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Modeling The Physical World

By Oscar Gordon

Also, for the handful of you reading this who are more intimately familiar with the nuts and bolts of what I’m laying out here, keep in mind this is primarily for the uninitiated. Yes, I know my explanation for X is simplistic and/or incomplete, but do we really have time for a week long discussion as to why?

The goal of this post is to explore the complexities involved in building models. Since modeling the physical world is my bailiwick, I’ll be using that as my basis, but the complexities I hope to expose exist in any modeling paradigm, be in physical, economic, social, or anywhere in-between.

First, what is modeling? Modeling is, at its root, any attempt to build a representative system of something that you can then control and experiment with. Model train hobbyists build models of trains and explore the complexities of railway infrastructure (and claim to have fun at the same time; to each his own, I guess). Airplane designers build model airplanes and fly them or stick them in wind tunnels to explore their designs. Car makers, shipyards, civil engineers, etc. all build physical models to help explore systems. But physical models such as these have drawbacks, such as they are expensive to build, and somewhat limited in the information they can provide due to the limits of technology.

Luckily, we clever monkeys developed math and the scientific method, and we’ve spent centuries paying attention to how the world works. This means we can build mathematical models that (usually) do a respectable job simulating complex systems. We really have two ways to mathematically model the physical world; statistically and numerically. Statistical models are built upon observed data and are most useful when the system can not (yet) be represented with concise equations or systems of equations (usually due to an incomplete understanding of the system due to an inability to observe the system in detail). Statistical Mechanics is one example. Numerical methods are more analytical, in that there are specific equations that can be solved, or approximated, to reach a solution. Computational Fluid Dynamics (CFD) is an example of a numerical approach.

We’ve been using both for a long time, with varying degrees of success and predictive capability. However, when Isaac Newton put together Calculus in order to kick Robert Hooke in the metaphorical and intellectual ‘nads, he opened up a whole new capability to more accurately model our world. This is not to say we weren’t already doing it, since the likes of Boyle, Maxwell, Weber, Halley, Hooke, Newton, and their peers who came before had all developed equations to model specific physical situations, but what we could do was very limited. For instance, Hooke’s Law is fundamental to all students of mechanics, and is expressed simply as:

Force

Hooke’s Law is a first order linear approximation of the behavior of springs (or any spring like material), but prior to Newton’s Philosophiæ Naturalis Principia Mathematica, and the works of minds like Leibniz, no one had the mathematical tools to really explore further. Hooke’s law provided solid predictive ability, right up to the point that the material left the linear elastic displacement range. Once it became non-linear, Hooke’s law stopped being able to predict behavior. Thanks to Newton, et. al., we were better able to develop nonlinear models and now we can model the material through all behavior domains.

Another example is Bernoulli’s Equation, shown here:

bernoi

This is also a linear approximation of what would later become the Navier-Stokes equations, shown here in the general form*:

navierStokes

(*I’ll discuss this later)

The Navier-Stokes equations model the movement of a particle of fluid through a fluid system, and like many other equations that model physical systems, are very complex. Currently they exist without a closed form solution (a way to solve the equations analytically), and the development of such a solution is one way to win the Millennium Prize (recently a solution was presented, but it has yet to be translated, vetted, and shown to work for all conditions). The only way to approximate a solution is with numerical modeling and a lot of computing power. Thankfully, computing power gets cheaper and more abundant every day.

So, armed with mathematical models of physical systems, we can build the complex models representing these systems inside a computer, which is currently orders of magnitude cheaper, faster, more flexible, and more informative that the old physical models (we still build the physical models, but only as a final check). Simple enough, right? Go to school, understand the math, build the models, compute, solve, cover yourself in money, fame, and glory! WooHoo!

Yeah, you already know the answer to that. Complex systems are complex. Modeling them involves not just understanding how to work with statistics, calculus, linear algebra, and differential equations, it also requires the modeler to have a detailed, almost intimate understanding of everything happening in, and to, the system. Being good at this requires not just math skills, but expert programming skills in a variety of languages, a solid knowledge of the physics and chemistry involved, plus the mind of an investigator and troubleshooter. In short, it takes a team with a lot of skills overlap and cross-training to develop complex systems models correctly.

