Propelling Your Numerical Results Forward

Hi everyone. Welcome to our Academic Innovation live stream at Warframe Research. I'm Yi, the program manager here. Joining me today is John McNally, our solution developer and our guest, Andrew Higgins from McGill University. So I will hand over to John.

Thank you. So Andrew is a professor of Mechanical Engineering at McGill, also the principal investigator at the Internet, excuse me, Interstellar Flight Experimental Research Group, which we'll get to hear a lot of interesting stuff about. So Andrew, thanks very much for joining us today.

I'm glad to be here.

So I also understand that there's a connection to University of Illinois. For those of you who don't know, the Wolfram headquarters building is just off the University of Illinois campus. It's also where I happen to do my undergraduate degree. And so there's a little bit of a connection here that I'm curious. So everyone in the audience remember that if you want to ask questions, be sure to put those into the stream chat. And we'll start off with Andrew talking a little bit about some of his recent work and some of that, that connection to University of Illinois in a just a brief structured presentation and we'll get into a little bit more conversation about these topics. So Andrew, if you'd like to jump into that whenever you're ready.

Sure, yeah, go ahead. I'm going to share my screen here. So, engineering professor, we can't Talk without our PowerPoint slide, so I'll just launch into it here. So yeah, I'll tell you a few words about myself. I actually grew up, born and raised in Illinois. I grew up in the northern part of the state and was a total space and astronomy fanatic as a teenager. And a very influential thing in my life was in the mid-1980s, I read a science fiction novel called Flight of the Dragonfly by Robert L. Forward, who is a physicist, also a science fiction writer who posited the possibility of interstellar interstellar travel using laser driven light sails. And in this book what he proposed is to build a giant laser, hundreds of gigawatts and then you'd need a giant lens tens of thousands of kilometers in diameter and you'd focus that laser onto a giant sail and the photons, when they reflect off the sail and part momentum on it and would push that sail to, to another solar system. And not only would forward write science fiction novels, he'd also write scientific papers that were published in the aerospace literature around the technical background. So this just when I was 15, 16 years old, just totally electrified my mind and led me to go to attend University of Illinois, Urbana Champaign in aerospace engineering. But I always kind of carried this message with me from, from Robert O. Forward's work that interstellar travel, it's imagine doing, but it can't be said to be impossible. So I sort of had that in the back of my mind through much of my career. When I got to urbana champaign in 1987, it was an interesting time. This is right when Mathematica was launching. So I remember a friend of mine was taking Calculus two from Jerry Uhl, who was I think worked with Theodore Gray and was quite instrumental in integrating the use of Mathematica and teaching. And I remember my friend meeting me in Elkhart hall and say, oh, my calculus professor just, just told us about this new software that's coming out that can actually do integrals symbolically and how, how such a thing was even possible. You know, just blew my mind at the time. And I then got to use Mathematica. When I was in Urbana Champaign. A lot of my aerospace courses we started using it. And first time I ever numerically integrated an ODE was with ND solve. And I have practiced, probably used Mathematica, I would say over the last 36 years, more days than not I use Mathematica. So I use it sometimes just to do simple arithmetic calculations that can be done in a calculator, plotting functions, and then also more sophisticated things as well. So it's been, you know, a hugely instrumental tool in my career all the way through graduate school and then onto being, onto being a professor. So where this now reintersects with this long interest I had an interstellar flight was in 2016, it was announced that a initiative would be funded to the tune of $100 million to develop the technology behind interstellar flight. And the concepts that most of this was built around is this individual here on the far right. So let me see if I can get my pointer to come up here on the far right. This is Philip Lubin, he's actually astronomer astrophysicist at University of California, Santa Barbara. You'll probably recognize some of the other famous people in this picture as well. There's Freeman, Dyson, Stephen Hawking was still alive at the time. And this made big news. You see the COVID of Scientific American here. And this just totally again electrified me because something I had dreamed about all my life might actually be possible. And in our century, whether whether I'm going to be around still to see this happen or not remains to be seen. But all of a sudden this idea, you know, kind of bounced to the, to the mainstream. And I was up for sabbatical the next year. So I contacted Philip Lubin and said, could I come and spend a year with you in Santa Barbara? Which if you used to spending your winters here in Montreal and Canada, spending a winter in Santa Barbara is a nice, nice place to spend your winter. So. And actually they have fantastic little cottages for visiting professors here on this little peninsula. And I showed up, bought a surfboard, learned to surf while I was there, and then spent my days at UC Santa Barbara trying to engage this problem to see if there's some way I, as a humble mechanical engineer, could contribute to this work that involves building giant phased arrays of lasers. So the key insight that Philip Lubin had in Santa Barbara was we don't need to build a single giant 100 gigawatt laser. We could build the laser the way we build our supercomputers. Now. We could, we could build it in parallel using small laser amplifiers that are made themselves from fiber optics. So these would all be joined together fiber optically, but the laser itself would be made up of a loop of fiber optic cable. So it could be done very inexpensively and done massively in parallel. So I found out pretty quickly there wasn't much I could contribute to figuring out how to phase lock billions of individual lasers. But the Breakthrough foundation, when they announced this initiative, did, I think, a really exemplary job of identifying all of the problems that need to be solved for laser driven interstellar travel. And a couple of the ones that piqued my interest as a mechanical aerospace engineer had to do with the structure and the stability of the light sail, which, when it's hit by this 100 gigawatt laser. So the idea now is we're going to scale all of this down. So rather than building a giant light sail thousands of kilometers, the sail may only be a few meters in diameter, and it's going to accelerate very, very quickly. So this won't be for people. It would be for a small robotic probe. The mass would maybe only be one gram. Okay. And