On Tuesday I had the privilege of attending a lecture by Thomas J.R. Hughes, who is somewhat of a legend in numerical methods circles. The man has over 300 academic publications and is pioneer of a new method of solving hitherto-intractable systems of engineering equations, called “isogeometric analysis”, or IGA. Three-quarters of the way through his lecture, I experienced a moment of transcendent, overwhelming wonder that I want to share with you.
No, it’s not for the reason you think, but it will require a bit more math than you were hoping for. Because to understand just why I experienced such profound wonder, you first need to understand Finite Element Analysis.
If you stop and think, it’s a little bizarre that everything around us works. That interstate overpass you take to work hasn’t collapsed under the weight of eighty gridlocked semis even once! Your office building hasn’t tilted or sunken into its foundations (even after that one office party involving a rave). And you literally never worry about your car killing you, not only because you consider yourself an above-average driver (which, hate to tell you, is unlikely), but also because some suits with bifocals made it that way. But we engineers can’t just test bridge capacity by building a bunch of full-size test bridges and driving a Peterbilt over them, so how are those engineers so smugly confident that you won’t die a thousand horrible deaths in the course of a single workday?
The answer is Finite Element Analysis, or FEA. It takes Newton’s Laws (“equal and opposite reaction” and all that), and applies them to the complex bodies of computer-aided design (CAD) files. Ordinarily, these equations would be nearly unsolvable. (There’s a reason we all learned physics using block-like approximations of reality, rather than real-shaped cars and people and stuff. It’s not because our teachers were lazy; it’s because without all our convenient block-shaped approximations, the equations get so complicated that even teachers can’t solve them!)
Imagine the following illustration: Take a CAD file of an airplane, and figure out whether its wings will break during a 3-g turn. Do NOT apply Newton’s Law just yet, because you’ll have to obtain like 4 doctorates to solve that by yourself. Instead, use FEA to “discretize” the CAD file, or split up the smooth aerodynamic surfaces of that airplane into 150,000 tiny triangular chunks.
Now apply Newton’s Laws to each one of those tiny triangles, via a handy computer algorithm. Yeah, it’s true that each triangle’s solution depends on the ones next to it, but using “boundary conditions” (the engineering term for “magic”), you can solve the triangles at the edges first and propagate your solution throughout the entire network. In this way, you can break up 1 huge unsolvable problem into 150,000 tiny solvable problems, and this is the heart of FEA. Sure, it might take 2,000 computer cores 72 hours to run the calculations, but it will be reasonably accurate, and it can handle geometries as complex as Maseratis and Frank Gehry’s architectural mistakes, so it’s an extremely powerful and versatile tool.
On the strength of this problem-solving capability, FEA has dominated industry for nearly 60 years — because of this technology, we have safer airplanes, stable bridges, and probably (though I can’t prove it) moon landings. And Prof. Hughes, when he published his novel method of IGA in 2005, was a direct competitor to the long-reigning champ…
Well, this is where the bulk of the lecture happened. Hughes discussed how his method uses b-splines (read: curves) and NURBS (read: curves) to construct a CAD file that is already discretized. IGA methods can solve the same problems as FEA, but because they don’t spend 60% of their uptime discretizing, IGA methods reach solution up to 1,000 times faster (so Hughes claimed — I didn’t want to comb through his novel-sized publication history to find the proof). What’s more, IGA methods enable engineers to make tweaks to their design without having to begin the discretization process all over, which seamlessly integrates the analysis/redesign phase of the design process. I was extremely impressed by the paradigm-shifting achievement here, and that was even before Hughes gave examples of IGA’s success.
One such example of an IGA simulation was of a human heart valve:
Understandably, the heart is a pretty complex process — you’ve got the fluid flow of blood complete with eddies and currents, the artery/vein walls expanding and contracting under blood pressure, and the little white one-way valves that allow your heart to pump oxygen to your body approximately 115,000 times a day.
Dr. Hughes’ IGA was able to simulate a single heart valve to study its fatigue life, which is critical for improving the design of artificial hearts and valve replacements. Unarguably, this is a scientific/engineering development that will greatly improve human flourishing. But that’s not what filled me with wonder.
In over 2,000 years of recorded civilization, mankind has split the atom, photographed black holes, and decoded our own genome. There is more digital information in the city of Philadelphia than what all of humanity’s brains, past and present, could ever hope to contain. That information has culminated in processes like FEA and IGA, which let us comprehend and design the world around us.
And yet… IGA, a pinnacle of scientific achievement, building off three centuries of industrialization and five decades of Moore’s-law-paced computing, can do no more than replicate a single valve of a naturally occurring four-valved pump. That pump actualizes unsolvable mathematical equations in fluid dynamics, elasticity, biology, and mechanics every time it beats. That pump, the human heart, with nary a conscious thought of instruction, defies comprehension 115,000 times a day in every one of 7 billion souls on this planet.
I appreciate Dr. Hughes’ contributions to science, and even more, I cherish the opportunity to hear him deliver a compelling lecture on the living history of engineering computing. But, almost by accident, his lecture crystallized in me a revelation: all of us, from Einstein to Elizabeth II, are but schoolchildren making finger paintings, oblivious to the masterpieces who are breathing and painting right alongside us.