So lets take a look at how a model is developed. For example, let’s model Tod’s golf swing (Tod does play golf, right? If not, this will be his hypothetical golf swing). In the interest of world peace and to combat global… something, it is imperative that we be able to understand and model Tod’s work with a driver. To that end, we’ve invited Tod to an ultra modern driving range (one of those ones that is inside a giant, inflatable building), where we can acquire relevant data.

Tod is fitted with a motion capture suit, given a motion capture club, and buckets of brightly colored golf balls. The driving range is surrounded by an array of cameras that will track the flight, impact location, and final resting place of each colored ball. Tod has a number of high speed cameras watching him (this is for combating global peace… or something, so spare no expense! – besides, digital high speed cameras are relatively cheap these days). In short, each swing will be recorded in detail. Tod will then spend the day swinging that club as many times as he can stand.

All of this data will serve two purposes. The first will be to aid in the development of the complete model. The second will be to validate the model.

Building the Model:

Since the model we want to build is a motion problem of an object on a (generally) ballistic trajectory, we will start with the ballistics equations. The ballistics equations are all derived from Newton’s First Law, which is expressed with the force equation:

Force2

This site at NASA does a nice job explaining the derivation of the ballistics equations from Newton’s First Law. This wiki page has the generalized equations (the NASA site is for vertical launch only). The equation we are going to use first is the distance traveled equation:

ballistic

This will tell us how far the ball travels from the tee, assuming we were hitting the ball in a vacuum. We do have to take air resistance into account, and the wiki page shows a derivation that would be fine if we just needed a rough estimate of the effects of drag on the ball. However, golf balls, thanks to their dimples, don’t follow that derivation very well. Still, for the moment, the distance traveled equation is a good place to start. We’ll also need the Height at X and Velocity at X equations later on, but for now let’s move on.

As you can see when you look at the distance traveled equation, you have three unknowns, initial velocity, angle of launch, and initial height. Initial height is easy; if the driving range is flat, it’s the height of the tee; if there is a hill or valley, we can estimate about where on the hill or in the valley it will hit and adjust the initial height accordingly. Initial velocity and angle of launch, however, depend on Tod.

But, before we get to Tod’s swing, we have to understand the material mechanics of the ball itself. A golf ball certainly feels solid and firm to the touch, but it’s actually quite deformable, and as such we’ll need an equation that describes how the ball will deform at impact with the club head, and how much energy it will expend as it returns to it’s original shape. This will affect the initial velocity of the ball. If the ball did not deform, would could just use the momentum equation and me done with it.

momentum

So the mass and velocity of the first object will equal the mass and velocity of the second, or the slow moving heavy driver head will make sure the lightweight golf ball is gonna leave at a pretty good clip. Of course, this equation assumes that objects 1 and 2 are perfectly elastic, i.e. they do not deform, or if they do, the deformation energy is perfectly returned. If there is any deformation, part of the transferred kinetic energy will be consumed as the material deforms and returns to it’s original shape (and expended as heat – yes, hitting a golf ball warms it up a smidge). So golf balls have an elasticity value (e) that is less than 1 and (usually) determined experimentally. Factoring e into the momentum equation, and doing a bit of algebra to solve for the final velocity, we get this.

velMomentum

So we treat the mass of the golf ball as a bit less than it is in order to accommodate the energy lost due to deformation.

GolfBallCompression

Now we can get to Tod’s swing. I honestly don’t know if there is an existing equation or model for a golf swing, but for the sake of this discussion, let’s say there isn’t. How do we model the swing? Well, we are taking tons of HD video of Tod swinging his club, so we can use that video to develop a statistical model of the swing. Why a statistical model? Because despite claims to the contrary, Tod is neither a literal, nor a metaphorical machine, so his swing will vary each time. Even pro golfers have measurable variation in their swings, although less so than others. That said, we don’t have to look at the whole swing, just the last bit, where he hits the ball. We need the speed of the club at impact, the angle of impact to the head, and how long the ball stays in contact with the club. If we use the video to get all this information for each swing, we can develop a statistical model of the swing that, in combination with the elastic model of the ball, will allow us to formulate a useful range of potential initial velocities and launch angles.

It won’t tell us exactly how Tod will hit the ball every time, but it will tell us what the chances are that Tod will hit a ball in a certain way during any given swing. Unless, of course, Tod has a couple of pints of Deschutes during lunch, which could add a fun wrinkle to the model.

So, we’ve figured out how the ball is to be hit, and we have a rough idea as to the path the ball will take, so what’s next? Well, landing and roll out. If the ball is somewhat deformable and elastic, the fairway or green is even more so. They are also rough (fairways more than greens, obviously). The elasticity of the ground will affect how much bounce the ball will have when it lands, and the characteristic of the grass will affect how long the ball will roll once it stops bouncing (coefficient of friction is factored into the equation of motion as an opposing force to the motion). This is pretty much the same equations we used up above, but on the other end of the ballistic equations.

We use the statistical model of Tods swing to inform (i.e. give us the unknowns) of the elastic momentum equation, which gives us the initial values for the ballistic equation. The ballistic equation then provides the values we’ll use to fill in the unknowns for the landing bounce and roll out, with each bounce being another ballistic calculation.

Remember how I said that golf balls are dimpled, and that matters? There was a reason I picked golf, and that is because golf balls don’t conform exactly to the ballistic equations, certainly not the way a heavy artillery shell, or even a rifle bullet would. Those dimples affect how a golf ball flies, and while the ballistic equations are a good approximation, an accurate flight path would need to be informed by the local weather.

Thus, once we have the initial velocity, angle, and any spin values of the ball as it leaves the face of Tod’s club, we can either develop another statistical model of the flight path, or we can use a numerical method, such as CFD, to determine the flight path, especially if we have weather data (although that isn’t necessary, since we can create whatever weather we want in the simulation). First, however, let me get back to the CFD model and the equations at the heart of it. Earlier, I showed the general form of the Navier Stokes equations. That single expression doesn’t really show just how complex the system is. Here they are in an expanded form:

nseqs

Navier Stokes is a system of five partial differential equations that computes rates of change for multiple unknowns across four dimensions (three spatial, one temporal). For those who didn’t get that far in math, here is what the notations mean:

deltaRho

So as time advances, the density of the fluid may change as well. If the fluid is water, this term is likely constant, since the density of water, absent other fluids or a change in temperature near a phase transition, will remain constant. Air, on the other hand, can change it’s density quite a bit, so this term is in flux. The various velocity and shear stress (turbulence) terms are all very much in flux almost all the time, and across the whole temporal and spatial domain. As you can imagine, with so many terms changing across such a wide set of dimensions, solving this is not a trivial task. So how do we solve it?

Well, first off, we don’t solve it, per se, we compute an approximate solution. To do that, we employ a technique called discretization. The specifics of how things are discretized can vary depending on the scheme, or the approach to the solution, but the basic idea is to break the problem up into teeny, tiny pieces and solve for it on each piece. If our piece is the 250 meters of the drive, a whole lot of change can happen from start to finish, but if our piece is a space about 1 cm across, that’s a different story. The change there will be very small, and, more importantly, very predictable. Thus:

discret

Where the denominator equals 1 cm, and the numerator will equal some very small change that we can compute based upon our known conditions of how aerodynamic drag affects a golf ball, plus whatever effect the weather might be having. The computed change is then applied to the value of u and the next centimeter of space is computed using the new value of u. This is then done for each term that is changing, for each little piece of space. The actual math, at this point, is not difficult. It’s a bit of a grind, but it’s basic algebra at this point. Which is why computers have been such a boon, because while it would take a person all day to do the math for this problem, a computer can do it in seconds, or even very small fractions of a second.

The capability of computers to do these kinds of repetitive tasks at such incredible speed opened up a whole world of simulations. Whereas previously, we’d only compute the path of the ball and treat the weather as merely inputs along that path, now we can overlay a mesh across the whole of the driving range, and factor in not just wind, but varying air densities and humidities, as well as thermal anomalies (such as the ball passing over a section of asphalt on a hot day and experiencing the thermal updraft that results) and whatever else we can dream up. The solutions are computed at points along the mesh (which points are used is something else that varies based upon the scheme in use). Of course, meshing a whole domain like that has all sorts of other issues, such as resolving the spatial and temporal changes, as well as requiring multiple passes, or iterations, of the problem domain in order to reduce the errors that the various schemes experience. However, the ability to expand the domain opens up whole new avenues for exploration.

