At school, my daughter needed to calculate how long it would take for a car in neutral to roll to the bottom of a hill. She knew how high the hill was, and how much distance the car would cover in the process. She worked out how long it would take, and noticed something troubling. Her answer didn't depend on any details about the car, and saying that the speed doesn't even depend on whether you push the brakes or not felt a bit too silly! (Of course the problem implicitly assumed that you aren't using brakes, but I was still proud of her for realizing it needed to be specified!)
We were talking about what kind of details the question would actually need, and started discussing things like bearing friction and the rotational resistance of the wheels and axle (we meant the rotational moment of inertia here). We ended up concluding that we needed at least a model of the car and a model of friction to get enough details. (We've had similar discussions before, so she knows what a model is and that I love bringing them up all the time). The interesting thing is, by the time dinner was over, she was totally fine with assuming that the hill had a straight, constant incline and the tires weren't slipping on the road. Things had gotten complicated enough as it is.
This conversation got me thinking, a big part of what we do at Noumenon is also deciding how much a problem should (or shouldn't) be simplified. Modeling physical phenomena is all about combining just the right amounts of approximation and precision.
So how should model creation be approached? What's the right recipe here? If there's too much approximation — too much simplicity — the model is going to work fine. That is to say, it will produce an answer. There's just the small problem that the answer probably won't match the real world too well. (My daughter's first answer was absolutely fine. She got the grade she was hoping for. It's just that it simply wasn't true.)
What about if there's too much detail? What if, to solve the rolling car problem, we actually built a quantum level model of all the physical particles that were interacting with each other? (There's a rumor flying around that anything's possible these days.)
Well there are a few problems there, too. First off, despite the rumors, no, the computing hardware to do that doesn't exist. Even if a computer that can do anywhere near this kind of simulation is ever going to exist in the future, that future is a very distant one (think wild science-fiction realm). Second, even if we did have a particle model to solve our problem, we can't really measure the initial conditions for a problem we want to simulate to that level of detail. So we're really wasting a lot of our ‘details’, by giving the model bogus initial conditions. Third, even if we did perfectly measure the initial conditions we want (and this is drifting from science-fiction to fantasy territory now), our model would only work for one specific set of initial conditions. So a simulation is replacing one single test, but what we really want is a simulation which replaces a set of tests. Again, this means our model has details that we don't really want.
So the appropriate model should have some simplification — but not too much.
And here we come to an interesting point — one of our objections to more detail was that the computing hardware simply doesn't exist. This is a very real constraint on what kind of problems we can or cannot solve. So even if there's a certain level of detail that we are interested in, the computing power available might forbid us from getting there. So it's not too much detail, it's just too much detail for now.
From this point of view, how much simplification our models should have isn't just a theoretical question, but also a practical one. And the answer changes with time. At one point of time, the best models we could hope to simulate were too simple. Engineers have often gotten the most out of those models, using ad-hoc assumptions to approximate specific problems. But as computing technology is improving, using appropriate models instead of ad-hoc assumptions is becoming more and more feasible.
Models are often created and popularized only when computer hardware permits their use. But this means that with great compute power comes great responsibility to develop new, more appropriate models to harness it. Simulation engineers often tend to continue using simplistic models, not because of computational constraints, but because the appropriate models simply don't exist yet!
Consider Noumenon's bulk scattering optics model. Our engineers were interested in modeling the scattering of light while passing through a translucent object, with the end goal of achieving flat (uniform) illumination of said object. The existing models of scattering assumed uniform energy dispersion in all directions when a ray of light is scattered. These models were useful for applications in interstellar optics — and they were simple, not very computationally demanding. But they were too simple for modeling optical scattering at the scale that a light source needs. With better computer hardware available, this was a classic example of an inappropriate model fulfilling an anachronistic role when the resources actually merited the re-evaluation of theory.
To fill this gap, Noumenon developed its own bulk scattering model, which successfully handles non-uniform energy dispersion from scattering events. What's more, this model (unlike its predecessors) takes into account the partial polarization of light post scattering, to create varying energy dispersion patterns from successive scatter events. The model successfully created a bulk level representation of the scattering phenomenon, taking these dynamic properties into consideration. With the computing hardware to handle this level of detail available, this was the new, appropriate model for the job. It handles small scale optical scattering much better than its simpler ancestors, matching laboratory data very accurately. This is the kind of jump in modeling theory that Noumenon is always after.
Examples like these suggests that the search for appropriate models always needs to be re-evaluated, taking into consideration the current state of hardware capabilities. Take for instance the model of electromagnetics — Maxwell's equations. These linear PDEs have done very well to explain a lot of electromagnetic observations. But again, it's no secret that non-linear electromagnetic materials and phenomena do exist. Though many models of electromagnetic nonlinearity exist, a general model to capture these phenomena is out there, waiting to be found.
At Noumenon, simulation engineers and mathematicians make it a point to keep in mind that hardware constraints are always changing, and the most appropriate model for a problem can change with them. In creating such models, Noumenon doesn't just work at the cutting edge of applied physics, but actually advances it.
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Aniruddha Khadke is the COO of Noumenon Multiphysics. With an M.S. in Computational Mechanics from Ohio State University and 16 years of experience in developing simulation software, Aniruddha leads simulation projects at Noumenon. In his previous roles, Aniruddha developed solvers, meshers, and post processing and visualization capabilities for industry leading simulation softwares.
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EXAMPLE PROJECTS
Each project at Noumenon Multiphysics has an aspect of modeling to it. Following are some projects showcasing a strong modeling component:
