WEBVTT 1 00:00:00.080 --> 00:00:03.680 So if you take a modern supercomputer, something with just 2 00:00:03.799 --> 00:00:06.360 terabytes of RAM, and you ask it to do some 3 00:00:06.400 --> 00:00:10.880 basic math I'm talking like elementary arithmetic, it will fundamentally 4 00:00:10.880 --> 00:00:11.919 get it slightly wrong. 5 00:00:12.080 --> 00:00:14.640 Yeah, it's a bit of a shock the first time 6 00:00:14.679 --> 00:00:15.320 you realize that. 7 00:00:15.640 --> 00:00:20.320 Right, today we're looking at why engineering software is mathematically broken, 8 00:00:20.440 --> 00:00:25.399 and how civil and structural engineers managed to build skyscrapers 9 00:00:25.440 --> 00:00:28.839 and suspension bridges and massive fluid networks. 10 00:00:28.879 --> 00:00:33.399 Anyway, it's honestly the ultimate compromise. Like for centuries we 11 00:00:33.560 --> 00:00:36.880 relied on pure analytical math. You have a quadratic equation, 12 00:00:37.200 --> 00:00:39.600 you use the quadratic formula, and you get a perfect, 13 00:00:39.640 --> 00:00:40.479 pristine answer. 14 00:00:40.520 --> 00:00:42.520 You get the exact number exactly yea. 15 00:00:42.759 --> 00:00:45.159 But the moment you step outside the classroom and try 16 00:00:45.200 --> 00:00:47.840 to model, say the turbulent wind forces on a high 17 00:00:47.920 --> 00:00:50.600 rise or the chaotic dissipation of heat through a steel beam, 18 00:00:50.880 --> 00:00:53.960 those perfect equations just they simply stop execting. 19 00:00:53.759 --> 00:00:57.520 Right, because physical reality is messy, it's totally nonlinear, and 20 00:00:57.600 --> 00:01:00.520 to handle that mess you have to undergo this massive 21 00:01:00.520 --> 00:01:01.359 paradigm shift. 22 00:01:01.479 --> 00:01:02.000 You really do. 23 00:01:02.280 --> 00:01:05.159 So if you are a civil engineering student tuning in 24 00:01:05.400 --> 00:01:08.920 or a young professional trying to master your structural analysis software. 25 00:01:09.480 --> 00:01:12.200 This deep dive is for you. We're getting into the 26 00:01:12.239 --> 00:01:14.719 core logic running beneath your screen, and. 27 00:01:14.680 --> 00:01:17.680 We're basing this on the principles from Rajesh Kamar Gupta's 28 00:01:17.719 --> 00:01:22.519 textbook Numerical Methods, Fundamentals and Applications. 29 00:01:22.879 --> 00:01:25.599 Yeah, the mission today is to figure out how we 30 00:01:25.680 --> 00:01:30.280 solve the mathematically unsolvable. We are trading absolute mathematical perfection 31 00:01:30.560 --> 00:01:34.480 for real world utility. We'll start with computational errors, move 32 00:01:34.480 --> 00:01:38.359 through solving complex equations, handle raw data, and finally look 33 00:01:38.400 --> 00:01:40.040 at predicting dynamic systems. 34 00:01:40.120 --> 00:01:42.439 And to do that we really have to start at 35 00:01:42.480 --> 00:01:44.120 the bedrock of computing. 36 00:01:43.840 --> 00:01:45.560 Right, numbers and errors. 37 00:01:45.680 --> 00:01:47.840 Yeah, before you can trust a computer to simulate a 38 00:01:47.840 --> 00:01:50.959 Category five hurricane hitting a bridge, you have to understand 39 00:01:51.000 --> 00:01:53.879 why it's basic arithmetic as flawed. I always tell the 40 00:01:53.879 --> 00:01:57.040 students in my educational content. Humans operate in a base 41 00:01:57.120 --> 00:01:58.319 ten decimal world. 42 00:01:58.239 --> 00:01:59.519 Right, yes, zero through nine. 43 00:02:00.000 --> 00:02:03.799 Computers operate in a base two binary world of zeros 44 00:02:03.799 --> 00:02:07.280 in ones or sometimes octaal or hexitecimal. So just the 45 00:02:07.319 --> 00:02:11.439 act of translating a simple decimal fraction into binary taper 46 00:02:12.159 --> 00:02:14.840 it creates a slight mismatch. I always pictured that like 47 00:02:15.919 --> 00:02:19.199 building a scale model of a bridge for a wind 48 00:02:19.240 --> 00:02:21.280 tunnel test. Oh, that's a good at it, like it's 49 00:02:21.280 --> 00:02:23.879 an incredible tool, but it's not the actual bridge. There 50 00:02:23.960 --> 00:02:27.439 is an inherent modeling error because you have to simplify 51 00:02:27.520 --> 00:02:31.599 the chaos of real wind into basic proportions just to 52 00:02:31.639 --> 00:02:32.639 make the math workable. 