WEBVTT 1 00:00:00.080 --> 00:00:02.799 If I asked you to calculate the exact stress on 2 00:00:02.839 --> 00:00:07.200 one hundred story skyscraper holding up against like a Category 3 00:00:07.240 --> 00:00:11.560 five hurricane, you'd probably assume I need a massive supercomputer 4 00:00:11.720 --> 00:00:12.720 to even begin the math. 5 00:00:12.880 --> 00:00:15.080 Oh absolutely, I mean most people would, right. 6 00:00:15.439 --> 00:00:18.760 But the fundamental mathematical trick that actually keeps that building 7 00:00:18.800 --> 00:00:21.800 from collapsing, it wasn't born in a high tech lab. 8 00:00:21.920 --> 00:00:22.800 No, it really wasn't. 9 00:00:22.960 --> 00:00:26.719 It was invented by ancient philosophers who were literally just 10 00:00:26.760 --> 00:00:28.559 trying to figure out the value of pi. 11 00:00:28.920 --> 00:00:32.799 It is a pretty profound realization, honestly, and well that's 12 00:00:32.840 --> 00:00:35.119 exactly why we are doing this deep dive today. We 13 00:00:35.159 --> 00:00:38.759 are looking at a textbook called Introductory Finite Element Method 14 00:00:39.079 --> 00:00:42.520 by Chandra Khan, S. Desai and Tribikram Kundu. 15 00:00:42.600 --> 00:00:44.200 Yeah, it's a fantastic source. 16 00:00:44.280 --> 00:00:47.479 It really is for anyone studying civil engineering, or stepping 17 00:00:47.479 --> 00:00:50.600 into professional structural design, or you know, even just teaching 18 00:00:50.640 --> 00:00:53.520 yourself the mechanics of how the physical world operates. This 19 00:00:53.560 --> 00:00:56.960 is a cornerstone text definitely. Our mission here is to 20 00:00:57.000 --> 00:01:00.759 take the really dense, complex differential equations presented in this 21 00:01:00.799 --> 00:01:05.400 book and map them directly onto practical real world field analysis. 22 00:01:05.599 --> 00:01:07.920 And I think a great place to start is chapter one, 23 00:01:08.159 --> 00:01:11.280 because I found the historical context there just fascinating. The 24 00:01:11.280 --> 00:01:14.640 whole concept of the finite element method, or FEM as 25 00:01:14.680 --> 00:01:17.920 we'll call it, It stems from a very ancient human 26 00:01:17.959 --> 00:01:19.760 necessity called discretization. 27 00:01:20.159 --> 00:01:21.239 Right, discretization. 28 00:01:21.400 --> 00:01:23.400 Yeah, it goes all the way back to early thinkers 29 00:01:23.439 --> 00:01:26.599 realizing that they couldn't just you know, measure a perfect 30 00:01:26.719 --> 00:01:29.519 continuous curve to find pi. They couldn't do it now, 31 00:01:29.560 --> 00:01:31.719 the mac just wasn't there exactly, so they had to 32 00:01:31.840 --> 00:01:37.120 draw progressively smaller and smaller straight sided polygons inside a 33 00:01:37.159 --> 00:01:42.400 circle to approximate that curve. Or consider Zeno's ancient argument 34 00:01:42.719 --> 00:01:46.280 that space is infinitely divisible. Oh yeah, it highlights this 35 00:01:46.400 --> 00:01:50.840 basic biological limitation. Right, The human brain simply cannot compute 36 00:01:50.920 --> 00:01:53.000 true infinity, So we have to chop it up into 37 00:01:53.000 --> 00:01:54.319 pieces we can actually handle. 38 00:01:54.439 --> 00:01:57.599 And to understand the really intense mathematics of FEM, you 39 00:01:57.680 --> 00:02:01.599 first have to grasp that philosophy. Why behind why we 40 00:02:01.760 --> 00:02:03.640 discritize the physical. 41 00:02:03.200 --> 00:02:03.640 World, right? 42 00:02:03.680 --> 00:02:04.799 Why do we need to chop it up? 43 00:02:04.879 --> 00:02:05.200 Well? 44 00:02:05.400 --> 00:02:09.240 Chapter one establishes a stark reality about continuous natural systems. 45 00:02:09.719 --> 00:02:12.319 Think of a solid steel beam spanning a bridge. 46 00:02:12.479 --> 00:02:14.599 Okay, picturing it when a heavy. 47 00:02:14.319 --> 00:02:17.120 Truck drives over it, that beam doesn't just have one 48 00:02:17.199 --> 00:02:20.800 or two points of stress. It has infinite points of stress. 49 00:02:20.599 --> 00:02:22.879 Because every single atom is involved. 