1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:03 The following content is provided under a Creative 3 00:00:03 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:05 Your support will help MIT OpenCourseWare continue to 5 00:00:05 --> 00:00:09 offer high-quality educational resources for free. 6 00:00:09 --> 00:00:11 To make a donation, or to view additional materials from 7 00:00:11 --> 00:00:15 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15 --> 00:00:20 at ocw.mit.edu. 9 00:00:20 --> 00:00:22 PROFESSOR STRANG: OK, so. 10 00:00:22 --> 00:00:24 This is a big day. 11 00:00:24 --> 00:00:28 Part One of the course is completed, and I have your 12 00:00:28 --> 00:00:31 quizzes for you, and that was a very successful result, 13 00:00:31 --> 00:00:33 I'm very pleased. 14 00:00:33 --> 00:00:35 I hope you are, too. 15 00:00:35 --> 00:00:39 Quiz average of 85, that's on the first part of the course. 16 00:00:39 --> 00:00:45 And then the second part so this is, Chapter 3 now. 17 00:00:45 --> 00:00:49 Starts in one dimension with an equation of a type that we've 18 00:00:49 --> 00:00:52 already seen a little bit. 19 00:00:52 --> 00:00:56 So there's some more things to say about the equation and the 20 00:00:56 --> 00:01:04 framework, but then we get to make a start on the finite 21 00:01:04 --> 00:01:06 element approach to solving it. 22 00:01:06 --> 00:01:12 We could of course in 1-D finite differences are probably 23 00:01:12 --> 00:01:14 the way to go, actually. 24 00:01:14 --> 00:01:21 In one dimension the special success of finite elements 25 00:01:21 --> 00:01:26 doesn't really show up that much because finite elements 26 00:01:26 --> 00:01:30 have been, I mean one great reason for their success 27 00:01:30 --> 00:01:33 is that they handle different geometries. 28 00:01:33 --> 00:01:36 They're flexible; you could have regions in the plane, 29 00:01:36 --> 00:01:39 three dimensional bodies of different shapes. 30 00:01:39 --> 00:01:42 Finite differences doesn't really know what to do on 31 00:01:42 --> 00:01:47 a curved boundary in in 2- or 3-D. 32 00:01:47 --> 00:01:50 Finite elements cope much better. 33 00:01:50 --> 00:01:56 So, we'll make a start today, more Friday on one dimensional 34 00:01:56 --> 00:02:02 finite elements and then, a couple of weeks later will be 35 00:02:02 --> 00:02:05 the real thing, 2-D and 3-D. 36 00:02:05 --> 00:02:09 OK, so, ready to go on Chapter 3? 37 00:02:09 --> 00:02:12 So, that's our equation and everybody sees right away 38 00:02:12 --> 00:02:17 what's the framework that that's A transpose in some way; 39 00:02:17 --> 00:02:21 this is A transpose C A, but what's new of course is 40 00:02:21 --> 00:02:25 that we're dealing with functions, not vectors. 41 00:02:25 --> 00:02:28 So we're dealing with, you could say, operators, 42 00:02:28 --> 00:02:30 not matrices. 43 00:02:30 --> 00:02:35 And nevertheless, the big picture is still as it was. 44 00:02:35 --> 00:02:40 So let me take u(x) to be the displacements again. 45 00:02:40 --> 00:02:43 So I'm thinking more of mechanics than 46 00:02:43 --> 00:02:45 electronics here. 47 00:02:45 --> 00:02:52 Displacements, and then we have du, the e(x) will be du/dx, 48 00:02:52 --> 00:02:59 that'll be the stretching, the elongation and, of course at 49 00:02:59 --> 00:03:05 that step you already see the big new item, the fact that the 50 00:03:05 --> 00:03:11 A, the one that gets us from u to du/dx, instead of being a 51 00:03:11 --> 00:03:15 difference matrix which it has been, our matrix A is 52 00:03:15 --> 00:03:18 now a derivative. 53 00:03:18 --> 00:03:19 A is d/dx. 54 00:03:21 --> 00:03:27 So maybe I'll just take out that arrow. 55 00:03:27 --> 00:03:28 So A is d/dx. 56 00:03:29 --> 00:03:37 OK, but if we dealt OK with difference matrices, 57 00:03:37 --> 00:03:39 we're going to deal OK with derivatives. 58 00:03:39 --> 00:03:45 Then, of course, this is the C part, that produces w(x). 59 00:03:46 --> 00:03:53 And it's a multiplication by this possibly varying, 60 00:03:53 --> 00:03:58 possibly jumping, stiffness constancy of x. 61 00:03:58 --> 00:04:03 So w(x) is c(x) c(x), that's our old w=Ce, 62 00:04:03 --> 00:04:05 this is Hooke's Law. 63 00:04:05 --> 00:04:09 I'll put Hooke's Law, but that's, or who's 64 00:04:09 --> 00:04:13 ever law it is. 65 00:04:13 --> 00:04:16 It's like a diagonal matrix; I hope you see that it's 66 00:04:16 --> 00:04:18 like a diagonal matrix. 67 00:04:18 --> 00:04:25 This function u is kind of like of a vector but a continuum 68 00:04:25 --> 00:04:30 vector instead of just a fixed, finite number of values. 69 00:04:30 --> 00:04:35 Then at each value we used to multiply by c_i, now our 70 00:04:35 --> 00:04:39 values are continuous with x, so we multiply by c(x). 71 00:04:40 --> 00:04:44 And then you're going to expect that, going up here, there's 72 00:04:44 --> 00:04:53 going to be an A transpose w f, and of course that A transpose 73 00:04:53 --> 00:04:57 we have to identify and that's the first point 74 00:04:57 --> 00:04:58 of the lecture, really. 75 00:04:58 --> 00:05:01 To identify what is A transpose. 76 00:05:01 --> 00:05:05 What do I mean by A transpose? 77 00:05:05 --> 00:05:11 And I've got to say right away that I'm a little, the 78 00:05:11 --> 00:05:16 notation, writing a transpose of a derivative is like, 79 00:05:16 --> 00:05:18 that's not legal. 80 00:05:18 --> 00:05:21 Because we think of the transpose of a matrix; you sort 81 00:05:21 --> 00:05:27 of flip it over the main diagonal, but obviously 82 00:05:27 --> 00:05:30 it's got to be something more to it than that. 83 00:05:30 --> 00:05:35 And so that's a central math part of this lecture is 84 00:05:35 --> 00:05:39 what's really going on when you transpose? 85 00:05:39 --> 00:05:41 Because then we can copy what's going on and it's quite 86 00:05:41 --> 00:05:43 important to get it. 87 00:05:43 --> 00:05:47 Because the transpose, well, other notations and other words 88 00:05:47 --> 00:05:51 for it would be notation might be a star. 89 00:05:51 --> 00:05:56 Star would be way more common than transpose, I'll just stay 90 00:05:56 --> 00:06:00 with transpose because I want to keep pressing the parallel 91 00:06:00 --> 00:06:04 with A transpose C A. 92 00:06:04 --> 00:06:08 And the name for it would be the adjoint. 93 00:06:08 --> 00:06:11 And the adjoint method, and adjoint operator, 94 00:06:11 --> 00:06:13 those appear a lot. 95 00:06:13 --> 00:06:16 And you'll see them up here in finite elements. 96 00:06:16 --> 00:06:24 So this is a good thing to catch on to. 97 00:06:24 --> 00:06:25 Why? 98 00:06:25 --> 00:06:30 Why should the transpose or the adjoint of the derivative 99 00:06:30 --> 00:06:32 be minus the derivative? 100 00:06:32 --> 00:06:37 And by the way, just while we're fixing this, this is a 101 00:06:37 --> 00:06:45 key factor which is certainly, we have a very strong hint from 102 00:06:45 --> 00:06:47 center difference, right? 