53 00:02:32.759 --> 00:02:35.319 Yeah, that covers the physical assumptions we make, for sure, 54 00:02:35.879 --> 00:02:39.240 But the computer itself introduces structural errors. 55 00:02:38.960 --> 00:02:40.919 Just by doing the math the actual calculation. 56 00:02:41.039 --> 00:02:44.319 Right, take an infinitely long mathematical process, like a tailor series. 57 00:02:44.360 --> 00:02:47.240 It mathematically describes a function using an endless string of. 58 00:02:47.319 --> 00:02:49.639 Terms, and a computer can't do endless exactly. 59 00:02:49.639 --> 00:02:52.240 It doesn't have infinite processing power or time, so it 60 00:02:52.240 --> 00:02:54.120 eventually just has to stop. It chops off the end 61 00:02:54.159 --> 00:02:58.039 of the series, and we call that truncation error. Gotcha. 62 00:02:58.159 --> 00:03:00.800 And then there's round off er, right, because it only 63 00:03:00.840 --> 00:03:03.520 has so much memory to hold the decimal places. But 64 00:03:04.199 --> 00:03:07.039 in the book I was reading about this specific phenomenon 65 00:03:07.120 --> 00:03:10.400 that stems from this, called smearing. It sounds like, I 66 00:03:10.400 --> 00:03:12.439 don't know, ink bleeding on a blueprint. 67 00:03:13.319 --> 00:03:17.120 The term is evocative, definitely, but that visual metaphor doesn't 68 00:03:17.159 --> 00:03:21.919 quite capture the danger. Smearing, which is also called catastrophic cancelation, 69 00:03:22.360 --> 00:03:25.919 is much more insidious. Insidious how well, imagine you want 70 00:03:25.919 --> 00:03:28.719 to weigh a single feather, but you only have an 71 00:03:28.759 --> 00:03:32.000 industrial truck scale. So you drive a massive truck onto 72 00:03:32.000 --> 00:03:35.039 the scale, record the weight, then place the feather in 73 00:03:35.080 --> 00:03:38.080 the truck, weigh it again, and subtract the two massive 74 00:03:38.159 --> 00:03:39.560 numbers to find the feather's weight. 75 00:03:39.759 --> 00:03:42.280 Oh wow, the scale just isn't sensitive enough for that. 76 00:03:42.479 --> 00:03:45.560 Right, The tiny weight of the feather is completely swallowed 77 00:03:45.560 --> 00:03:47.159 by the margin of error of the truck's weight. 78 00:03:47.280 --> 00:03:48.199 That makes total sense. 79 00:03:48.439 --> 00:03:52.360 And in mathematical series with mixed signs, your software is 80 00:03:52.400 --> 00:03:56.280 often adding and subtracting incredibly massive numbers to find a 81 00:03:56.319 --> 00:04:00.479 tiny resulting value. The computer just loses that true, tiny 82 00:04:00.599 --> 00:04:04.599 value in the noise of subtracting those giants. It destroys 83 00:04:04.599 --> 00:04:06.840 your precision with how you even realizing it. 84 00:04:07.080 --> 00:04:10.879 Man, let's ground this a bit for the civil engineers listening. 85 00:04:11.159 --> 00:04:13.919 Say you're calculating the volume of a right circular cylinder. 86 00:04:14.039 --> 00:04:17.759 That's geometry right hi times radio squared times height a classic. 87 00:04:17.800 --> 00:04:19.920 But if you're out in the field and your measurements 88 00:04:20.079 --> 00:04:22.839 for the radius in height, have say a two percent 89 00:04:22.920 --> 00:04:25.639 data error. That doesn't just stem a flat two percent, 90 00:04:25.720 --> 00:04:26.040 does it? 91 00:04:26.240 --> 00:04:28.600 Far from it? Because the radius is squared in the 92 00:04:28.680 --> 00:04:33.079 volume formula, that initial two percent error multiplies the uncertainty 93 00:04:33.160 --> 00:04:34.240 completely compounds, So. 94 00:04:34.279 --> 00:04:34.920 It's no balls. 95 00:04:35.040 --> 00:04:37.879 Yeah. If you are calculating the volume of a massive 96 00:04:37.879 --> 00:04:41.079 concrete pillar to determine its dead load for a building foundation, 97 00:04:41.720 --> 00:04:45.600 that compounding error dramatically shifts your entire structural analysis. 98 00:04:45.639 --> 00:04:47.160 Okay, but let me push back on this a little. 99 00:04:47.199 --> 00:04:49.639 If round off errors are caused by a lack of memory, 100 00:04:49.800 --> 00:04:53.000 why is this still an issue today? What do you mean, like, 101 00:04:53.480 --> 00:04:56.480 we have insanely powerful processors, now, why can't we just 102 00:04:56.519 --> 00:05:00.319 tell the software to use infinite decimal places and just 103 00:05:00.600 --> 00:05:02.000 root force our way out of the error? 