50 00:02:22.560 --> 00:02:25.919 Exactly, every single atom in that beam is interacting with 51 00:02:25.960 --> 00:02:29.759 the load. And because humans and frankly, even our most 52 00:02:29.800 --> 00:02:35.159 advanced modern processors cannot compute absolute totality, we have to approximate. 53 00:02:35.240 --> 00:02:36.199 We have to cut corners. 54 00:02:36.199 --> 00:02:41.199 Basically, well, mathematically, yes, we take that continuous physical body 55 00:02:41.479 --> 00:02:45.919 and divide it into a patchwork of smaller, manageable finite units, 56 00:02:45.960 --> 00:02:47.319 hence finite elements. 57 00:02:47.360 --> 00:02:49.680 Okay, let's unpack this for a second. It makes me 58 00:02:49.759 --> 00:02:52.800 think of a high definition digital photograph. 59 00:02:52.800 --> 00:02:54.199 Oh that's a good analogy, right. 60 00:02:54.199 --> 00:02:56.080 Because if you look at a photo of a smooth, 61 00:02:56.280 --> 00:03:00.599 sweeping architectural arch it looks perfectly continuous, But if you 62 00:03:00.680 --> 00:03:02.919 zoom in close enough, you see that the curve is 63 00:03:02.960 --> 00:03:06.080 actually an illusion, totally pixelated exactly. It's made up of 64 00:03:06.120 --> 00:03:10.240 thousands of tiny individual square pixels. And FEM is doing 65 00:03:10.240 --> 00:03:13.240 this exact same thing, but instead of mapping color data 66 00:03:13.280 --> 00:03:16.639 for a computer screen, it is mapping the structural integrity 67 00:03:16.680 --> 00:03:18.680 of concrete soil and steel. 68 00:03:19.039 --> 00:03:22.000 That's spot on, And what's fascinating here is that by 69 00:03:22.039 --> 00:03:26.080 surveying the data inside those small finite pixels or elements 70 00:03:26.520 --> 00:03:30.080 and stitching them together, an engineer can essentially model the 71 00:03:30.240 --> 00:03:31.919 entire cause and effect chain, the. 72 00:03:31.919 --> 00:03:34.759 Cause and effect of the physical loads acting on the structure. 73 00:03:34.879 --> 00:03:35.439 Precisely. 74 00:03:36.000 --> 00:03:39.280 That cause and effect analysis is the entire ballgame. It's 75 00:03:39.319 --> 00:03:42.599 what allows us to predict structural fatigue before a crack 76 00:03:42.639 --> 00:03:45.879 ever forms wow, or to prevent a material from yielding 77 00:03:45.960 --> 00:03:51.240 under pressure, which ultimately stops catastrophic failures. The authors actually 78 00:03:51.240 --> 00:03:54.719 compare it to matching together dozens of smaller aerial photographs 79 00:03:54.719 --> 00:03:57.280 to accurately survey a massive tract of land. 80 00:03:57.400 --> 00:04:00.000 Oh, like mapping a continent. You can't capture the whole 81 00:04:00.120 --> 00:04:01.280 thing in one shot from. 82 00:04:01.120 --> 00:04:03.599 A plane, right, But you can build a highly accurate 83 00:04:03.639 --> 00:04:05.439 mosaic by piecing it all together. 84 00:04:05.560 --> 00:04:07.960 Okay, So knowing that we are chopping the physical world 85 00:04:08.039 --> 00:04:10.560 up into mathematical pixels is one thing, But the real 86 00:04:10.599 --> 00:04:13.199 puzzle for me is how we actually do it. How 87 00:04:13.199 --> 00:04:16.639 do we turn a three dimensional physical concrete pillar into 88 00:04:16.759 --> 00:04:18.759 a solvable math problem. 89 00:04:18.879 --> 00:04:21.319 Well, Desigh and Kundu lay this out in a very 90 00:04:21.360 --> 00:04:22.800 strict eight step blueprint. 91 00:04:22.920 --> 00:04:24.439 Eight steps to solve anything. 92 00:04:24.720 --> 00:04:28.279 Basically, yeah, this sequence acts as the universal engine for 93 00:04:28.319 --> 00:04:32.439 solving virtually anything with FEM. Step one is to discretize 94 00:04:32.480 --> 00:04:33.839 and select your element. 95 00:04:33.519 --> 00:04:37.160 Configuration, meaning I have to decide what shape my specific 96 00:04:37.199 --> 00:04:39.319 pixels are going to take depending on the object I'm 97 00:04:39.319 --> 00:04:39.759 looking at. 98 00:04:39.839 --> 00:04:43.920 Yes, your geometry dictates your tools. If you are modeling 99 00:04:43.959 --> 00:04:48.040 a long, simple cable, you use one dimensional line elements. 100 00:04:48.160 --> 00:04:48.639 Makes sense. 101 00:04:48.839 --> 00:04:52.160 If you are analyzing a flat, thin retaining wall, you 102 00:04:52.199 --> 00:04:55.480 move up to two dimensional shapes like triangles or quadrilaterals, 103 00:04:56.000 --> 00:05:00.480 and for a complex, thick foundation, you use three dimensional hexahuge. 