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How Math and Engineering Interact

(A note to accompany my NYT op-ed of Apr 25, 2016.)

According to Google Scholar, Ivo Babu┼íka’s most cited paper (as of April, 2016), was “The partition of unity finite element method: Basic theory and applications,” written jointly with his student J.M. Melenk (and published in 1996, when Ivo was 70 years old*).

What is the idea behind this paper? In particular, what does it have to do with engineering?

Here’s the gist. A lot of engineering design is now done using computer simulations. For instance, everything from small machine parts to large airplane components might be designed entirely on a computer screen, so that kinks are worked out and shapes and sizes optimized before such objects are actually built and physically tested. A commonly used method used for such design is called the “finite element method (FEM),” which was invented by engineers, but has been analyzed by mathematicians in a rich series of engineering/mathematician interactions. This is Ivo’s primary field of research (and mine as well).

What the FEM does is to approximately solve the systems of equations that determine how the machine parts (or airplane components or other mechanical objects) will deform when subject to loads. This is crucial: one wants to design objects which will not fail under stress (think airplane wings subjected to strong weather conditions, for instance – you want them to remain attached!). The solution of these equations gets particularly complicated in areas such as corners, joints, small holes, etc. These are generally areas which can be particularly susceptible to high stresses, and where cracks or other problems could easily develop. So the usual strategy is to put in a lot of computer power to analyze such sections (essentially, one “zooms in” on these sections by really cranking up the degree of approximation). This can be expensive, inefficient, and sometimes ineffective.

But here’s the thing: mathematicians can actually use their paper and pencil formulas to predict the underlying structure of the solution in such areas! Remember that everything’s governed by equations, after all. While these equations are too complicated to solve completely, they do yield some of their basic secrets – secrets that mathematicians have managed to carefully coax out through classical modes of study. For instance, they can predict that at any corner, the mathematical formula determining the solution will be of a certain special kind (let’s call this special formula type a “singularity”). They might not know the exact strength of such “singularities,” but they can come up with a bunch of them and then assert that the solution will be largely determined by some (unknown) combination of them.

As a consequence, several methods have been developed to incorporate this already-determined knowledge of singularities into the solution process. Instead of an unknown combination of these singularities, the calculations find (almost) exactly what combination is present. This can be a smarter way to attack the problem, rather than subjecting it to sheer “shock and awe” computer power. However, there are some problems: inserting these special “singularity” formulas can be very messy, and give rise to matching and compatibility problems that reverberate through the rest of the calculation.

This is where the above paper comes in. The MAIN IDEA is to present a very simple method (the so-called “partition of unity” FEM) that easily facilitates the insertion of such singularity formulas (or any other solution features one knows in advance) into select localized areas of the problem. Finite element software can be modified to easily allow such insertions, thereby giving engineers a smooth and efficient way of practically utilizing the intuition that mathematics provides. As a result, the computer simulations are much more effective, making the whole design process more efficient and reliable. This idea can be applied to such objects as gears and bolts, just as it can be made to work for joints between (say) the fuselage components of a plane.

Let me also mention that there’s another aspect to this symbiosis between engineering and mathematics. Strains and stresses in machine parts, crack formation, component failure, etc, all have been mathematically modeled. These models, when abstracted, give rise to some deep and complex questions in mathematics – which can often lead to some very elegant solutions. These solutions, in turn, generate several more “what if?” games – the kinds of questions mathematicians love to play with. And some of these games, when “solved,” end up having practical applications, which give rise to more questions, and so on. The wonderful cycle of interactions between mathematics’ beauty and its utility continues.

*NOTE: Ivo first published his “Partition of Unity FEM” idea in a 1994 paper with collaborators Caloz and Osborn – this was later elaborated upon in Melenk’s Ph.D. thesis and in various other papers. The mathematics in these papers was frequently cited to explain the related “XFEM” method, developed later by engineer Ted Belytschko and his group.