104 00:05:02.160 --> 00:05:04.759 I get that question a lot from young professionals. It's 105 00:05:04.800 --> 00:05:08.920 because you run into a hard physical ceiling. In computer architecture, 106 00:05:09.199 --> 00:05:12.720 it's a concept known as machine epsilon. Machine epsilon, Yeah, 107 00:05:13.040 --> 00:05:15.480 it doesn't matter if your computer costs five hundred dollars 108 00:05:15.759 --> 00:05:20.000 or fifty million dollars. Software uses floating point arithmetic, which 109 00:05:20.040 --> 00:05:23.040 allocates a fixed number of bits to represent any number. 110 00:05:23.639 --> 00:05:27.639 Machine epsilon is the absolute smallest difference between two numbers 111 00:05:27.920 --> 00:05:30.199 that the processor can physically recognize. 112 00:05:30.319 --> 00:05:32.680 Wait, so if a calculation results in a different smaller 113 00:05:32.720 --> 00:05:33.199 than that. 114 00:05:33.480 --> 00:05:36.439 The computer literally cannot see it. It just registers as zero. 115 00:05:36.519 --> 00:05:39.160 You can't outcompute it, you know, you can only design 116 00:05:39.199 --> 00:05:40.680 algorithms that manage it safely. 117 00:05:40.920 --> 00:05:43.279 Wow. Okay, so we just have to accept that our 118 00:05:43.319 --> 00:05:47.199 digital tools have physical boundaries. The baseline math will always 119 00:05:47.199 --> 00:05:50.160 have tiny errors. So given that, how do we actually 120 00:05:50.199 --> 00:05:52.839 start solving these complex engineering equations. 121 00:05:52.879 --> 00:05:56.639 Well, we basically abandon traditional algebra and move to iterative. 122 00:05:56.279 --> 00:05:58.480 Rootfinding finding the zeros right. 123 00:05:58.839 --> 00:06:00.800 The goal is to find the roots of an equation 124 00:06:01.319 --> 00:06:04.600 the exact point where the function equals zero. If you 125 00:06:04.639 --> 00:06:08.439 have a basic polynomial sure, direct analytical methods work fine, 126 00:06:09.199 --> 00:06:13.600 but real engineering problems often generate transcendental equations. 127 00:06:13.720 --> 00:06:18.199 Meaning an equation where your variable is trapped inside a 128 00:06:18.240 --> 00:06:21.759 function you can't easily undo, like e to the power 129 00:06:21.920 --> 00:06:24.399 of x minus cosign of x equals zero. 130 00:06:24.680 --> 00:06:27.519 Exactly, you can't just use standard algebra to isolate x 131 00:06:27.560 --> 00:06:30.199 on one side of the equal sign. The analytical toolkit 132 00:06:30.240 --> 00:06:30.959 is useless there. 133 00:06:31.040 --> 00:06:31.759 So what do we do? 134 00:06:32.240 --> 00:06:36.079 We use iterative methods we make educated guesses. The simplest 135 00:06:36.120 --> 00:06:38.839 one is the bisection method. You define a range that 136 00:06:38.879 --> 00:06:41.439 you know contains the answer, and you just keep chopping 137 00:06:41.480 --> 00:06:43.480 that interval in half until you track the route. 138 00:06:43.720 --> 00:06:45.959 It kind of reminds me of tuning a guitar string 139 00:06:46.000 --> 00:06:49.920 by ear. You don't instantly know the exact mathematical tension 140 00:06:49.959 --> 00:06:51.920 needed for a perfect A note. 141 00:06:51.800 --> 00:06:52.759 Right, You have to feel it out. 142 00:06:52.920 --> 00:06:55.279 Yeah, you pluck the string here that it's too flat, 143 00:06:55.319 --> 00:06:57.279 tighten it too much, and it goes sharp, and you 144 00:06:57.360 --> 00:07:00.160 iteratively have the distance between flat and sharp and till 145 00:07:00.160 --> 00:07:02.040 you zero in on the perfect pitch. 146 00:07:02.360 --> 00:07:07.480 That analogy works perfectly for bisection, But bisection is slow. 147 00:07:08.680 --> 00:07:11.839 If you want to find that route incredibly fast, you 148 00:07:11.959 --> 00:07:13.480 use the Newton Raftson method. 149 00:07:13.759 --> 00:07:14.720 How does that one work? 150 00:07:15.040 --> 00:07:18.199 Instead of just guessing blindly, Newton rafts and uses calculus. 151 00:07:18.839 --> 00:07:21.720 You take an initial guess on the curve, calculate the 152 00:07:21.720 --> 00:07:23.680 derivative which gives you the slope of the curve at 153 00:07:23.680 --> 00:07:27.160 that exact point, and you draw a straight tangent line 154 00:07:27.199 --> 00:07:29.360 from your guess down to the X axis. 155 00:07:29.399 --> 00:07:30.480 Okay, I'm picturing it. 156 00:07:30.759 --> 00:07:33.439 Where that line hits the axis becomes your next guess. 