104 00:05:00.680 --> 00:05:04.240 Which function like uh, interlocking mathematical bricks. 105 00:05:04.279 --> 00:05:04.800 Exactly. 106 00:05:04.879 --> 00:05:07.240 So, once your geometry is chopped up, you arrive at 107 00:05:07.240 --> 00:05:09.920 step two, which is selecting your approximation models. 108 00:05:09.959 --> 00:05:10.800 Okay, what does that mean? 109 00:05:11.040 --> 00:05:14.839 You choose mathematical functions, typically pollnomials, to describe the shape 110 00:05:14.879 --> 00:05:17.399 of the unknown distribution within that specific element. 111 00:05:17.480 --> 00:05:18.920 Wait, hold on, I have to push back on this 112 00:05:18.959 --> 00:05:19.639 step a little bit. 113 00:05:19.680 --> 00:05:20.040 Go ahead. 114 00:05:20.079 --> 00:05:22.959 If we are choosing a pollinomial to describe the deformation, 115 00:05:23.079 --> 00:05:26.399 aren't we basically guessing the shape of the deformation in advance? 116 00:05:26.600 --> 00:05:28.800 It can definitely sound like that, because how can a 117 00:05:28.800 --> 00:05:32.199 mathematical model be safe enough to build an actual suspension 118 00:05:32.240 --> 00:05:36.759 bridge on if the fundamental premise is just an approximation 119 00:05:37.480 --> 00:05:38.160 a guess. 120 00:05:38.360 --> 00:05:41.839 That is a completely valid concern, But you are not 121 00:05:41.959 --> 00:05:45.199 guessing the final stress value. You are just assuming the 122 00:05:45.240 --> 00:05:48.199 pattern of how the material behaves between the corners of 123 00:05:48.240 --> 00:05:49.360 your elements. 124 00:05:48.920 --> 00:05:51.720 And those corners are called nodes, right, yes, yeah, exactly. 125 00:05:52.079 --> 00:05:55.800 The safety of this whole method comes from two intertwined concepts, 126 00:05:56.120 --> 00:05:57.399 bounds and convergence. 127 00:05:57.519 --> 00:05:59.720 Bounds and converbence. Okay, break that down for me. 128 00:06:00.000 --> 00:06:03.759 As long as your mathematical functions meet strict continuity requirements, 129 00:06:04.120 --> 00:06:08.439 meaning they don't apply impossible physical behaviors like overlapping matter, 130 00:06:08.879 --> 00:06:11.240 you establish a bound of potential. 131 00:06:10.879 --> 00:06:13.360 Error, you box the error in right, and. 132 00:06:13.360 --> 00:06:15.839 As you refine the mesh by making your elements smaller 133 00:06:15.879 --> 00:06:19.120 and more numerous, the approximate solution mathematically converges. 134 00:06:19.279 --> 00:06:22.600 It converges toward the exact continuous physical reality. 135 00:06:22.759 --> 00:06:23.240 Exactly. 136 00:06:23.319 --> 00:06:26.120 So the smaller the pixels, the tighter the bounds, and 137 00:06:26.160 --> 00:06:28.279 the closer you get to the true picture of the. 138 00:06:28.240 --> 00:06:32.360 Stress, which leads directly into step three, defining the relationships. 139 00:06:33.120 --> 00:06:36.399 This is where the actual physics of the material enters 140 00:06:36.439 --> 00:06:36.920 the equation. 141 00:06:37.199 --> 00:06:39.839 Oh, because up until now we've just been doing geometry. 142 00:06:40.040 --> 00:06:43.439 Right now, you have to establish the strained displacement relationships 143 00:06:43.439 --> 00:06:47.360 and the stress strain relationships. For solid materials, you apply 144 00:06:47.480 --> 00:06:49.040 principles like Hook's law. 145 00:06:49.480 --> 00:06:51.839 Hook's law being the rule that tells you how much 146 00:06:51.920 --> 00:06:56.079 a specific material will stretch when a specific force is 147 00:06:56.079 --> 00:06:59.439 applied to it. Yes, because obviously a steel beam is 148 00:06:59.439 --> 00:07:03.079 going to resist deformation very differently than say a block 149 00:07:03.120 --> 00:07:06.920 of hard rubber or a column of reinforced concrete. The 150 00:07:06.959 --> 00:07:09.600 math needs to know what it is simulating exactly. 151 00:07:09.720 --> 00:07:12.720 And once it knows the material properties, we hit step four, 152 00:07:12.879 --> 00:07:16.839 which is the mathematical heavy lifting of driving the element equation. 