going to get called.

483
00:28:21,780 --> 00:28:24,820
What's the quantity there?

484
00:28:24,820 --> 00:28:29,250
It's going to be, let's see.

485
00:28:29,250 --> 00:28:30,210
What happens?

486
00:28:30,210 --> 00:28:33,760
So this is just geometric now.

487
00:28:33,760 --> 00:28:38,400
Let me put in the
easy part, C. That'll

488
00:28:38,400 --> 00:28:40,800
be sort of the
bending stiffness.

489
00:28:40,800 --> 00:28:41,300
Right?

490
00:28:41,300 --> 00:28:44,480
This'll be the
bending stiffness.

491
00:28:44,480 --> 00:28:49,670
Because the beam
is going to bend.

492
00:28:49,670 --> 00:28:54,283
And over here I'll
get a suitable w.

493
00:28:54,283 --> 00:28:59,820
What I get here-- So
when the beam bends...

494
00:28:59,820 --> 00:29:01,780
Its curvature.

495
00:29:01,780 --> 00:29:04,360
Curvature of the beam
is what's produced.

496
00:29:04,360 --> 00:29:08,090
It's not stretching of the
beam; it's curving of the beam.

497
00:29:08,090 --> 00:29:12,580
So this quantity, e, will
be the curvature that

498
00:29:12,580 --> 00:29:16,700
comes from the displacement.

499
00:29:16,700 --> 00:29:22,660
If I displace these beams,
suppose I put a load here,

500
00:29:22,660 --> 00:29:24,900
it's going to bend that down.

501
00:29:24,900 --> 00:29:26,580
The bar will curve.

502
00:29:26,580 --> 00:29:29,080
So the curvature e.

503
00:29:29,080 --> 00:29:30,960
Now then, the
question is, what is

504
00:29:30,960 --> 00:29:33,600
this A; what is the curvature?

505
00:29:33,600 --> 00:29:38,060
Well, do you remember?

506
00:29:38,060 --> 00:29:39,590
You're on the ball
if you remember

507
00:29:39,590 --> 00:29:42,180
the formula for curvature.

508
00:29:42,180 --> 00:29:44,830
It's a horrible
formula, actually.

509
00:29:44,830 --> 00:29:47,430
But that's only-- We're
going to make it better.

510
00:29:47,430 --> 00:29:52,390
Do you remember the
curvature, it was in calculus.

511
00:29:52,390 --> 00:29:54,860
Yes, you all remember this.

512
00:29:54,860 --> 00:29:58,590
Suppose I have a graph.

513
00:29:58,590 --> 00:30:02,260
I know its slope, that's
become easy now, right?

514
00:30:02,260 --> 00:30:02,770
Calculus.

515
00:30:02,770 --> 00:30:08,180
But the curvature of it, what
derivative did it involve?

516
00:30:08,180 --> 00:30:09,590
Second derivative.

517
00:30:09,590 --> 00:30:13,210
And was it the second
derivative exactly?

518
00:30:13,210 --> 00:30:18,070
No, unfortunately there's
some term which is horrible.

519
00:30:18,070 --> 00:30:24,980
One plus the first derivative
squared, all square root.

520
00:30:24,980 --> 00:30:33,010
But I'm just going to
take u double prime.

521
00:30:33,010 --> 00:30:34,790
So this is an approximation.

522
00:30:34,790 --> 00:30:37,740
So what is A now?

523
00:30:37,740 --> 00:30:41,900
What's my matrix A, that gets
me from u-- or my operator

524
00:30:41,900 --> 00:30:45,270
A, that gets me from u to e?

525
00:30:45,270 --> 00:30:48,450
What's the e=Au equation?

526
00:30:48,450 --> 00:30:52,210
A is just second derivative.

527
00:30:52,210 --> 00:30:53,620
That's something new.

528
00:30:53,620 --> 00:30:58,070
A is second derivative.

529
00:30:58,070 --> 00:31:00,610
And why do I do that?