157 00:07:33.879 --> 00:07:36.680 It basically rides the slope of the math directly toward 158 00:07:36.720 --> 00:07:37.720 the true answer. 159 00:07:37.439 --> 00:07:40.639 Which sounds amazing, But there is a massive warning attached 160 00:07:40.639 --> 00:07:43.439 to this in the book, Right, the difference between convergence 161 00:07:43.439 --> 00:07:44.079 and divergence. 162 00:07:44.120 --> 00:07:46.879 Yeah, convergence is the goal. It's how fast your guess 163 00:07:46.879 --> 00:07:49.680 is zero in on the truth. Newton rafts In has 164 00:07:49.759 --> 00:07:53.800 quadratic convergence, meaning the number of correct decimal places you 165 00:07:53.839 --> 00:07:56.199 get roughly doubles with every single guess. 166 00:07:56.240 --> 00:07:57.639 That is blindingly fast. 167 00:07:57.720 --> 00:08:00.839 It really is. But if you make a bad initial guess, 168 00:08:00.959 --> 00:08:02.360 or if you happen to guess at a point where 169 00:08:02.399 --> 00:08:05.120 the curve flattens out, meaning the slope is near zero. 170 00:08:04.920 --> 00:08:08.519 Then your tangent line is totally horizontal. It just shoots 171 00:08:08.519 --> 00:08:11.920 you way out into mathematical space, miles away from the 172 00:08:11.959 --> 00:08:12.480 real root. 173 00:08:12.680 --> 00:08:17.279 Yes, your guesses get progressively worse, spiraling away from the solution. 174 00:08:18.160 --> 00:08:21.839 That is divergence. The algorithm fails, and if you aren't 175 00:08:21.879 --> 00:08:25.920 monitoring it, your software might output a catastrophic structural value 176 00:08:25.920 --> 00:08:27.360 that looks totally legitimate. 177 00:08:27.480 --> 00:08:31.560 That's terrifying. Okay, so we can iteratively solve a single 178 00:08:31.639 --> 00:08:34.639 nightmare equation, But you know, real civil engineering is never 179 00:08:34.679 --> 00:08:37.600 just one equation never. If you're designing a trust bridge 180 00:08:37.799 --> 00:08:41.360 or modeling a city's water distribution network, you have thousands 181 00:08:41.360 --> 00:08:45.600 of interconnected nodes. The pressure at joint A directly impacts 182 00:08:45.600 --> 00:08:48.799 the flow at jon't b, which impacts joint C, which. 183 00:08:48.600 --> 00:08:50.919 Means we're no longer looking for a single route. We 184 00:08:50.960 --> 00:08:55.159 are solving massive systems of linear equations simultaneously, and the 185 00:08:55.200 --> 00:08:59.200 computational methods for this divide into two distinct cans, direct 186 00:08:59.200 --> 00:09:00.759 methods and methods. 187 00:09:00.799 --> 00:09:03.039 Okay, Direct methods are things like Gass elimination. 188 00:09:03.440 --> 00:09:07.399 Right direct methods aim to find the exact solution in 189 00:09:07.480 --> 00:09:12.919 a predetermined number of steps. Gass elimination systematically manipulates the equations, 190 00:09:13.240 --> 00:09:16.559 eliminating variables row by row until you are left with 191 00:09:16.679 --> 00:09:18.159 one simple equation at the. 192 00:09:18.120 --> 00:09:19.879 Bottom, and then you just solve that one. 193 00:09:20.000 --> 00:09:22.879 Yeah, you solve that, and then back substitute that answer 194 00:09:22.960 --> 00:09:24.279 up the chain to solve the rest. 195 00:09:24.519 --> 00:09:27.039 A practical example of this from the text is balancing 196 00:09:27.039 --> 00:09:30.320 a complex chemical reaction, Like if you're analyzing the combustion 197 00:09:30.440 --> 00:09:33.080 of ethane. You have to figure out exactly how many 198 00:09:33.120 --> 00:09:36.840 molecules of each you need so the atoms balance perfectly right, You. 199 00:09:36.799 --> 00:09:38.799 Set that up as a linear system, feed it into 200 00:09:38.840 --> 00:09:42.039 a matrix, and use a direct method to eliminate variables. 201 00:09:41.759 --> 00:09:46.399 Until it's balanced and beyond solving for static values. Analyzing 202 00:09:46.440 --> 00:09:50.559 these matrices lets us extract eigenvalues and eigenvectors. 