103 00:06:47 --> 00:06:51 If I think of derivatives, if I associate them with 104 00:06:51 --> 00:06:54 differences, the center difference matrix, so 105 00:06:54 --> 00:06:59 the a matrix may be centered, would be. 106 00:06:59 --> 00:07:03 Just to remind us, it's a center difference has onws 107 00:07:03 --> 00:07:08 and minus one, one, zeroes on the diagonal, right? 108 00:07:08 --> 00:07:10 Minus one, one. 109 00:07:10 --> 00:07:16 Takes that difference at every row. 110 00:07:16 --> 00:07:18 Except possibly boundary rows. 111 00:07:18 --> 00:07:21 And of course as soon as you look at that matrix you see, 112 00:07:21 --> 00:07:23 yeah, it's anti symmetric. 113 00:07:23 --> 00:07:26 It's an anti symmetric matrix. 114 00:07:26 --> 00:07:32 So a transpose is minus a for center differences and 115 00:07:32 --> 00:07:36 therefore we're not so surprised to see a minus sign 116 00:07:36 --> 00:07:39 up here when we go to the continuous case, 117 00:07:39 --> 00:07:40 the derivative. 118 00:07:40 --> 00:07:45 But, we still have to say what it means. 119 00:07:45 --> 00:07:47 So that's what I'll do next, OK? 120 00:07:47 --> 00:07:49 So this is a good thing to know. 121 00:07:49 --> 00:07:52 And I was just going to comment, what would be 122 00:07:52 --> 00:07:55 the transpose of the second derivative? 123 00:07:55 --> 00:07:57 I won't even write this down. 124 00:07:57 --> 00:08:02 If the derivative transpose sort of flips its sign to 125 00:08:02 --> 00:08:07 minus, what would you guess for this x transpose of second 126 00:08:07 --> 00:08:12 derivative, our more familiar d second by dx squared? 127 00:08:12 --> 00:08:14 Well we'll have two minus signs. 128 00:08:14 --> 00:08:16 So it'll come out fine. 129 00:08:16 --> 00:08:21 So second derivatives, even order derivatives are sort 130 00:08:21 --> 00:08:24 of like symmetric guys. 131 00:08:24 --> 00:08:27 Odd order derivatives, first and third and fifth 132 00:08:27 --> 00:08:30 derivatives, well, God forbid we ever meet a fifth 133 00:08:30 --> 00:08:34 derivative, but first derivative anyway, 134 00:08:34 --> 00:08:36 is anti symmetric. 135 00:08:36 --> 00:08:38 Except for boundary conditions. 136 00:08:38 --> 00:08:42 So I really have to emphasize that the boundary 137 00:08:42 --> 00:08:44 conditions come in. 138 00:08:44 --> 00:08:45 And you'll see them come in. 139 00:08:45 --> 00:08:47 They have to come in. 140 00:08:47 --> 00:08:53 OK, so what meaning can I assign to the transpose, or 141 00:08:53 --> 00:08:58 what was the real thing happening when we flipped the 142 00:08:58 --> 00:09:00 matrix across its diagonal? 143 00:09:00 --> 00:09:14 I claim that we really define the transpose by this rule. 144 00:09:14 --> 00:09:17 By we know what inner products are. 145 00:09:17 --> 00:09:20 I'll do vectors first, we know about inner products, dot 146 00:09:20 --> 00:09:23 products, we know what the dot product of two vectors is. 147 00:09:23 --> 00:09:27 So, this is the transpose of A. 148 00:09:27 --> 00:09:29 How am I going to define the transpose of A? 149 00:09:29 --> 00:09:36 Well, I look at the dot product of Au with w. 150 00:09:36 --> 00:09:40 I'll use a dot here for once, I may erase it and replace it. 151 00:09:40 --> 00:09:47 If at the dot product of Au with w, then that equals for 152 00:09:47 --> 00:09:51 all u and w, all vectors u and w, that equals the dot 153 00:09:51 --> 00:09:56 product of u with something. 154 00:09:56 --> 00:09:59 Because u is coming. 155 00:09:59 --> 00:10:02 If I write out what the dot product is, I see u_1 156 00:10:02 --> 00:10:05 multiplies something, u_2 multiplies something. 157 00:10:05 --> 00:10:13 And what goes in that little space? 158 00:10:13 --> 00:10:15 This is just an identity. 159 00:10:15 --> 00:10:18 I mean, it's like, you'll say no big deal. 160 00:10:18 --> 00:10:22 But I'm saying there is at least a small deal. 161 00:10:22 --> 00:10:22 OK. 162 00:10:22 --> 00:10:30 So if I write it this way, you'll tell me right away this 163 00:10:30 --> 00:10:33 should be the same as u transpose times something. 164 00:10:33 --> 00:10:36 And again, so I'm asking for the same something 165 00:10:36 --> 00:10:38 on both lines. 166 00:10:38 --> 00:10:41 What is that something? 167 00:10:41 --> 00:10:44 A transpose w. 168 00:10:44 --> 00:10:48 Whatever A transpose is, it's the matrix that 169 00:10:48 --> 00:10:50 makes this right. 170 00:10:50 --> 00:10:51 That's really my message. 171 00:10:51 --> 00:10:55 That A transpose is the reason we flipped the matrix across 172 00:10:55 --> 00:11:01 the diagonal, is that it makes that equation correct. 173 00:11:01 --> 00:11:03 And I'm writing the same thing here. 174 00:11:03 --> 00:11:04 OK. 175 00:11:04 --> 00:11:10 So again, if we knew what dot products were, what inner 176 00:11:10 --> 00:11:15 product of vectors were, then A transpose is the matrix that 177 00:11:15 --> 00:11:18 makes this identity correct. 178 00:11:18 --> 00:11:23 And of course if you write it all out in terms of i, j every 179 00:11:23 --> 00:11:26 component, you find it is correct. 180 00:11:26 --> 00:11:31 So that defines the transpose of a matrix. 181 00:11:31 --> 00:11:34 And of course it coincides with flipping across the diagonal. 182 00:11:34 --> 00:11:39 Now, how about the transpose of a derivative. 183 00:11:39 --> 00:11:43 OK, so I'm going to follow the same rule. 184 00:11:43 --> 00:11:47 Here A is now going to be the derivative, and A transpose 185 00:11:47 --> 00:11:50 is going to be whatever it takes to make this true. 186 00:11:50 --> 00:11:51 But what do I mean? 187 00:11:51 --> 00:11:55 Now I have functions, so I have to think again, what do I mean 188 00:11:55 --> 00:11:57 by the inner product, the dot product? 189 00:11:57 --> 00:12:03 So for this to make sense I need to say, and it's a very 190 00:12:03 --> 00:12:07 important thing anyway, and it's the right natural choice, 191 00:12:07 --> 00:12:14 I need to say the dot product, or the inner product is a 192 00:12:14 --> 00:12:19 better word, of functions. 193 00:12:19 --> 00:12:21 Of two functions. 194 00:12:21 --> 00:12:28 A, e(x), and w. if I have two functions, what do I mean 195 00:12:28 --> 00:12:30 by their inner products? 196 00:12:30 --> 00:12:35 Well, really I just think back what did we mean in the finite 197 00:12:35 --> 00:12:39 dimensional case, I multiplied each e by a w, each component 198 00:12:39 --> 00:12:43 of e by w, and I added, so what am I going to do here? 199 00:12:43 --> 00:12:50 Maybe my notation should be parentheses with a comma 200 00:12:50 --> 00:12:54 would be better than a dot, for function. 201 00:12:54 --> 00:12:55 So I have a function. 202 00:12:55 --> 00:12:59 I'm in function space now. 203 00:12:59 --> 00:13:04 We moved out of our n, today, into function space. 204 00:13:04 --> 00:13:06 Our vectors have become functions. 205 00:13:06 --> 00:13:09 And now what's the dot product of two vectors? 206 00:13:09 --> 00:13:13 Well, what am I going to do? 207 00:13:13 --> 00:13:15 I'm going to do what I have to do. 208 00:13:15 --> 00:13:21 I'm going to multiply each e by its corresponding w, and now 209 00:13:21 --> 00:13:26 they depend on this continuous variable x, so that's 210 00:13:26 --> 00:13:27 e(x) times w(x). 