the giant laser would actually be built on the ground we built here on Earth. And the acceleration would happen very, very quickly. So you don't need a giant lens in space. Basically, when you run out the end of the focal length of the laser, you're already going to be at about 30% the speed of light. And that only takes about three minutes. So you turn on this hundred gigawatt laser, focus it on this sail that's already in space, it accelerates at 20,000 GS, 30,000 GS. For a few minutes and it's going 30% the speed of light. But how do you know that sail just doesn't get crumpled up? So how do you know that that sail doesn't just let me advance my slide? Instead of being nice and smooth like this, how do you know it doesn't start to wrinkle or just get crumpled up like a sheet of tissue paper? And that was a problem that hadn't been previously addressed. I'm still kind of amazed Lord Rayleigh or someone didn't solve this problem. What happens when you have electromagnetic waves imparting momentum on a flexible membrane? What happens to it? But it hadn't been solved. This is the basis of a paper which at the time was an undergraduate student. I had a really remarkable undergraduate student. I got interested in this problem, Dan Cornelius Sahu. And this is the paper we published. And the entire analysis in this paper was done using Mathematica. So the idea is we're going to take a sail, we're going to perturb it, introduce some sort of sinusoidal wrinkle on the sail, and then see if that wrinkle grows or does it. Does it just sort of oscillate about that oscillation? And so we did this with essentially a finite element method. These are well used in engineering. But we decided to try and do all this from scratch rather than using a commercial finite element software. So we just broke up the sail into elements to mimic the bending stiffness of the sail. We used little torsional springs to treat the elasticity of the sail. We use linear springs. And then you can write down the equations of motion using Lagrangian dynamics. So just write down the kinetic and potential energy in terms of these, you know, the theta orientation of the sail and its X and Y position, and you can formulate the Lagrangian. So it ends up being a pretty impressive set of equations. Someday, I hope I have a house big enough, I have an extra bathroom, because I would maybe wallpaper it with the equations from this paper because it is one of the more mathematically heavy papers I published. Usually an experimentalist, but this was during the pandemic, so we had a lot of time on our hands to manipulate with equations. So from the Lagrangian, you then use the Euler Lagrange equations to formulate the odes. And what we found from the literature was that this is best formulated as a differential algebraic system of equations. I would have treated this just as pure odes. But in going through the literature Dan Cornelius said, you know, really, this is better treated using these DAE solvers, which I wasn't familiar with. But interestingly, and this is a real testament to the power of ND Solve is if you just load this set of odes into NDSolve, it actually identified for us. You know, it generated a message saying, this appears to be a system of differential algebraic equations, and then it uses the correct solver to do that. So that was. You know, many times ndsolve has amazed me in my career. And this is. This is another one of them. So this is what it looks like if the sail is stable. So that the sort of pink light you see is the laser coming from below. The little red dots are the little finite elements that we use to discretize the sail into. And this is a good result. So it's kind of oscillating here. This is. Now we've greatly magnified the Y axis, so it's in. It's in millimeters, and the sail is, you know, a full. A meter wide. So these are tiny little oscillations so that we can live with. But I won't play you the animations of what looks like when the sail gets crumpled up because it's kind of. Kind of disheartening to the. To the concept. So then what Dan did is an entire parametric study. So each one of these little squares that you see was an individual simulation done in ND solve to see if the oscillations grow out of control or does it just stay and oscillate. And Dan had also derived an analytic theory. So we used sort of buckling theory, classic buckling theory, to derive an analytic solution. That's the black line you see here. And the colored simulations is basically how quickly the sail went unstable. So if it went unstable right away, that would be like this dark orange color. If it took it a while, like on the order of a second to go unstable, it would be this light blue color. And remarkably, Dan Cornelius analytic theory does a pretty good job identifying the boundary between stable and unstable. So here we're varying the elastic modulus. And this is a little concerning because this is in gigapascals. And you can see to make it stable, we would need to get into the hundreds of thousands of gigapascals. And there's no material that is that. That stiff. But the good news is that if we put a little bit of tension on the sail, so maybe you just have an inflatable ring that goes around the outside, a little bit of Tension or the other thing you can do is you can spin the sail. So if you spin it, then there's a little bit of centrifugal tension put in the sail. It turns out to be pretty easy to keep the sail flat. So you only need millinewtons of tension. So if you don't want to go through all the equations, the too long, didn't read version of the paper is there's no material stiff enough. Even if you made out of diamond, it wouldn't be stiff enough. But if you spin the sail just at a few rpm and you may want to spin it likewise for other stability reasons, that's enough to keep it smooth. Smooth and flat. So that was this paper again, entirely enabled by the simulations, entirely enabled by Mathematica and indiesolve. And now we're back in the lab, so look forward to what's coming up next is we're going to do experiments on this. So this is Dan Cornelius in the lab. Now, we don't have 100 gigawatt laser in our lab, but my background is in, is in fluid dynamics, gas dynamics, shock waves. And we can simulate the effect of 100 gigawatts of laser power, because 100 gigawatts of photons hitting you turns out to only be about 660 pa of pressure, which is. Which is nothing. Right. Remember, atmospheric pressure is a hundred thousand pa, so in the lab, it's relatively easy for us to create a puff of gas to simulate what the loading is on a membrane. So this apparatus you're seeing is simply called a shock tube that we're now using to dynamically load films that would represent the light sail and experimentally going to study what happens to them. So we got some awesome movies, but that's Dan Cornelius's work, so I'm not going to steal his thunder, but look forward to papers coming in the future that will hopefully give some experimental validation of this work. So, yeah, that's an overview of what the work was. Yeah. So if you have any questions about that or anything else about my journey with Mathematica, I'd be happy to.