203 00:09:50.639 --> 00:09:55.000 Oh. Absolutely. If you are a structural engineer, listening eigenvalues 204 00:09:55.039 --> 00:09:58.080 are your best friend. They are the mathematical backphone of 205 00:09:58.120 --> 00:10:02.879 analyzing oscillatory system vibrations. Yes, when you want to know 206 00:10:02.960 --> 00:10:06.679 the natural resonant frequencies of a skyscraper vibrating an earthquake, 207 00:10:07.279 --> 00:10:10.759 or how a suspension bridge will sway under rhythmic windloads, 208 00:10:11.399 --> 00:10:14.000 you use methods like the power method to extract the 209 00:10:14.080 --> 00:10:16.559 dominant eigenvalue from your system's matrix. 210 00:10:16.759 --> 00:10:20.039 It tells you exactly where the structure's critical vibrational limits 211 00:10:20.080 --> 00:10:24.240 are precisely. Here's what I don't get, though. If gass 212 00:10:24.279 --> 00:10:27.399 elimination gives you the exact answer and a guaranteed number 213 00:10:27.440 --> 00:10:30.679 of steps, why would I ever use an iterative method 214 00:10:30.799 --> 00:10:33.759 like goss title If I'm coding a fluid network with 215 00:10:33.799 --> 00:10:36.679 ten thousand nodes, shouldn't I just use the direct method 216 00:10:36.679 --> 00:10:37.360 and get it over with. 217 00:10:37.600 --> 00:10:40.000 You'd think so, but you have to remember the machine 218 00:10:40.039 --> 00:10:42.799 epsilon and round off errors we just talked about. If 219 00:10:42.799 --> 00:10:45.200 you have a matrix of ten thousand equations, a direct 220 00:10:45.200 --> 00:10:48.799 method like gass elimination requires a staggering number of arithmetic 221 00:10:48.840 --> 00:10:53.159 operations multiplications, additions, subtractions, millions of them. 222 00:10:53.240 --> 00:10:54.480 Oh, I see where this is going. 223 00:10:54.759 --> 00:10:58.759 Every single operation introduces a tiny round off err. By 224 00:10:58.799 --> 00:11:01.840 the time you reach the final back substitution, those tiny 225 00:11:01.919 --> 00:11:06.200 errors has snowballed into a massive structural inaccuracy. Ah. 226 00:11:06.240 --> 00:11:09.440 So the direct method literally crushes itself under the weight 227 00:11:09.440 --> 00:11:11.960 of its own flawed arithmetic exactly. 228 00:11:12.440 --> 00:11:16.480 Plus, consider the physical reality of a fluid network. Pipe 229 00:11:16.519 --> 00:11:19.320 A is only connected to pipes B, C, and D. 230 00:11:19.639 --> 00:11:21.799 It has zero direct interaction with pipe Z on the 231 00:11:21.799 --> 00:11:23.879 other side of the city. When you put that into 232 00:11:23.919 --> 00:11:26.600 a matrix, almost all the entries are zeros. We call 233 00:11:26.639 --> 00:11:27.720 this a sparse. 234 00:11:27.360 --> 00:11:31.559 Matrix, and doing millions of complex mathematical operations on a 235 00:11:31.559 --> 00:11:34.360 bunch of zeros is a massive waste of computer memory. 236 00:11:34.440 --> 00:11:38.159 It's hugely inefficient. That's why we use iterative methods like 237 00:11:38.159 --> 00:11:41.159 Gauss Seitl for large networks. Instead of trying to solve 238 00:11:41.159 --> 00:11:43.679 the whole thing perfectly, you just guess the starting pressure 239 00:11:43.679 --> 00:11:47.000 for every pipe. Then you use the equations to sequentially 240 00:11:47.039 --> 00:11:49.919 update each guess based on its immediate neighbors. 241 00:11:49.960 --> 00:11:51.240 So it's constantly refining. 242 00:11:51.399 --> 00:11:53.919 Yeah, it requires a fraction of the memory. And because 243 00:11:53.919 --> 00:11:57.320 you are constantly updating the guesses, the round off errors 244 00:11:57.360 --> 00:12:00.840 don't accumulate in a destructive snowball. You just stop the 245 00:12:00.879 --> 00:12:03.279 loop when the numbers stop changing significantly. 246 00:12:03.519 --> 00:12:06.480 You just let the algorithm gently settle into the right answer. 247 00:12:07.399 --> 00:12:11.519 I love that. Okay, so we've covered solving equations, but 248 00:12:11.600 --> 00:12:14.799 what if you are in the lab conducting tensile tests 249 00:12:14.799 --> 00:12:18.519 on concrete cylinders and you don't actually have an equation? Uh, Like, 250 00:12:18.600 --> 00:12:21.159 all you have is raw discrete data points on a clipboard. 251 00:12:21.799 --> 00:12:25.919 This brings us to interpolation and curve fitting. We shift 252 00:12:25.960 --> 00:12:29.720 from solving functions to creating functions from raw field data. 253 00:12:29.960 --> 00:12:32.399 Interpolation is when you want to build a mathematical curve 254 00:12:32.440 --> 00:12:35.759 that passes perfectly through every single data point you collect, 255 00:12:35.799 --> 00:12:38.919 it like using the Newton divided difference method. 