530
00:31:00,610 --> 00:31:05,730
Because I assume small
curvature, small displacements.

531
00:31:05,730 --> 00:31:11,650
I assume that u' squared is
very small compared to one.

532
00:31:11,650 --> 00:31:16,850
So it's just slightly bent.

533
00:31:16,850 --> 00:31:21,560
A beam that goes way down
here, I'd have to go nonlinear.

534
00:31:21,560 --> 00:31:26,780
But if I want to keep things
linear, I approximate.

535
00:31:26,780 --> 00:31:29,380
This will be much
smaller than this,

536
00:31:29,380 --> 00:31:36,460
so one is fine, that term goes.

537
00:31:36,460 --> 00:31:39,870
And now the next
step will be easy.

538
00:31:39,870 --> 00:31:43,110
What's the bending moment?

539
00:31:43,110 --> 00:31:47,320
This will be called
the bending moment.

540
00:31:47,320 --> 00:31:50,430
And let me use the
letter w again.

541
00:31:50,430 --> 00:31:54,340
I should really use capital
M for bending moment.

542
00:31:54,340 --> 00:32:04,640
That will be the stiffness
times the curvature.

543
00:32:04,640 --> 00:32:10,700
So that's the force,
the way the spring

544
00:32:10,700 --> 00:32:13,610
had a restoring
force by Hooke's law.

545
00:32:13,610 --> 00:32:15,710
This is the equivalent
of Hooke's law.

546
00:32:15,710 --> 00:32:19,260
But this restoring force
is not pulling back,

547
00:32:19,260 --> 00:32:20,570
it's bending back.

548
00:32:20,570 --> 00:32:24,220
It's torquing back.

549
00:32:24,220 --> 00:32:26,690
And then, of course
you know, there'll

550
00:32:26,690 --> 00:32:30,660
be an equilibrium equation
to balance the load.

551
00:32:30,660 --> 00:32:32,030
So this load is the f(x).

552
00:32:32,030 --> 00:32:34,760


553
00:32:34,760 --> 00:32:36,570
And you know what that'll be.

554
00:32:36,570 --> 00:32:38,550
Because you know it'll
involve A transpose.

555
00:32:38,550 --> 00:32:46,080
And the transpose of that, do
you want to just make a guess?

556
00:32:46,080 --> 00:32:49,800
I shouldn't use transpose, of
course, but I'll use it again.

557
00:32:49,800 --> 00:32:53,300
It'll be the same; it'll
be second derivative.

558
00:32:53,300 --> 00:32:57,730
I mentioned we had a minus
sign with the first derivative,

559
00:32:57,730 --> 00:32:59,580
but now we're going to
have two minus signs,

560
00:32:59,580 --> 00:33:02,100
so it's going to come out
symmetric, second derivative,

561
00:33:02,100 --> 00:33:07,740
and the equation here
will be w''=f(x).

562
00:33:07,740 --> 00:33:13,380


563
00:33:13,380 --> 00:33:16,520
Our framework is working.

564
00:33:16,520 --> 00:33:25,510
We have plus boundary
conditions on u.

565
00:33:25,510 --> 00:33:32,930
And here we have plus
boundary conditions on w.

566
00:33:32,930 --> 00:33:40,220
So those parts we have
not yet mentioned.

567
00:33:40,220 --> 00:33:43,650
And of course, that
depends on my picture.

568
00:33:43,650 --> 00:33:46,700
While we're at it, why don't we
figure out, what do you think

569
00:33:46,700 --> 00:33:49,180
is the boundary condition here?

570
00:33:49,180 --> 00:33:51,010
And how many boundary
conditions am I

571
00:33:51,010 --> 00:33:53,960
going to look for altogether?

572
00:33:53,960 --> 00:33:55,850
Four altogether.

573
00:33:55,850 --> 00:33:58,120
Because I have a
fourth order equation.

574
00:33:58,120 --> 00:34:00,260
There will be four
arbitrary constants

575
00:34:00,260 --> 00:34:02,740
until I plug in
boundary conditions.