Sure. No, I mean, I'm certainly looking forward to seeing the experimental angle as well, because it's also interesting that. So your background studying fluid flow and shock waves and these sorts of things is also relevant for doing the experiments to sort of scale this down and get a third angle. Not just the theoretical result and the numerical results, but also tying in experimental results, which is always, I think, a sign of a good problem when you can really do all three of these kinds?

Absolutely. That's when I'm at my happiest. Even some experiments in the lab, you're scratching out some kind of what you think is a theory, and then you got some simulations running on a desktop someplace or on a cluster someplace. And yeah, when all three of those start to grind against each other, you know it's going to be good. Right. No matter what happens, you know, if two of them agree and one of them doesn't, you know, then you know where to put your thinking.

So I do have sort of two directions of questions, both of which I want to make sure that we touch on. One is sort of other research directions that you're. You're into with this interstellar flight experimental research group because you also studied wind pellet shear scaling recently.

Yeah.

And you mentioned dust was another one of the. So in that long list of problems that was broken down, there were a couple that caught your eye. So I definitely want to ask about that. And then the other direction is sort of your own introduction to using Mathematica and your current practice and using it to teach. So both of these directions, we definitely want to make sure that we touch on. So does one of those sound like you want to jump on it first in terms of where we should go?

Yeah, let me. Yeah, I think that the dust problem you brought up is a good one, because every time I give talks now on interstellar travel, there's a lot of public interest on this. And whenever I talk about this, people, they usually really light up. But I can also, I can even tell from your guests here that people are saying, wait a minute. I can see where there's a problem with this approach. So one of the big ones is always what happens when you hit a dust grain. So let me, let me address that right on that. That is an important problem. I actually lose a lot of sleep over that problem. And whenever I give presentations to the, to the, you know, this Breakthrough Star Shot foundation or anyone else that has an interest in this, I'm always waving the flag. You know, you're going to get hit a lot of dust grains. You know, interstellar space is very, very dilute. Okay? There's not a lot out there, but if you travel four light years, you know, it's like collecting all the dust between you and that nearest star. And, and just imagine collapsing that, you know, over a period of years. But you're gonna, you're gonna take hits. So one thing is getting hit with a dust grain when you're going 20 or 30% the speed of light. It's not like an atomic bomb going, so sleep. I say, oh my gosh. Everything will be immediately vaporized. So when you're going at those speeds, getting hit by a dust grain. So we believe the dust grains in interstellar space are about micron sized and smaller. So astronomers study this a lot because dust is a major nuisance for them. It interferes with their measurements. Famously, one of the reasons why Hubble got the Hubble constant wrong was the starlight he was measuring was sort of red shifted by some of the dust. So they study this a lot, and their best understanding is most of the dust is micron sized and smaller. We don't know if there's anything bigger out there. We don't have any evidence of it. So getting hit by a dust grain when you're going 10% the speed of light, you know, it's like one joule of energy. And that's like, like a little starter pistol going off or maybe like a camera, like a little disposable camera flash. When that flash goes flash, that's about one joule of energy. So it will likely do some local damage to the sail, but it's not going to vaporize it. But yeah, it's a concern. And the thing I'm particularly concerned about with the laser light sail is what happens when you hit a dust grain in the solar system when the laser is on. Okay, now this is pretty scary. Okay. There's a lot more dust in the solar system. It's a more dusty place. And that sail, as you can imagine, it has to be highly reflective. But the more important thing is it has to have very low absorptivity. If it absorbs even a fraction of a percent of, of that 100 gigawatts of laser power, it will just, it'll just vaporize the sail. Okay, so the, the people that are working on the development of the sail, the materials for that, they're looking at, for example, the materials that make up our fiber optic networks, which is a glass called Zblad, which has very low absorptivity. It's like parts per billion of the light gets absorbed. And there's ways you can turn a reflective glass into a reflector by making something called the dielectric mirror or Bragg mirror. So, you know, they've identified materials that look promising. They have the low absorptivity, but they have to be, you know, perfect, pure materials. And if they were to get hit by a dust grain when the laser is on and they get damaged and some of that laser light now gets absorbed in the cracks again. That could propagate across the sail and just, just, just vaporize it. So another project we've been working on is, is it possible to clear out a bit of the dust in front of the sail? Well, at least while the laser's on, we can't clear out the whole path to Alpha Centauri, but at least it's about one tenth of an astronomical unit is how far the sail is going to travel while the laser is on. And so we looked at, you know, if the layout, if the sail was a little bit transparent, which would likely be. Anyway, again, it doesn't need to be perfectly reflective. It just can't absorb any of the light. So if the sail is a little bit reflective, it might actually vaporize some of the grains just before they hit the sail. So I think of it sort of like the old steam locomotives that had the cow catcher in front, right? If a cow wandered out of the train tracks, they just get pushed aside by that plow shape on the front of the locomotive. So that looks pretty promising. And that actually was something we did also with a lot of mathematical calculations. Looking at the optics, I think it's pretty straightforward optic calculations, but as a mechanical engineer, I don't do a lot of optics. So doing all the calculations there, because it's all done with complex variables. Right. So anything you do in optics, you have to have the real and imaginary part of the index of refraction. And I think that's something where Mathematica really shined as well. Being able to work very natively with complex variables was pretty helpful. So, yeah, that looks promising that we might be able to clear out some of the dust grains ahead of the sale, at least while the laser is on.

This is a really fascinating area of research. So we have a question, actually, from the audience. So the question has to do with when you're doing your experimental check of your simulation and theoretical results in these experiments, will you be tracking the individual elements? So when you're solving the problem numerically for the audience, the finite element method, you, you know, break the material into little discrete pieces and then simulate each of those. So will you be able to experimentally see down to that level of detail, or will you be looking at sort of more bulk quantities?

Yeah, we'll look at, you know, just commercially available films that are as thin as we can get them. Lately, the experiments we've been doing, we've been looking at elastic films, just because the theory is a Little bit easier. And we can monitor those with high speed cameras. So again, I have a lab that deals with high speed gas dynamics and shockwaves. So I have a lot of nice high speed cameras. And we can also monitor them with something called photonic Doppler velocimetry, which is a new technique that's been developed just the last 15 years. It's kind of a revolution. Interestingly, it also comes from the world of photonics. So this idea that we could build the giant laser photonically, there's a way now we can use commercial telecommunications lasers and detectors to measure the Doppler shift of laser light when it's reflected off a moving surface. So this is called PDV Photonic Doppler velocimetry. It was developed at Lawrence Livermore about 15, 20 years ago. And so you buy a commercial telecommunication laser and you buy a commercial telecommunication detector and you just shine the laser off, in our case, this sail that's moving and we can then measure the. You blend that reflected light with the original laser and you pick up the beats. So the actual frequency of the light is too fast for me to measure, but when you blend the Doppler ship's light with the original light, you can pick up the beat frequency. And it's essentially like the police officer's radar gone. Only because it's working at laser wavelengths, it's much more spatially accurate. And we have now a whole array of these fiber optics looking at the sail as it comes towards us so we can measure the amplitude growth rate of the sail photonically. It involves doing a Fourier transform. And that's where Mathematica helps as well. So my PDV script that I use to process data, I use the built in FFT of Mathematica. And I'm aware that national labs do as well. I know I have colleagues at Los Alamos who have a Mathematica script when they use this technique.

So you're interested not only in doing your simulation and some of your theoretical calculation Mathematica, but you also use it for analyzing your experimental data.

Experimental data, yeah. Yeah.

That's great. So in terms of when you had also a background in studying mechanical engineering before getting into specifically interstellar travel, you mentioned that you were originally interested in. This is one of the sort of the things that got you into aerospace and mechanical engineering. And then recently you've been able to come back directly to it. But you've also done work that relates to shock waves and studying flows with systems that are high energy. So in terms of this work what are some of the connections between what you've done previously and now these other directions that you're looking at with light sails and interstellar dust and these sorts of things?

Yeah, yeah. I mean my background, all my degrees from Urbana Champaign and I did my graduate studies, University of Washington, Seattle. They're all in aerospace and propulsion. So I was interested in, you know, things in space going really, really fast. And I was always kind of drawn to these kind of far out ideas. I wasn't quite ready to work on interstellar flight in graduate school, but I worked on other pretty far out there ideas. So I did my graduate study working on a concept that might be used for gun launch to space. Like the old Jules Verne de la Terre a la Lune, you know, build the giant space gun. And the concept there was to fill a long tube with an explosive gas and fire a projectile into it that would sort of ride a wave of combustion or explosion down the tube and accelerate down the tube. So this was developed by my advisors at the University of Washington, Seattle. And I went and worked on that and that's what got me into this kind of field of high speed reacting flows, shockwaves, detonations, and that's most of what my academic career has been on. And yeah, I mean I did an awful lot of work with Mathematica then as well, you know, so I find particularly working with nonlinear equations and you know, wave tracking, wave solutions and so on, you know, mathematics has been really integral tool for that as well.

We had a follow up question from the audience which are so especially in the scaled down experimental version, what's the typical timescale that you're looking at? And then sort of related to this or what's the laser frequencies that you'll be dealing with in the, you know, lab version? Not necessarily the scale version.

Yeah, in the lab when we do these experiments where we use a sort of gas dynamic simulation loading on the, on the sail, where we're trying to match the accelerations that the real laser driven lightsail would experience. So they're accelerating at 30, 40, 50,000 GS which, which really isn't all that much. I've in the past accelerated things in my lab at, at millions or tens of millions of GS. Incredibly, things can survive those levels of acceleration. And so the experiment lasts on the order of a millisecond. So in that time it's going to go from rest to maybe 100 meters a second. But that's enough time for us to see whether the sail material Remains stable or it starts to, starts to get, you know, goes unstable and oscillates so that it's enough to do something meaningful in the lab. As far as the laser wavelengths, the plan is for this laser driven interstellar sail would probably operate at the same wavelengths that telecommunication lasers do. Just because of economics, just because of the billions of dollars of R and D that have gone into developing telecommunication lasers. They're really optimized around either 1062 nanometers or 1550 nanometers is kind of where these erbium and ytterbium doped fiber optic lasers really, really optimize. And so just for, for economic reasons, they'll probably be operating in that, you know, it's kind of the near infrared. And the Earth's atmosphere is more or less transparent at that wavelength. So you could build the laser on the ground and beam it up to, up to your sail in space.

That was another question that I was curious about is that. So the attenuation at those wavelengths is going to be small enough that, you know, it's definitely worth doing as a ground based thing, not, not trying to do some sort of, you know, intermediate, you know, orbital thing.

Anything you can do. Yeah, anything you can do on the ground is going to be cheaper than doing it in space. Right. So I think that that's certainly the near, the near term tests that people are discussing doing would be with the ground based laser. You know, maybe eventually you'd want to build it in space because solar power is plentiful in space and people may have some concerns about what else you could do with a giant laser like this. So maybe building it on the backside of the moon or something might address some, some people's unease about having 100 gigawatt laser available. Yeah, but that's been a big breakthrough is adaptive optics. So essentially, astronomers have developed techniques now where they, you know, they distort the mirror of the telescope and they can basically remove the Earth's turbulence. So there's really impressive things that have been done with these giant observatories in Chile where with just sort of one click of the mouse, they turn on the adaptive optics and all of a sudden the images get, get. Not as good as the Hubble Space Telescope, but better. You know, they can actually, I've seen pictures of Neptune and so on with adaptive optics that are taken from the ground, with telescopes in Chile that are better than Hubble. And the idea is you can run those same algorithms backwards. So when you send the laser out of the atmosphere, you would Intentionally distort the laser so that the turbulence of the Earth's atmosphere has the, the beam as it leaves the atmosphere. It's sort of the ideal beam that you want to put on your sail.