No longer are we limited to modeling the flow around the cross section of a wing and extrapolating from that the performance of the whole wing. Now we can model the wing itself, with all the control surfaces and the wing-body interface, and using a Design of Experiments approach, rapidly explore how changes in various design parameters alters the performance, and even automatically optimize the design by allowing the computer to seek out performance targets. It really is an exciting time to be an engineer.

But, back to the golf ball. We have the swing, the impact, the flight path, and the terminal performance. All put together, that is the mathematical model of Tod’s golf game. Now, we test the model against real world data!

If the model is accurate and robust, we should be able to predict where Tod’s golf ball will go within a certain probability envelope. So what happens if our model falls outside of that envelope? What then? This is where the investigator/troubleshooter skills come into play. It could be a mathematical error (usually easy to spot and fix), or an incorrect assumption (less easy to spot and fix since this can have a large helping of human foible attached to it), or an erroneous input (less easy to spot, very easy to fix), or the original data is bad (hoo boy…). If Tod did have a couple of Black Buttes at lunch on the day of data collection, and no one noticed and wrote it down, then there could be an hour or two worth of data that is inaccurate, especially if he is stone cold sober on the day of validation.

Which finally brings us to error bars. Error bars are a modelers way of saying, “we are sure of our results, but only this sure”. They are a simple thing, one that you’ll find in some form or another in every research paper that plots data, something that is critical and incredibly informative with regard to the quality of information being presented, and usually the first thing to vanish from media reports of scientific research results (to my great annoyance). If the researchers didn’t know Tod was buzzed after lunch, their error bars would expand considerably, as opposed to if they did know, and could adjust the data, or just exclude that data from the set (with an attached explanation of why the data was excluded, e.g. Tod was soused). Finally, error bars, even large error bars, do not suggest that the research was bad, or done incorrectly, or that the conclusions are necessarily wrong, it only speaks to the quality of the data (which can often be low through no fault of the researchers) or the accuracy of the applied model. Usually what large error bars mean is that the research will need to be repeated or replicated once a way to get better data is determined.

So, that is the basics of what goes into a mathematical model. As you can see, even simple models can be very complex, and complex models can be a nightmare to develop and untangle. So next time you see a report on the results of a mathematical model (or a computer model), you’ll hopefully have a better understanding of what potentially went into it. And if you see modeling data plotted without error bars or some notation regarding the error of the data, keep your hand on your wallet, because someone is trying to sell you something.

Rules For Modeling

1) The model is wrong (for certain values of wrong). This is usually the hardest thing for new engineers, and most everyone else, to understand and accept. When you put in a lot of effort creating a model, and the computer spits out an answer that looks correct at first glance, it can be difficult to check that enthusiasm for a more pragmatic response.
2) You have to know, in detail, why the model is wrong. If you can’t explain why the model is wrong, you don’t understand the physics involved.  Assumptions abound during the construction of a model, and being able to explain the assumptions made & how those assumptions can cause trouble is part of the process.
3) You have to know precisely how wrong the model is. If you don’t know your error bars, then you don’t understand your model.  It’s not enough to just know the assumptions & their limitations, you also have to have a good idea how badly those assumptions can throw off the answer.

If you want more like this, let me know. I can talk about modeling all day long.

{Feature Image from the CD-Adepco Image Gallery

Golf Ball image from Titleist Golf Ball Education}

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59 thoughts on “Modeling The Physical World

    • I’m general fluids & heat transfer, although I dabble in structural mechanics & I’ve done kinetics & kinematics, but not anytime recently.

      I understand the basics of chemical transport, but I’ve never built or used a model of it.

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      • Cool.

        Just a couple small quibbles. I’d hesitate to say that statistical mechanics relies on a “statistical model” even though they both have the word statistical in them. It’s more about the statistics of having a large number of individual particles that can all carry different (but random) states such as the spin of an electron, etc. Many thermodynamic properties that we’re used to talking about such as temperature, enthalpy, etc. are simply the average of an ensemble of many, many states. This differs quite significantly from a “statistical model” of a system where you observe x over and over and use the average to predict x in the future. These kinds of models lack explanatory power in a way that statistical mechanics (quite elegantly imo) has in spades.