211 00:13:28 --> 00:13:32 And what do I do now? 212 00:13:32 --> 00:13:33 Integrate. 213 00:13:33 --> 00:13:38 Here, I added e_i times w_i, of course. 214 00:13:38 --> 00:13:41 Over here I have functions. 215 00:13:41 --> 00:13:42 I integrate dx. 216 00:13:43 --> 00:13:47 Over whatever the region of the problem is. 217 00:13:47 --> 00:13:52 And then our example's in 1-D, be zero to one. 218 00:13:52 --> 00:13:55 If these are functions of two variables I'd be integrating 219 00:13:55 --> 00:13:58 over some 2-D region, but we're in 1-D today. 220 00:13:58 --> 00:14:05 OK, so you see that I'm prepared to say this 221 00:14:05 --> 00:14:15 now makes sense. 222 00:14:15 --> 00:14:20 I now want to say, I'm going to let A be the derivative, and 223 00:14:20 --> 00:14:24 I'm going to figure out what A transpose has to be. 224 00:14:24 --> 00:14:29 So if A is the derivative, so now is this key step. 225 00:14:29 --> 00:14:32 y is the transpose x, OK? 226 00:14:32 --> 00:14:41 So I look at the derivative, du/dx, with w, so that's this 227 00:14:41 --> 00:14:52 integral, zero to one of du/dx*w(x)dx, so 228 00:14:52 --> 00:14:54 that's my left side. 229 00:14:54 --> 00:14:58 Now I want to get u by itself. 230 00:14:58 --> 00:15:01 I want to get the dot product, so I want to get another 231 00:15:01 --> 00:15:07 integral here that has u(x) by itself. 232 00:15:07 --> 00:15:09 Times something, and that something is 233 00:15:09 --> 00:15:10 what I'm looking for. 234 00:15:10 --> 00:15:15 That something will be A transpose w. 235 00:15:15 --> 00:15:17 Right? 236 00:15:17 --> 00:15:19 Do you see what I'm doing? 237 00:15:19 --> 00:15:25 This is is the dot product, this Auw, so I've written 238 00:15:25 --> 00:15:28 out what a u in a product with w is. 239 00:15:28 --> 00:15:32 And now I want to get u out by itself and what it multiplies 240 00:15:32 --> 00:15:38 here will be the a transpose w, and my rule will be extended to 241 00:15:38 --> 00:15:42 the function case and I'll be ready to go. 242 00:15:42 --> 00:15:49 Now do you recognize, this is a basic calculus step, what rule 243 00:15:49 --> 00:15:51 of calculus am I going to use? 244 00:15:51 --> 00:15:53 We're back to 18.01. 245 00:15:53 --> 00:15:56 I have the integral of a derivative times w, and 246 00:15:56 --> 00:15:57 what do I want to do? 247 00:15:57 --> 00:16:01 I want to get the derivative off of u. 248 00:16:01 --> 00:16:02 What happens? 249 00:16:02 --> 00:16:04 What's it called? 250 00:16:04 --> 00:16:06 Integration by part. 251 00:16:06 --> 00:16:08 Very important thing. 252 00:16:08 --> 00:16:09 Very important. 253 00:16:09 --> 00:16:13 If you miss it's important in calculus. 254 00:16:13 --> 00:16:17 It gets sometimes introduced as a rule, or a trick to find some 255 00:16:17 --> 00:16:21 goofy integral, but it's really the real thing. 256 00:16:21 --> 00:16:23 So what is integration by parts? 257 00:16:23 --> 00:16:24 What's the rule? 258 00:16:24 --> 00:16:29 You take the derivative off of u, you put it on to the other 259 00:16:29 --> 00:16:34 one just what we hope for, and then you also have to remember 260 00:16:34 --> 00:16:40 that there is a minus. 261 00:16:40 --> 00:16:42 Integration by parts has a minus. 262 00:16:42 --> 00:16:45 And usually you'd see it out there but here I've left 263 00:16:45 --> 00:16:49 more room for it there. 264 00:16:49 --> 00:16:54 So I have identified now, A transpose w. 265 00:16:54 --> 00:17:00 A transpose w has, if this is Au, in a product with w, then 266 00:17:00 --> 00:17:03 this is u in a product with A transpose w, it had 267 00:17:03 --> 00:17:04 to be what was. 268 00:17:04 --> 00:17:08 And so that one integration by parts brought out a minus sign. 269 00:17:08 --> 00:17:11 If I was looking at second derivatives there would 270 00:17:11 --> 00:17:14 probably be somewhere two integration by parts; 271 00:17:14 --> 00:17:17 I'd have minus twice, I'd be back to plus. 272 00:17:17 --> 00:17:20 And you're going to ask about boundary conditions. 273 00:17:20 --> 00:17:22 And you're right to ask about boundary conditions. 274 00:17:22 --> 00:17:26 I even circled that, because that is so important. 275 00:17:26 --> 00:17:33 So what we've done so far is to get the interior 276 00:17:33 --> 00:17:35 of the interval right. 277 00:17:35 --> 00:17:39 Between zero and one, if A is the derivative, then A 278 00:17:39 --> 00:17:42 transpose is minus the derivative. 279 00:17:42 --> 00:17:43 That's all we've done. 280 00:17:43 --> 00:17:46 We have not got the boundary conditions yet. 281 00:17:46 --> 00:17:50 And we can't go on without that. 282 00:17:50 --> 00:17:54 OK, so I'm ready now to say something about 283 00:17:54 --> 00:17:57 boundary conditions. 284 00:17:57 --> 00:18:00 And it will bring up this square versus rectangular 285 00:18:00 --> 00:18:07 also, so we're getting the rules straight before we 286 00:18:07 --> 00:18:09 tackle finite elements. 287 00:18:09 --> 00:18:15 OK, let me take an example of a matrix and its transpose. 288 00:18:15 --> 00:18:18 Just so you see how boundary conditions. 289 00:18:18 --> 00:18:23 Suppose I have a free fixed problem. 290 00:18:23 --> 00:18:26 Suppose I have a free fixed line of springs. 291 00:18:26 --> 00:18:30 What's the matrix A for that? 292 00:18:30 --> 00:18:32 Well - question? 293 00:18:32 --> 00:18:33 Yes. 294 00:18:33 --> 00:18:36 AUDIENCE: [INAUDIBLE] 295 00:18:36 --> 00:18:40 PROFESSOR STRANG: Yeah. that's Yes. 296 00:18:40 --> 00:18:44 When I learned it, it was also that stupid trick. 297 00:18:44 --> 00:18:50 So you would like me to put plus, can I put plus, whatever. 298 00:18:50 --> 00:18:51 What do you want me to call that? 299 00:18:51 --> 00:18:52 An integrated term? 300 00:18:52 --> 00:18:54 It would be, yeah. 301 00:18:54 --> 00:18:55 I even remember what it is. 302 00:18:55 --> 00:19:03 As you do better than me. u times w at the, is that good? 303 00:19:03 --> 00:19:06 Yeah, I think. 304 00:19:06 --> 00:19:11 So it's really this part that I'm now coming to. 305 00:19:11 --> 00:19:14 It's the boundary part that I'm now coming to. 306 00:19:14 --> 00:19:18 And let me say, so I'm glad you asked that question because 307 00:19:18 --> 00:19:23 I made it seem unimportant, where that's not true at all. 308 00:19:23 --> 00:19:27 The boundary condition is part of the definition of A, and 309 00:19:27 --> 00:19:30 part of the definition of A transpose. 310 00:19:30 --> 00:19:34 Just the way I'm about to say free fixed, I had to tell 311 00:19:34 --> 00:19:38 you that for you to know what a was. 312 00:19:38 --> 00:19:41 Until I tell you the boundary condition, you don't know 313 00:19:41 --> 00:19:42 what the boundary rows are. 314 00:19:42 --> 00:19:45 You only know the inside of the matrix. 315 00:19:45 --> 00:19:47 Or one possible inside. 316 00:19:47 --> 00:19:51 So I'm thinking my inside is going to be minus one, 317 00:19:51 --> 00:19:53 one, minus one, one. 318 00:19:53 --> 00:19:59 So on, as given we find the differences. 