256 00:12:39.320 --> 00:12:42.320 A highly relevant engineering scenario for this is testing the 257 00:12:42.360 --> 00:12:45.759 tensile strength of stainless steel cables. Let's say you test 258 00:12:45.840 --> 00:12:47.679 a one point two inch cable a one point six 259 00:12:47.759 --> 00:12:49.679 inch cable and a two point four inch cable. In 260 00:12:49.720 --> 00:12:51.879 the lab, you record exactly when they snap. 261 00:12:51.919 --> 00:12:54.159 But your bridge design calls for a two point zero 262 00:12:54.200 --> 00:12:56.679 inch cable, which you didn't test exactly. 263 00:12:57.120 --> 00:12:59.600 So you take your test data, use interpolation to build 264 00:12:59.600 --> 00:13:02.159 a customer polynomial equation, and then you just plug two 265 00:13:02.200 --> 00:13:04.799 point zero into that new formula to predict its strength. 266 00:13:05.159 --> 00:13:07.799 You do, but the book mentions you have to be 267 00:13:07.919 --> 00:13:11.240 very careful with this. Interpolation forces the math to hit 268 00:13:11.320 --> 00:13:12.600 every data point. 269 00:13:12.360 --> 00:13:15.200 Exactly yes, which can be a problem if. 270 00:13:15.039 --> 00:13:19.480 Your lab sensors were slightly miscalibrated, or there's natural noise 271 00:13:19.480 --> 00:13:22.799 in your data. Forcing a high degree polynomial through those 272 00:13:22.879 --> 00:13:26.200 noisy points causes the curve to whip wildly up and 273 00:13:26.279 --> 00:13:30.919 down between the points. It's a phenomenon called polynomial wiggle. 274 00:13:31.480 --> 00:13:32.480 It's a great term. 275 00:13:32.720 --> 00:13:35.480 I picture that like trying to connect a bunch of 276 00:13:35.480 --> 00:13:38.120 dots on a wooden peg board using a really stiff, 277 00:13:38.240 --> 00:13:41.080 rigid wire. You have to kink and bend the wire 278 00:13:41.200 --> 00:13:43.679 violently to make sure it touches every single peg. 279 00:13:44.000 --> 00:13:46.679 That's a really vivid way to think about it, right. 280 00:13:46.440 --> 00:13:48.080 But if some of those pegs are in the wrong 281 00:13:48.080 --> 00:13:50.159 spot because of sensor error. You don't want your wire 282 00:13:50.240 --> 00:13:53.480 jerking around. It doesn't represent reality. That's where curve fitting 283 00:13:53.600 --> 00:13:56.519 or the least squares method comes in, and how would. 284 00:13:56.279 --> 00:13:58.919 You adapt your peg board analogy for. 285 00:13:58.919 --> 00:14:02.159 Least squares instead of a stiff wire. Curve fitting is 286 00:14:02.200 --> 00:14:05.120 like stretching a thick, straight rubber band through that cloud 287 00:14:05.159 --> 00:14:08.200 of pegs. The rubber band doesn't force itself to touch 288 00:14:08.279 --> 00:14:11.759 every single peg. It naturally snaps to the center of mass. 289 00:14:12.120 --> 00:14:15.080 It finds the true trend line without getting violently thrown 290 00:14:15.120 --> 00:14:16.879 off course by a few outliers. 291 00:14:17.159 --> 00:14:20.200 That perfectly captures the mechanics of it. You use lese 292 00:14:20.279 --> 00:14:24.039 squares when you want the overall structural trend of noisy data. 293 00:14:24.679 --> 00:14:28.240 We also use specialized curves like splines and baser curves, 294 00:14:28.519 --> 00:14:30.159 particularly in computer aided design. 295 00:14:30.240 --> 00:14:32.519 Oh like an AutoCAD or solid. 296 00:14:32.159 --> 00:14:36.039 Works exactly if you're designing the aerodynamic sweep of a 297 00:14:36.120 --> 00:14:40.399 car's body, A baser curve uses your data points differently. 298 00:14:40.960 --> 00:14:42.919 The curve doesn't pass through the points at all. The 299 00:14:42.960 --> 00:14:46.240 points act like magnetic gravity wells, pulling the curve into 300 00:14:46.240 --> 00:14:50.879 a beautifully smooth shape. It prioritizes aesthetic and aerodynamic fluidity 301 00:14:51.039 --> 00:14:52.480 over rigid data compliance. 