So the other thing that you mentioned is that you really like studying systems with non linearities in them. And so was this something that you initially, from the start of being introduced to Mathematica, you know, you said, wow, you know, this is a tool that I want to do these sorts of problems or was that something that emerged later in graduate school or. I kind of want to go back to your initial introduction and hear about when it was a new thing.

Yeah, so I mean, I always, I always kind of interesting. I, I actually like math, but I just don't know if I have the, the, the intellectual capacity to do math at the level that you probably have to be successful in doing math. So when I started out in Urbana Champaign, I was actually trying to double major in physics and engineering because I felt if I really want to get into this interstellar flight, I need the physics background as well. And it just was clear to me that I, you know, well, I like physics. I couldn't do math at the level you need to do math to be, to do well in physics. And I was confessing to John just before the interview here that linear algebra is something that I just really don't enjoy all that much. I think linear algebra appeals to people who are very meticulous with their accounting and bookkeeping, whereas I like to be a little more heuristic in my thinking. And I find that when you get into nonlinear problems, all the usual bag of tricks in mathematics, using linear superposition and so on, they kind of break down and the methods become the sort of methods I like, very heuristic and ad hoc. And my favorite textbook in university was Bender and Orzog, if anybody knows that book. It's kind of a textbook on how to solve nonlinear odes that, that can't be solved using the techniques that you, you learn in your, in your ode courses. And then I just sort of, I think there's just a nice intersection because they had an interest in high speed flows and combustion and detonation and shock waves. And then the mathematics of those tend to lend themselves. You know, it's all highly nonlinear problems. So you end up, you know, solving a lot of nonlinear odes and I think just meshed well. And so I think Mathematica is a tool that works very, very well with those kind of problems.

You also, I don't remember Whether it was actually at the beginning of when we went live or beforehand, you were talking about how it was nice to have tools that could solve things symbolically, so that when you were going through problems and learning, you know, you get sort of a check on, you know, things that you might not have automatic in terms of your own memory yet, but eventually you're building toward it, and you still use Mathematica as part of your teaching. So I'm curious, what are some of the things that you like to do in terms of making things interactive for people, in terms of what, what. What do you find is effective methodology to use when not only teaching? You know, these are, how do you solve problems? But this is how to use a tool that can help you solve other problems.

Yeah. Yeah. I think when I teach class, I almost always have a Mathematica notebook open on my. On my desk, on my laptop. If nothing else, if I need to do a quick calculation as we're solving a problem in class, you know, it's. It's great for that, or plotting a quick function. And I always like to joke, you know, that, oh, I don't know how to do this. Let me pull down one of these palettes, and there's a palette there called, you know, classroom. Classroom Assistant, I think. Right. And I always say, oh, well, I'm in a classroom. This is. This is, I guess, the appropriate palette to use. So, yeah, you know, and just being able to plot functions quickly, I mean, that was a real breakthrough. When that capability with Mathematica came out in the late 1980s, early 1990s for the younger listeners here, it used to be, you know, to just plot a function used to be really onerous. You know, you have to write a separate code, you know, output a data file, import a data file, and just the ability to just quickly visualize a function like that was really a huge, huge thing. So I think that's, you know, the symbolic nature of Mathematic is, of course, incredible. You know, the number of times I've had an ode or an integral, and I say, okay, there's just no way a symbolic solution for this thing exists. And then you enter it into Mathematica, and then all of a sudden it pops out a Lambert function, which I didn't know what it was, but, wow, there's apparently a solution for this. And then I had to educate myself on what the Lambert function is. And then you come to find out that there's this function that shows up in all kinds of interesting applications. So that's always a Revelation when it just pops out an analytic solution when you never imagined one. But then the idea just to visualize things quickly. I just can't probably convey to younger people what that was like. Just to have a function and then be able to plot it or make contours or 3D surface maps was really a breakthrough when that appeared in the late 1980s.

Do you find in your teaching that you make use of this ability to quickly plot things in order to respond to student questions or what does that look like for you when you're.

Yeah, yeah. I think usually students are pretty amazed when they see this. So I got John's permission. It's okay for me to mention this. So in engineering programs these days, the, the sort of default software that's, that's mainly used is, is matlab. So most of our students are, are trained in, in using matlab. This is now shifting more and more to Python, which we can get into as well. A little bit concerns me because there's quite a bit of learning curve and overhead to, you know, you know, Python is not likely to, you know, magically identify what the type of ode you have or magically tell you, you know, oh, there's a Lambert function solution to your, to your ode. But I think usually when I do a demonstration, students see me using Mathematica, they often come up and ask me, you know, a little bit more about it. I mean, most students are aware of Wolfram, you know, the website, this magical website that can solve even the most difficult integrals for you. But Mathematica has not made huge inroads in undergraduate engineering education. When graduate students come to work with me and we're working side by side closely together, then I have more of an opportunity to say, I think this is a tool that would really help you with this problem. And then usually students are quite receptive to it. But yeah, I think, and as I mentioned to John, I think one of the challenges for Mathematica might be in its linear algebra and matrix manipulation abilities. I think that's maybe one area where it, it might, some improvements might really impact its ability to make inroads into undergraduate engineering education. Because my personal taste aside, there's a lot of linear algebra in engineering, especially my colleagues that do control theory and dynamics. It's really applied linear algebra.

So you were saying in particular, the thing you thought that would be very helpful would be sort of more graphic interfaces for people to be able to set up these sort of problems, am I correctly?