153 00:07:16.639 --> 00:07:17.519 The actual algebra. 154 00:07:17.720 --> 00:07:20.560 Right, we need a localized equation for every single puzzle piece. 155 00:07:20.800 --> 00:07:23.519 The classic formulation basically states that the stiffness of the 156 00:07:23.560 --> 00:07:27.279 element multiplied by its displacement equals the applied force. 157 00:07:27.480 --> 00:07:29.439 Stiffness times displacement equals force. 158 00:07:29.839 --> 00:07:30.199 Okay. 159 00:07:30.399 --> 00:07:33.040 In matrix algebra, this is written as k time's q 160 00:07:33.199 --> 00:07:37.720 equals q. To drive these localized matrices, engineers use energy 161 00:07:37.759 --> 00:07:41.439 methods such as the principle of minimum potential energy, or 162 00:07:41.480 --> 00:07:42.800 they use residual methods. 163 00:07:43.000 --> 00:07:45.319 Okay, I want to bookmark the residual methods for a second, 164 00:07:45.360 --> 00:07:48.199 because that concept shifts the whole paradigm. 165 00:07:47.759 --> 00:07:49.439 Later on in the book, it definitely does. 166 00:07:49.519 --> 00:07:52.600 But sticking to our virtual bridge construction, for now, we 167 00:07:52.680 --> 00:07:56.800 have thousands of localized equations for thousands of individual pieces. 168 00:07:57.240 --> 00:07:58.800 But if we just leave them like that, we just 169 00:07:58.839 --> 00:08:02.439 have a pile of disconnected mathematical bricks. They aren't a 170 00:08:02.439 --> 00:08:03.519 bridge yet, which. 171 00:08:03.279 --> 00:08:06.519 Is why step five solves that. By assembling the equations, 172 00:08:07.000 --> 00:08:10.120 you stitch all those individual element equations together to form 173 00:08:10.319 --> 00:08:15.639 one massive global set of algebraic equations for the entire structure. 174 00:08:15.839 --> 00:08:19.160 And the governing rule here is compatibility. Yes, compatibility is 175 00:08:19.160 --> 00:08:22.600 crucial because if we treat these elements individually, what stops 176 00:08:22.600 --> 00:08:26.079 them from overlapping or tearing apart? In the simulation, if 177 00:08:26.120 --> 00:08:28.800 element moves two inches to the left under a load, 178 00:08:29.439 --> 00:08:32.000 the edge of element B that is physically attached to 179 00:08:32.039 --> 00:08:34.159 it must also move two inches to the left. 180 00:08:34.440 --> 00:08:39.080 Exactly, the structure can't magically rip itself apart. Inside the computer, 181 00:08:39.720 --> 00:08:42.840 compatibility ensures the nodes stay locked together. 182 00:08:43.000 --> 00:08:44.360 That makes total sense. 183 00:08:44.120 --> 00:08:47.320 And during this assembly phase. You also apply your boundary. 184 00:08:47.000 --> 00:08:49.879 Conditions, which are what exactly. 185 00:08:49.519 --> 00:08:52.360 Well, think of a skyscraper. It is bolted down to 186 00:08:52.440 --> 00:08:55.879 a foundation. The nodes at the very bottom cannot move 187 00:08:55.919 --> 00:08:59.080 in any direction, no matter what forces are applied. 188 00:08:58.720 --> 00:08:59.159 At the top. 189 00:08:59.279 --> 00:09:02.240 Oh, I see you input those fixed realities into the 190 00:09:02.320 --> 00:09:03.360 matrix exactly. 191 00:09:03.440 --> 00:09:07.240 Step six is solving for the primary unknowns. You execute 192 00:09:07.279 --> 00:09:09.759 the math to find out exactly how much every single 193 00:09:09.799 --> 00:09:12.120 node displaces across the entire mesh. 194 00:09:12.240 --> 00:09:14.960 And once I know how far my nod'es moved, let's 195 00:09:14.960 --> 00:09:17.480 say the center of my beam sag by a fraction 196 00:09:17.559 --> 00:09:19.759 of an inch, I can move to stake seven. 197 00:09:19.559 --> 00:09:21.399 Which is solving for the secondary quantities. 198 00:09:21.600 --> 00:09:24.679 Right. I use that displacement data to calculate the resulting 199 00:09:24.799 --> 00:09:28.000 internal strains and stresses ripping through the material. 200 00:09:27.840 --> 00:09:30.799 Leaving us with step eight interpretation of results. 201 00:09:30.480 --> 00:09:31.240 The human element. 202 00:09:31.399 --> 00:09:31.879 Exactly. 