302 00:14:52.679 --> 00:14:54.759 That is so cool. So we take our raw data, 303 00:14:55.000 --> 00:14:57.120 we filter the noise, and we fit a clean curve 304 00:14:57.159 --> 00:15:00.279 to it. The next logical step in engineering analysis is 305 00:15:00.320 --> 00:15:02.840 to figure out how that data is changing or how 306 00:15:02.840 --> 00:15:04.720 it accumulates over an area. 307 00:15:04.600 --> 00:15:08.320 Right, which brings us to numerical differentiation and integration. 308 00:15:08.799 --> 00:15:12.919 In classical calculus, you need a continuous, perfectly defined equation 309 00:15:13.440 --> 00:15:17.480 to find a derivative or an integral, but numerical methods 310 00:15:17.519 --> 00:15:21.600 allow us to approximate calculus directly from our discrete data points. 311 00:15:21.840 --> 00:15:25.159 This is massive for mechanical and civil engineers. Let's say 312 00:15:25.159 --> 00:15:28.159 you're looking at heat flow through a concrete slab. The 313 00:15:28.200 --> 00:15:31.559 physics say heat flow relies on the temperature gradient the 314 00:15:31.639 --> 00:15:32.799 rate of change. 315 00:15:32.600 --> 00:15:34.120 The DT over dx. 316 00:15:34.200 --> 00:15:36.879 Right. If you only have temperature readings from five sensors 317 00:15:36.919 --> 00:15:39.320 based out across the slab, you don't have an equation 318 00:15:39.399 --> 00:15:43.440 to differentiate, but you can use finite difference operators to 319 00:15:43.480 --> 00:15:47.679 calculate that exact gradient just by comparing the discrete sensor values. 320 00:15:47.799 --> 00:15:51.000 Wow. And the inverse is true for numerical integration, which 321 00:15:51.039 --> 00:15:53.679 is also known as quadrature. When you need to calculate 322 00:15:53.720 --> 00:15:55.840 an area but the boundary is too complex for an 323 00:15:55.840 --> 00:15:59.840 analytical integral We use algorithms, and the foundational approach here 324 00:15:59.879 --> 00:16:00.960 is the trapezoidal rule. 325 00:16:01.240 --> 00:16:03.919 The classic example is calculating the surface area of an 326 00:16:03.919 --> 00:16:05.240 irregularly shaped lake. 327 00:16:05.399 --> 00:16:09.000 Because you can't use a basic geometry formula for a natural. 328 00:16:08.759 --> 00:16:11.320 Lake exactly, so you measure the width of the lake 329 00:16:11.360 --> 00:16:15.559 at equal ten foot intervals. The trapezoidal rule essentially draws 330 00:16:15.600 --> 00:16:19.320 straight lines between those width measurements, slicing the lake into 331 00:16:19.320 --> 00:16:23.000 a series of chunky trapezoids, and just adds up their areas. 332 00:16:23.240 --> 00:16:25.639 It is incredibly practical, but I've got to challenge this 333 00:16:25.679 --> 00:16:30.360 a little. The entire magic of integral calculus is infinite limits. 334 00:16:30.919 --> 00:16:34.960 The slices are supposed to become infinitely thin, creating perfect 335 00:16:35.000 --> 00:16:39.240 smooth precision. Doesn't chopping a lake or a structural stress 336 00:16:39.279 --> 00:16:42.799 curve into chunky trapezoids feel like we are just giving 337 00:16:42.879 --> 00:16:44.759 up and stepping backward mathematically. 338 00:16:44.879 --> 00:16:47.519 I mean, it might feel blunt intuitively, but we don't 339 00:16:47.519 --> 00:16:50.720 stop at basic trapezoids. This is where the sheer brilliance 340 00:16:50.720 --> 00:16:54.600 of numerical mathematics really shines. If you need extreme precision 341 00:16:54.679 --> 00:16:58.960 without infinitely thin slices, you use advanced methods like gass quadrature. 342 00:16:59.159 --> 00:17:01.399 How does that bypass that's the whole chunky problem. 343 00:17:01.080 --> 00:17:04.279 By working smarter, not harder. The trapezoidal rule forces you 344 00:17:04.319 --> 00:17:07.960 to take measurements at equally spaced intervals. Goss quadrature abandons 345 00:17:08.000 --> 00:17:13.279 equal spacing. Yeah, it uses complex algorithms to intelligently optimize 346 00:17:13.279 --> 00:17:17.119 exactly where you take your samples and assign specific mathematical 347 00:17:17.119 --> 00:17:21.200 weights to those points. By choosing the absolute optimal points 348 00:17:21.240 --> 00:17:25.559 to sample, goass quidature can achieve accuracy that heavily rivals 349 00:17:25.599 --> 00:17:29.759 perfect analytical calculus using a shockingly small number of data points. 