Yeah, yeah. I mean, I think the few times I've tried to do things that involve somewhat heavy linear algebra in Mathematic. I do get a little lost in nesting the curly brackets. Okay. And I think if I haven't tried in a while. So maybe there's, there's, you know, in version 14, there's better tools there. But, you know, it's. I just find my sort of distaste for linear algebra matches very well with where I think Mathematica's strength strengths are at.

One of the functions that for me was a game changer was learning that you can use things like array and table to set up matrices with symbolic entries where you can sort of programmatically generate the subscripts or indices however you want to represent those. Because prior to finding how to do those, typing in these large things by hand. I agree, it was totally onerous. And then it turns out that, yeah, there are some things to help with that.

So, yeah, I mean, this is maybe interesting thing we can talk about for a minute. Here is again, I've been a Mathematica user. I did the math here just before the interview. For 36 years I've been using Mathematica. Is that right? Yeah, since 19, let's say 1990. So that would be 34 years pretty heavily. But I've never. And you probably hear this a lot. I don't feel like I ever found. Formerly been trained to use it correctly. And it reminds me a bit, my, my wife actually is a little bit of a tangent, but my wife, both of her parents are hearing impaired. So my wife has totally fine hearing, but she grew up learning to sign essentially simultaneously with learning to speak. And so she, she very intuitively knows no sign language, but she never formally taught sign language. So when she meets people who are also in the hearing impaired community and she signs with them, they say, you know, wow, you sign very naturally and fluidly, but you also use all kinds of nonstandard sign language. And I sort of feel like that's my story with Mathematica. So I started using it, you know, when I was an undergraduate and I just sort of found ways to make things work. But I've always wondered if I'm doing things quite right or not. You know, like when I get a solution out of Indie Solve and you use the backslash dot to extract the solution you want. And then I always have to find. I have to put a double square bracket, one close double square bracket because I think that solution is embedded within a. Within an array, you know. So I sort of found things that work. But I've Always been a little bit concerned that I'm not doing things quite right. So I don't know when you encounter people like that is, you know, do I need to undergo some kind of, some kind of re education or just keep doing what I'm doing?

No, I mean, so when it comes to those sorts of things, there are a couple of different idioms that people, I think, typically learn, right, where, you know, you could either get, as you say, if you just want the first thing out of that list, you could use the double square brackets. You could use first. You could many times, if this is a replacement rule, because I think I don't actually know when. So like ndsolve versus ndsol values, right? You'll get, you know, either the replacement rule that gives you what the function should be or you'll get. Yeah, so here as you see that the X goes to interpolating function. So I'm not sure what year they added ndsol value and then like sort of similar functions for that. But I do remember when I was first learning, you know, sort of wrapping my head, my mind around like, okay, so this, this is giving me a rule that tells me what the variable in question. So here X should be, rather than just giving me the function directly. And then there's another function now that if I just want the function directly, I can use, you know, the, the ones that. It's the same sort of base name of the function, but then either value or values, depending on which one you're talking about. So, yeah, so there, there are things like that, which I think.

Yeah, yeah, this is a simulation I running today, actually. It's not the lightsail thing I was showing. It's some other work we're doing related to interstellar travel, looking at how charged particles interact with magnetic fields. So, yeah, I have a huge set of odes here. And again, I'm plugging into my old friend Andy solve here to solve the equations of motion. And then what I usually do is when I get this, I store that into something I call solution. And then when I want to visualize the output, what I do is I define a function which I call X solution, which is X evaluated, you know, using the slash dot that solution. Then I have to put the double bracket one, you know, to extract that value out. And then I can. Then I can plot it. But I've always wondered if that's the right way.

So I believe that, yeah, I believe that double bracket one isn't strictly necessary. It doesn't break anything. If you do it. But I think if you.

Yeah, but then when I go to, I think when I go to plot it, it won't plot if I don't have that. If I don't have that here, if I want to evaluate the value numerically, it works. But if you go to plot it, I don't think it plots because it needs to be. I don't know. Anyway, I'll look forward to when this goes up on YouTube. I'm sure in the comments people will say, oh, this poor late middle aged engineering professor, you just would do this, it would work out. But again, I have a set of tools that really work for me and probably at this point I don't know if I have the neuroplasticity left to learn something new. So I've always found a way to make things work.

Yeah, yeah, no, I mean people, you know, people of course find idioms that both work for them and then some of them I think kind of catch on through which actually speaking of. So Dan Cornelius, Savu, the student that you mentioned that did this great, great notebook on light sales, he made an excellent post on Wolfram Community showing, you know, both part of the work that was in that, that paper as well as how to make some, some videos and things that are related to that. And I was saying to myself that this is a very high quality notebook. And so if people are curious, you know, what, what sort of idioms should I learn for how to do things? I was a big fan of that notebook. So, so I would definitely say that, you know, whatever, whatever tips and tricks have, have made their way from, you know, where you first learned them to, than students that you're working with. You know, I liked that notebook a lot.

So. Okay, great. Yeah, I'll pass it on to Dan Cornelius. Yeah.

Another thing that you had mentioned previously was, so I think offline, you had elaborated on specifically why, you know, you felt that things like Mathematica or even compared to Python, you know, Matlab, you know, there's some advantages there instead of having to go and get, you know, individual Python packages. So you'd said a little bit more about that offline. So I was curious if you wanted to elaborate more.

Yeah, yeah, it's.

You briefly mentioned the fact that, yeah.

It'S an interesting, interesting trade off I can see here. So, yeah, we're under a lot of emphasis these days on trying to get students to do everything in Python because it's, you know, it's free and it's becoming kind of the de facto standard. But I do. See, when I, in these projects we've been talking about, and you need to work with Fourier transformer and ode solver, or you need to implement a Lambert function, or you need to implement complex variables to do optics calculations. I see students that work in Python, it takes them a long time to get, get the packages together that they need and it can be kind of overwhelming. Right. So if, if you need to implement a free transform and you go look for what's available in Python, you're probably going to find 50 packages out there and well, now what do you do? You probably download 10 of them and try different ones and see, you know, and, and in a way that's a good skill. I mean, it's, it's, it's, it's, it's a plus that students learn to work in an, in an open environment like this and learn to go on GitHub and download packages, but at the same time it also consumes a lot of time and you can, you can lose your enthusiasm for the project if, if, you know, you come to meet with the professor. And what have we got to show for this week? Well, I played with seven different Fourier Transform packages, whereas with something like Mathematica, it's just built in and you just go, you know, it's. So I think there's a trade off there. I mean, ultimately you could argue that the students that go the Python route, they're going to be stronger because they know how to work with packages and things. But I think in my own case, if I think back to when I was an undergraduate, I might have gotten overwhelmed or lost my enthusiasm for the project. Whereas when you use a tool like Mathematica, you can just focus on getting the results, just getting the work done.

Yeah. As you say, going and collecting packages from different places is sort of its own skill. Right. So like, if, if that's the skill that you're wanting to teach, then okay, great. But you know, in many applications that isn't actually going to be what you're wanting to focus on. And so I definitely agree with you. In fact, that's actually the reason why I started using Mathematic again after a while. I had been working with a group of students in Python and they, exactly as you say. Right. A very common experience is like, okay, well what did you work toward in your problem this week? And it was like, well, I figured out something about these packages and okay, sure, but that wasn't what I was intending to teach. So I definitely, from firsthand experience, sort of agree with Your analysis of that whole trade off there. So another thing that we recently have a audience question coming in. Let me take a look here. So it looks like this is more of a comment about nonlinear dynamics. So riffing off that more comment than question to come in. So when you are teaching courses, what sort of courses are you teaching these days?

Okay, so I teach a course on statistical thermodynamics. So I'm greatly enjoying your colleague Stephen Wolfram's new book on thermodynamics. It's been fascinating reading for me and I'm going to probably be incorporating some of the ideas from that into my course. And so that course lends itself very well to using Mathematica to do a lot of little statistical demonstrations, coin toss simulations and things like that. And I also teach course on combustion and I use Mathematica very heavily for that as well there. We do a lot of chemical kinetics, which is again coupled sets of nonlinear odes that you integrate through time. And I teach undergraduate courses in fluid dynamics, specifically high speed compressible flow thermodynamics, things like that. So really in all of those courses, you know, Mathematic is definitely, definitely used. And you know, I use it in the classroom to teach, but because it hasn't really taken off with undergraduate students. And I usually do not assign assignments that require Mathematica. You know, if somebody uses it, that's fine, but I always kind of leave it. I think this is a sort of a general trend now in engineering. We don't, unless it's for a specific programming course or something, we don't really require students to use any given package. We just sort of tell them, you know, it's your responsibility to, to get familiar with a, with a tool that can solve these kind of problems, which I think is probably the right, the right choice.

That is interesting. It actually is in a bit of contrast with my experience in undergraduate was that, you know, it tended to be that specific courses. And I know there still are some courses that are like this where they tie themselves to specific tools. But I think that tended to be, be very common when I was an undergraduate. For example, I first learned Mathematica through a specific course. There would be specific courses where one might learn matlab or one might learn Java or Python or what have you. And so that is an interesting difference in mindset and philosophy of saying there are tools out there. Here's the problem. You're responsible for learning a tool to solve the problem. I think that is sort of new from my past experience.

Yeah.

So another area that I'm quite curious about are when you are working on these experimental results, what are some other things that you think will come next? Right. So the immediate next steps are looking at the experimental checks on some of the simulations you've done with light cells before. What are some of the other ideas that you'd be interested to put out there that your research is going to be looking at in the near future?

Yeah, so what we want to start doing is working with a real high power laser. And so I just got installed this year a 3 kilowatt laser in our lab, which is a little scary. So again, Nowhere near the 100 gigawatts that you would need for interstellar travel, but that's quite a step up. The previous laser I had was about 1 watt. So getting a 3 watt for 3 kilowatt laser installed and certified and letting make sure the university laser safety officer didn't shut us down took. It took a bit of time. But now that we have that, I think there's quite a bit of things we can, we can explore. One thing is we're looking at sort of some more near term applications of this idea of using directed energy to a spacecraft. So before we go interstellar, we need to find other ways we can use this, this concept of beaming energy to a spacecraft. So the other thing we're doing experiments on in our lab is something called laser thermal propulsion. So you use the laser to heat up a more conventional propellant. So hydrogen is actually ends up being the best because it has a low molecular weight. So it doesn't weigh a lot. But if you heat that up and then expand it through a nozzle, it's kind of like a laser powered steam kettle that can not get us to the stars. It won't be able to do interstellar missions, but it can do, for example, rapid missions within the solar system. So we published a paper about two years ago on using this laser thermal propulsion to do a rapid transit to Mars mission. We could show you can get to Mars in about six weeks if you have a, you wouldn't need 100 gigawatt, probably 100 megawatt laser on the ground. You can use that laser to again deliver energy to the spacecraft to heat propellant. So we're now setting up to do a little laser thermal rocket in the lab. So that's one thing we have experiment going and then the other thing, this is pretty ambitious, but I'm trying to convince Dan Cornelius to take this on for his PhD is can you actually, for the Laser light sail, can you recycle the photons? So when a photon bounces off the laser, it hasn't given up much of its energy to the. Or bounces off the sail, it hasn't given up much of its energy to the sail. What energy it has given up is what you see in the Doppler shift of the light. So if the sail is moving and the photon bounces off, it gets Doppler shifted to a longer wavelength, which means lower energy. But initially, when the sail's first starting out, it's moving so slowly that the efficiency is very, very poor. You have to put this big, powerful laser on there for a long time to get it accelerating. But if you could have that laser light bounce back and forth a bit like a laser cavity itself, you could recycle those photons over and over again. And there's a little bit of controversy in the literature as to whether that would work in a passive cavity or not. So, again, we're not experts on optics, but I think that might be an interesting experiment to try in the lab is have a moving mirror moving at high speed. But I know a little bit about having things move at high speed is to see if we could measure this photon recycling effect between two moving mirrors. That's something we hope to gear up to do in the next couple of years.