203 00:09:32.279 --> 00:09:35.320 You now have a mountain of numerical data. The engineer 204 00:09:35.360 --> 00:09:37.960 must look at those calculator stresses and decide if the 205 00:09:38.000 --> 00:09:39.279 design is actually safe. 206 00:09:39.440 --> 00:09:42.519 Will it withstand the wind loads? Will it support the 207 00:09:42.559 --> 00:09:43.000 dead weight? 208 00:09:43.159 --> 00:09:43.320 Right? 209 00:09:43.440 --> 00:09:46.440 It is a highly logical progression. You divide the object, 210 00:09:46.759 --> 00:09:50.440 define its physics, assemble the global puzzle, solve for movement 211 00:09:50.879 --> 00:09:52.600 and analyze the resulting stress. 212 00:09:52.960 --> 00:09:56.759 It's brilliant. And you know, we have been visualizing solid 213 00:09:56.759 --> 00:10:01.279 structures so far, columns, beams, materials that fit physically bend underweight. 214 00:10:01.879 --> 00:10:05.320 But the textbook pivots here and makes a massive conceptual 215 00:10:05.360 --> 00:10:06.960 point about the scope of fem. 216 00:10:07.039 --> 00:10:09.120 Yes, it isn't just a structural tool. 217 00:10:09.399 --> 00:10:10.919 Here is where it gets really interesting. 218 00:10:11.000 --> 00:10:14.720 It is entirely universal. The finite element method is fundamentally 219 00:10:14.759 --> 00:10:16.679 a general mathematical procedure. 220 00:10:16.799 --> 00:10:17.879 This blew my mind. 221 00:10:17.960 --> 00:10:21.600 It's amazing. The governing differential equations for a one dimensional 222 00:10:21.600 --> 00:10:25.759 column undergoing physical stress deformation are well. They are essentially 223 00:10:25.799 --> 00:10:29.200 identical to the equations governing fluid flow through a porous 224 00:10:29.200 --> 00:10:32.720 soil wow, or even steady heat flow radiating through a 225 00:10:32.720 --> 00:10:33.720 metal engine block. 226 00:10:34.000 --> 00:10:37.200 I just love when different disciplines suddenly overlap like this. 227 00:10:37.799 --> 00:10:41.639 The exact same matrix algebra used by a structural engineer 228 00:10:41.679 --> 00:10:45.120 to find how a load displaces a steel beam is 229 00:10:45.240 --> 00:10:48.320 used by a geotechnical engineer to track the pressure of 230 00:10:48.399 --> 00:10:50.600 groundwater seeping underneath a dam. 231 00:10:50.759 --> 00:10:51.960 It's the exact same math. 232 00:10:52.120 --> 00:10:55.799 It requires you to completely shift your mental backdrop instead 233 00:10:55.799 --> 00:11:00.279 of visualizing steel bending under gravity, you're visualizing water pressing 234 00:11:00.279 --> 00:11:04.480 through microscopic gaps in rock or thermal energy diffusing through 235 00:11:04.480 --> 00:11:05.000 a barrier. 236 00:11:05.200 --> 00:11:08.159 Because the math doesn't care about the physical medium. It 237 00:11:08.240 --> 00:11:10.919 only cares about the gradient, the flow, and the resistance. 238 00:11:11.519 --> 00:11:15.320 But that universality introduces a mathematical problem, which brings us 239 00:11:15.360 --> 00:11:17.080 back to your bookmark about step four. 240 00:11:17.320 --> 00:11:20.720 Ah. Yes, the energy methods versus residual. 241 00:11:20.279 --> 00:11:23.840 Methods right, because when we derive equations for a physical beam, 242 00:11:24.000 --> 00:11:27.320 we use the principle of minimum potential energy. A physical 243 00:11:27.320 --> 00:11:30.039 structure bends and naturally wants to rest at a state 244 00:11:30.080 --> 00:11:31.559 of lowest internal energy. 245 00:11:31.639 --> 00:11:35.159 But groundwater seeping through dirt or heat radiating through a wall, 246 00:11:35.519 --> 00:11:37.840 while they don't possess a physical potential energy in that 247 00:11:37.919 --> 00:11:39.039 same structural sense. 248 00:11:39.200 --> 00:11:41.679 So the structural metaphor just breaks down completely. 249 00:11:41.840 --> 00:11:43.799 And this is where the Gallakin method, which is a 250 00:11:43.840 --> 00:11:46.639 method of weighted residuals, really proves its genius. 251 00:11:46.720 --> 00:11:48.639 Okay, tell me about Gallerkin's method. 252 00:11:48.879 --> 00:11:51.919 Instead of relying on a physical principle like potential energy, 253 00:11:52.240 --> 00:11:56.639 Gaalakin's method attacks the differential equation directly. You assume an 254 00:11:56.679 --> 00:11:59.720 approximate solution and plug it into the governing equation. 