319 00:19:59 --> 00:20:01 Minus one, one. 320 00:20:01 --> 00:20:03 But. 321 00:20:03 --> 00:20:04 Oh no, let's see. 322 00:20:04 --> 00:20:07 So I'm doing free fixed. 323 00:20:07 --> 00:20:08 Is that right? 324 00:20:08 --> 00:20:10 Am I doing free fixed? 325 00:20:10 --> 00:20:16 OK, so am I taking free at the left end? 326 00:20:16 --> 00:20:17 Yes. 327 00:20:17 --> 00:20:21 Alright, so if I'm free at the left and fixed at the 328 00:20:21 --> 00:20:23 right end, what's my A? 329 00:20:23 --> 00:20:27 We're getting better at this, right? 330 00:20:27 --> 00:20:28 Minus one, one. 331 00:20:28 --> 00:20:29 Minus one, one. 332 00:20:29 --> 00:20:30 Minus one, one. 333 00:20:30 --> 00:20:37 Minus one, and the one here gets chopped off. 334 00:20:37 --> 00:20:41 You could say if you want the fifth row of A_0, remembering 335 00:20:41 --> 00:20:46 A_0 as the hint on the quiz, where it had five rows for 336 00:20:46 --> 00:20:48 the full thing, free free. 337 00:20:48 --> 00:20:55 And then when an n got fixed, the fifth column got removed, 338 00:20:55 --> 00:20:59 and that's my free fixed matrix. 339 00:20:59 --> 00:21:02 At the left hand end, at the zero end, it's got 340 00:21:02 --> 00:21:04 the difference in there. 341 00:21:04 --> 00:21:05 Difference, difference, difference. 342 00:21:05 --> 00:21:11 But here at the right hand end, it's the fixing, the setting u, 343 00:21:11 --> 00:21:15 whatever it would be. u_5 to zero, or maybe it's u_4, 344 00:21:15 --> 00:21:17 because it's like one, two, three. 345 00:21:17 --> 00:21:20 Setting u_4 to zero knocked that out. 346 00:21:20 --> 00:21:22 OK. 347 00:21:22 --> 00:21:26 All I want to do is transpose that. 348 00:21:26 --> 00:21:30 And you'll see something that we maybe didn't notice before. 349 00:21:30 --> 00:21:33 So I transpose it, that's minus one, one becomes 350 00:21:33 --> 00:21:36 a column, minus one, one becomes a column. 351 00:21:36 --> 00:21:38 Minus one, one becomes a column. 352 00:21:38 --> 00:21:41 Minus one, all there is. 353 00:21:41 --> 00:21:50 That row becomes, so this was, so, have I got it right? 354 00:21:50 --> 00:21:52 Yes. 355 00:21:52 --> 00:21:55 What's happened? 356 00:21:55 --> 00:21:58 A transpose, what are the boundary conditions 357 00:21:58 --> 00:22:00 going with A transpose? 358 00:22:00 --> 00:22:05 The boundary conditions that went with a were, let me 359 00:22:05 --> 00:22:08 say first, what were the boundary conditions with A? 360 00:22:08 --> 00:22:12 Those are going to be boundary conditions on u. 361 00:22:12 --> 00:22:19 So A has boundary conditions on u. 362 00:22:19 --> 00:22:26 And A transpose has boundary conditions on w. 363 00:22:26 --> 00:22:31 Because A transpose acts on w, and A acts on u. 364 00:22:31 --> 00:22:33 So there was no choice. 365 00:22:33 --> 00:22:36 So now what was the boundary condition here? 366 00:22:36 --> 00:22:40 The boundary condition was u_4=0, right? 367 00:22:40 --> 00:22:44 That was what I meant by that guy getting fixed. 368 00:22:44 --> 00:22:49 Now, and no boundary condition at u_0, it was free. 369 00:22:49 --> 00:22:51 Now, what are the boundary conditions that go 370 00:22:51 --> 00:22:52 with A transpose? 371 00:22:52 --> 00:22:55 And remember, A transpose is multiplying w. 372 00:22:55 --> 00:22:59 I'm going to put w here, so what are the boundary 373 00:22:59 --> 00:23:02 conditions that go with A transpose? 374 00:23:02 --> 00:23:05 This thing, nothing got knocked off. 375 00:23:05 --> 00:23:09 The boundary condition came up here for A transpose, the 376 00:23:09 --> 00:23:13 battery condition was w_0=0. 377 00:23:13 --> 00:23:21 That got knocked out, w_0 to be zero. 378 00:23:21 --> 00:23:22 No surprise. 379 00:23:22 --> 00:23:26 Free fixed, this is the free end at the left, this is the 380 00:23:26 --> 00:23:29 fixed end at the right. 381 00:23:29 --> 00:23:32 Did you ever notice that the matrix does it for you? 382 00:23:32 --> 00:23:37 I mean, when you transpose that matrix it just automatically 383 00:23:37 --> 00:23:44 built in the correct boundary conditions on w by, you started 384 00:23:44 --> 00:23:48 with the conditions on u, you transpose the matrix and you've 385 00:23:48 --> 00:23:51 discovered what the boundary conditions on w are. 386 00:23:51 --> 00:23:58 And I'm going to do the same for the continuous problem. 387 00:23:58 --> 00:24:01 I'm going to do the same for the continuous problem, so 388 00:24:01 --> 00:24:08 the continuous free fixed. 389 00:24:08 --> 00:24:14 OK, what's the boundary condition on u? 390 00:24:14 --> 00:24:18 If it's free fixed, I just want you to repeat this, on the 391 00:24:18 --> 00:24:22 integral zero to one for functions u(x), w(x) 392 00:24:22 --> 00:24:24 instead of for vectors. 393 00:24:24 --> 00:24:27 What's the boundary condition on u, if I have a free 394 00:24:27 --> 00:24:30 fixed problem? u(1)=0. 395 00:24:30 --> 00:24:35 396 00:24:35 --> 00:24:38 u over one equals zero. 397 00:24:38 --> 00:24:40 So this is the boundary conditions that goes with 398 00:24:40 --> 00:24:42 A in the free fixed case. 399 00:24:42 --> 00:24:52 And this is part of A. 400 00:24:52 --> 00:24:57 That is part of A. 401 00:24:57 --> 00:25:01 I don't know what a is until I know its boundary condition. 402 00:25:01 --> 00:25:04 Just the way I don't know what this matrix is. 403 00:25:04 --> 00:25:06 It could have been A_0, it could have lost one column, 404 00:25:06 --> 00:25:08 it could have lost two columns, whatever. 405 00:25:08 --> 00:25:11 I don't know until I've told you the boundary 406 00:25:11 --> 00:25:12 condition on u. 407 00:25:12 --> 00:25:16 And then transposing is going to tell me, automatically, 408 00:25:16 --> 00:25:19 without any further input, the boundary condition that 409 00:25:19 --> 00:25:21 goes on the adjoint. 410 00:25:21 --> 00:25:26 So what's the boundary condition on w that goes 411 00:25:26 --> 00:25:28 as part of A transpose? 412 00:25:28 --> 00:25:30 Well, you're going to tell me. 413 00:25:30 --> 00:25:36 Tell me. w(0) should be zero. 414 00:25:36 --> 00:25:39 It came out automatically, naturally. 415 00:25:39 --> 00:25:43 This is a big distinction between boundary conditions. 416 00:25:43 --> 00:25:48 I would call that an essential boundary condition. 417 00:25:48 --> 00:25:51 I had to start with it, I had to decide on that. 418 00:25:51 --> 00:25:54 And then this, I call a natural boundary condition. 419 00:25:54 --> 00:25:59 Or there are even two guys' names, which are associated 420 00:25:59 --> 00:26:01 with these two types. 421 00:26:01 --> 00:26:05 So maybe a first chance to just mention these names. 422 00:26:05 --> 00:26:14 Because you'll often see, reading some paper, maybe a 423 00:26:14 --> 00:26:16 little on the mathematical side, you'll see 424 00:26:16 --> 00:26:18 that word used. 425 00:26:18 --> 00:26:20 The guy's name with this sort of boundary condition 426 00:26:20 --> 00:26:22 is a French name. 427 00:26:22 --> 00:26:25 Not so easy to say, Dirichlet. 