        Also, this is really, really minor, but some of your equations are uuuuugly. I know they’re just shown for shock and awe purposes, but something about italicized sine and cosine makes me shudder.

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        • Perhaps I’m wrong but I was thinking statistical mechanics is along the same vein as turbulence modeling, in that observed behaviors are used to form a model with significant predictive capabilities, but the model itself is an imperfect representation of the phenomena. Which is why there are numerous turbulence models, each being quite useful within certain domains, but not perfect & even less so outside of the preferred domain. Of course, direct numerical simulation of the turbulent structures is possible, just massively expensive.

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  1. Note that CAD models can be wrong, as well.

    Obviously there are machining rounds (the rounded insides of corners cut by an end mill), but a sufficiently-detailed CAD model can imply a level of design maturity that doesn’t exist. It’s a bit of a Catch-22; people always say they want to see a fully-detailed model, but they’re used to only seeing that level of detail in the final fully-approved model, so when you show it to them they respond to Obvious Mistakes as though you’d already sent the part down to the shop and were cutting metal.

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    • When it comes to modeling the interaction of CAD to a physical situation, sometimes the detail is important, sometimes not.

      If I’m modeling an airplane in flight, all the joints and seams are not important to the overall analysis, as I can apply a drag factor to the analysis to account for their assumed presence (the factor which is usually determined by experiment). If I’m doing a structural analysis of the geometry, then the fact that a given corner is rounded instead of a defined angle is potentially critical to the analysis & has to be represented in the CAD.

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  2. Oscar,

    Interesting stuff. I think, anyway. :) I wish I could make a substantive comment about it.

    keep in mind this is primarily for the uninitiated.

    {{Gulp!}}

    One thing I can say is that some talking heads on the TV the other day were discussing whether a robot could play a round of golf better than a human. I found that interesting, actually, since a robot would be less likely to make egregious errors off the tee, which means on balance the Bot would be hitting successive shots from +/- uniformly flat lies and eliminating most of the “problem” shots where human creativity and feel are required.

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    • Is the robot constrained to a fixed configuration? If I were designing a robot to hit the ball consistently well, I would design the lower part to allow for reconfiguration on the spot. In effect, the robot would never have to adjust its swing for having the ball sit higher or lower than its feet, or with one foot higher or lower than the other. Just reconfigure so that the part that strikes the ball is always level and always in the same position relative to the ball.

      How much of the total game is the robot responsible for? Good golfers play the hole backwards in their head — I want to putt from here, and the best chance of getting the ball in that position is to hit a fairway shot from here, and that means I have to hit something other than driver from the tee. Is the software going to be responsible for that overall planning?

      And of course there’s the whole issue of playing against an opponent, not the course. The summer after my junior year in college I was in a position to hustle some doctors and dentists on Wednesday afternoons. On more than one occasion, with confidence in my ability to hit certain shots and the situation one of the other players was in, I hit the ball into a sand trap on purpose. Or missed a putt intentionally. There’s a lot more to the game than the mechanics of hitting the ball.

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      • They didn’t really get into it all that deeply. The conversation started when one of em said that robots have perfect swings everytime and the conversation just sorta took off, briefly, from there. But if I were to design a bot golfer, I’d certainly allow it to adjust it’s “feet” so’s it’s “torso” was level and plumb every swing. I mean, why not?

        You bring up an interesting point about shot selection tho. My guess is that advanced enough programming could allow for a robot to compute the odds-to-par/birdie correlated with the right miss given the contours of a green and bunkers, etc. But really, you’d know more about that than I do. It just seems like something that could certainly be done.

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        • Certainly this falls within the scope of AI planning problems. Those can get “hard” in a hurry, especially if you need an optimal solution and it turns into a search problem. For this, easy approximations are probably sufficient.

          Sometimes even the best players fail to do this. Some years back, a guy came to the tee of the last hole of the British Open leading by three strokes. It was a nasty par four, but all he needed was a five to win (according to rumor at the time, the jeweler had already started engraving his name on the trophy). The right strategy was hit a five-iron into the fairway, hit another five-iron to the front of the green, chip on, two-putt. Hell, I could have played it for five when I was younger. Instead the guy hits driver into trouble, then tries to make a very difficult shot to recover and gets into even more trouble, ends up with a seven and loses the tournament when the last player in the group behind him makes a lucky three.