255 00:12:00.120 --> 00:12:03.240 But because it is only an approximation, the equation will 256 00:12:03.240 --> 00:12:04.679 not balance perfectly to zero. 257 00:12:04.799 --> 00:12:07.559 Right exactly, There is an error or remainder, and we 258 00:12:07.639 --> 00:12:09.200 call that remainder the residual. 259 00:12:09.559 --> 00:12:12.279 The math is slightly lopsided because we are using our 260 00:12:12.320 --> 00:12:16.080 pixelated approximation instead of the true infinite reality. 261 00:12:16.320 --> 00:12:16.519 Right. 262 00:12:16.919 --> 00:12:20.559 So, Gallikin's method introduces waiting functions to handle that residual. 263 00:12:21.000 --> 00:12:24.320 It forces that error to average out to zero across 264 00:12:24.320 --> 00:12:25.879 the entire domain or the element. 265 00:12:26.159 --> 00:12:29.240 Let me try to visualize this. Imagine trying to level 266 00:12:29.279 --> 00:12:32.679 a wobbly wooden table on an uneven stone floor. 267 00:12:32.879 --> 00:12:33.879 Okay, I can picture that. 268 00:12:34.200 --> 00:12:36.960 You can't make the floor perfectly flat. That is our 269 00:12:36.960 --> 00:12:42.559 inability to compute the true complex continuous domain. But you 270 00:12:42.600 --> 00:12:46.879 can strategically distribute little wooden shims under the table legs. 271 00:12:47.200 --> 00:12:48.240 Ah, I see where you're going. 272 00:12:48.320 --> 00:12:50.919 Those shims are the waiting functions. You push and pull 273 00:12:50.960 --> 00:12:54.320 the balance until the average wobble basically cancels out to zero, 274 00:12:55.039 --> 00:12:57.720 giving you a completely stable surface to work on. 275 00:12:58.000 --> 00:13:00.639 That is a highly effective way to concen victualize it. 276 00:13:01.279 --> 00:13:04.960 By averaging out the residual error, engineers can solve incredibly 277 00:13:05.000 --> 00:13:09.960 complex differential equations for fluid and thermal dynamics, where traditional 278 00:13:10.000 --> 00:13:12.919 potential energy theorems simply do not exist. 279 00:13:13.120 --> 00:13:16.440 It really demonstrates the ultimate adaptability of the math. 280 00:13:16.600 --> 00:13:20.080 It does, but we have built this entire foundation today 281 00:13:20.200 --> 00:13:23.120 using simple one dimensional examples lines and columns. 282 00:13:23.279 --> 00:13:25.000 Yeah, but the physical world. 283 00:13:24.879 --> 00:13:28.720 Operates in three dimensions. Moving from the theory to actual 284 00:13:28.879 --> 00:13:33.600 complex civil engineering projects requires scaling this process up mass. 285 00:13:33.360 --> 00:13:34.799 Which has to be incredibly difficult. 286 00:13:34.840 --> 00:13:38.360 Scaling from one dimension to two dimensions is conceptually straightforward, 287 00:13:38.600 --> 00:13:43.120 but computationally immense. The textbook transitions the reader into two 288 00:13:43.200 --> 00:13:46.759 D elements, specifically focusing on four. Note isoperimetric. 289 00:13:46.799 --> 00:13:52.759 Quadrilaterals isoperimetric, meaning the mathematics map of perfect idealized square 290 00:13:52.840 --> 00:13:56.039 from the software's memory onto the highly distorted real world 291 00:13:56.080 --> 00:13:59.240 shape of the element functioning in the physical structure exactly. 292 00:13:59.639 --> 00:14:05.600 This apping capability allows engineers to model highly irregular organic geometries. 293 00:14:05.240 --> 00:14:06.440 Not just perfect boxes. 294 00:14:06.559 --> 00:14:06.720 Right. 295 00:14:07.080 --> 00:14:11.399 The later chapters demonstrate this with some incredible practical applications. 