428 00:26:25 --> 00:26:28 I'll say it more often in the future. 429 00:26:28 --> 00:26:33 Anyway, I would call that a Dirichlet condition and you 430 00:26:33 --> 00:26:35 would say it's a fixed boundary condition. 431 00:26:35 --> 00:26:37 And if you were doing heat flow you would say it's 432 00:26:37 --> 00:26:39 a fixed temperature. 433 00:26:39 --> 00:26:40 Whatever. 434 00:26:40 --> 00:26:45 Fixed, is really the word to remember there. 435 00:26:45 --> 00:26:51 OK, and then I guess I better give Germany a shot here, too. 436 00:26:51 --> 00:26:56 So the boundary condition is associated with 437 00:26:56 --> 00:27:04 the name of Neumann. 438 00:27:04 --> 00:27:09 So if I said a Dirichlet problem, a total Dirichlet 439 00:27:09 --> 00:27:13 problem, I would be speaking about fixed fixed. 440 00:27:13 --> 00:27:15 And if I spoke about a Neumann problem I would be 441 00:27:15 --> 00:27:17 talking about free free. 442 00:27:17 --> 00:27:21 And this problem is Dirichlet at one end, Neumann 443 00:27:21 --> 00:27:22 at the other. 444 00:27:22 --> 00:27:26 Anyway, so essential and natural. 445 00:27:26 --> 00:27:31 And now of course I'm hoping that that's going to make this 446 00:27:31 --> 00:27:38 this boundary term go away. 447 00:27:38 --> 00:27:41 OK, now I'm paying attention to this thing that you 448 00:27:41 --> 00:27:42 made me write. uw. 449 00:27:43 --> 00:27:45 OK, what happens there? 450 00:27:46 --> 00:27:48 uw? 451 00:27:48 --> 00:27:53 Oh, right this isn't bad because it shows that there's a 452 00:27:53 --> 00:27:59 boundary condition, I've got some little deals going. 453 00:27:59 --> 00:28:06 But do you see that that becomes zero? 454 00:28:06 --> 00:28:10 Why isn't zero at the top end, at one? 455 00:28:10 --> 00:28:14 When I take u times w at one, why do I get zero? 456 00:28:14 --> 00:28:17 Because u(1) is zero. 457 00:28:17 --> 00:28:18 Good. 458 00:28:18 --> 00:28:20 And it's the bottom end. 459 00:28:20 --> 00:28:25 When I take uw at the other boundary, why do I get zero? 460 00:28:25 --> 00:28:27 Because of w. 461 00:28:27 --> 00:28:30 You see that w was needed. 462 00:28:30 --> 00:28:34 That w(0) was needed because there was no controlling u(0). 463 00:28:35 --> 00:28:38 I had no control of u at the left hand end, 464 00:28:38 --> 00:28:39 because it was free. 465 00:28:39 --> 00:28:43 So the control has to come from w. 466 00:28:43 --> 00:28:47 And so w naturally had to be zero, because I wasn't 467 00:28:47 --> 00:28:51 controlling u at that left hand, free end. 468 00:28:51 --> 00:28:57 So one way or the other, the integration by 469 00:28:57 --> 00:28:59 parts is the key. 470 00:28:59 --> 00:29:09 So that said, what I critically wanted to say about 471 00:29:09 --> 00:29:14 transposing, taking the adjoint, except I was just 472 00:29:14 --> 00:29:17 going to add a comment about this square A versus 473 00:29:17 --> 00:29:20 rectangular. 474 00:29:20 --> 00:29:24 And this was a case of square, right? 475 00:29:24 --> 00:29:27 This was a case, this free fixed case, this example I 476 00:29:27 --> 00:29:29 happened to pick was squared. 477 00:29:29 --> 00:29:32 A_0, the free-free guy that was a hint on the 478 00:29:32 --> 00:29:34 quiz, was rectangular. 479 00:29:34 --> 00:29:38 The fixed fixed, which was also on the quiz, was 480 00:29:38 --> 00:29:40 also rectangular. 481 00:29:40 --> 00:29:44 It was what, four by three or something. 482 00:29:44 --> 00:29:47 This A is four by four. 483 00:29:47 --> 00:29:57 And what is especially nice when it's squared? 484 00:29:57 --> 00:30:01 If our problem happens to give a square matrix, in the trust 485 00:30:01 --> 00:30:05 case if the number of displacement unknowns happens 486 00:30:05 --> 00:30:09 to equal the number of bars, so m equals n, I have 487 00:30:09 --> 00:30:11 a square matrix A. 488 00:30:11 --> 00:30:16 And this guy's invertible, so it's all good. 489 00:30:16 --> 00:30:18 Oh, that may be the whole point. 490 00:30:18 --> 00:30:21 That if it's rectangular I wouldn't talk 491 00:30:21 --> 00:30:22 about its inverse. 492 00:30:22 --> 00:30:27 But this is a square matrix, so A itself has an inverse. 493 00:30:27 --> 00:30:32 Instead of having, as I usually have, to deal with A transpose 494 00:30:32 --> 00:30:36 C A all at once, let me put this comment because it's 495 00:30:36 --> 00:30:40 just a small one up here. 496 00:30:40 --> 00:30:41 Right under these words. 497 00:30:41 --> 00:30:43 Square versus rectangular. 498 00:30:43 --> 00:30:50 Square A, and let's say invertible, otherwise we're in 499 00:30:50 --> 00:30:55 the unstable K, so you know what I mean, in the network 500 00:30:55 --> 00:31:02 problems the number of nodes matches the number of edges? 501 00:31:02 --> 00:31:10 In the spring problem we have free fixed situations. 502 00:31:10 --> 00:31:12 Anyway, A comes out square. 503 00:31:12 --> 00:31:15 Whatever the application. 504 00:31:15 --> 00:31:19 If it comes out square, what is especially good? 505 00:31:19 --> 00:31:22 It comes out square, what's especially good is that 506 00:31:22 --> 00:31:23 it has an inverse. 507 00:31:23 --> 00:31:29 So that in this square case, this K inverse is A transpose 508 00:31:29 --> 00:31:35 C A inverse, can be split. 509 00:31:35 --> 00:31:40 This allows us to separate, to do what you better 510 00:31:40 --> 00:31:45 not do otherwise. 511 00:31:45 --> 00:31:50 In other words, we are three steps, which usually mash 512 00:31:50 --> 00:31:53 together and we can't separate them and we have to deal with 513 00:31:53 --> 00:31:55 the whole matrix at once. 514 00:31:55 --> 00:31:58 In this square case, they do separate. 515 00:31:58 --> 00:32:02 And so that's worth noticing. 516 00:32:02 --> 00:32:10 It means that we can solve backwards, we can solve 517 00:32:10 --> 00:32:13 these three one at a time. 518 00:32:13 --> 00:32:15 The inverses can be done separately. 519 00:32:15 --> 00:32:20 When A and A transpose are square, then from this 520 00:32:20 --> 00:32:23 equation I can find w. 521 00:32:23 --> 00:32:27 From knowing w I can find e, just by inverting C. 522 00:32:27 --> 00:32:30 By knowing e I can find u, just by inverting A. 523 00:32:30 --> 00:32:32 You see the three steps? 524 00:32:32 --> 00:32:34 You could invert that, and then you can invert the middle 525 00:32:34 --> 00:32:36 step, and you can invert A. 526 00:32:36 --> 00:32:37 And you've got u. 527 00:32:37 --> 00:32:41 So the square case is worth noticing. 528 00:32:41 --> 00:32:46 It special enough that in this case we would 529 00:32:46 --> 00:32:48 have an easy problem. 530 00:32:48 --> 00:32:54 And this case is called, for trusses and mechanics, 531 00:32:54 --> 00:32:55 there's a name for this. 532 00:32:55 --> 00:32:57 And unfortunately it's a little long. 533 00:32:57 --> 00:33:02 Statically, that's not the key word, determinant 534 00:33:02 --> 00:33:07 is the key word. 535 00:33:07 --> 00:33:13 Statically determinate, that goes with squares. 