That is interesting. As someone who always had a bit more of an interest on the theory side of things, I have to admit that when people start disagreeing about theory in the literature, you know, a good experiment is really helpful for helping people clarify which assumptions of other authors they've either taken on or not taken on.

So that's exactly. Yeah, yeah. Very, very good advice. Career advice I would give to younger researchers is anytime you're at a conference and you ask one expert a question, you ask another expert and you get a different answer or different opinion on things. That's where you should put your emphasis on. You know, if everybody is just adding praise and more decimal points beyond or more digits past the decimal point onto a measurement, you know, you're really not making advance. The idea is to try to find something where there's a little bit of controversy and see if you can, you can resolve it, is how you really advance your career.

That's great. And so speaking of, you know, career programs, Yi is also involved in, you know, some career development workshops that we do with students all over the place. And so I'm curious for your perspective on sort of this aspect versus having to do packages in Python, versus something that is built into one place. I know in your background, this is also something that came up as well.

Yeah, I mean, I think, Ben, you were going to ask me, do I do everything in Mathematica, Do I ever go beyond? So I should probably put this out there, is that the few times I get into a problem that I really need to do some number crunching, I mean, I really need to. Need to grind the tool I go to beyond that is get ready for this is Fortran. So I was taught Fortran as an undergraduate engineering student and got to be quite good in coding and Fortran. And I just have a nice library of. It's actually all the subroutines that are in the classic numerical recipes textbook. When I was a grad student, I just typed all those out. So I have a nice library of solvers that I use. So it's probably too late for me because I just don't have the neuroplasticity left. I've tried to teach myself C and Python and these things, but when you have a set of tools that you're comfortable with, it's really hard to convince yourself to learn something new. So what I would encourage in working with students is, is maybe recognize that fact that if you're gonna develop mastery of something, it's gonna happen in your. In your 20s. So it's good to, you know, expose yourself to at least a couple of different tools that are gonna be able to do what you need to do to. To get through your career, because it just gets harder and harder the older you get to learn something new. So, yeah, I always enjoy, you know, when I sometimes assign projects and I'll show students how I implemented it in Fortran, and. And I open up my Fortran code, I always have it at the top there, you know, numbers going from 1 to 80. And I say, you never, never go beyond the 80th column in Fortran. And does anybody know why that is? You know, and just go to blank stares. And then I tell students it's because past the 80th character, you've gone off the edge of the punch card and your Fortran compiler just doesn't recognize it anymore. And then, you know, most students have never even heard of a punch card, so I missed that. I never coded on punch cards, but I had an older brother in Urbana Champaign. Apparently, until 1983, they were still teaching coding on punch cards in Urbana Champagne, so I just missed that. But, yeah, I think that's probably the main message is, you know, try to get a tool Or a nice, what's the word they use now? You know, a talent stack of tools that is going to serve you through your career and then try to keep those up. So I am a little bit proud that as old as I am, I have still kept up my coding skills. You know, I'm not a computational person, I'm mostly an experimentalist. But I have kept up my skills to be able to code and I would encourage people to, you know, put in the, always keep a little project going. You know, if you're conversant in something, try to keep a little bit of effort going in that so you don't, that skill doesn't, doesn't atrophy away. So it's a little embarrassing. But I actually haven't done any CAD for decades now and when I need to build a new piece of equipment in my lab, I may draw it in like you know, Illustrator Inkscape or something like a first iteration of it and then ask one of my graduates, you know, could you, could you please please, you know, CAD this up for me because I just have lost, lost the ability to do cad, use CAD software. That's a little bit embarrassing as an engineering professor. But yeah, my Mathematica skills and my Fortran coding are still pretty sharp. So. But you know, that's message I give a lot to students is your time to really obtain mastery of these things is, you know, your 20s, maybe early 30s is kind of your last chance. And then after that you tend to really calcify and just go with the tools that are in your toolbox.

Yeah, I think no matter what language do you use, I would suggest student to post their code on their website or GitHub somewhere so people can know what they have done and it can serve as their portfolio. Maybe we should invite more graduate students or undergraduate students to our live stream so they can show their work.

Yeah, I served on a faculty hiring committee last year and I was impressed. Candidates that put their GitHub page right on their CV there and we do look at those, you know, that actually I think is very good advice that students should, should view that as part of their CV, their GitHub profile.

Yeah. So we're about to hit the time. If we don't receive any additional questions, this will be the end of this live stream.

Yeah, while we're waiting, just a moment for the audience to maybe ask a last minute question or two, a thank you for being here and you mentioned students showing their work. Definitely remember that if you're watching this and you know any students that might be interested in a Wolfram Summer program. We have our Wolfram Summer School that is a great opportunity to both learn some of the tips and tricks and idioms that we were talking about, as well as your show, an actual project that you've done as part of your portfolio. So. But, Andrew, thanks so much for being here, and I know we both really appreciate it. It's a fascinating topic of research that I really look forward to seeing what comes out of this in the future.

Okay, yeah, stay tuned. We got a lot of stuff coming, coming down the pipeline.

Great. Thanks, everyone watching this.