576
00:34:02,740 --> 00:34:04,780
So I'm looking for four
boundary conditions,

577
00:34:04,780 --> 00:34:07,340
two at this end,
two at that end now.

578
00:34:07,340 --> 00:34:10,750
And what will be the
two at the fixed end?

579
00:34:10,750 --> 00:34:15,490
At the fixed end,
obviously, it's built in.

580
00:34:15,490 --> 00:34:18,180
Built in.

581
00:34:18,180 --> 00:34:22,110
Slightly different words
sometimes for the beam problem.

582
00:34:22,110 --> 00:34:25,910
Here I'll have u=0 and u'=0.

583
00:34:25,910 --> 00:34:28,920


584
00:34:28,920 --> 00:34:36,290
Those apply with
A; those go with A.

585
00:34:36,290 --> 00:34:38,300
Those are the
essential conditions,

586
00:34:38,300 --> 00:34:43,270
the Dirichlet conditions, the
ones I must impose all the way.

587
00:34:43,270 --> 00:34:47,220
And now at this free
end, what do you think?

588
00:34:47,220 --> 00:34:50,200
Well, it's great.

589
00:34:50,200 --> 00:34:52,520
It's w=0, and w'=0.

590
00:34:52,520 --> 00:34:55,070


591
00:34:55,070 --> 00:34:58,880
It's just beautiful,
the way it all works.

592
00:34:58,880 --> 00:35:01,980
So that's a
completely fixed-free.

593
00:35:01,980 --> 00:35:04,580
Then why don't I draw
in, just while we're

594
00:35:04,580 --> 00:35:08,410
talking about boundary
conditions, an alternative.

595
00:35:08,410 --> 00:35:10,960
So here's my beam.

596
00:35:10,960 --> 00:35:23,000
And now you see, it's under a
load, f(x), transverse load.

597
00:35:23,000 --> 00:35:25,530
Now that would be different
boundary conditions.

598
00:35:25,530 --> 00:35:32,270
Anybody know the name of a
beam that's set up like that?

599
00:35:32,270 --> 00:35:35,580
Simply supported.

600
00:35:35,580 --> 00:35:37,660
You don't need
beam theory, and I

601
00:35:37,660 --> 00:35:42,190
don't know beam theory,
to tell the truth,

602
00:35:42,190 --> 00:35:43,980
to do these problems.

603
00:35:43,980 --> 00:35:48,610
So that's a simply
supported beam.

604
00:35:48,610 --> 00:35:51,120
And what are the boundary
conditions that go with that?

605
00:35:51,120 --> 00:35:53,400
Well this is u=0.

606
00:35:53,400 --> 00:35:57,610


607
00:35:57,610 --> 00:36:03,870
No displacement, it sits
there on that support.

608
00:36:03,870 --> 00:36:07,660
And what else is
happening at that support?

609
00:36:07,660 --> 00:36:10,500
There's no bending moment.

610
00:36:10,500 --> 00:36:11,830
Nobody's here.

611
00:36:11,830 --> 00:36:13,420
Right?

612
00:36:13,420 --> 00:36:14,470
So it's w=0.

613
00:36:14,470 --> 00:36:16,980


614
00:36:16,980 --> 00:36:18,890
And at this end, too.

615
00:36:18,890 --> 00:36:22,860
Also at this end, u at
one is sitting there,

616
00:36:22,860 --> 00:36:25,560
and w(1) is zero.

617
00:36:25,560 --> 00:36:26,290
Yeah.

618
00:36:26,290 --> 00:36:30,900
That would be the boundary
conditions, four of them,

619
00:36:30,900 --> 00:36:33,380
for a fourth order
equation that we'll just

620
00:36:33,380 --> 00:36:36,360
write down in a minute,
for simply supported.

621
00:36:36,360 --> 00:36:38,050
And we could have a mix.

622
00:36:38,050 --> 00:36:41,890
This could be simply
supported here, free here.

623
00:36:41,890 --> 00:36:42,540
I think.