536 00:33:13 --> 00:33:16 And you can guess what the rectangular matrix a, what 537 00:33:16 --> 00:33:19 word would I use, what's the opposite of determinate? 538 00:33:19 --> 00:33:23 It's got to be indeterminate. 539 00:33:23 --> 00:33:26 Rectangular a will be indeterminate. 540 00:33:26 --> 00:33:32 And all that is referring to is the fact that in the 541 00:33:32 --> 00:33:40 determinate case the forces determine the stresses. 542 00:33:40 --> 00:33:45 You don't have to take that, we'd get all three together; 543 00:33:45 --> 00:33:48 mix them, invert, go backwards. 544 00:33:48 --> 00:33:48 All that. 545 00:33:48 --> 00:33:53 You just can do them one at the time in this determinate case. 546 00:33:53 --> 00:33:58 And now I guess I'd better say, so here's the matrix case. 547 00:33:58 --> 00:34:00 But now in this chapter I always have to do 548 00:34:00 --> 00:34:02 the continuous part. 549 00:34:02 --> 00:34:09 So let me just stay with free fixed, and what is 550 00:34:09 --> 00:34:11 this balance equation? 551 00:34:11 --> 00:34:17 So this is my force balance. 552 00:34:17 --> 00:34:23 I didn't give it its moment but its moment has come now. 553 00:34:23 --> 00:34:29 So the force balance equation is -dw/dx, because A transpose 554 00:34:29 --> 00:34:35 is minus a derivative equal f(x). and my free boundary 555 00:34:35 --> 00:34:43 condition, my free n, gave me w of what was it? 556 00:34:43 --> 00:34:45 The Neumann guy gave me w(0)=0. 557 00:34:45 --> 00:34:49 558 00:34:49 --> 00:34:52 And what's the point? 559 00:34:52 --> 00:34:54 Do you see what I'm saying? 560 00:34:54 --> 00:34:57 I'm saying that this free fixed is a beautiful 561 00:34:57 --> 00:34:59 example of determinate. 562 00:34:59 --> 00:35:05 Square matrix A in the matrix case and the parallel and 563 00:35:05 --> 00:35:08 the continuous cases. 564 00:35:08 --> 00:35:09 I can solve that for w(x). 565 00:35:09 --> 00:35:12 566 00:35:12 --> 00:35:14 I can solve directly for w(x). 567 00:35:14 --> 00:35:18 568 00:35:18 --> 00:35:23 Without involving, you see I didn't have to know c(x). 569 00:35:24 --> 00:35:26 I hadn't even got that far. 570 00:35:26 --> 00:35:28 I'm just going backwards now. 571 00:35:28 --> 00:35:35 I can solve this, just the way I can invert that matrix. 572 00:35:35 --> 00:35:37 Inverting the matrix here is the same as 573 00:35:37 --> 00:35:39 solving the equation. 574 00:35:39 --> 00:35:41 You see I have a first order first derivative? 575 00:35:41 --> 00:35:44 I mean, it's so trivial, right? 576 00:35:44 --> 00:35:48 It's the equation you solved in the final problem of the quiz, 577 00:35:48 --> 00:35:50 where an f was a delta function. 578 00:35:50 --> 00:35:55 It was simple because it was a square determinate problem 579 00:35:55 --> 00:35:58 with one condition on w. 580 00:35:58 --> 00:36:08 When both conditions are on u, then it's not square any more. 581 00:36:08 --> 00:36:10 OK for that point? 582 00:36:10 --> 00:36:13 Determinate versus indeterminate. 583 00:36:13 --> 00:36:14 OK. 584 00:36:14 --> 00:36:19 So that's sort of, and I could do examples. 585 00:36:19 --> 00:36:21 Maybe I've asked you on the homework to take 586 00:36:21 --> 00:36:22 a particular f(x). 587 00:36:23 --> 00:36:26 I hope it was a free fixed problem, if I was feeling good 588 00:36:26 --> 00:36:29 that day, because free fixed you'll be able to get 589 00:36:29 --> 00:36:31 w(x) right away. 590 00:36:31 --> 00:36:34 If it was fixed-fixed then I apologize it's going to 591 00:36:34 --> 00:36:39 take you a little bit longer to get to w. 592 00:36:39 --> 00:36:41 To get to u. 593 00:36:41 --> 00:36:41 OK. 594 00:36:41 --> 00:36:47 But this, of course- I just integrate. 595 00:36:47 --> 00:36:55 Inverting a difference matrix is just integrating a function. 596 00:36:55 --> 00:36:56 Good. 597 00:36:56 --> 00:37:07 OK, so this lecture so far was the transition from vectors 598 00:37:07 --> 00:37:11 and matrices to functions and continuous problems. 599 00:37:11 --> 00:37:15 And then, of course we're going to get deep into that because 600 00:37:15 --> 00:37:18 we got partial differential equations ahead. 601 00:37:18 --> 00:37:22 But today let's stay in one dimension and introduce 602 00:37:22 --> 00:37:23 finite elements. 603 00:37:23 --> 00:37:26 OK. 604 00:37:26 --> 00:37:34 Ready for finite elements, so that's now a major step. 605 00:37:34 --> 00:37:38 Finite differences, maybe I'll mention this, probably in this 606 00:37:38 --> 00:37:41 afternoon's review session, where I'll just be open 607 00:37:41 --> 00:37:43 to homework problems. 608 00:37:43 --> 00:37:47 I'll say something more about truss examples, and I might 609 00:37:47 --> 00:37:50 say something about finite differences for this. 610 00:37:50 --> 00:37:53 But really, it's finite elements that get 611 00:37:53 --> 00:37:55 introduced right now. 612 00:37:55 --> 00:37:57 So let me do that. 613 00:37:57 --> 00:38:02 Finite elements and introducing them. 614 00:38:02 --> 00:38:06 OK. 615 00:38:06 --> 00:38:10 So the prep, the getting ready for finite elements is to get 616 00:38:10 --> 00:38:23 hold of something called the weak form of the equation. 617 00:38:23 --> 00:38:28 So that's going to be a statement of, the finite 618 00:38:28 --> 00:38:30 elements aren't appearing yet. 619 00:38:30 --> 00:38:32 Matrices are not appearing yet. 620 00:38:32 --> 00:38:38 I'm talking about the differential equation. 621 00:38:38 --> 00:38:41 But what do I mean by this weak form? 622 00:38:41 --> 00:38:44 OK, let me just go for it directly. 623 00:38:44 --> 00:38:46 You see the equation up there? 624 00:38:46 --> 00:38:47 Let me copy it. 625 00:38:47 --> 00:38:52 So here's the strong form. 626 00:38:52 --> 00:38:55 The strong form is, you would say, the ordinary equation. 627 00:38:55 --> 00:39:01 Strong form is what our equation is, minus d/dx 628 00:39:01 --> 00:39:03 of c(x), du/dx=f(x). 629 00:39:03 --> 00:39:11 630 00:39:11 --> 00:39:14 OK, that's the strong form. 631 00:39:14 --> 00:39:15 That's the equation. 632 00:39:15 --> 00:39:19 Now, how do I get to the weak form? 633 00:39:19 --> 00:39:22 Let me just go to it directly and then over the next 634 00:39:22 --> 00:39:27 days we'll see why it's so natural and important. 635 00:39:27 --> 00:39:31 If I go for it directly, what I do is this. 636 00:39:31 --> 00:39:35 I multiply both sides of the equation by something I'll 637 00:39:35 --> 00:39:37 call a test function. 638 00:39:37 --> 00:39:41 And I'll try to systematically use the letter v for the test 639 00:39:41 --> 00:39:48 function. u will be the solution. v isn't the solution, 640 00:39:48 --> 00:39:54 v is like any function that I test this equation this way. 641 00:39:54 --> 00:39:57 I'm just multiplying both sides by the same thing, some v(x). 642 00:39:58 --> 00:39:58 Any v(x). 643 00:40:00 --> 00:40:03 We'll see if there are any limitations, OK? 644 00:40:03 --> 00:40:14 And I integrate. 645 00:40:14 --> 00:40:15 OK. 646 00:40:15 --> 00:40:18 So you're going to say nope, no problem. 