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    • There is a pretty good mathematical model out there for golf ball flight. It’s much more complicated than simple ballistics, but it is easily programmed into software.

      Of course, for a robot to hit the mark consistently every time, it needs to know more than the target. It also needs current, real time weather conditions. We do that by watching the flags on the course, and trees blowing, etc, so the robot would need a similar sensor suite and the software to process that data, along with game strategy.

      Oh, and thanks, I hope it was at least informative. If I get good feedback, I’ll dive into this all a bit further.

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      • Oh, I love the basic idea here, Oscar. (Emphasis on “basic”…)

        Wind was one thing I thought of too. Usually the caddy provides that type of info to the golfer to limit the amount of calculations the person swinging the club has to do, and I don’t know exactly where the caddy gets that information from. Presumably, by reading the weather reports before the round and looking at flags and tree-tops and such. Seems to me it’d be harder for a bot to make correct adjustments based on wind than on “correct miss” given a pin placement and contours of the fairway, etc. It may be one of those things a human can intuit based on years of experience in varied conditions.

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  3. Reading this made me think primarily “i r dum” but, after that, various questions about how, given the limitations of models, how that sort of thing would best be gamable in practice.

    I mean, I grew up being taught that the ancient Hebrews had a tradition with regards to prophets where a prophet who got something wrong would then be stoned to death as a false prophet. Fair enough. This, of course, resulted in prophets who specialized in saying that stuff would happen years and generations in the future. Okay, that makes sense. So the religious leaders wrote this stuff down. Well, the scrolls that wrote down false prophecies were then burned.

    As such, the prophecies that were really oblique and relied heavily on metaphor were the ones that stuck around.

    I suppose arguments could be made that these prophecies came true, of course.

    Anyway, in that sort of vein, we know that models that are proven incorrect in short order are then scrapped as not being accurate.

    Wouldn’t this reward the models that can’t be falsified for a good long while or the stuff modeling stuff that isn’t easily verified?

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    • Yeah probably. Except nobody FUNDS models for years upon years in the future.
      Climatology (the study of “what will happen this year” rather than what will happen tommorrow) is all about predicting next year (and the next century, but if you can’t predict next year to a decent degree, you throw out the whole model).

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    • Note my three rules at the bottom. One of the reasons I wrote this post is because of climate change. I’ve talked with lots of climate modelers, and while I can often be critical of the assumptions made in their models, I can be critical of them because they are clear about what assumptions were made and how problematic those assumptions potentially are. They are also, outside of the climate darlings, annoyed with how often their own discussions of the limitations are ignored or glossed over in the media. From what I hear, a lot of medical researchers have similar issues with media reporting on studies.

      So for everyone here, there are always assumptions, always room for error. We address that error through additional studies and by playing with the variables & constraints. Modeling gives us a lot more flexibility for those variables & constraints at extremely low cost.

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      • Oscar,
        I’m certainly quite furious when people report that “it was the hottest year on record” and forget to mention “not statistically significant”. That’s Just Wrong.

        “The worst thing about real-time modeling is when you do it too well…”
        “How’s that?”
        “My colleagues say, ‘it’s running wonderfully, but there’s all these heat alarms going off…so many warnings…'”

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    • The limitations you’re speaking of often give modeling a somewhat bad reputation in the scientific community. As such, new models are often treated with a large dose of skepticism (with good reason). Anyone who actually knows what they’re doing will carefully spend time trying to validate their model against several different experiments under different conditions. Even then, scientists or engineers are still human. Sometimes people tend to fall in love with their model and forget that reality is the ultimate arbitrator of right and wrong.

