Large-scale Pareto-optimal topologies, or: how to describe a hexahedron

I got to meet Karen Caswelch, the CEO of Madison startup SciArtSoft last week. The company is based on tech developed by my colleague Krishnan Suresh. When I looked at one of his papers about this stuff I was happy to find there was a lovely piece of classical solid geometry hidden in it!

Here’s the deal. You want to build some component out of metal, which metal is to be contained in a solid block. So you can think of the problem as: you start with a region V in R^3, and your component is going to be some subregion W in R^3. For each choice of W there’s some measure of “compliance” which you want to minimize; maybe it’s fragility, maybe it’s flexibility, I dunno, depends on the problem. (Sidenote: I think lay English speakers would want “compliance” to refer to something you’d like to maximize, but I’m told this usage is standard in engineering.) (Subsidenote: I looked into this and now I get it — compliance literally refers to flexibility; it is the inverse of stiffness, just like in the lay sense. If you’re a doctor you want your patient to comply to their medication schedule, thus bending to outside pressure, but bending to outside pressure is precisely what you do not want your metal widget to do.)

So you want to minimize compliance, but you also want to minimize the weight of your component, which means you want vol(W) to be as small as possible. These goals are in conflict. Little lacy structures are highly compliant.

It turns out you can estimate compliance by breaking W up into a bunch of little hexahedral regions, computing compliance on each one, and summing. For reasons beyond my knowledge you definitely don’t want to restrict to chopping uniformly into cubes. So a priori you have millions and millions of differently shaped hexahedra. And part of the source of Suresh’s speedup is to gather these into approximate congruence classes so you can do a compliance computation for a whole bunch of nearly congruent hexahedra at once. And here’s where the solid geometry comes in; an old theorem of Cauchy tells you that if you know what a convex polyhedron’s 1-skeleton looks like as a graph, and you know the congruence classes of all the faces, you know the polyhedron up to rigid motion. In partiuclar, you can just triangulate each face of the hexahedron with a diagonal, and record the congruence class by 18 numbers, which you can then record in a hash table. You sort the hashes and then you can instantly see your equivalence classes of hexahedra.

(Related: the edge lengths of a tetrahedron determine its volume but the areas of the faces don’t.)

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