647 00:40:18 --> 00:40:22 I integrate from zero to one. 648 00:40:22 --> 00:40:30 Alright, this would be true for f. 649 00:40:30 --> 00:40:34 So now I'll erase the word strong form, because the 650 00:40:34 --> 00:40:37 strong form isn't on the board anymore. 651 00:40:37 --> 00:40:40 It's the weak form now that we're looking at. 652 00:40:40 --> 00:40:44 And this is for any, and I'll put "any" in quotes just 653 00:40:44 --> 00:40:51 because eventually I'll say a little more about this. 654 00:40:51 --> 00:40:56 I'll write the equation this way. 655 00:40:56 --> 00:41:02 And you might think, OK, if this has to hold for every 656 00:41:02 --> 00:41:09 v(x), I could let v(x) be concentrated in a little area. 657 00:41:09 --> 00:41:11 And this would have to hold, then I could try another 658 00:41:11 --> 00:41:16 v(x), concentrated around other points. 659 00:41:16 --> 00:41:23 You can maybe feel that if this holds for every v(x), then 660 00:41:23 --> 00:41:25 I can get back to the strong form. 661 00:41:25 --> 00:41:29 If this holds for every v(x), then somehow that had 662 00:41:29 --> 00:41:31 better be the same as that. 663 00:41:31 --> 00:41:39 Because if this was f(x+1) and this is f(x), then I wouldn't 664 00:41:39 --> 00:41:41 have the equality any more. 665 00:41:41 --> 00:41:43 Should I just say that again? 666 00:41:43 --> 00:41:46 It's just like, at this point it's just a feeling, that 667 00:41:46 --> 00:41:52 if this is true for every v(x), then that part had 668 00:41:52 --> 00:41:55 better equal that part. 669 00:41:55 --> 00:41:57 That'll be my way back to the strong form. 670 00:41:57 --> 00:41:59 It's a little bit like climbing a hill. 671 00:41:59 --> 00:42:03 Going downhill was easy, I just multiplied by v and integrated. 672 00:42:03 --> 00:42:05 Nobody objected to that. 673 00:42:05 --> 00:42:10 I'm saying I'll be able to get back to the strong form 674 00:42:10 --> 00:42:11 with a little patience. 675 00:42:11 --> 00:42:14 But I like the week form. 676 00:42:14 --> 00:42:15 That's the whole point. 677 00:42:15 --> 00:42:19 You've got to begin to like the week form. 678 00:42:19 --> 00:42:22 If you begin to take it in and think OK. 679 00:42:22 --> 00:42:25 Now, why do I like it? 680 00:42:25 --> 00:42:28 What am I going to do to that left side? 681 00:42:28 --> 00:42:30 The right side's cool, right? 682 00:42:30 --> 00:42:31 It looks good. 683 00:42:31 --> 00:42:35 Left side does not look good to me. 684 00:42:35 --> 00:42:38 When you see something like that, what do you think? 685 00:42:38 --> 00:42:42 Today's lecture has already said what to think. 686 00:42:42 --> 00:42:46 What should I do to make that look better? 687 00:42:46 --> 00:42:48 I should, yep. 688 00:42:48 --> 00:42:50 Integrate by parts. 689 00:42:50 --> 00:42:54 If I integrate by parts, you see what I don't like about it 690 00:42:54 --> 00:42:58 as it is, is two derivatives are hitting u, and 691 00:42:58 --> 00:43:01 v is by itself. 692 00:43:01 --> 00:43:05 And I want it to be symmetric. 693 00:43:05 --> 00:43:08 I'm going to integrate this by parts, this is minus the 694 00:43:08 --> 00:43:10 derivative of something. 695 00:43:10 --> 00:43:12 Times v. 696 00:43:12 --> 00:43:15 And when I integrate by parts, I'm going to have, 697 00:43:15 --> 00:43:17 it'll be an integral. 698 00:43:17 --> 00:43:19 And can you integrate by parts now? 699 00:43:19 --> 00:43:24 I mean, you probably haven't thought about integration 700 00:43:24 --> 00:43:26 by parts for a while. 701 00:43:26 --> 00:43:29 Just think of it as taking the derivative off of this, so 702 00:43:29 --> 00:43:32 it leaves that by itself. 703 00:43:32 --> 00:43:41 And putting it onto v, so it's dv/dx, and remembering the 704 00:43:41 --> 00:43:43 minus sign, but we have a minus sign so now it's 705 00:43:43 --> 00:43:45 coming up plus. 706 00:43:45 --> 00:43:46 That's the weak form. 707 00:43:46 --> 00:43:50 Can I put a circle around the weak form? 708 00:43:50 --> 00:43:53 Well, that wasn't exactly a circle. 709 00:43:53 --> 00:43:54 OK. 710 00:43:54 --> 00:43:57 But that's the weak form. 711 00:43:57 --> 00:44:01 For every v, this is, I could give you a physical 712 00:44:01 --> 00:44:10 interpretation but I won't do it just this minute. 713 00:44:10 --> 00:44:14 This is going to hold for any v. 714 00:44:14 --> 00:44:16 That's the weak form. 715 00:44:16 --> 00:44:17 OK. 716 00:44:17 --> 00:44:18 Good. 717 00:44:18 --> 00:44:25 Now, why did I want to do that? 718 00:44:25 --> 00:44:29 The person who reminds me about boundary conditions 719 00:44:29 --> 00:44:31 should remind me again. 720 00:44:31 --> 00:44:33 That when I did this integration by parts, there 721 00:44:33 --> 00:44:36 should have been also. 722 00:44:36 --> 00:44:39 What's the integrated part now, that has to be 723 00:44:39 --> 00:44:43 evaluated at zero and one? 724 00:44:43 --> 00:44:47 This c, so it's that times that, right? 725 00:44:47 --> 00:44:53 It's that c(x)du/dv times v(x). 726 00:44:53 --> 00:44:56 727 00:44:56 --> 00:44:57 Maybe minus. 728 00:44:57 --> 00:44:58 Yeah, you're right. 729 00:44:58 --> 00:44:59 Minus. 730 00:44:59 --> 00:45:01 Good. 731 00:45:01 --> 00:45:03 What do I want this to come out? 732 00:45:03 --> 00:45:05 Zero, of course. 733 00:45:05 --> 00:45:08 I don't want to think that the same. 734 00:45:08 --> 00:45:11 Alright, so now I'm doing this free fixed problem still. 735 00:45:11 --> 00:45:16 So what's the deal on the free fixed problem? 736 00:45:16 --> 00:45:22 Well, let's see. 737 00:45:22 --> 00:45:26 OK, I got the two ends and I want them to be zero. 738 00:45:26 --> 00:45:38 OK, now at the free end, I'm not controlling v. 739 00:45:38 --> 00:45:40 I wasn't controlling u and I'm not going to be 740 00:45:40 --> 00:45:43 controlling its friend v. 741 00:45:43 --> 00:45:45 So this had to be zero. 742 00:45:45 --> 00:45:52 So this part will be zero at the free end. 743 00:45:52 --> 00:45:56 That boundary condition has just appeared again naturally. 744 00:45:56 --> 00:45:59 I had to have it because I had no control over b. 745 00:45:59 --> 00:46:01 And what about at the fixed end. 746 00:46:01 --> 00:46:07 At the fixed end, which is that, at the free end. 747 00:46:07 --> 00:46:11 Now, what's up at the fixed end? 748 00:46:11 --> 00:46:13 What was the fixed end? 749 00:46:13 --> 00:46:17 That's where u was zero. 750 00:46:17 --> 00:46:20 I'm going to make v also zero. 751 00:46:20 --> 00:46:27 So there's, when I said any v(x), I better put in with v=0 752 00:46:27 --> 00:46:37 at the Dirichlet point, at fixed point, at fixed end. 753 00:46:37 --> 00:46:40 I need that. 754 00:46:40 --> 00:46:44 I need to know that v is zero at that end. 755 00:46:44 --> 00:46:45 I had u=0. 756 00:46:46 --> 00:46:50 Here's why I'm fine. 757 00:46:50 --> 00:46:55 So I'm saying that any time I have a Dirichlet condition, a 758 00:46:55 --> 00:46:59 fixed condition that tells me u, I think of v and you'll 759 00:46:59 --> 00:47:02 begin to think of v as a little movement away from 760 00:47:02 --> 00:47:13 u. u is the solution. 