624
00:36:42,540 --> 00:36:48,010
Or maybe, could it be, or
maybe that's too risky.

625
00:36:48,010 --> 00:36:52,820
Would that be a singular
case, simply supported?

626
00:36:52,820 --> 00:36:54,190
Huh.

627
00:36:54,190 --> 00:37:00,910
So as always with boundary
conditions, some are unstable.

628
00:37:00,910 --> 00:37:04,350
Some are not going to
determine all four constants.

629
00:37:04,350 --> 00:37:06,740
Just the way free-free
didn't work, right?

630
00:37:06,740 --> 00:37:11,460
Free-free for a rod
didn't determine anything,

631
00:37:11,460 --> 00:37:15,720
it left a whole rigid motion.

632
00:37:15,720 --> 00:37:20,940
Maybe u=0, w=0 at one end,
and free at the other end,

633
00:37:20,940 --> 00:37:22,300
is that--?

634
00:37:22,300 --> 00:37:24,590
It sounds risky to me.

635
00:37:24,590 --> 00:37:27,010
But we can see.

636
00:37:27,010 --> 00:37:27,910
Okay.

637
00:37:27,910 --> 00:37:31,540
So, do you get the general
picture of the beam?

638
00:37:31,540 --> 00:37:32,970
So what's the equation?

639
00:37:32,970 --> 00:37:36,680
What's A transpose C A,
when I put it all together?

640
00:37:36,680 --> 00:37:43,370
I'll use that space to put in
A transpose C A. Continuous,

641
00:37:43,370 --> 00:37:45,490
we're talking here.

642
00:37:45,490 --> 00:37:48,670
Right now we've got
differential equations.

643
00:37:48,670 --> 00:37:51,610
So what's the differential
equation, A transpose C A?

644
00:37:51,610 --> 00:37:54,580
So it's the second derivative.

645
00:37:54,580 --> 00:37:59,280
I'm just going backwards around
the framework, as always.

646
00:37:59,280 --> 00:38:05,970
The second derivative of this,
and this is c(x) times e(x),

647
00:38:05,970 --> 00:38:19,670
and e(x) is second
derivative of u, equal f(x).

648
00:38:19,670 --> 00:38:21,850
Good.

649
00:38:21,850 --> 00:38:23,910
In ten minutes,
we've written down

650
00:38:23,910 --> 00:38:29,920
the framework, some possible
boundary conditions,

651
00:38:29,920 --> 00:38:34,500
and the combined A
transpose C A equation.

652
00:38:34,500 --> 00:38:37,340
I mean, we're ready to go.

653
00:38:37,340 --> 00:38:43,760
We've got the pattern
to think about this.

654
00:38:43,760 --> 00:38:47,340
So let's see, what
should we do first?

655
00:38:47,340 --> 00:38:53,300
I would say the first thing
to do is, let c be one

656
00:38:53,300 --> 00:38:55,350
and solve some problems.

657
00:38:55,350 --> 00:39:00,620
Let c be one, and
consider-- So if c is one,

658
00:39:00,620 --> 00:39:05,620
it's a fourth
derivative equation.

659
00:39:05,620 --> 00:39:08,420
Should we take uniform load?

660
00:39:08,420 --> 00:39:10,330
Yeah.

661
00:39:10,330 --> 00:39:14,660
How does a beam bend
under its own weight?

662
00:39:14,660 --> 00:39:18,350
So it's just one, or
whatever constant.

663
00:39:18,350 --> 00:39:21,200
So it's constant load,
it's just its own weight,

664
00:39:21,200 --> 00:39:24,360
it's going to sag a
little in the middle.

665
00:39:24,360 --> 00:39:26,860
What's the solution
to that equation?

666
00:39:26,860 --> 00:39:32,790
And what shall I take
as boundary conditions?

667
00:39:32,790 --> 00:39:35,480
Let me do the simply
supported one.

668
00:39:35,480 --> 00:39:37,210
Because that would
be kind of nice.