      For example, I reviewed a paper once where they only measured one thing in an experiment (the loss of a chemical from the surrounding gas). However, the associated model had 15 to 16 different parameters they had to fit to get the “right” answer in several different spatial areas within the experiment. They used this complicated model because they had in several other papers that had more measurements to test it against. Despite being somewhat based on an experiment, several parts of their model were also based off assumed physical quantities that you could never verify with an experiment. (In fact, they referenced their own models (!!) as justification for the values they used.) Suffice to say I reviewed it (in brief) as “This is over fit, can a simpler model capture the experiment without so many assumptions?” That one didn’t go over very well. :)

      Anyway, the point of that long-winded anecdote is that you’re right that some modelers do make assumptions that can never be proven wrong as part of their models. Sometimes it’s a necessary evil (see this entire post above), but good modelers know that they’re making assumptions. Thus, they are much more careful about claiming those assumptions ARE assumptions, and not letting their ego get in the way of recognizing when they’re flawed. And knowing the kinds of compromises you have to make to get a model of a physical system to work, I can’t even imagine trying to do something like an economic model, which has way too much chance of subtle bias creeping in.

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  4. This is a bit OT, but I have a sciencey question I am hoping someone can answer.

    Due to a set of circumstances I won’t get in to, we had a room at the old house that was full of baking soda and moisture (via humidifier). Somehow, the combination of the two (along with the friction in my pocket) just ripped the paint off my ecigarette batteries. Most of my other batteries have nicks of paint missing here and there, but that one has almost all the paint off of it. (Didn’t all come off immediately, but fell off very quickly).

    Anyway, I would actually kind of like to be able to reproduce this, but I can’t seem to. So it’s not just the baking soda and moisture, but I can’t think of what else it is.

    Anyway, can anyone give me some “idiot’s guide” insight on the chemical reactions of baking soda, moisture, and paint?

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    • Just a guess, but paints usually contain ionic compounds that serve as pigments (http://www.compoundchem.com/2014/03/21/inorganic-pigment-compounds-the-chemistry-of-paint/). What color was your paint? I can probably guess exactly what reaction occurred if you tell me what color.

      In any case, the pigment probably reacted with the baking soda. Ionic double displacement reactions usually occur in water, since this medium allows for the interactions that allow the reaction to take place, which explains why the moisture accelerated it.

      Actually, “baking soda blasting” is a technique that’s sometimes used to remove paint. Baking soda’s chemical formula is NaHCO3. “Washing soda”, more commonly used for paint removal, is Na2CO3 – so, just one hydrogen different.

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        • Let’s assume the pigment is Co2SnO4 (cerulean blue), since that’s the easiest to work with, the reaction would go something like this:

          NaHCO3 (aq) + Co2SnO4 (s) –> Na2SnO4 (aq) + Co(HCO3)3 (s)

          I’m guessing that this reaction or an analogous one is probably somewhat like what happened, where the ions just switched with each other. (I’m putting a lot of qualifiers in there since I haven’t really looked any of these chemicals up – I’m just extrapolating from your account and some general chemical principles.)

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    • Baking soda in water is actually a great paint remover, especially from metal. When I was a teenager, I helped a friend of mine’s parents renovate their historic home, which included restoring a bunch of painted metal fixtures. We would boil baking soda in water and then soak the metal in it for a few hours, and the paint could then just be brushed off.

      I don’t know why this works, chemically, but a quick tour of Google suggests it’s a common method, particularly for antiques.

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  5. Dude, I’ve been looking forward to this post for the, what, 2 or 3 years you’ve been hinting at it? ;) Can’t read it now, but I’m excited to read it later.

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  6. Well, I tried to read the post anyway. It became a bit too complex for me about halfway through, so all I can say is “D-a-a-a-amn.” Both in response to the author’s obvious intelligence, care for, and love for the subject matter, and in response to my relief that I did not pursue an academic career path of the hard sciences because I’d have washed out.

    I can say that while I don’t know if plays golf, I do, and the mathematical model of what happens when club strikes ball is so arcanely removed from the physical discipline required to cause contact with the ball to happen just right on a reliable basis that I must admit of little ability to understand whether, and if so how, my puzzling through the technique described in the post will help me keep my head down and hold my left arm straight during my swing.

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    • Burt,

      You’ll have noticed that I completely avoided the whole bit about the swing itself.

      If we put you in for Tod, all the HD video we have of your swing would allow us to look at your swing, find the error in it, and help you adjust. After that, it’s practice & presence of mind until you’ve acquired the relevant muscle memory.

      I suppose we could wire the cameras up to a computer that looks for your head held too high and your arm bent, and then gives you a zap every time you fail to meet the criteria. Pain is a wonderful motivator.