761 00:47:13 --> 00:47:16 Now, remind me, this was free fixed. 762 00:47:16 --> 00:47:21 So the u might have been something like this. 763 00:47:21 --> 00:47:22 I just draw that. 764 00:47:22 --> 00:47:24 That's my u. 765 00:47:24 --> 00:47:27 This guy was fixed, right? 766 00:47:27 --> 00:47:29 By u. 767 00:47:29 --> 00:47:35 Now, I'm thinking of v's as, the letter v is very fortunate 768 00:47:35 --> 00:47:38 because it stands for virtual displacement. 769 00:47:38 --> 00:47:41 A virtual displacement is a little displacement away from 770 00:47:41 --> 00:47:46 u, but it has to satisfy the zero, the fixed condition 771 00:47:46 --> 00:47:48 that u satisfied. 772 00:47:48 --> 00:47:51 In other words, the little virtual v can't move 773 00:47:51 --> 00:47:53 away from zero. 774 00:47:53 --> 00:48:02 So I get this term is zero at the fixed end. 775 00:48:02 --> 00:48:11 OK. that's the little five minute time out state to check 776 00:48:11 --> 00:48:13 the boundary condition part. 777 00:48:13 --> 00:48:19 The net result is that that term's gone and I've got the 778 00:48:19 --> 00:48:21 weak format I've wanted. 779 00:48:21 --> 00:48:26 OK, three minutes to start to tell you how 780 00:48:26 --> 00:48:30 to use the weak form. 781 00:48:30 --> 00:48:39 So this is called Galerkin's method. 782 00:48:39 --> 00:48:50 And it starts with the weak form. 783 00:48:50 --> 00:48:51 So he's Russian. 784 00:48:51 --> 00:48:54 Russia gets into the picture now. 785 00:48:54 --> 00:48:56 We had France and Germany with the boundary conditions, now 786 00:48:56 --> 00:49:01 we've got Russia with this fundamental principle of how to 787 00:49:01 --> 00:49:06 turn a continuous problem into a discrete problem. 788 00:49:06 --> 00:49:09 That's what Galerkin's idea does. 789 00:49:09 --> 00:49:13 Instead of a function unknown I want to have n unknowns. 790 00:49:13 --> 00:49:17 I want to get a discrete equation which will 791 00:49:17 --> 00:49:19 eventually be kKU=F. 792 00:49:20 --> 00:49:27 So I'm going to get to an equation KU=F, but not by 793 00:49:27 --> 00:49:29 finite difference, right? 794 00:49:29 --> 00:49:31 I could but, I'm not. 795 00:49:31 --> 00:49:36 I'm doing it this weak Galerkin finite element way. 796 00:49:36 --> 00:49:42 OK, so if I tell you the Galerkin idea then next time we 797 00:49:42 --> 00:49:46 bring in, we have libraries of finite elements. 798 00:49:46 --> 00:49:48 But you have to get the principle straight. 799 00:49:48 --> 00:49:51 So it's Galerkin's idea. 800 00:49:51 --> 00:50:01 Galerkin's idea was was choose trial functions. 801 00:50:01 --> 00:50:11 Let me call them call them t? 802 00:50:11 --> 00:50:16 Have to get the names right. 803 00:50:16 --> 00:50:21 Phi. 804 00:50:21 --> 00:50:24 OK, the Greeks get a shot ok. 805 00:50:24 --> 00:50:29 Trial functions, phi_1(x) to phi_n(x). 806 00:50:31 --> 00:50:35 OK, so that's a choice you make. 807 00:50:35 --> 00:50:37 And we have a free choice. 808 00:50:37 --> 00:50:40 And it's a fundamental choice for all of applied math here. 809 00:50:40 --> 00:50:43 You choose some functions, and if you choose them well you get 810 00:50:43 --> 00:50:45 a great method, if you choose them badly you got 811 00:50:45 --> 00:50:47 a lousy method. 812 00:50:47 --> 00:50:49 OK, so you choose trial functions, and now what's 813 00:50:49 --> 00:50:51 the idea going to be? 814 00:50:51 --> 00:50:59 Your approximate U, approximate solution will be some 815 00:50:59 --> 00:51:03 combination of this. 816 00:51:03 --> 00:51:09 So combinations of those, let me call the coefficients U's, 817 00:51:09 --> 00:51:10 because those are the unknowns. 818 00:51:10 --> 00:51:12 Plus U_n*phi_n. 819 00:51:12 --> 00:51:15 820 00:51:15 --> 00:51:16 So those are the unknowns. 821 00:51:16 --> 00:51:23 The n unknowns. 822 00:51:23 --> 00:51:28 I'll even remove that for the moment. 823 00:51:28 --> 00:51:31 You see, these are functions of x. 824 00:51:31 --> 00:51:34 And these are numbers. 825 00:51:34 --> 00:51:39 So our unknown, our n unknown numbers are the coefficients 826 00:51:39 --> 00:51:43 to be decided of the functions we chose. 827 00:51:43 --> 00:51:47 OK., now I need n equations. 828 00:51:47 --> 00:51:49 I've got n unknowns now, they're the unknown 829 00:51:49 --> 00:51:52 coefficients of these functions. 830 00:51:52 --> 00:51:55 I need an equation so I get n equations by 831 00:51:55 --> 00:52:01 choose test functions. 832 00:52:01 --> 00:52:08 V_1, V_2, up to V(x). 833 00:52:08 --> 00:52:11 Each V will give me an equation. 834 00:52:11 --> 00:52:14 So I'll have n equations at the end, I have n unknowns, 835 00:52:14 --> 00:52:16 I'll have a square matrix. 836 00:52:16 --> 00:52:20 And that'll be a linear system. 837 00:52:20 --> 00:52:21 I'll get to KU=F. 838 00:52:22 --> 00:52:24 But do you see how I'm getting there? 839 00:52:24 --> 00:52:30 I'm getting there by using the weak form, By using Galerkin's 840 00:52:30 --> 00:52:33 idea of picking some trial functions, and some test 841 00:52:33 --> 00:52:37 functions, and putting them into the weak form. 842 00:52:37 --> 00:52:41 So Galerkin's idea is, take these functions 843 00:52:41 --> 00:52:43 and these functions. 844 00:52:43 --> 00:52:47 And apply the weak form just to those guys. 845 00:52:47 --> 00:52:50 Not to, the real weak form, the continuous weak form, 846 00:52:50 --> 00:52:53 was for a whole lot of V. 847 00:52:53 --> 00:52:57 We'll get n equations by picking n V's, and we'll get n 848 00:52:57 --> 00:53:01 unknowns by picking n phis. 849 00:53:01 --> 00:53:04 So this method, this idea, was a hundred years older 850 00:53:04 --> 00:53:06 than finite elements. 851 00:53:06 --> 00:53:11 The finite element idea was a particular choice of these 852 00:53:11 --> 00:53:18 guys, a particular choice of the phis and the V's as 853 00:53:18 --> 00:53:21 simple polynomials. 854 00:53:21 --> 00:53:25 And you might think well, why didn't Galerkin try those 855 00:53:25 --> 00:53:26 first, maybe he did. 856 00:53:26 --> 00:53:35 But key is that now with the computing power we now have 857 00:53:35 --> 00:53:40 compared to Galerkin, we can choose thousands of functions. 858 00:53:40 --> 00:53:41 If we keep them simple. 859 00:53:41 --> 00:53:46 So that's really what the finite element brought, 860 00:53:46 --> 00:53:48 finite element brought is. 861 00:53:48 --> 00:53:51 Keep the functions as simple polynomials and 862 00:53:51 --> 00:53:53 take many of them. 863 00:53:53 --> 00:53:59 Where Galerkin, who didn't have MATLAB, he probably didn't even 864 00:53:59 --> 00:54:02 have a desk computer, he used pencil and paper, he 865 00:54:02 --> 00:54:04 took one function. 866 00:54:04 --> 00:54:05 Or maybe two. 867 00:54:05 --> 00:54:08 I mean, that took him a day. 868 00:54:08 --> 00:54:12 But we take thousands of functions, simple functions, 869 00:54:12 --> 00:54:17 and we'll see on Friday the steps that get us to KU=F. 870 00:54:17 --> 00:54:21 So this is the prep for finite elements.