669
00:39:37,210 --> 00:39:41,060
So it's simply supported, it's
sagging under its own weight,

670
00:39:41,060 --> 00:39:49,550
with u(0)=0, u''(0)=0,
because that's the w.

671
00:39:49,550 --> 00:39:55,840
And u(1)=0, u''(1)=0.

672
00:39:55,840 --> 00:39:59,340
Whatever.

673
00:39:59,340 --> 00:40:01,860
I don't know that
I'll have the patience

674
00:40:01,860 --> 00:40:04,760
to go through and plug in
all four boundary conditions

675
00:40:04,760 --> 00:40:07,090
to determine all four constants.

676
00:40:07,090 --> 00:40:09,250
Just get me to that point.

677
00:40:09,250 --> 00:40:12,590
Get me to a solution
u(x), the general solution

678
00:40:12,590 --> 00:40:20,210
here that's got four
constants in it is what?

679
00:40:20,210 --> 00:40:21,900
Okay.

680
00:40:21,900 --> 00:40:23,792
Okay, think again.

681
00:40:23,792 --> 00:40:24,750
What are we looking at?

682
00:40:24,750 --> 00:40:28,590
We're looking at a linear
differential equation.

683
00:40:28,590 --> 00:40:31,710
Linear problem.

684
00:40:31,710 --> 00:40:35,490
I'm asking for the general
solution to a linear equation.

685
00:40:35,490 --> 00:40:37,590
What's the general set up?

686
00:40:37,590 --> 00:40:40,780
General set up is,
particular solution

687
00:40:40,780 --> 00:40:42,550
plus nullspace solution.

688
00:40:42,550 --> 00:40:43,330
Right?

689
00:40:43,330 --> 00:40:46,035
You see an equation
like that, you're

690
00:40:46,035 --> 00:40:47,940
looking for the
general solution,

691
00:40:47,940 --> 00:40:51,250
tell me one particular
solution and then tell me

692
00:40:51,250 --> 00:40:54,470
all the solutions when
it has zero on the right,

693
00:40:54,470 --> 00:40:56,320
and we've got everybody.

694
00:40:56,320 --> 00:41:00,520
So that was true for matrices,
it was true for Ax=b,

695
00:41:00,520 --> 00:41:03,870
it's just as true for
differential equations.

696
00:41:03,870 --> 00:41:06,660
So what's one
particular solution?

697
00:41:06,660 --> 00:41:13,680
What's one function whose
fourth derivative is one?

698
00:41:13,680 --> 00:41:16,350
Yes?

699
00:41:16,350 --> 00:41:18,730
What am I looking for here?

700
00:41:18,730 --> 00:41:22,640
1/4 of x to the-- No, what?

701
00:41:22,640 --> 00:41:23,790
1/24, is it?

702
00:41:23,790 --> 00:41:31,032
1/24 of x to the fourth?
x to the fourth over 24.

703
00:41:31,032 --> 00:41:32,490
Because four
derivatives-- So we're

704
00:41:32,490 --> 00:41:35,320
thinking we're in the
polynomial world here.

705
00:41:35,320 --> 00:41:39,090
Just as we were with u''.

706
00:41:39,090 --> 00:41:42,060
With the bar it was
x squared over two,

707
00:41:42,060 --> 00:41:46,280
the particular solution, now
we're up to x fourth over 24.

708
00:41:46,280 --> 00:41:50,410
So that's the fourth
derivative is one, good.

709
00:41:50,410 --> 00:41:55,250
So I'm seeing a fourth
degree bending there.

710
00:41:55,250 --> 00:41:58,480
And now what about the
null space solutions,

711
00:41:58,480 --> 00:42:02,140
the homogeneous solutions.

712
00:42:02,140 --> 00:42:06,590
This accounts for the one,
now what are the possibilities

713
00:42:06,590 --> 00:42:11,190
if it was a zero?

714
00:42:11,190 --> 00:42:13,390
You're going to tell me
the whole bunch, right?