296 00:14:11.759 --> 00:14:16.519 For example, they analyze steady confined seepage flowing through a 297 00:14:16.559 --> 00:14:21.080 two layered soil foundation beneath a massive retaining dam Wow. 298 00:14:21.360 --> 00:14:25.279 They also calculate the intense stress concentrations that invisibly gather 299 00:14:25.440 --> 00:14:28.279 around the sharp corners of windows and doors in a 300 00:14:28.320 --> 00:14:29.440 concrete shear wall. 301 00:14:29.559 --> 00:14:32.519 Oh that's so practical. They even delve into multi component systems, 302 00:14:32.519 --> 00:14:34.360 don't they. They do like you can bottle how an 303 00:14:34.519 --> 00:14:38.039 entire building frame interacts with the concrete floor slabs, and 304 00:14:38.080 --> 00:14:40.759 then model how that entire assembly interacts with the soil 305 00:14:40.759 --> 00:14:41.840 foundation underneath it. 306 00:14:41.960 --> 00:14:44.960 Yes, they idealize the dirt under the building as a 307 00:14:45.000 --> 00:14:48.120 series of spring elements to simulate how the massive weight 308 00:14:48.519 --> 00:14:50.559 causes the whole system to settle over time. 309 00:14:50.720 --> 00:14:53.480 But going from a simple one dimensional line to a 310 00:14:53.600 --> 00:14:57.960 sprawling two dimensional mesh or a full three D model, 311 00:14:58.120 --> 00:15:00.480 I mean that seems like going from playing a single 312 00:15:00.519 --> 00:15:03.639 note on a piano to conducting a full symphony orchestra. 313 00:15:04.120 --> 00:15:07.679 The sheer volume of simultaneous calculations must be staggering. 314 00:15:07.799 --> 00:15:10.000 Oh, it reaches a scale that no human could ever 315 00:15:10.120 --> 00:15:13.120 process manually. You could literally be looking at tens of 316 00:15:13.159 --> 00:15:15.320 thousands of simultaneous equations. 317 00:15:14.879 --> 00:15:15.879 Which is insane. 318 00:15:16.039 --> 00:15:18.879 It is This is exactly why the finite element method 319 00:15:18.919 --> 00:15:22.720 is inextricably tied to the evelation of computer science. The 320 00:15:22.799 --> 00:15:27.279 textbook recognizes this reality and actually provides educational computer codes 321 00:15:27.320 --> 00:15:31.120 for the reader, such as DFT slash c DAH one 322 00:15:31.200 --> 00:15:33.360 DFE for one dimensional. 323 00:15:32.879 --> 00:15:36.120 Problems and plane TV for two dimensional setups, yes, which 324 00:15:36.159 --> 00:15:38.320 I love because it actually forces you to look under 325 00:15:38.360 --> 00:15:40.799 the hood of the software rather than just treating the 326 00:15:40.799 --> 00:15:42.679 program like a magic box where you put in a 327 00:15:42.759 --> 00:15:44.320 drawing and somehow get out of safety. 328 00:15:44.320 --> 00:15:47.639 Reading right, it outlines exactly how these codes use Gaussian 329 00:15:47.679 --> 00:15:50.440 elimination to solve the massive algebraic matrices. 330 00:15:50.679 --> 00:15:53.600 I always find Gaussian elimination fascinating to picture. It is 331 00:15:53.720 --> 00:15:56.559 essentially a massive game of mathematical Pseudoku. 332 00:15:56.840 --> 00:15:57.879 That's a fun way to put it. 333 00:15:57.960 --> 00:16:00.399 You can't solve the whole board at once. You have 334 00:16:00.480 --> 00:16:04.440 this giant grid of interconnected equations, so you use the 335 00:16:04.440 --> 00:16:07.440 first equation to systematically eliminate a variable from all the 336 00:16:07.480 --> 00:16:10.159 equations below it. Right, then you move to the second 337 00:16:10.159 --> 00:16:14.200 equation and eliminate another variable from the remaining rows. You 338 00:16:14.240 --> 00:16:18.279 sweep down the matrix row by row, isolating the unknowns 339 00:16:18.440 --> 00:16:21.919 until you find one definitive value at the very bottom, 340 00:16:22.000 --> 00:16:25.440 and then then you back substitute that value all the 341 00:16:25.440 --> 00:16:27.480 way up the chain. To solve the rest of the board. 342 00:16:27.799 --> 00:16:31.759 It is a brute force algebraic assembly line that only 343 00:16:31.799 --> 00:16:37.039 a microchip can execute efficiently at scale. But you know 344 00:16:37.159 --> 00:16:40.240 this computational power brings a profound responsibility. 345 00:16:40.519 --> 00:16:41.559 How so well? 346 00:16:42.080 --> 00:16:46.000 The software handling the Gaushan elimination is ultimately blind. It 347 00:16:46.039 --> 00:16:47.960 does not know if it is calculating the stress on 348 00:16:48.000 --> 00:16:51.080 a ninety story skyscraper or the plastic casing of a 349 00:16:51.200 --> 00:16:51.759 kitchen toaster. 