350 00:17:29.920 --> 00:17:33.119 That's probably we aren't abandoning calculus. We are optimizing it 351 00:17:33.160 --> 00:17:35.279 for the reality of discrete data. 352 00:17:35.119 --> 00:17:37.480 Which perfectly sets the stage for the final piece of 353 00:17:37.480 --> 00:17:41.039 the puzzle. If differentiation tells us how things are changing 354 00:17:41.079 --> 00:17:44.559 in a single moment, differential equations are the tool engineers 355 00:17:44.680 --> 00:17:48.240 used to predict how dynamics systems will evolve over time 356 00:17:48.599 --> 00:17:49.119 and space. 357 00:17:49.400 --> 00:17:52.480 This is the ultimate synthesis of every method we've discussed, 358 00:17:52.559 --> 00:17:55.880 there really is. Differential equations are basically the operating system 359 00:17:55.920 --> 00:17:59.759 of the physical universe. We categorize them into two main types. First, 360 00:18:00.079 --> 00:18:04.480 ordinary differential equations or odes. These track changes dependent on 361 00:18:04.519 --> 00:18:07.960 a single variable, usually time. Think of the swinging of 362 00:18:07.960 --> 00:18:11.599 a pendulum or how a radioactive isotope decays over a week. 363 00:18:11.920 --> 00:18:15.480 And the heavyweight champion for solving odes seems to be 364 00:18:15.519 --> 00:18:19.079 the run Kuda fourth order method. It's universally used. 365 00:18:19.200 --> 00:18:22.319 It is mostly because of its mechanism. Instead of just 366 00:18:22.400 --> 00:18:24.240 looking at the slope at the beginning of a time 367 00:18:24.279 --> 00:18:26.720 step and guessing where the pendulum will be a second later, 368 00:18:27.119 --> 00:18:30.200 runge Coudis samples the slope at the beginning, takes two 369 00:18:30.359 --> 00:18:32.759 sample slopes in the middle of the time step and one. 370 00:18:32.640 --> 00:18:34.799 At the end, so it's taking multiple readings, right. 371 00:18:35.359 --> 00:18:38.319 It averages those slopes together, giving more weight to the 372 00:18:38.319 --> 00:18:42.240 middle ones to project a highly accurate curve into the future. 373 00:18:42.960 --> 00:18:46.359 It balances computational efficiency with tremendous accuracy. 374 00:18:46.920 --> 00:18:50.039 But civil engineering structures don't just change over time. They 375 00:18:50.119 --> 00:18:53.279 change across three dimensional space, and that brings us to 376 00:18:53.400 --> 00:18:55.839 partial differential equations or PDEs. 377 00:18:56.119 --> 00:18:59.039 These are the beasts that model quantities depending on multiple 378 00:18:59.119 --> 00:19:00.200 variables exactly. 379 00:19:00.440 --> 00:19:04.799 The text categorizes second order linear PDEs into three essential types. 380 00:19:05.160 --> 00:19:10.119 Elliptic equations model steady state systems, like groundwater slowly seeping 381 00:19:10.119 --> 00:19:11.000 through an earth and dam. 382 00:19:11.240 --> 00:19:14.960 Parabolic equations handle transient states like how a blast of 383 00:19:15.000 --> 00:19:18.359 heat conducts and diffuses through a speel ibeam over time. 384 00:19:18.440 --> 00:19:22.920 And hyperbolic equations model wave propagation, which is critical if 385 00:19:22.920 --> 00:19:26.839 you are analyzing how seismic vibrations travel through a building frame. 386 00:19:26.960 --> 00:19:29.160 But you can't just plug these into a basic solver. 387 00:19:29.799 --> 00:19:33.519 To actually compute a PDE, you use finite difference methods. 388 00:19:33.880 --> 00:19:37.720 You essentially lay a massive, invisible three D grid over 389 00:19:37.759 --> 00:19:39.559 the physical object you are analyzing. 390 00:19:39.599 --> 00:19:40.359 Okay, I'm tracking. 391 00:19:40.640 --> 00:19:43.839 Yes. You take the continuous calculus of the PDE, and 392 00:19:43.880 --> 00:19:47.480 at every single intersecting node on that grid, you replace 393 00:19:47.559 --> 00:19:51.279 the calculus with discrete algebraic equations. You convert the physical 394 00:19:51.319 --> 00:19:53.640 object into a massive web of algebra. 395 00:19:54.000 --> 00:19:55.079 That sounds intense. 396 00:19:55.519 --> 00:19:58.359 It is. And the critical factor here is stability. If 397 00:19:58.359 --> 00:20:01.000 you use certain methods to update the grid, the round 398 00:20:01.000 --> 00:20:03.279 off errors we talked about in the very beginning can 399 00:20:03.319 --> 00:20:06.680 amplify with every timestep until the mass blows up into infinity. 400 00:20:06.799 --> 00:20:07.359 Oh wow. 401 00:20:07.519 --> 00:20:10.599