715
00:42:13,390 --> 00:42:23,910
A plus Bx plus Cx
squared plus Dx cubed.

716
00:42:23,910 --> 00:42:28,230
Because all of those have
fourth derivatives equal zero.

717
00:42:28,230 --> 00:42:31,570
So that's the general solution.

718
00:42:31,570 --> 00:42:33,080
Okay.

719
00:42:33,080 --> 00:42:36,730
So whatever the
boundary conditions are,

720
00:42:36,730 --> 00:42:43,360
they determine A,
B, C, D. We're not

721
00:42:43,360 --> 00:42:47,970
that far away from Monday's
lecture on fitting cubics.

722
00:42:47,970 --> 00:42:50,760
Actually we're
really close to it.

723
00:42:50,760 --> 00:42:52,640
When we use finite
elements, we're

724
00:42:52,640 --> 00:42:57,140
going to use exactly
those cubics.

725
00:42:57,140 --> 00:43:01,510
I'll get to that point.

726
00:43:01,510 --> 00:43:04,290
Let me take the
other model problem,

727
00:43:04,290 --> 00:43:07,420
that everybody
knows what's coming.

728
00:43:07,420 --> 00:43:12,890
What's the other right-hand side
that this course lives and dies

729
00:43:12,890 --> 00:43:14,760
on-- lives on.

730
00:43:14,760 --> 00:43:15,590
Delta function.

731
00:43:15,590 --> 00:43:16,170
Right.

732
00:43:16,170 --> 00:43:18,970
Delta at some point.

733
00:43:18,970 --> 00:43:21,890
So that's a point load then.

734
00:43:21,890 --> 00:43:29,690
I can make it-- Whatever
the boundary conditions are.

735
00:43:29,690 --> 00:43:30,960
Right.

736
00:43:30,960 --> 00:43:32,550
Good point.

737
00:43:32,550 --> 00:43:35,520
This boring stuff will
just repeat, right?

738
00:43:35,520 --> 00:43:37,810
That's the null space solution.

739
00:43:37,810 --> 00:43:41,430
But now, what is a
particular solution?

740
00:43:41,430 --> 00:43:44,370
The particular solution
has become interesting.

741
00:43:44,370 --> 00:43:48,870
The particular solution here
was straightforward, simple,

742
00:43:48,870 --> 00:43:50,400
a good one to do first.

743
00:43:50,400 --> 00:43:55,110
What's a particular
solution to a point load?

744
00:43:55,110 --> 00:44:01,420
So instead of having
distributed load here,

745
00:44:01,420 --> 00:44:05,680
I'm putting a heavy weight
here at this point, a.

746
00:44:05,680 --> 00:44:12,110
And it's heavy weight, I'm
multiplying the delta function

747
00:44:12,110 --> 00:44:16,190
by one, I could multiply by
some L for load or something,

748
00:44:16,190 --> 00:44:18,590
but let's just keep it simple.

749
00:44:18,590 --> 00:44:23,920
What's a solution to that?

750
00:44:23,920 --> 00:44:27,370
Fourth derivative equals delta.

751
00:44:27,370 --> 00:44:30,500
So that means I've
now got to integrate,

752
00:44:30,500 --> 00:44:32,820
one way to get the
answer here would

753
00:44:32,820 --> 00:44:34,650
be to integrate four times.

754
00:44:34,650 --> 00:44:35,660
Right?

755
00:44:35,660 --> 00:44:39,570
If I integrate delta
four times, then

756
00:44:39,570 --> 00:44:42,940
I've got something whose fourth
derivative will match delta.

757
00:44:42,940 --> 00:44:46,720
So do you remember the
integrals of delta?

758
00:44:46,720 --> 00:44:48,050
Okay, so I integrate.

759
00:44:48,050 --> 00:44:51,520
First integral is: step.

760
00:44:51,520 --> 00:44:54,860
Second integral is: ramp.

761
00:44:54,860 --> 00:44:59,290
Third integral is:
quadratic, right?