350 00:16:52.000 --> 00:16:54.840 It only knows numbers exactly which circles us all the 351 00:16:54.840 --> 00:16:57.039 way back to step eight of the blueprint, the interpretation 352 00:16:57.080 --> 00:17:00.480 of results. Yes, all this computational power is totally useless 353 00:17:00.480 --> 00:17:03.679 and potentially highly dangerous if the human sitting at the 354 00:17:03.720 --> 00:17:07.400 keyboard doesn't understand the underlying physics. Absolutely a computer will 355 00:17:07.440 --> 00:17:11.000 happily give you a highly precise, beautifully rendered, utterly wrong 356 00:17:11.039 --> 00:17:14.480 answer if you fet it the wrong boundary conditions back 357 00:17:14.480 --> 00:17:15.200 in step five. 358 00:17:15.759 --> 00:17:17.799 It happens all the time in structural analysis. If you 359 00:17:17.799 --> 00:17:19.720 aren't careful, you have to be able to look at 360 00:17:19.720 --> 00:17:23.519 the colorful stress maps generated by the software and intuitively 361 00:17:23.640 --> 00:17:26.799 know if the structural output actually makes physical sense. 362 00:17:27.079 --> 00:17:29.599 So what does this all mean for the listener? 363 00:17:29.759 --> 00:17:33.039 It means the software is simply a tool. Understanding the 364 00:17:33.079 --> 00:17:35.880 physical behavior is the actual craft of engineering. 365 00:17:35.920 --> 00:17:36.400 Well said. 366 00:17:37.119 --> 00:17:39.960 And to test that understanding, I actually have a short 367 00:17:40.000 --> 00:17:43.039 review exercise for you to visualize. It bridges the theory 368 00:17:43.079 --> 00:17:44.480 with the mechanics we've discussed today. 369 00:17:44.519 --> 00:17:45.920 Okay, I'm ready. Let's hear it all right. 370 00:17:46.319 --> 00:17:50.440 Imagine you are modeling a simple one dimensional concrete column 371 00:17:50.839 --> 00:17:54.000 subjected to a heavy axial load pressing down on it. 372 00:17:54.119 --> 00:17:56.640 Okay, got a concrete column in my head, pressing down. 373 00:17:56.880 --> 00:17:57.160 Good. 374 00:17:57.720 --> 00:18:01.039 Now, if you decide to refine your mesh by increasing 375 00:18:01.079 --> 00:18:03.720 the number of elements in your models from two to four, 376 00:18:04.160 --> 00:18:08.160 what happens to the physical size of your global stiffness matrix? 377 00:18:08.640 --> 00:18:09.279 And how does this. 378 00:18:09.279 --> 00:18:11.880 Affect the convergence of your primary unknowns? 379 00:18:12.039 --> 00:18:14.799 Ooh, okay, let me reason through the mechanics of that. 380 00:18:14.839 --> 00:18:15.319 Take your time. 381 00:18:15.480 --> 00:18:17.720 So, if I increase the number of elements from two 382 00:18:17.720 --> 00:18:21.359 to four, I am physically cutting the column into smaller pieces. 383 00:18:21.480 --> 00:18:21.680 Right. 384 00:18:22.039 --> 00:18:24.920 That means I am adding more nodes to connect those pieces, 385 00:18:25.279 --> 00:18:28.319 and more nodes mean more degrees of freedom in the 386 00:18:28.319 --> 00:18:31.960 overall system. Exactly because my global stiffness matrix, the big 387 00:18:32.079 --> 00:18:35.119 K in our K times Q equals q equation is 388 00:18:35.160 --> 00:18:38.599 built to account for every single degree of freedom. Adding 389 00:18:38.640 --> 00:18:41.440 more nodes causes the physical size of the matrix to 390 00:18:41.519 --> 00:18:42.720 expand significantly. 391 00:18:42.880 --> 00:18:45.319 Yes, the computer will have to solve a much larger 392 00:18:45.319 --> 00:18:47.000 set of simultaneous equations. 393 00:18:47.200 --> 00:18:49.440 Okay, so that's the first part. As for the second 394 00:18:49.440 --> 00:18:53.200 part about convergence, Because my mesh is now finer and 395 00:18:53.240 --> 00:18:56.480 the pieces are smaller, the mathematical bounds of my approximation 396 00:18:56.680 --> 00:19:00.640 must tighten. The assumed polynomial curves have less distance to 397 00:19:00.680 --> 00:19:05.519 cover between the nodes. Therefore, my primary unknowns, which are 398 00:19:05.799 --> 00:19:09.799 the physical displacements of the concrete, should converge much closer 399 00:19:09.839 --> 00:19:14.200 to the exact, true mathematical solution of the continuous column. 400 00:19:14.319 --> 00:19:16.319 The logic holds perfectly. You nailed it. 401 00:19:16.359 --> 00:19:16.759