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  7. There was a reason I picked golf, and that is because golf balls don’t conform exactly to the ballistic equations, certainly not the way a heavy artillery shell, or even a rifle bullet would. Those dimples affect how a golf ball flies, and while the ballistic equations are a good approximation, an accurate flight path would need to be informed by the local weather.

    One thing to note for this. The shape of the object in flight is important, but so is the object’s other motions.

    So a cube in flight behaves differently from a sphere in flight, which behaves differently from a bullet shape in flight. Set the items spinning and things change radically. A sphere spinning on an axis perpendicular to the path of flight will acquire some lift and push it off the ballistic flight path (Magnus Effect). If it spins on an axis aligned with the flight path, it will resist the effects of a cross wind and tumbling (spin stabilization – which is why gun barrels are rifled, so the exiting round is spin stabilized).

    All of this is to say that when a model is being built, missing a detail like this can throw your model off, usually by quite a lot. Which is where the skills of an investigator/troubleshooter come into play – figure out what was missed or assumed incorrectly & correct for it.

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  8. Just wow.

    Before I bashed my head in, I would probably have been able to learn and understand this stuff; math was my thing. After (and it’s 40 years after,) I’m still constantly relearning how to speak; which is probably why I gravitated to writing.

    My kid does some modeling as he designs quad copters; particularly important for calculating propeller angles and programming the flight controllers. He’s working on a new one now, the last model dropped in a lake last week; battery problems. The radio he used has a battery level monitor, but the user interface has it several pages in, and (I don’t understand this,) it only works in a short range but not over the range that the radio actually works to control the copter.

    It was a painful (and expensive) learning experience; one that will probably lead to him learning how to program the radios himself.

    Good to have a mad rocket scientist around the house.

    Thank you, , I’ll re-read this tomorrow, and work the wiki, see if I can better understand it; tonight, it’s all greek to me.

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  9. Good stuff Oscar, there is some particular trickiness with social science modelling, but I recognise a lot of the general principals you’re describing.

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    • Oh I imagine the variables & assumptions at play with social science modeling are a lot more ambiguous than what I deal with, even if the equations are usually more straightforward.

      And thanks!

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  10. What I do for a living is, in fact, modelling software. (Mind you, I’m not an engineer. I help implement their crazy mad engineering and science into software that does said mad science for another engineer who then decides if they can do X without it breaking off and killing someone).

    One tidbit I’ve learned over the years is there are different TYPES of models for even the most basic problem. Say, for instance “Part of my engine has a crack in it.”. There are models that you can use to help determine how the crack started (complex ones that model all the bits and slowly — oh god so slowly — map out stresses in ridiculous detail and show you exactly what pressures did what). There are ones that ignore HOW it started and instead tell you whether it will grow — or how fast it will grow. Others will tell you how long you can do “something” to a specific bit of metal before the odds of it developing a threatening crack happen.

    So I’ve seen guys model stresses on parachute parts for re-entry capsules (answering critical questions like “If we do this, will this piece of metal snap like a twig?”) and guys basically saying “If there’s a crack on this bit of an airplane, but that’s too tiny for our inspections to see, how long can we fly this part without worrying about that crack growing to the point it can break it?”

    I’ve also seen guys get really, really, really excited about aluminum. I mean really excited. I wonder if that’s how I sound when I find a new tweak for a machine learner.

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    • PS for those interested, is talking about fatigue analysis & crack or fracture propagation. Things structural Aero is always concerned about & is devilishly tricky to model (although we have good ways to predict it on structures we have lots of experience with).

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      • I’m only in it for the fun videos of stuff failing under load. :) Double points if the thing is filled with water so there’s a massive splash.

        One of the NASA branches (don’t remember which) was testing some fire extinguisher for use on Station (water with compressed air as a driver) and as part of the test process, they tested it to burst. (I think it required something like 3 times the mandated maximum load to produce a burst vessel).

        The video of the burst is…impressive. The bottom of the vessel leaves an impression in a concrete wall ten feet away.

        Pressurized vessels are nothing to fool around with!

        But yep. Lately it’s been crack growth, metal fatigue, and other fun stuff like that. (And man, some of the software is expensive! Do you know what an Abaqus or StressCheck license runs? Good god!)

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