762
00:44:59,290 --> 00:45:02,710
This was linear, boop
boop, linear pieces.

763
00:45:02,710 --> 00:45:09,070
The next integral is going to
bring me up to quadratic ramp.

764
00:45:09,070 --> 00:45:11,220
And the next,
fourth one, is going

765
00:45:11,220 --> 00:45:14,661
to bring me up to cubic ramp.

766
00:45:14,661 --> 00:45:15,160
Cubic.

767
00:45:15,160 --> 00:45:22,330
So it's going to be cubic
ramp, is what I get there.

768
00:45:22,330 --> 00:45:27,990
So one particular solution
would be a function that's zero,

769
00:45:27,990 --> 00:45:33,540
and then at the point a, it
suddenly goes up cubically.

770
00:45:33,540 --> 00:45:39,840
So it's zero here, and it's x
cubed over six there, I think.

771
00:45:39,840 --> 00:45:45,340
If I integrate three times,
I'll be up to x cubed over six.

772
00:45:45,340 --> 00:45:46,910
Is that right?

773
00:45:46,910 --> 00:45:49,370
Yes, cubic.

774
00:45:49,370 --> 00:45:53,540
So that's an
interesting function.

775
00:45:53,540 --> 00:46:05,590
Of course, these parts will tilt
the function, will change it.

776
00:46:05,590 --> 00:46:08,860
So our solution
won't look like this,

777
00:46:08,860 --> 00:46:12,320
because I've only got
one particular function,

778
00:46:12,320 --> 00:46:16,050
and I'll need these to satisfy
the boundary conditions.

779
00:46:16,050 --> 00:46:18,470
So there is one
particular solution.

780
00:46:18,470 --> 00:46:22,460
The general solution-- Yes,
good for us to think out

781
00:46:22,460 --> 00:46:24,450
the general solution.

782
00:46:24,450 --> 00:46:32,960
What does that picture look
like when I add in this stuff?

783
00:46:32,960 --> 00:46:34,260
Very, very important.

784
00:46:34,260 --> 00:46:36,600
I'm sorry I don't have
more space for this highly

785
00:46:36,600 --> 00:46:37,960
important picture.

786
00:46:37,960 --> 00:46:42,950
Okay, so here's my point a.

787
00:46:42,950 --> 00:46:44,930
Keep your eye on that point a.

788
00:46:44,930 --> 00:46:48,970
Okay, so to the left of it,
I've got some curve, whatever,

789
00:46:48,970 --> 00:46:51,990
dut dut dut dut dut, the beam.

790
00:46:51,990 --> 00:46:55,120
And to the right of it, I've
got some other curve, whatever

791
00:46:55,120 --> 00:46:57,530
it is, dut dut dut dut dut.

792
00:46:57,530 --> 00:47:01,280
And what's cooking at point a?

793
00:47:01,280 --> 00:47:03,590
What's the jump
condition at point a?

794
00:47:03,590 --> 00:47:05,630
That's the critical question.

795
00:47:05,630 --> 00:47:08,070
What changes at point a?

796
00:47:08,070 --> 00:47:13,790
You remember, this is
the corresponding thing,

797
00:47:13,790 --> 00:47:16,130
the analog of our ramp.

798
00:47:16,130 --> 00:47:18,560
So what changed for
the ramp at point a?

799
00:47:18,560 --> 00:47:20,420
What jumped?

800
00:47:20,420 --> 00:47:21,780
The slope.

801
00:47:21,780 --> 00:47:25,650
Okay, now the question is:
what's going to jump here?

802
00:47:25,650 --> 00:47:27,580
What jumped there?

803
00:47:27,580 --> 00:47:31,010
Here, did the function jump?

804
00:47:31,010 --> 00:47:31,840
Certainly not.

805
00:47:31,840 --> 00:47:32,830
Did the slope jump?

806
00:47:32,830 --> 00:47:33,560
Certainly not.

807
00:47:33,560 --> 00:47:35,040
Did the second derivative jump?

