Suppose there is a cylindrical object in a round hole in a block and you want to model diffusion from inside the object to outside the hole. For example, suppose the object is hot (a piston perhaps) and it is losing heat to a block it’s been inserted into, or suppose the object is an electrical conductor and there’s a flow of current across the gap and into the surrounding block. Mechanically, in such a circumstance, the objects aren’t perfectly concentric (the object rests against the side of the hole along a line or at two diagonally opposite points). For that matter, things are never perfectly round. Suppose the hole is what keeps the object in position; the object is resting in the hole in other words.
The finite element method for modeling is pretty typical, and where we can define the gap reasonably well, it would work fine. That is, if you picture a crescent shape in which the points of the crescent touch each other, that would be the shape you’d model diffusion across.
There are two difficulties:
In a mathematical sense there ought to be either a mathematical line of contact (if the hole and object axes are parallel) or two mathematical points of contact (if the axes aren’t parallel). The diffusion rate there would be unlimited by the locally nonexistant gap.
The crescent shaped gap grows thinner and thinner the closer you get to the mathematical contacts, and it takes more and more finite elements to model what happens in that thinner and thinner gap.
I think this problem would have come up any number of times in finite element modeling, and perhaps there’s some well worked out approach to it, some kind of empirical formula or something. Anybody?
The diffusion rate would be limited by the conductivity/diffusivity of the materials from which the piston and bore are constructed. If they’re brought into contact with a large temperature differential, then at the instant of contact you’d have an extremely large rate of transfer per-unit-area (within the very small area of contact), but this condition wouldn’t last for long; the temperatures of the two parts in the region surrounding the point of contact would quickly come into agreement. This is true in real life and in FEA modeling.
“All models are wrong; some are useful.”
A finite-element model can never match reality with absolutely perfect accuracy, but depending on what you’re doing, it can come close. The level of detail you choose for your mesh will depend on what sort of accuracy you’re looking for from your model, and what computing resources are available. If you have limited computing power, and you’re just interested in measuring a global heat transfer rate, then you don’t care about the local details in the very small region where they make contact, so you’ll model the crescent region with one or two triangular/tetrahedral elements. The more accuracy and detail you want from your model, the more elements you’ll use in this region. When you get to the point where the size of your mesh element is very small compared to the radius of the cylinder/bore, then one mesh element can model that last crescent region with very high accuracy; at this point, dividing this small region into more mesh elements won’t significantly enhance the accuracy of your model. If you use CAD software to draw two circles of nearly the same diameter that are internally tangent, and then zoom way in on the point of tangency, you’ll see that this is true.