808
00:47:35,040 --> 00:47:36,810
No, no.

809
00:47:36,810 --> 00:47:39,020
The second derivative
was zero and then zero.

810
00:47:39,020 --> 00:47:40,870
What jumped?

811
00:47:40,870 --> 00:47:42,880
Third derivative.

812
00:47:42,880 --> 00:47:45,090
The third derivative
is allowed to jump.

813
00:47:45,090 --> 00:47:46,410
And of course.

814
00:47:46,410 --> 00:47:51,770
A jump in the third derivative
produces a delta in the fourth.

815
00:47:51,770 --> 00:47:52,340
Right?

816
00:47:52,340 --> 00:47:55,070
It just works.

817
00:47:55,070 --> 00:48:01,400
So this is a cubic of some
sort, coming from that jump.

818
00:48:01,400 --> 00:48:06,290
This is another, a different
cubic, coming from this sort,

819
00:48:06,290 --> 00:48:09,140
from this junk, and this.

820
00:48:09,140 --> 00:48:11,630
So it's cubic in each piece.

821
00:48:11,630 --> 00:48:15,380
Why is it cubic in each piece?

822
00:48:15,380 --> 00:48:20,960
Because, what's the equation
in the middle of that piece?

823
00:48:20,960 --> 00:48:25,630
What's our differential
equation if I look here?

824
00:48:25,630 --> 00:48:26,650
Here's my equation.

825
00:48:26,650 --> 00:48:33,960
What is it in the middle of
that piece? u fourth equal zero.

826
00:48:33,960 --> 00:48:36,310
The delta function
is zero there.

827
00:48:36,310 --> 00:48:37,620
And u fourth is zero here.

828
00:48:37,620 --> 00:48:41,140
So of course this
is a cubic spline.

829
00:48:41,140 --> 00:48:43,710
We're meeting that neat
word, cubic spline.

830
00:48:43,710 --> 00:48:46,490
Those turn out to be
very, very handy functions

831
00:48:46,490 --> 00:48:48,140
for other things, too.

832
00:48:48,140 --> 00:48:51,070
So we see them
here as a solution

833
00:48:51,070 --> 00:48:55,640
to fourth order equations with
point loads are cubic splines.

834
00:48:55,640 --> 00:49:04,140
Because the big key point is
that there's a jump here in u

835
00:49:04,140 --> 00:49:05,980
triple prime, the
third derivative.

836
00:49:05,980 --> 00:49:10,670
A jump in the third derivative,
that's what we saw here.

837
00:49:10,670 --> 00:49:17,360
And we'll see it if we have
any cubic meeting any cubic.

838
00:49:17,360 --> 00:49:21,210
Let me just say, a jump
in the third derivative,

839
00:49:21,210 --> 00:49:24,350
your eye probably
won't notice it.

840
00:49:24,350 --> 00:49:28,000
I mean, it's a
discontinuity, somehow.

841
00:49:28,000 --> 00:49:30,640
We don't have the same
polynomial from here to here.

842
00:49:30,640 --> 00:49:34,190
But that discontinuity
in u triple prime,

843
00:49:34,190 --> 00:49:37,000
it's pretty darn smooth still.

844
00:49:37,000 --> 00:49:41,100
The slope is continuous, so
your eye doesn't see a ramp.

845
00:49:41,100 --> 00:49:44,940
And even more, the
curving is continuous,

846
00:49:44,940 --> 00:49:50,450
the curvature is continuous.
e and w are good.

847
00:49:50,450 --> 00:49:55,290
It's just a jump in
the third derivative.

848
00:49:55,290 --> 00:50:01,180
Okay, so I want to speak
about splines, and more

849
00:50:01,180 --> 00:50:05,880
about this, and about finite
elements for beam problems

850
00:50:05,880 --> 00:50:07,010
on Friday.

851
00:50:07,010 --> 00:50:11,900
And then that will take care
of 1-D and we'll move into 2-D.

852
00:50:11,900 --> 00:50:13,150
Okay.